11affiliationtext: School of Mathematics, University of Edinburgh, Edinburgh
EH9 3FD, UK22affiliationtext: Mathematical Institute, University of Oxford,
Oxford, OX2 6GG, UK33affiliationtext: Isaac Newton Institute, Cambridge CB3
0EH, UK

# A Stochastic Model of Chemorepulsion with Additive Noise and Nonlinear
Sensitivity

Ilya Chevyrev _Email:_<EMAIL_ADDRESS>Ben Hambly _Email:_
<EMAIL_ADDRESS>Avi Mayorcas _Email:_<EMAIL_ADDRESS>
_Acknowledgments._ AM gratefully acknowledges support from the EPSRC Centre
For Doctoral Training in Partial Differential Equations: Analysis and
Applications [grant number EP/L015811/1]

###### Abstract

We consider a stochastic partial differential equation (SPDE) model for
chemorepulsion, with non-linear sensitivity on the one-dimensional torus. By
establishing an a priori estimate independent of the initial data, we show
that there exists a pathwise unique, global solution to the SPDE. Furthermore,
we show that the associated semi-group is Markov and possesses a unique
invariant measure, supported on a Hölder–Besov space of positive regularity,
which the solution law converges to exponentially fast. The a priori bound
also allows us to establish tail estimates on the $L^{p}$ norm of the
invariant measure which are heavier than Gaussian.

Keywords: Chemotaxis; exponential ergodicity; advection-diffusion SPDE
AMS 2020 Mathematics Subject Classification: Primary: 60H15, Secondary: 35K45;
37A25; 92B05

##  1 Introduction

We study the Cauchy problem and establish exponential ergodicity for the SPDE

$\begin{cases}\partial_{t}u-\partial_{xx}u=\chi\partial_{x}(u^{2}\partial_{x}\rho_{u})+\xi,&\text{
on }\mathbb{R}_{+}\times\mathbb{T},\\\
-\partial_{xx}\rho_{u}=u-\bar{u},&\text{ on }\mathbb{T},\\\
u|_{t=0}=\zeta,&\text{ on }\mathbb{T},\end{cases}$ (1.1)

where $\xi$ is a space-time white noise, $\mathbb{T}:=\mathbb{R}/\mathbb{Z}$
is the one dimensional torus with unit volume and $\chi>0$ is a positive
constant. The spatial average, $\bar{u}:=\langle
u,1\rangle_{L^{2}(\mathbb{T})}$, is equal to the $0^{\text{th}}$ Fourier mode
and we will take $\zeta$ to be a Hölder–Besov distribution of possibly
negative regularity. Without loss of generality we assume that the solution to
the second equation of (1.1) is mean free.

Without noise, (1.1) is a Keller–Segel model of chemorepulsion with nonlinear
sensitivity and is an example from a wide family of parabolic PDE models of
chemotaxis written, with some generality on the $d$-dimensional torus,
$\mathbb{T}^{d}$, for $a\geq 0$, as

$\begin{cases}\partial_{t}u-\Delta u=\nabla\cdot(f(u,\rho)\nabla\rho),&\text{
on }\mathbb{R}_{+}\times\mathbb{T}^{d},\\\
a\partial_{t}\rho-\Delta\rho=u-\bar{u},&\text{ on
}\mathbb{R}_{+}\times\mathbb{T}^{d},\\\ u\big{|}_{t=0}=u_{0}\geq
0,\quad\rho\big{|}_{t=0}=\rho_{0},&\text{ on }\mathbb{T}^{d}.\end{cases}$
(1.2)

Chemotaxis refers to the directed movement of cells or bacteria in response to
a chemical field. In (1.2), $u$ represents the cell or bacteria density,
$\rho$ the strength of the chemical field and $f\in
C^{\infty}(\mathbb{R}^{2};\mathbb{R})$ is a density dependent sensitivity
function such that $f(0,y)=0$ for all $y\in\mathbb{R}$. The viscous terms are
included to account for microscopic fluctuations in the cell and chemical
diffusions. Chemorepulsion refers to a situation where cells are biased to
move down the chemical gradient, modelled by choosing $f(x,y)\geq 0$ for
$x\geq 0$, while chemoattraction refers to the situation where cells are
biased to move up the gradient, modelled by choosing $f(x,y)\leq 0$ for $x\geq
0$. In situations of typical interest, the chemical field will be generated by
the cell population itself, which leads to the nonlinearity in (1.2). When
$f(u,\rho)=\chi u$, for $\chi\in\mathbb{R}$, (1.2) is referred to as the
Patlak–Keller–Segel model and was first introduced by Keller and Segel in
[15], building on earlier work by Patlak, [28]. When $a=0$, (1.2) is known as
a parabolic-elliptic model and when $a>0$, as a parabolic-parabolic model. A
common feature of chemotaxis models is strongly differing behaviour in the
chemorepulsive and chemoattractive settings. For example, in the parabolic-
elliptic Patlak–Keller–Segel model ($f(u,\rho)=\chi u$, $a=0$), global well-
posedness holds in all dimensions in the chemorepulsive regime ($\chi>0$) and
in both regimes (any $\chi\in\mathbb{R}$) when $d=1$. On the other hand, if
$d\geq 2$, in the chemoattractive regime ($\chi<0$) there exists a finite
threshold $-\infty<\bar{\chi}_{d}<0$ such that, under some additional
conditions when $d>2$, the model exhibits finite time blow-up for all
$\chi<\bar{\chi}_{d}$. We refer to [11, 29] for a more detailed exposition of
these phenomena and discussion of other models. For the one dimensional
setting in particular see [12, 25] and for the chemorepulsive setting see
[33].

In this paper we study (1.1), a one-dimensional additive noise, stochastic
version of (1.2) in the chemorepulsive and parabolic–elliptic ($a=0$) regime,
with nonlinear sensitivity function, $f(u,\rho)=\chi u^{2}$. Deterministic
models including nonlinear sensitivity have been studied since the early work
of Keller and Segel, see for example [7, 16, 18, 26]. These works focused on
modelling nonlinear sensitivity with respect to the chemical signal, i.e.
$f(u,\rho)=\chi u\tilde{f}(\rho)$. More recently there has been interest in
modelling nonlinear sensitivity with respect to the cell density, i.e.
$f(u,\rho)=\tilde{f}(u)$. This interest has been motivated, in part, by
desires to include volume filling, saturation and density dependent (quorum
sensing) regularisation effects into chemotaxis models, [9, 10, 13, 17, 31,
32]. The nonlinearity that we treat in (1.1) was already considered in the
deterministic context in [17, 31]; in [17] it was shown that in the
chemorepulsive setting the parabolic-elliptic model exhibits global existence
and convergence to equilibrium when $\tilde{f}(u)\sim u^{m}$ for all $m>0$. In
this case, the nonlinear sensitivity models an increased response to the
chemorepulsant when cell density is high.

While it is common to study chemotaxis using PDE models, SPDE approaches can
be used to model exogenous processes, study meta-stability or replicate
physically relevant fluctuations around the continuum limit. In this work we
focus on an additive noise model for two reasons. Firstly, it is a relatively
simple model for which we can accommodate space-time white noise and establish
a full ergodic analysis. Secondly, it allows us to make meaningful comparisons
with additive noise stochastic reaction-diffusion equations which have been
extensively studied; we refer to [4, 20, 23, 27, 34] for an incomplete list of
works concerning these models. In particular, we highlight a comparison
between our stochastic model, (1.1), and those considered in [20]. As in the
above works, the main technical step we require is to obtain sufficiently
strong a priori bounds on the pathwise solution which are independent of the
initial data. In our context we divide the solution into a regular and an
irregular part and the key step is a careful analysis of the more regular,
remainder equation exploiting the repulsive nature of our nonlinearity to
compensate for lower order terms without definitive sign. This is the content
of Theorem 4.4. Furthermore, we keep careful track of the exponents and
constants through the proof, which in particular allows us to obtain a tail
bound, which is heavier than Gaussian, on the $L^{p}$ norm of the invariant
measure. We give a detailed discussion on the implications of this bound and
comparisons to the literature on stochastic reaction-diffusion equations in
Remarks 4.5, 4.6 and 4.7.

There are many natural questions that arise from our work. For this model it
would be interesting to understand if the heavier than Gaussian tails that we
establish are in fact optimal. Furthermore, applying our current methods, it
appears that the low integrability of the solution, as compared with
stochastic reaction-diffusion equations, imposes a significant barrier to
considering $m>2$ in (1.1). Understanding whether this is a genuine issue or
simply one of methodology would be interesting. Going further, one could
naturally ask if it is possible to extend the well-known global well-posedness
of attractive chemotaxis models in one dimension, [12, 25], to the stochastic
case with white noise forcing. In addition, since chemotaxis is typically
observed in two or three dimensions, extending the above study to higher
dimensional versions would certainly be of interest. Using simple power
counting one would expect (1.1) to be sub-critical in the sense of regularity
structures, [8], on $\mathbb{R}_{+}\times\mathbb{T}^{d}$ for $d<4$. While
local analysis is tractable in the presence of white-noise forcing,
establishing global well-posedness for the repulsive model in dimension two
seems challenging by our current methods, see [19, Ch. 7]. Finally, beyond the
additive noise case, both multiplicative and conservative noise models are
highly relevant in the context of chemotaxis. Extending our results to those
cases, especially in a two or three dimensional setting, is a natural goal.

In the remainder of the introduction we summarise some notation in Section
1.1, and present our main results in Section 1.2. In Section 2 we recap the
definitions and properties of the linear stochastic heat equation, which plays
a key role in our analysis. In Section 3 we obtain local well-posedness of
(1.1) and some ancillary properties of the solution. Section 4 contains the
main contribution of this paper, establishing an a priori bound that is
independent of the initial data and constitutes a coming down from infinity
property. This a priori bound enables us to establish global well-posedness,
existence of invariant measures and the tail bound, (1.5), below. In Section 5
we establish the existence of invariant measures to (1.1) and closely
following the approach of [34], we establish the strong Feller property,
irreducibility, and exponential ergodicity of the associated semi-group. The
appendices, A and B respectively contain summaries and useful properties of
the inhomogeneous Hölder–Besov spaces and the stochastic heat equation on
$\mathbb{T}$.

###  1.1 Notation

Let $\mathbb{N}=\\{0,1,\ldots,\\}$ denote the set of non-negative integers.
For $k\in\mathbb{N}$, we denote by $C^{k}(\mathbb{T})$ the space of $k$ times
continuously differentiable, $1$-periodic, real functions. We denote by
$C^{\infty}(\mathbb{T})$ the space of smooth, periodic functions with values
in $\mathbb{R}$ and $\mathcal{S}^{\prime}(\mathbb{T})$ for its dual. For
$k\in\mathbb{N},\,p\in[1,\infty)$ (resp. $p=\infty$) we write
$W^{k,p}(\mathbb{T})$ for the spaces of periodic functions with $p$-integrable
(resp. essentially bounded) weak derivatives up to $k^{\text{th}}$ order, and
write $L^{p}(\mathbb{T})=W^{0,p}(\mathbb{T})$. We write
$\mathcal{B}^{\alpha}_{p,q}(\mathbb{T})$ for the Besov space associated to
$\alpha\in\mathbb{R}$, $p,q\in[1,\infty]$ and use the shorthand
$\mathcal{C}^{\alpha}(\mathbb{T})=\mathcal{B}^{\alpha}_{\infty,\infty}(\mathbb{T})$.
Note that for $\alpha\in\mathbb{N}$ the spaces $\mathcal{C}^{\alpha}$ and
$C^{\alpha}$ are not equal. See Appendix A for a full definition.

For $f\in\mathcal{S}^{\prime}(\mathbb{T})$ we denote its spatial mean by
$\bar{f}:=\langle 1,f\rangle_{L^{2}(\mathbb{T})}$. For
$\mathfrak{m}\in\mathbb{R}$ we write, for example,
$\mathcal{S}^{\prime}_{\mathfrak{m}}(\mathbb{T}),C^{k}_{\mathfrak{m}}(\mathbb{T}),\mathcal{C}^{\alpha}_{\mathfrak{m}}(\mathbb{T}),L^{p}_{\mathfrak{m}}(\mathbb{T})$,
for the corresponding spaces with the additional constraint that
$\bar{f}=\mathfrak{m}$. When the context is clear we drop dependence on the
domain in order to lighten notation.

Given a Banach space $E$, a subset $I\subseteq[0,\infty)$ and
$\kappa\in(0,1)$, we write $C_{I}E:=C(I;E)$ (resp.
$\mathcal{C}^{\kappa}_{I}E:=\mathcal{C}^{\kappa}(I;E)$) for the space of
continuous (resp. $\kappa$-Hölder continuous) maps $f\colon I\rightarrow E$
equipped with the norm $\|f\|_{C_{I}E}:=\sup_{t\in I}\|f_{t}\|_{E}$ (resp.
$\|f\|_{\mathcal{C}_{I}^{\kappa}E}:=\|f\|_{C_{I}E}+\sup_{t\neq s\in
I}\frac{\|f_{t}-f_{s}\|_{E}}{|t-s|^{\kappa}}$). For $T>0$, we use the
shorthand $C_{T}E=C_{[0,T]}E$ and
$\mathcal{C}_{T}^{\kappa}E=\mathcal{C}_{[0,T]}^{\kappa}E$. Note that the norm
$\|f\|_{\mathcal{C}_{T}^{\kappa}E}$ is equivalent to
$\|f_{0}\|_{E}+\sup_{t\neq
s\in[0,T]}\frac{\|f_{t}-f_{s}\|_{E}}{|t-s|^{\kappa}}$. For $\eta>0$ we let
$C_{\eta;T}E:=C_{\eta}((0,T];E)$ be the set of continuous functions
$f:(0,T]\rightarrow E$ such that

$\|f\|_{C_{\eta;T}E}:=\sup_{t\in(0,T]}(t^{\eta}\wedge 1)\|f_{t}\|_{E}<\infty.$

For a Banach space, $E$, equipped with its Borel $\sigma$-algebra,
$\mathcal{E}$, we use the notation $\mathcal{B}_{b}(E)$, $\mathcal{C}_{b}(E)$
and $\mathcal{C}^{1}_{b}(E)$ respectively for the sets of bounded Borel
measurable, continuous and continuously Fréchet differentiable maps
$\Phi:E\rightarrow\mathbb{R}$. We write $\mathcal{P}(E)$ for the set of
probability measures on $E$ which we equip with the topology of weak
convergence. For a sequence $(\mu_{n})_{n\geq 1}\subset\mathcal{P}(E)$ we
write $\mu_{n}\rightharpoonup\mu$ to indicate the weak convergence of
$(\mu_{n})_{n\geq 1}$ to $\mu\in\mathcal{P}(E)$. Using $\Pi_{\mu,\nu}$ to
denote the set of all couplings between $\mu,\,\nu\in\mathcal{P}(E)$, we write
the total variation distance as,

$\|\mu-\nu\|_{\mathrm{TV}}:=\sup_{A\in\mathcal{E}}|\mu(A)-\nu(A)|=\inf_{\pi\in\Pi_{\mu,\nu}}\iint_{E^{2}}\mathds{1}_{x=y}\mathop{}\\!\mathrm{d}\pi(x,y).$

We write $\lesssim$ to indicate that an inequality holds up to a constant
depending on quantities that we do not keep track of or are fixed throughout.
When we do wish to emphasise the dependence on certain quantities, we either
write $\lesssim_{K,v}$ or define $C:=C(\alpha,p,d)>0$ and write $\leq C$.

###  1.2 Main Results

We fix a filtered probability space
$(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq 0},\mathbb{P})$ carrying a mean-
free, space-time white noise, $\xi$, defined in Section 2 below. We fix
$\alpha_{0}\in\left(-\frac{1}{2},0\right)$,
$\alpha\in\left(0,\alpha_{0}+\frac{1}{2}\right)$ and $\eta>0$ such that,

$\frac{\alpha-\alpha_{0}}{2}<\eta<\frac{1}{4}.$ (1.3)

######  Theorem 1.1 (Global Well-Posedness).

Let $T>0$, $\mathfrak{m}\in\mathbb{R}$, $\chi>0$ and
$\zeta\in\mathcal{C}^{\alpha_{0}}_{\mathfrak{m}}(\mathbb{T})$. Then there
exists a unique, probabilistically strong, mild solution, $u(\zeta)$, to (1.1)
such that $\mathbb{P}$-a.s. $u(\zeta)\in
C_{\eta;T}\mathcal{C}_{\mathfrak{m}}^{\alpha}(\mathbb{T})$.

######  Remark 1.2.

While we frame Theorem 1.1 as a probabilistic statement, the proof is mainly
deterministic and based on PDE techniques. In the course of the proof we
additionally establish local Lipschitz continuity of the solution in both
initial data and the noise, see Proposition 4.8. This will be important in
Section 5.3 when we establish full support of the law on
$\mathcal{C}^{\alpha}_{\mathfrak{m}}(\mathbb{T})$.

######  Remark 1.3.

The statement of Theorem 1.1 remains valid for
$\eta=\frac{\alpha-\alpha_{0}}{2}$. However, we take
$\eta>\frac{\alpha-\alpha_{0}}{2}$ to simplify some statements and proofs
below. In general we may think of $\eta$ as being arbitrarily close to
$\frac{\alpha-\alpha_{0}}{2}$.

Let
$\mathcal{L}(u_{t}(\zeta)):=u_{t}(\zeta)\\#\mathbb{P}\in\mathcal{P}\big{(}\mathcal{C}_{\mathfrak{m}}^{\alpha}(\mathbb{T})\big{)}$
denote the law of the solution, started from
$\zeta\in\mathcal{C}^{\alpha_{0}}_{\mathfrak{m}}(\mathbb{T})$, at time $t>0$.

######  Theorem 1.4 (Exponential Ergodicity).

Let $\mathfrak{m}\in\mathbb{R}$ and $\delta\in(0,1/2)$. Then there exists a
unique measure
$\nu\in\mathcal{P}\big{(}\mathcal{C}^{1/2-\delta}_{\mathfrak{m}}(\mathbb{T})\big{)}$
which is invariant for the semi-group associated to (1.1). Furthermore, $\nu$
has full support in $\mathcal{C}^{1/2-\delta}_{\mathfrak{m}}$ and there exists
$c>0$ such that for all $t>1$ and
$\zeta\in\mathcal{C}^{\alpha_{0}}_{\mathfrak{m}}$,

$\|\mathcal{L}(u_{t}(\zeta))-\nu\|_{\mathrm{TV}}\,\leq e^{-ct}.$ (1.4)

Finally, for any $p\in[1,\infty)$ there exists $\Lambda:=\Lambda(p,\delta)>0$
such that

$\int_{\mathcal{C}^{1/2-\delta}_{\mathfrak{m}}}\exp\left(\Lambda\|f\|^{1-2\delta}_{L^{p}}\right)\,\nu(\mathop{}\\!\mathrm{d}f)<\infty.$
(1.5)

The proof of Theorem 1.1 is completed at the end of Section 4 and the proof of
Theorem 1.4 is completed at the end of Section 5.

##  2 Stochastic Heat Equation

In what follows we will decompose the solution $u$ to (1.1) into a regular and
irregular part, with the irregular part being the solution to a linear
stochastic heat equation (SHE). We briefly recap some necessary definitions of
space-time white noise, the solution to the SHE and some of its properties.

######  Definition 2.1 (White Noise).

Given an abstract probability space $(\Omega,\mathcal{F},\mathbb{P})$ we say
that an $\mathbb{R}$-valued, stochastic process, indexed by
$L^{2}(\mathbb{R}_{+}\times\mathbb{T})$, $\\{\xi(\varphi)\,:\,\varphi\in
L^{2}(\mathbb{R}_{+}\times\mathbb{T})\\}$, defines a spatially mean-free
space-time white noise if

  1. 1.

$\mathbb{E}[\xi(\varphi)]=0$,
$\mathbb{E}[\xi(\varphi)\xi(\varphi^{\prime})]=\langle\varphi,\varphi^{\prime}\rangle_{L^{2}(\mathbb{R}_{+}\times\mathbb{T})}$
for all $\varphi,\,\varphi^{\prime}\in L^{2}(\mathbb{R}_{+}\times\mathbb{T})$,

  2. 2.

$\xi(\psi\otimes 1)=0$ for all $\psi\in L^{2}(\mathbb{R}_{+})$,
$\mathbb{P}$-a.s.

For $\varphi\in L^{2}(\mathbb{R}_{+}\times\mathbb{T})$ we write $\xi(\varphi)$
in the convenient form of a stochastic integral,

$\xi(\varphi)=\int_{\mathbb{R}_{+}}\int_{\mathbb{T}}\varphi(t,x)\xi(\mathop{}\\!\mathrm{d}t,\,\mathop{}\\!\mathrm{d}x),$
(2.1)

even though $\xi$ is almost surely not a measure. We refer to [6, Ch. 4] for
further details and a proper construction of (2.1).

We define the filtration

$(\tilde{\mathcal{F}}_{t})_{t\geq
0}:=\sigma\left(\left\\{\xi(\varphi)\,:\,\varphi\in
L^{2}(\mathbb{R}_{+}\times\mathbb{T}),\,\varphi|_{(t,\infty)}\equiv
0\right\\}\right),$

and let $(\mathcal{F}_{t})_{t\geq 0}$ denote its usual augmentation.

######  Definition 2.2.

Let $0\leq t_{0}<T$ and $\mathcal{H}$ be the periodic heat kernel on
$\mathbb{T}$ defined in (A.9). We say that the
$\mathcal{S}^{\prime}(\mathbb{T})$ valued process,

$(t_{0},T]\ni t\mapsto
v_{t_{0},t}(\phi):=\int_{t_{0}}^{t}\int_{\mathbb{T}}(\mathcal{H}_{t-r}\ast\phi)(x)\,\xi(\mathop{}\\!\mathrm{d}r,\mathop{}\\!\mathrm{d}x)\quad\,\forall\phi\in
C^{\infty}(\mathbb{T}),$ (2.2)

is a mild solution to the SHE, with zero initial condition at $t=t_{0}$.

######  Theorem 2.3.

The process $[t_{0},T]\ni t\mapsto v_{t_{0},t}$ is, continuous, Markov,
$(\mathcal{F}_{t})_{t\in[t_{0},T]}$ adapted and for any $\alpha<1/2$,
$\kappa\in[0,1/2)$ and $p\geq 1$ there exists $C:=C(T,\alpha,\kappa,p)>0$ such
that,

$\mathbb{E}\left[\sup_{t\,\neq\,s\in[t_{0},T]}\frac{\|v_{t_{0},t}-v_{t_{0},s}\|^{p}_{\mathcal{C}^{\alpha-2\kappa}}}{|t-s|^{p\kappa}}\right]<C.$
(2.3)

Furthermore, for any $0\leq t_{0}<t\leq T$, the law of $v_{t_{0},t}$ depends
only on $|t-t_{0}|$.

###### Proof.

See Appendix B. ∎

##  3 Local Well-Posedness

Throughout this section, we fix $T>0$ and $\chi\in\mathbb{R}$ (not necessarily
positive). From Theorem 2.3 we see that $\mathbb{P}$-a.s. the map $t\mapsto
v_{0,t}$ is finite only in $C_{T}\mathcal{C}_{0}^{\alpha}$, for $\alpha<1/2$.
We therefore cannot expect to find solutions to (1.1) of any higher
regularity. Obtaining the a priori estimates in Section 4 requires at least
one degree of spatial regularity. Therefore, although the equation is not
singular itself, we solve for a remainder process with higher regularity using
a Da Prato–Debussche trick, [4]. Concretely we decompose the solution as
$u_{t}:=w_{t}+Z_{t}$, where $t\mapsto Z_{t}$ is a deterministic function
taking values in the Hölder–Besov space $\mathcal{C}_{0}^{\alpha}$ with zero
spatial mean. Subsequently we will take $Z_{t}$ to be a $\mathbb{P}$-a.s.
realisation of the stochastic heat equation $t\mapsto v_{0,t}$. The unknown,
$w$, solves,

$\begin{cases}\partial_{t}w-\partial_{xx}w=\chi\partial_{x}((w+Z)^{2}\partial_{x}\rho_{w+Z}),&\text{
on }\mathbb{R}_{+}\times\mathbb{T},\\\
-\partial_{xx}\rho_{w+Z}=w-\bar{w}+Z,&\text{ on }\mathbb{T},\\\
w|_{t=0}=\zeta,&\text{ on }\mathbb{T}.\end{cases}$ (3.1)

For the rest of this section, we fix
$\zeta\in\mathcal{C}^{\alpha_{0}}(\mathbb{T})$ and $Z\in
C_{T}\mathcal{C}^{\alpha}_{0}$.

We first show that under suitable regularity assumptions on $w$, the right
hand side of the first equation in (3.1) is a well-defined element of
$C_{\eta;T}\mathcal{C}^{\alpha}(\mathbb{T})$.

######  Lemma 3.1.

Let $w\in C_{\eta;T}\mathcal{C}^{\alpha}$. Then the map $w\mapsto\Psi w$,
defined for any $t\in(0,T]$, by

$(\Psi
w)_{t}:=e^{t\Delta}\zeta+\int_{0}^{t}e^{(t-s)\Delta}\chi\partial_{x}((w_{s}+Z_{s})^{2}\partial_{x}\rho_{w_{s}+Z_{s}})\mathop{}\\!\mathrm{d}s,$
(3.2)

is well-defined from $C_{\eta;T}\mathcal{C}^{\alpha}$ to itself.

###### Proof.

Applying (A.10), for any $t>0$, we have that

$\|e^{t\Delta}\zeta\|_{\mathcal{C}^{\alpha}}\lesssim
t^{-\frac{\alpha-\alpha_{0}}{2}}\|\zeta\|_{\mathcal{C}^{\alpha_{0}}}.$ (3.3)

Concerning the integral term, expanding the square and applying Theorem A.5,
along with (A.12) we obtain the bounds

$\displaystyle\|w^{2}_{s}\partial_{x}\rho_{w_{s}+Z_{s}}\|_{\mathcal{C}^{\alpha}}$
$\displaystyle\leq\|w\|^{2}_{\mathcal{C}^{\alpha}}\|\partial_{x}\rho_{w_{s}+Z_{s}}\|_{\mathcal{C}^{\alpha}}\lesssim\|w_{s}\|^{3}_{\mathcal{C}^{\alpha}}+\|w_{s}\|^{2}_{\mathcal{C}^{\alpha}}\|Z_{s}\|_{\mathcal{C}^{\alpha}}$
$\displaystyle
2\|w_{s}Z_{s}\partial_{x}\rho_{w_{s}+Z_{s}}\|_{\mathcal{C}^{\alpha}}$
$\displaystyle\leq
2\|w\|_{\mathcal{C}^{\alpha}}\|Z_{s}\|_{\mathcal{C}^{\alpha}}\|\partial_{x}\rho_{w_{s}+Z_{s}}\|_{\mathcal{C}^{\alpha}}\lesssim
2\|w_{s}\|^{2}_{\mathcal{C}^{\alpha}}\|Z_{s}\|_{\mathcal{C}^{\alpha}}+2\|w_{s}\|_{\mathcal{C}^{\alpha}}\|Z_{s}\|^{2}_{\mathcal{C}^{\alpha}}$
$\displaystyle\|Z^{2}_{s}\partial_{x}\rho_{w_{s}+Z_{s}}\|_{\mathcal{C}^{\alpha}}$
$\displaystyle\leq\|Z_{s}\|^{2}_{\mathcal{C}^{\alpha}}\|\partial_{x}\rho_{w_{s}+Z_{s}}\|_{\mathcal{C}^{\alpha}}\lesssim\|w_{s}\|_{\mathcal{C}^{\alpha}}\|Z_{s}\|^{2}_{\mathcal{C}^{\alpha}}+\|Z_{s}\|^{3}_{\mathcal{C}^{\alpha}}.$

Combining these yields that

$\|(w_{s}+Z_{s})^{2}\partial_{x}\rho_{w_{s}+Z_{s}}\|_{\mathcal{C}^{\alpha}}\lesssim\sum_{k=0}^{3}\binom{3}{k}\|w_{s}\|^{3-k}_{\mathcal{C}^{\alpha}}\|Z_{s}\|^{k}_{\mathcal{C}^{\alpha}}.$
(3.4)

Therefore, applying (A.10), (A.5) for any $s<t\in(0,T\wedge 1]$ we have

$\displaystyle\|e^{(t-s)\Delta}\partial_{x}((w_{s}+Z_{s})^{2}\partial_{x}\rho_{w_{s}+Z_{s}})\|_{\mathcal{C}^{\alpha}}$
$\displaystyle\lesssim(t-s)^{-\frac{1}{2}}\|(w_{s}+Z_{s})^{2}\partial_{x}\rho_{w_{s}+Z_{s}}\|_{\mathcal{C}^{\alpha}}$
$\displaystyle\lesssim(t-s)^{-\frac{1}{2}}s^{-3\eta}\max_{k=0,\ldots,3}\left\\{\|w\|^{3-k}_{C_{\eta;T}\mathcal{C}^{\alpha}}\|Z\|^{k}_{C_{T}\mathcal{C}^{\alpha}}\right\\}.$

Since $\eta<\frac{1}{4}$ we may integrate $s^{-3\eta}$ near $0$ and so for
$t\in(0,T\wedge 1]$ we have

$t^{\eta}\|(\Psi w)_{t}\|_{\mathcal{C}^{\alpha}}\lesssim
t^{\eta-\frac{\alpha-\alpha_{0}}{2}}\|\zeta\|_{\mathcal{C}^{\alpha_{0}}}+t^{\frac{1}{2}-2\eta}\max_{k=0,\ldots,3}\left\\{\|w\|^{3-k}_{C_{\eta;T}\mathcal{C}^{\alpha}}\|Z\|^{k}_{C_{T}\mathcal{C}^{\alpha}}\right\\}.$
(3.5)

where both exponents are positive due to (1.3) and the norms on the right hand
side are finite by assumption. For $t>1$ one may argue in almost exactly the
same way, only splitting the time integral at $t=1$ and replacing the
multiplication by $t^{\eta}$ with $(t\wedge 1)^{\eta}$. ∎

######  Definition 3.2 (Mild Solutions to (3.1)).

We say that $w\in C_{\eta;T}\mathcal{C}^{\alpha}$ is a mild solution to (3.1)
on $[0,T]$ (started from $\zeta$ and driven by $Z$) if for every $t\in(0,T]$,

$w_{t}=e^{t\Delta}\zeta+\int_{0}^{t}e^{(t-s)\Delta}\chi\partial_{x}\left((w_{s}+Z_{s})^{2}\partial_{x}\rho_{w_{s}+Z_{s}}\right)\,\mathop{}\\!\mathrm{d}s.$
(3.6)

######  Remark 3.3.

Lemma 3.1 demonstrates that for any solution, the right hand side of (3.6) is
well-defined.

######  Theorem 3.4 (Local Well-Posedness of (3.1)).

Let $\mathfrak{R}\geq 1$ be such that
$\|Z\|_{C_{T}\mathcal{C}^{\alpha}}^{3}+\|\zeta\|_{\mathcal{C}^{\alpha_{0}}(\mathbb{T})}<\mathfrak{R}$.
Then there exists $C>0$, independent of $\mathfrak{R}$, $\zeta$, and $Z$, such
that (3.1) has a unique mild solution $w\in
C_{\eta;T_{*}}\mathcal{C}^{\alpha}$ where

$T_{*}=\left(\frac{1}{C\mathfrak{R}}\right)^{\frac{1}{\theta}}\wedge
T,\,\,\text{ with
}\,\,\theta:=\left(\eta-\frac{\alpha-\alpha_{0}}{2}\right)\wedge\left(\frac{1}{2}-2\eta\right).$
(3.7)

Furthermore

$\sup_{t\in(0,T_{*}]}t^{\eta}\|w_{t}\|_{\mathcal{C}^{\alpha}}\leq 1$ (3.8)

and

$\lim_{t\to 0}\|w_{t}-\zeta\|_{\mathcal{C}^{\alpha_{0}}}=0.$ (3.9)

###### Proof.

Denoting

$B_{T_{*}}:=\Big{\\{}w\in
C((0,T_{*}];\mathcal{C}^{\alpha}(\mathbb{T}))\,:\,\sup_{t\in(0,T_{*}]}t^{\eta}\|w_{t}\|_{\mathcal{C}^{\alpha}}\leq
1\Big{\\}},$

we will show that

$w\mapsto(\Psi
w)_{t}:=e^{t\Delta}\zeta+\int_{0}^{t}e^{(t-s)\Delta}\chi\partial_{x}\left((w_{s}+Z_{s})^{2}\partial_{x}\rho_{w_{s}+Z_{s}}\right)\,\mathop{}\\!\mathrm{d}s,$

is a contraction on $B_{T_{*}}$ for $T_{*}$ defined by (3.7) for $C>0$
sufficiently large. By (3.5), for $w\in B_{T_{*}}$ and $t\in(0,T_{*}]$, there
exists $C>0$ such that

$\displaystyle t^{\eta}\|(\Psi w)_{t}\|_{\mathcal{C}^{\alpha}}$
$\displaystyle\leq
Ct^{\left(\eta-\frac{\alpha-\alpha_{0}}{2}\right)\wedge\left(\frac{1}{2}-2\eta\right)}\mathfrak{R},$

and so $\Psi$ maps $B_{T_{*}}$ into itself for $T_{*}$ defined by (3.7). To
show that $\Psi$ is a contraction we let $w,\,\tilde{w}\in B_{T_{*}}$. For any
$s\in(0,T_{*}]$, using similar steps to those in the proof of (3.5), we have
that

$\displaystyle\|(w_{s}+Z_{s})^{2}\partial_{x}\rho_{w_{s}+Z_{s}}-(\tilde{w}_{s}+Z_{s})^{2}$
$\displaystyle\partial_{x}\rho_{\tilde{w}_{s}+Z_{s}}\|_{\mathcal{C}^{\alpha}}$
$\displaystyle\lesssim\|w_{s}-\tilde{w}_{s}\|_{\mathcal{C}^{\alpha}}\sum_{k=0}^{2}\binom{2}{k}\left(\|w_{s}\|_{\mathcal{C}^{\alpha}}\vee\|\tilde{w}_{s}\|_{\mathcal{C}^{\alpha}}\right)^{2-k}\|Z_{s}\|^{k}_{\mathcal{C}^{\alpha}}$
$\displaystyle\lesssim\mathfrak{R}s^{-2\eta}\|w_{s}-\tilde{w}_{s}\|_{\mathcal{C}^{\alpha}}.$

So then, for any $t\in(0,T_{*}]$, we have

$\displaystyle t^{\eta}\|(\Psi
w)_{t}-(\Psi\tilde{w})_{t}\|_{\mathcal{C}^{\alpha}}$ $\displaystyle\leq
t^{\eta}\int_{0}^{t}\|e^{(t-s)\Delta}\chi\partial_{x}\big{(}(w_{s}+Z_{s})^{2}\partial_{x}\rho_{w_{s}+Z_{s}}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad-(\tilde{w}_{s}+Z_{s})^{2}\partial_{x}\rho_{\tilde{w}_{s}+Z_{s}}\big{)}\|_{\mathcal{C}^{\alpha}}\,\mathop{}\\!\mathrm{d}s$
$\displaystyle\lesssim
t^{\eta}\int_{0}^{t}(t-s)^{-\frac{1}{2}}\|(w_{s}+Z_{s})^{2}\partial_{x}\rho_{w_{s}+Z_{s}}-(\tilde{w}_{s}+Z_{s})^{2}\partial_{x}\rho_{\tilde{w}_{s}+Z_{s}}\|_{\mathcal{C}^{\alpha}}\,\mathop{}\\!\mathrm{d}s$
$\displaystyle\lesssim\mathfrak{R}t^{\frac{1}{2}-2\eta}\|w-\tilde{w}\|_{C_{\eta;t}\mathcal{C}^{\alpha}}.$

It follows that there exists $C>0$ such that, for $T_{*}$ given by (3.7),

$\displaystyle\|\Psi w-\Psi\tilde{w}\|_{C_{\eta;T_{*}}\mathcal{C}^{\alpha}}\,$
$\displaystyle\leq
C\mathfrak{R}(T_{*})^{\frac{1}{2}-2\eta}\|w-\tilde{w}\|_{C_{\eta;T_{*}}\mathcal{C}^{\alpha}}$
$\displaystyle\leq\frac{1}{2}\|w-\tilde{w}\|_{C_{\eta;T_{*}}\mathcal{C}^{\alpha}}.$

Hence $\Psi$ is a contraction on $B_{T_{*}}$, and therefore there exists a
unique fixed point $w\in B_{T_{*}}$ of $\Psi$ which, by construction, is a
mild solution to (3.1) in the sense of Definition 3.2 and satisfies (3.8).

To show that $w$ is the unique solution in all of
$C_{\eta;T_{*}}\mathcal{C}^{\alpha}$, let $\tilde{w}$ be another mild
solutions of (3.1). Then, by a similar argument to the above, there exists a
$\tilde{T}(\|w\|_{C_{\eta;T_{*}}\mathcal{C}^{\alpha}},\|\tilde{w}\|_{C_{\eta;T_{*}}\mathcal{C}^{\alpha}})=:\tilde{T}\in(0,T_{*}]$
such that $w,\tilde{w}\in B_{\tilde{T}}$. Since both must be fixed points of
$\Psi$ on $B_{\tilde{T}}$ we have that $w=\tilde{w}$ on $[0,\tilde{T}]$.
Iterating the argument, using the same $\tilde{T}$ at each step, shows that
$w=\tilde{w}$ on $[0,T_{*}]$.

To show (3.9), observe first that $\lim_{t\to
0}\|e^{t\Delta}\zeta-\zeta\|_{\mathcal{C}^{\alpha_{0}}}=0$ by Remark A.7 and
so it only remains to show that the integral term converges to zero in
$\mathcal{C}^{\alpha_{0}}$. Consider
$\tilde{\eta}\in\left(\frac{\alpha-\alpha_{0}}{2},\eta\right)$. Since
$C_{\tilde{\eta};T_{*}}\mathcal{C}^{\alpha}\hookrightarrow
C_{\eta;T_{*}}\mathcal{C}^{\alpha}$, applying what we have proved so far to
$(\alpha_{0},\alpha,\tilde{\eta})$ in place of $(\alpha_{0},\alpha,\eta)$, we
see that, for $S>0$ sufficiently small, $w$ is also the unique mild solution
to (3.1) in $C_{\tilde{\eta};S}\mathcal{C}^{\alpha}$, and that
$\sup_{t\in(0,S]}t^{\tilde{\eta}}\|w_{t}\|_{\mathcal{C}^{\alpha}}\leq 1$.
Applying (A.10) and (3.4), for all $t\in(0,S]$

$\displaystyle\int_{0}^{t}\|e^{(t-s)\Delta}\partial_{x}((w_{s}+Z_{s})^{2}\partial_{x}\rho_{w_{s}+Z_{s}}\|_{\mathcal{C}^{\alpha_{0}}}\mathop{}\\!\mathrm{d}s$
$\displaystyle\lesssim\int_{0}^{t}(t-s)^{-\frac{\alpha_{0}-\alpha+1}{2}}s^{-3\tilde{\eta}}\mathop{}\\!\mathrm{d}s$
$\displaystyle=t^{\frac{1}{2}+\frac{\alpha-\alpha_{0}}{2}-3\tilde{\eta}},$

where we used the fact that $\frac{\alpha_{0}-\alpha+1}{2}\vee
3\tilde{\eta}<1$ to evaluate the integral. We now choose $\tilde{\eta}$
sufficiently close to $\frac{\alpha-\alpha_{0}}{2}$ so that
$\frac{1}{2}+\frac{\alpha-\alpha_{0}}{2}-3\tilde{\eta}>0$, from which (3.9)
follows. ∎

######  Lemma 3.5.

Suppose that $w\in C_{\eta;T}\mathcal{C}^{\alpha}$ is a mild solution to
(3.1). Then for all $t_{0}\in(0,T)$, $\beta\in(\alpha,\alpha+1)$ and
$\kappa\in(0,1)$,

$\sup_{t\,\neq\,s\,\in[t_{0},T]}\frac{\|w_{t}-w_{s}\|_{\mathcal{C}^{\beta-2\kappa}}}{|t-s|^{\kappa}}<\infty.$
(3.10)

###### Proof.

Applying (A.10) gives

$\|e^{t\Delta}\zeta\|_{\mathcal{C}^{\beta}}\lesssim
t^{-\frac{\beta-\alpha_{0}}{2}}\|\zeta\|_{\mathcal{C}^{\alpha_{0}}}.$ (3.11)

Regarding the integral term, applying similar steps to those in the proof of
(3.5) and the assumption $\frac{1+\beta-\alpha}{2}<1$,

$\displaystyle\int_{0}^{t}\|e^{(t-s)\Delta}\partial_{x}\left((w_{s}+Z_{s})^{2}\partial_{x}\rho_{w_{s}+Z_{s}}\right)\|_{\mathcal{C}^{\beta}}\,\mathop{}\\!\mathrm{d}s$
$\displaystyle\lesssim\int_{0}^{t}(t-s)^{-\frac{1+\beta-\alpha}{2}}s^{-3\eta}\,\mathop{}\\!\mathrm{d}s$
$\displaystyle=t^{\frac{1}{2}-\frac{\beta-\alpha}{2}-3\eta}.$ (3.12)

It follows that

$\sup_{t\in[t_{0},T_{*}]}\|w_{t}\|_{\mathcal{C}^{\beta}}<\infty.$

Now, let $0<t_{0}<s<t\leq T$ and $\kappa\in[0,1)$. Applying the triangle
inequality and using the semi-group property, we have that

$\displaystyle\|w_{t}-w_{s}\|_{\mathcal{C}^{\beta-2\kappa}}$
$\displaystyle\leq\left\|(e^{(t-s)-1)\Delta}w_{s}\right\|_{\mathcal{C}^{\beta-2\kappa}}$
$\displaystyle\quad+\int_{s}^{t}\left\|e^{(t-r)\Delta}\partial_{x}((w_{r}+Z_{r})^{2}\partial_{x}\rho_{w_{r}+Z_{r}})\right\|_{\mathcal{C}^{\beta-2\kappa}}\,\mathop{}\\!\mathrm{d}r$
$\displaystyle=:I_{1}(s,t)+\int_{s}^{t}I_{2}(r,t)\,\mathop{}\\!\mathrm{d}r.$

Using (A.10) and (A.11),

$\displaystyle
I_{1}(s,t)\lesssim\left\|\left(e^{(t-s)\Delta}-1\right)w_{s}\right\|_{\mathcal{C}^{\beta-2\kappa}}\lesssim(t-s)^{\kappa}\|w_{s}\|_{\mathcal{C}^{\beta}}.$
(3.13)

Since $\beta-1<\alpha$ and
$\sup_{t\in[t_{0},T_{*}]}\|Z_{t}\|_{\mathcal{C}^{\alpha}}+\|w_{t}\|_{\mathcal{C}^{\alpha}}<\infty$,
applying (A.10) followed by (A.11) and (3.4), we obtain

$\displaystyle
I_{2}(r,t)\lesssim(t-r)^{-\frac{1+\beta-2\kappa-(\beta-1)}{2}}\left\|((w_{r}+Z_{r})^{2}\partial_{x}\rho_{w_{r}+Z_{r}})\right\|_{\mathcal{C}^{\beta-1}}\lesssim(t-r)^{-1+\kappa}.$

Therefore

$\int_{s}^{t}I_{2}(r,t)\,\mathop{}\\!\mathrm{d}r\lesssim(t-s)^{\kappa}.$
(3.14)

Combining (3.13) and (3.14) we obtain (3.10). ∎

######  Lemma 3.6 (Mild Solutions are Weak Solutions).

Let $t_{0}\in[0,T)$ and $w\in C_{\eta;T}\mathcal{C}^{\alpha}$ be a mild
solution to (3.1) on $[0,T]$. Then for any $\phi\in C^{\infty}(\mathbb{T})$
and $t\in[t_{0},T]$,

$\langle w_{t},\phi\rangle-\langle
w_{t_{0}},\phi\rangle=-\int_{t_{0}}^{t}\langle\partial_{x}w_{s}+\chi(w_{s}+Z_{s})^{2}\partial_{x}\rho_{w_{s}+Z_{s}},\partial_{x}\phi\rangle\,\mathop{}\\!\mathrm{d}s.$
(3.15)

Moreover, (3.15) holds for all $t_{0}\in(0,T)$ and $\phi\in
C^{1}(\mathbb{T})$.

###### Proof.

From the mild form of the equation started from data $w_{t_{0}}$ at $t=t_{0}$,
it follows that for $\phi\in C^{\infty}(\mathbb{T})$

$\displaystyle\int_{t_{0}}^{t}\langle
w_{s},\partial_{xx}\phi\rangle\,\mathop{}\\!\mathrm{d}s$
$\displaystyle=\int_{t_{0}}^{t}\left\langle
e^{s\Delta}w_{t_{0}},\partial_{xx}\phi\right\rangle\,\mathop{}\\!\mathrm{d}s$
$\displaystyle\qquad\qquad+\int_{t_{0}}^{t}\int_{t_{0}}^{s}\left\langle
e^{(s-r)\Delta}\chi\partial_{x}((w_{r}+Z_{r})^{2}\partial_{x}\rho_{w_{r}+Z_{r}}),\partial_{xx}\phi\right\rangle\,\mathop{}\\!\mathrm{d}r\,\mathop{}\\!\mathrm{d}s.$

Integrating by parts in the first term on the right hand side gives

$\int_{t_{0}}^{t}\left\langle
e^{s\Delta}w_{t_{0}},\partial_{xx}\phi\right\rangle\,\mathop{}\\!\mathrm{d}s=\int_{t_{0}}^{t}\langle\partial_{xx}e^{s\Delta}w_{t_{0}},\phi\rangle\,\mathop{}\\!\mathrm{d}s=\langle
e^{t\Delta}w_{t_{0}},\phi\rangle-\langle w_{t_{0}},\phi\rangle.$ (3.16)

Using the same argument for the second term on the right hand side we have
that

$\left\langle
e^{(s-r)\Delta}\partial_{x}((w_{r}+Z_{r})^{2}\partial_{x}\rho_{w_{r}+Z_{r}}),\partial_{xx}\phi\right\rangle=\partial_{s}\left\langle
e^{(s-r)\Delta}\partial_{x}((w_{r}+Z_{r})^{2}\partial_{x}\rho_{w_{r}+Z_{r}}),\phi\right\rangle.$

Changing the order of integration gives

$\displaystyle\int_{t_{0}}^{t}\int_{t_{0}}^{s}\langle
e^{(s-r)\Delta}\partial_{x}((w_{r}+Z_{r})^{2}$
$\displaystyle\partial_{x}\rho_{w_{r}+Z_{r}}),\partial_{xx}\phi\rangle\,\mathop{}\\!\mathrm{d}r\,\mathop{}\\!\mathrm{d}s$
$\displaystyle=\int_{t_{0}}^{t}\langle
e^{(t-r)\Delta}\partial_{x}((w_{t}+Z_{t})^{2}\partial_{x}\rho_{w_{t}+Z_{t}}),\phi\rangle\,\mathop{}\\!\mathrm{d}r$
$\displaystyle\quad\quad-\int_{t_{0}}^{t}\langle\partial_{x}((w_{r}+Z_{r})^{2}\partial_{x}\rho_{w_{r}+Z_{r}}),\phi\rangle\,\mathop{}\\!\mathrm{d}r.$

Putting this together with (3.16) gives that

$\displaystyle\int_{t_{0}}^{t}\langle
w_{s},\partial_{xx}\phi\rangle\,\mathop{}\\!\mathrm{d}s$
$\displaystyle=\langle e^{t\Delta}w_{t_{0}},\phi\rangle-\langle
w_{t_{0}},\phi\rangle+\int_{t_{0}}^{t}\langle
e^{(t-r)\Delta}\chi\partial_{x}((w_{r}+Z_{r})^{2}\partial_{x}\rho_{w_{r}+Z_{r}}),\phi\rangle\,\mathop{}\\!\mathrm{d}r$
$\displaystyle\quad-\int_{t_{0}}^{t}\langle\chi\partial_{x}((w_{r}+Z_{r})^{2}\partial_{x}\rho_{w_{r}+Z_{r}}),\phi\rangle\,\mathop{}\\!\mathrm{d}r.$

Rearranging and integrating the left hand side by parts once proves (3.15) for
any $\phi\in C^{\infty}(\mathbb{T})$. For $t_{0}\in(0,T)$, by Theorem 3.4, we
have that $\sup_{t\in[t_{0},T]}\|w_{t}\|_{C^{1}}<\infty$, so that the map
$C^{1}(\mathbb{T})\ni\phi\mapsto\int_{t_{0}}^{t}\langle\partial_{x}w_{s}+(w_{s}+Z_{s})^{2}\partial_{x}\rho_{w_{s}+Z_{s}},\partial_{x}\phi\rangle\mathop{}\\!\mathrm{d}s$
is continuous. Approximating $\phi\in C^{1}(\mathbb{T})$ by a smooth sequence
gives the result. ∎

It follows from the proof of Lemma 3.6 that any solution to (3.1) has constant
spatial mean.

######  Corollary 3.7.

Let $t\in[0,T]$ and $w\in C_{\eta;T}\mathcal{C}^{\alpha}$ be a mild solution
to (3.1). Then $\bar{w}_{t}=\bar{\zeta}$.

###### Proof.

Applying Lemma 3.6 with $\phi\equiv 1$ and $t_{0}=0$ gives
$\bar{w}_{t}=\langle w_{t},1\rangle=\langle\zeta,1\rangle=\bar{\zeta}$. ∎

##  4 A Priori Estimate and Global Well-Posedness

In this section we make use of the specific sign choice $\chi>0$ in (1.1) to
obtain an a priori estimate on the solution in Theorem 4.4. Throughout this
section we fix $T>0$, and a zero spatial mean function $Z\in
C_{T}\mathcal{C}^{\alpha}_{0}$.

######  Proposition 4.1 (Testing the Equation).

Let $0<t_{0}<t\leq T$, $p\geq 2$ an integer,
$\zeta\in\mathcal{C}^{\alpha_{0}}(\mathbb{T})$, and $w(\zeta)\in
C_{\eta;T}\mathcal{C}^{\alpha}$ be a mild solution to (3.1) on $[0,T]$. Then

$\displaystyle\frac{1}{p(p-1)}\left(\|w_{t}\|_{L^{p}}^{p}-\|w_{t_{0}}\|_{L^{p}}^{p}\right)$
$\displaystyle=-\int_{t_{0}}^{t}\langle
w_{s}^{p-2}\partial_{x}w_{s},\partial_{x}w_{s}\rangle\,\mathop{}\\!\mathrm{d}s$
(4.1)
$\displaystyle\quad-\int_{t_{0}}^{t}\langle\chi(w_{s}+Z_{s})^{2}\partial_{x}\rho_{w_{s}+Z_{s}},w_{s}^{p-2}\partial_{x}w_{s}\rangle\,\mathop{}\\!\mathrm{d}s.$

###### Proof.

Let $\beta\in(1,\alpha+1)$ and $\kappa\in\left(1/2,\beta/2\right)$. By Lemma
3.5,

$\sup_{s\,\neq\,r\in[t_{0},T]}\frac{\|w_{s}-w_{r}\|_{L^{\infty}}}{|s-r|^{\kappa}}\leq\sup_{s\,\neq\,r\in[t_{0},T]}\frac{\|w_{s}-w_{r}\|_{\mathcal{C}^{\beta-2\kappa}}}{|s-r|^{\kappa}}<\infty.$
(4.2)

Now, let $\varphi\in C^{\infty}(\mathbb{R})$ and observe that for any
$t_{0}\leq r<s\leq T$ we have the identity

$\displaystyle\langle w_{s},\varphi(w_{s})\rangle-\langle
w_{r},\varphi(w_{r})\rangle$ $\displaystyle=\langle
w_{s},\varphi(w_{r})\rangle-\langle w_{r},\varphi(w_{r})\rangle+\langle
w_{s},\varphi(w_{s})-\varphi(w_{r})\rangle.$

So for any $t\in(t_{0},T]$, and $n\geq 2$, defining a family of partitions, by
setting $t_{i}=t_{0}+i(t-t_{0})/n$, for $i=0,\ldots,n$ and applying Lemma 3.6
with the test function $\varphi(w_{t_{i}})$, we have

$\displaystyle\langle w_{t},\varphi(w_{t})\rangle-\langle
w_{t_{0}},\varphi(w_{t_{0}})\rangle$
$\displaystyle=-\sum_{i=0}^{n-1}\int_{t_{i}}^{t_{i+1}}\langle\partial_{x}w_{s}+\chi(w_{s}+Z_{s})^{2}\partial_{x}\rho_{w_{s}+Z_{s}},\partial_{x}\varphi(w_{t_{i}})\rangle\mathop{}\\!\mathrm{d}s$
$\displaystyle\quad+\sum_{i=0}^{n-1}\left\langle
w_{t_{i+1}},\varphi(w_{t_{i+1}})-\varphi(w_{t_{i}})\right\rangle$
$\displaystyle=:-I_{n}(t)+R_{n}(t).$

By continuity of $s\mapsto\partial_{x}\varphi(w_{s})$ we see that

$\lim_{n\rightarrow\infty}I_{n}(t)=\int_{0}^{t}\langle\partial_{x}w_{s}+\chi(w_{s}+Z_{s})^{2}\partial_{x}\rho_{w_{s}+Z_{s}},\partial_{x}\varphi(w_{s})\rangle\mathop{}\\!\mathrm{d}s.$
(4.3)

For $R^{n}(t)$, Taylor’s formula gives

$\displaystyle R_{n}(t)$ $\displaystyle=\sum_{i=0}^{n-1}\left[\langle
w_{t_{i+1}},(w_{t_{t+i}}-w_{t_{i}})\varphi^{\prime}(w_{t_{i}})\rangle+\Big{\langle}w_{t_{i+1}},\int_{w_{t_{i}}}^{w_{t_{i+1}}}\varphi^{\prime\prime}(y)(w_{t_{i+1}}-y)\,\mathop{}\\!\mathrm{d}y\Big{\rangle}\right]$
$\displaystyle=:R_{n;1}(t)+R_{n;2}(t).$

We show that $R^{n}_{2}(t)$ converges to zero. Using (4.2), we have the bound

$\displaystyle|R_{n;2}(t)|$
$\displaystyle\lesssim\sup_{s\in[t_{0},t]}\|w_{s}\|_{L^{\infty}}\sup_{\begin{subarray}{c}i=0,\ldots,n-1\\\
y\in[w_{t_{i}},w_{t_{i+1}}]\end{subarray}}|\varphi^{\prime\prime}(y)|\,\sum_{i=0}^{n-1}\|w_{t_{i+1}}-w_{t_{i}}\|^{2}_{L^{\infty}}\overset{\tiny\eqref{eq:SolTimeHolder}}{\lesssim}\sum_{i=0}^{n-1}|t_{i+1}-t_{i}|^{2\kappa},$

where we used that $\varphi^{\prime\prime}$ is continuous. Regarding
$R_{n;1}(t)$, we may apply Lemma 3.6 once again to each bracket, this time
with the test function $w_{t_{i+1}}\varphi^{\prime}(w_{t_{i}})$, to give that

$\displaystyle R_{n;1}(t)$
$\displaystyle=-\sum_{i=0}^{n-1}\int_{t_{i}}^{t_{i+1}}\langle\partial_{x}w_{s}+\chi(w_{s}+Z_{s})^{2}\partial_{x}\rho_{w_{s}+Z_{s}},\partial_{x}(w_{t_{i+1}}\varphi^{\prime}(w_{t_{i}}))\rangle\,\mathop{}\\!\mathrm{d}s.$

So again, by regularity of the map $s\mapsto\partial_{x}w_{s}$ and $\varphi$,
we have that

$\lim_{n\rightarrow\infty}R_{n}(t)=-\int_{t_{0}}^{t}\langle\partial_{x}w_{s}+\chi(w_{s}+Z_{s})^{2}\partial_{x}\rho_{w_{s}+Z_{s}},\partial_{x}(w_{s}\varphi^{\prime}(w_{s}))\rangle\,\mathop{}\\!\mathrm{d}s.$
(4.4)

So combining (4.3) and (4.4) gives,

$\displaystyle\langle w_{t},\varphi(w_{t})\rangle-\langle
w_{t_{0}},\varphi(w_{t_{0}})\rangle=-\int_{t_{0}}^{t}\langle\partial_{x}w_{r}$
$\displaystyle+\chi(w_{r}+Z_{r})^{2}\partial_{x}\rho_{w_{r}+Z_{r}},$
$\displaystyle\qquad\left(2\varphi^{\prime}(w_{r})+w_{r}\varphi^{\prime\prime}(w_{r})\right)\partial_{x}w_{r}\rangle\,\mathop{}\\!\mathrm{d}r.$

For $\varphi(x):=x^{p-1}$ we have
$2\varphi^{\prime}(x)+x\varphi^{\prime\prime}(x)=p(p-1)x^{p-2}$ and

$\langle w_{t},w_{t}^{p-1}\rangle-\langle
w_{t_{0}},w_{t_{0}}^{p-1}\rangle=\|w_{t}\|_{L^{p}}^{p}-\|w_{t_{0}}\|_{L^{p}}^{p},$

which gives (4.1). ∎

Observe that one of the terms appearing in the second term on the right hand
side of (4.1) is

$-\langle\partial_{x}\rho_{w},w^{p}\partial_{x}w\rangle=-\frac{1}{p+1}\langle\partial_{x}\rho_{w},\partial_{x}w^{p+1}\rangle=-\frac{1}{p+1}\|w^{p+2}\|_{L^{1}}+\frac{1}{p+2}\langle\bar{w},w^{p+1}\rangle.$

We will use this term in combination with the first term,
$-\|w_{t}^{p-2}|\partial_{x}w_{t}|^{2}\|_{L^{1}}$, coming from the Laplacian,
to obtain an a priori estimate in Theorem 4.4 on $\|w_{t}\|^{p}_{L^{p}}$ that
is independent of the initial data. To do so we make use of an ODE comparison
lemma which can be found with proof as [34, Lem. 3.8].

######  Lemma 4.2 (ODE Comparison).

Let $\lambda>1$, $c_{1},\,c_{2}>0$ and $f\colon[0,T]\rightarrow[0,\infty)$ be
differentiable and satisfying

$f^{\prime}(t)+c_{1}f(t)^{\lambda}\leq c_{2},$

for every $t\in[0,T]$. Then for all $t\in(0,T]$

$f(t)\leq\max\Big{\\{}\Big{(}\frac{tc_{1}(\lambda-1)}{2}\Big{)}^{-\frac{1}{\lambda-1}},\,\left(\frac{c_{2}}{c_{1}}\right)^{\frac{1}{\lambda}}\Big{\\}}.$

######  Remark 4.3.

For a given initial data $\zeta\in\mathcal{C}^{\alpha_{0}}(\mathbb{T})$, a
mild solution $w$ on $[0,T]$ to (3.1), and a constant $c\in\mathbb{R}$ with
$c\neq 0$, we have that $\tilde{w}_{t}=cw_{t}$ satisfies

$\tilde{w}_{t}=e^{t\Delta}\tilde{\zeta}+\int_{0}^{t}e^{(t-s)\Delta}c^{-2}\chi\partial_{x}\big{(}(\tilde{w}_{s}+\tilde{Z}_{s})^{2}\partial_{x}\rho_{\tilde{w}_{s}+\tilde{Z}_{s}}\big{)}\mathop{}\\!\mathrm{d}s\;,$

where $\tilde{\zeta}=c\zeta$ and $\tilde{Z}_{s}=cZ_{s}$. In particular, if
$\chi>0$, then setting $c:=\chi^{1/2}$ we see that any solution to (3.1) is
equal, up to a scalar multiple, to a solution to (3.1) with $(\zeta,Z,\chi)$
replaced by $(\tilde{\zeta},\tilde{Z},1)$. Conversely, if $\bar{\zeta}\neq 0$,
then setting $c:=1/\bar{\zeta}$ we see that any solution to (3.1) is equal, up
to a scalar multiple, to a solution to (3.1) with $(\zeta,Z,\chi)$ replaced by
$(\tilde{\zeta},Z,c^{2}\chi)$ and where now
$\tilde{\zeta}\in\mathcal{C}^{\alpha_{0}}_{1}(\mathbb{T})$.

In light of Remark 4.3, we phrase the following a priori estimate only for the
case $\bar{\zeta}\in\\{0,1\\}$.

######  Theorem 4.4 (A Priori Bound on the Remainder).

Let $p\geq 2$ be an even integer, $\chi>0$, $\mathfrak{m}\in\\{0,1\\}$,
$\zeta\in\mathcal{C}^{\alpha_{0}}_{\mathfrak{m}}(\mathbb{T})$ and $w(\zeta)\in
C_{\eta;T}\mathcal{C}^{\alpha}$ be a mild solution to (3.1) on $[0,T]$. Then
there exists a constant $C>0$, independent of $\zeta$, $Z$, $p$ and $\chi$,
such that

$\|w_{t}(\zeta)\|_{L^{p}}\leq\max\Big{\\{}(\chi
t/4)^{-\frac{1}{2}},C(p\vee\chi)^{\gamma}\|Z\|^{\frac{1}{\alpha}}_{C_{t}\mathcal{C}^{\alpha}},\,C(p\vee\chi)^{\frac{p+1}{p+2}}\mathfrak{m}\,\Big{\\}},$
(4.5)

where $\gamma=\frac{p(1+\alpha)+2+\alpha}{\alpha(p+2)}$.

###### Proof.

Since $t\mapsto\partial_{x}w_{t}$ and
$t\mapsto(w_{t}+Z_{t})^{2}\partial_{x}\rho_{w_{t}+Z_{t}}$ are both continuous
mappings from $(0,T]$ into $L^{\infty}(\mathbb{T})$ by Lemma 3.5, we may
differentiate (4.1) with respect to $t$ in order to obtain

$\frac{1}{p(p-1)}\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}t}\|w_{t}\|_{L^{p}}^{p}=-\|w_{t}^{p-2}|\partial_{x}w_{t}|^{2}\|_{L^{1}}-\chi\langle(w_{t}+Z_{t})^{2}\partial_{x}\rho_{w_{t}+Z_{t}},w_{t}^{p-2}\partial_{x}w_{t}\rangle,$

where we used that $p$ is even in the first term on the right hand side.
Regarding the second term, since $f\mapsto\partial_{x}\rho_{f}$ is linear, we
have

$\displaystyle\langle(w_{t}+Z_{t})^{2}\partial_{x}\rho_{w_{t}+Z_{t}},w_{t}^{p-2}\partial_{x}w_{t}\rangle$
$\displaystyle=\langle\partial_{x}\rho_{w_{t}},w^{p}_{t}\partial_{x}w_{t}\rangle+\langle\partial_{x}\rho_{Z_{t}},w^{p}_{t}\partial_{x}w_{t}\rangle$
$\displaystyle\quad+2\langle
Z_{t}\partial_{x}\rho_{w_{t}+Z_{t}},w^{p-1}_{t}\partial_{x}w_{t}\rangle+\langle
Z_{t}^{2}\partial_{x}\rho_{w_{t}+Z_{t}},w^{p-2}_{t}\partial_{x}w_{t}\rangle.$

Integrating the first term by parts, using that
$-\partial_{xx}\rho_{w_{t}}=w_{t}-\bar{\zeta}$ by Corollary 3.7 and recalling
that we have set $\bar{\zeta}=\mathfrak{m}$,

$\displaystyle\langle\partial_{x}\rho_{w_{t}},w^{p}_{t}\partial_{x}w_{t}\rangle=\frac{1}{p+1}\langle\partial_{x}\rho_{w_{t}},\partial_{x}w^{p+1}_{t}\rangle$
$\displaystyle=\frac{1}{p+1}\langle w_{t}-\mathfrak{m},w_{t}^{p+1}\rangle$
$\displaystyle=\frac{1}{p+1}\|w_{t}^{p+2}\|_{L^{1}}-\frac{1}{p+1}\langle\mathfrak{m},w_{t}^{p+1}\rangle,$

Therefore, applying the chain rule in the remaining terms gives

$\begin{split}\frac{1}{p(p-1)\chi}\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}t}\|w_{t}\|_{L^{p}}^{p}&=-\frac{1}{\chi}\|w_{t}^{p-2}|\partial_{x}w_{t}|^{2}\|_{L^{1}}-\frac{1}{p+1}\|w_{t}^{p+2}\|_{L^{1}}+\frac{1}{p+1}\langle\mathfrak{m},w_{t}^{p+1}\rangle\\\
&\quad-\frac{1}{p+1}\langle\partial_{x}\rho_{Z_{t}},\partial_{x}w^{p+1}_{t}\rangle-\frac{2}{p}\langle
Z_{t}\partial_{x}\rho_{Z_{t}},\partial_{x}w^{p}_{t}\rangle\\\
&\quad-\frac{1}{p-1}\langle
Z_{t}^{2}\partial_{x}\rho_{Z_{t}},\partial_{x}w^{p-1}_{t}\rangle-\frac{2}{p}\langle
Z_{t},\partial_{x}\rho_{w_{t}}\partial_{x}w^{p}_{t}\rangle\\\
&\quad-\frac{1}{p-1}\langle
Z_{t}^{2},\partial_{x}\rho_{w_{t}}\partial_{x}w^{p-1}_{t}\rangle.\end{split}$
(4.6)

We demonstrate that all terms without a definite sign can be bounded by a
constant multiple of the two negative terms,

$\frac{1}{\chi}\|w_{t}^{p-2}|\partial_{x}w_{t}|^{2}\|_{L^{1}}+\frac{1}{p+1}\|w_{t}^{p+2}\|_{L^{1}}=:\frac{1}{\chi}A_{t}+\frac{1}{p+1}B_{t}.$

From Theorems A.2 and A.8, for any $\alpha\in\mathbb{R}$ and
$p,q\in[1,\infty]$, we have that

$\|\partial_{x}\rho_{w}\|_{\mathcal{B}^{\alpha}_{p,q}}\lesssim\|w\|_{\mathcal{B}^{\alpha-\frac{1}{p}}_{1,q}},\quad\text{
and }\quad\|\partial_{x}\rho_{w}\|_{L^{\infty}}\lesssim\|w\|_{L^{1}}.$ (4.7)

For any $0<q\leq p+2$, by Jensen’s inequality we have

$\|w_{t}^{q}\|_{L^{1}}\leq B_{t}^{\frac{q}{p+2}}.$ (4.8)

Furthermore, by applying Cauchy–Schwarz followed by (4.8), for
$\frac{2}{p}<q\leq p+1$ we have

$\|\partial_{x}w^{q}_{t}\|_{L^{1}}=q\|w^{q-1}_{t}\partial_{x}w_{t}\|_{L^{1}}\leq
q\|w^{2q-p}_{t}\|^{\frac{1}{2}}_{L^{1}}\,A_{t}^{\frac{1}{2}}\leq
qB_{t}^{\frac{2q-p}{2(p+2)}}\,A_{t}^{\frac{1}{2}}.$ (4.9)

To keep track of dependence on $p$, we also make use of the following
inequality which readily follows from Young’s inequality for products: for
$\gamma_{1},\gamma_{2}\in(1,\infty)$ such that
$\frac{1}{\gamma_{1}}+\frac{1}{\gamma_{2}}=1$, $a,\,b>0$ and $c\geq 1$

$ab<\frac{c^{\gamma_{1}-1}}{\gamma_{1}-1}a^{\gamma_{1}}+\frac{1}{c}b^{\gamma_{2}}$
(4.10)

(in fact
$ab<\frac{c^{\gamma_{1}-1}}{e(\gamma_{1}-1)}a^{\gamma_{1}}+\frac{1}{c}b^{\gamma_{2}}$).
From now on we let $C>0$, $c\geq 1$ be constants, that are independent of
$\zeta,\,Z,\,p,$ and $\chi$. Later, we will fix $c\geq 1$ sufficiently large
at the end of the proof. If we write $\lesssim$ in an inequality below, the
implied proportionality constant is equal to $C>0$ which we take sufficiently
large so that the inequality holds.

We work through the terms of (4.6) in order. For the third term of (4.6),
applying Hölder, (4.8), (4.10) and using that $\mathfrak{m}\in\\{0,1\\}$, we
have,

$\frac{1}{p+1}|\langle\mathfrak{m},w^{p+1}_{t}\rangle|\leq\frac{\mathfrak{m}}{p+1}\|w^{p+1}_{t}\|_{L^{1}}\leq\frac{\mathfrak{m}}{p+1}B_{t}^{\frac{p+1}{p+2}}\leq\frac{\mathfrak{m}c^{p+1}}{(p+1)^{2}}+\frac{1}{c(p+1)}B_{t}.$
(4.11)

For the fourth term of (4.6), we integrate by parts to give

$\displaystyle\frac{1}{p+1}|\langle\partial_{x}\rho_{Z_{t}},\partial_{x}w_{t}^{p+1}\rangle|=\frac{1}{p+1}|\langle
Z_{t},w_{t}^{p+1}\rangle|\hskip
1.79997pt\overset{\tiny\eqref{eq:BesovDuality}\,\eqref{eq:Jensen1}}{\lesssim}$
$\displaystyle\frac{1}{p}\|Z\|_{C_{t}\mathcal{C}^{\alpha}}B_{t}^{\frac{p+1}{p+2}}$
(4.12) $\displaystyle\overset{\tiny\eqref{eq:YoungPeterAndPaul}}{\leq}$
$\displaystyle\frac{c^{p+1}}{p^{2}}\|Z\|_{C_{t}\mathcal{C}^{\alpha}}^{p+2}+\frac{1}{cp}B_{t}.$

Concerning the two remaining terms of (4.6) involving
$\partial_{x}\rho_{Z_{t}}$, we have for $k=1,2$

$\displaystyle\frac{1}{p+1-k}|\langle
Z^{k}_{t}\partial_{x}\rho_{Z_{t}},\partial_{x}w^{p+1-k}_{t}\rangle|\hskip
11.70004pt\leq$
$\displaystyle\frac{1}{p+1-k}\|Z^{k}_{t}\partial_{x}\rho_{Z_{t}}\|_{L^{\infty}}\,\|\partial_{x}w^{p+1-k}_{t}\|_{L^{1}}$
(4.13)
$\displaystyle\overset{\tiny\eqref{eq:BesovProduct},\,\eqref{eq:BesovGradElliptic},\,\eqref{eq:GradientOfPowerBound}}{\lesssim}$
$\displaystyle\|Z_{t}\|^{k+1}_{\mathcal{C}^{\alpha}}B_{t}^{\frac{p+2-2k}{2(p+2)}}A_{t}^{\frac{1}{2}}$
$\displaystyle\overset{\tiny\eqref{eq:YoungPeterAndPaul}}{\leq}$
$\displaystyle
c\|Z_{t}\|^{2(k+1)}_{\mathcal{C}^{\alpha}}\,B_{t}^{\frac{p+2-2k}{p+2}}+\frac{1}{c}A_{t}$
$\displaystyle\overset{\tiny\eqref{eq:YoungPeterAndPaul}}{\lesssim}$
$\displaystyle\frac{c^{\frac{p+2-k}{k}}}{p}\|Z\|^{\frac{k+1}{k}(p+2)}_{C_{t}\mathcal{C}^{\alpha}}+\frac{1}{c}B_{t}+\frac{1}{c}A_{t}.$

(In the case $p=k=2$, note that we do not need to apply (4.10) in the final
line.) Combining (4.12), (4.13) and noting that $\frac{3}{2}(p+2)$ is the
highest power of $\|Z\|_{C_{t}\mathcal{C}^{\alpha}}$ encountered so far, we
have that

$\displaystyle\sum_{k=0}^{2}\binom{2}{k}\frac{1}{p+1-k}|\langle
Z^{k}_{t}\partial_{x}\rho_{Z_{t}},\partial_{x}w^{p+1-k}_{t}\rangle|$
$\displaystyle\lesssim\frac{c^{p+1}}{p}\|Z\|^{\frac{3}{2}(p+2)}_{C_{t}\mathcal{C}^{\alpha}}+\frac{1}{c}B_{t}+\frac{1}{c}A_{t}.$
(4.14)

For the seventh term of (4.6), we first integrate by parts and apply the
triangle inequality to obtain

$\frac{1}{p}|\langle
Z_{t},\partial_{x}\rho_{w_{t}}\partial_{x}w_{t}^{p}\rangle|\,\leq\,\frac{1}{p}|\langle\partial_{x}Z_{t},w^{p}_{t}\partial_{x}\rho_{w_{t}}\rangle|+\frac{1}{p}|\langle
Z_{t},w_{t}^{p+1}\rangle|+\frac{1}{p}|\langle
Z_{t},w^{p}_{t}\mathfrak{m}\rangle|.$ (4.15)

Since $\alpha\in(0,\frac{1}{2})$, using Theorems A.5 and A.4, we have

$\displaystyle\|w_{t}^{p}\partial_{x}\rho_{w_{t}}\|_{\mathcal{B}^{1-\alpha}_{1,1}}\hskip
14.40007pt\overset{\tiny\eqref{eq:BesovPositiveProductSplit}}{\lesssim}\hskip
8.49998pt$
$\displaystyle\|w_{t}^{p}\|_{L^{\frac{1}{1-\alpha}}}\|\partial_{x}\rho_{w_{t}}\|_{\mathcal{B}^{1-\alpha}_{\frac{1}{\alpha},1}}+\|w^{p}_{t}\|_{\mathcal{B}^{1-\alpha}_{1,1}}\|\partial_{x}\rho_{w_{t}}\|_{L^{\infty}}$
$\displaystyle\overset{\tiny\eqref{eq:BesovGradElliptic2},\,\eqref{eq:BesovEmbedding},\,\eqref{eq:Besov_Lp_embedding}}{\lesssim}$
$\displaystyle\|w_{t}^{p}\|_{\mathcal{B}^{1-\alpha}_{1,1}}\|w_{t}\|_{L^{1}}$
$\displaystyle\overset{\tiny\eqref{eq:BesovPoincare}}{\lesssim}\hskip
8.49998pt$
$\displaystyle\left(\|\partial_{x}w_{t}^{p}\|^{1-\alpha}_{L^{1}}\|w_{t}^{p}\|^{\alpha}_{L^{1}}+\|w^{p}_{t}\|_{L^{1}}\right)\|w_{t}\|_{L^{1}}$
$\displaystyle\overset{\tiny\eqref{eq:Jensen1}}{\lesssim}\hskip 8.00003pt$
$\displaystyle\|\partial_{x}w_{t}^{p}\|^{1-\alpha}_{L^{1}}B_{t}^{\frac{p\alpha+1}{p+2}}+B_{t}^{\frac{p+1}{p+2}}$
$\displaystyle\overset{\tiny\eqref{eq:GradientOfPowerBound}}{\lesssim}\hskip
8.00003pt$ $\displaystyle
p^{1-\alpha}B_{t}^{\frac{p(1+\alpha)+2}{2(p+2)}}A_{t}^{\frac{1-\alpha}{2}}+B_{t}^{\frac{p+1}{p+2}}.$
(4.16)

Considering the first term of (4.15), applying (4.16) and then (4.10) twice
with $\gamma_{1}=\frac{2}{1+\alpha}$ and then with
$\gamma_{1}=\frac{(p+2)(1+\alpha)}{2\alpha}$, and using that $p^{-\alpha}\leq
1$,

$\displaystyle\frac{1}{p}|\langle\partial_{x}Z_{t},w^{p}_{t}\partial_{x}\rho_{w_{t}}\rangle|\hskip
6.29997pt\overset{\tiny\eqref{eq:BesovDuality}}{\lesssim}\hskip 3.49998pt$
$\displaystyle\frac{1}{p}\|\partial_{x}Z_{t}\|_{\mathcal{C}^{\alpha-1}}\|w_{t}^{p}\partial_{x}\rho_{w_{t}}\|_{\mathcal{B}^{1-\alpha}_{1,1}}$
$\displaystyle\overset{\tiny\eqref{eq:wPGradRhoWBound}}{\lesssim}\hskip
3.00003pt$
$\displaystyle\|Z_{t}\|_{\mathcal{C}^{\alpha}}\left(B_{t}^{\frac{p(1+\alpha)+2}{2(p+2)}}A_{t}^{\frac{1-\alpha}{2}}+\frac{1}{p}B_{t}^{\frac{p+1}{p+2}}\right)$
$\displaystyle\overset{\tiny\eqref{eq:YoungPeterAndPaul}}{\lesssim}\hskip
3.00003pt$ $\displaystyle
c\|Z\|^{\frac{2}{1+\alpha}}_{C_{t}\mathcal{C}^{\alpha}}B_{t}^{\frac{p(1+\alpha)+2}{(p+2)(1+\alpha)}}+\frac{c^{p+1}}{p^{2}}\|Z\|^{p+2}_{C_{t}\mathcal{C}^{\alpha}}$
$\displaystyle\quad+\frac{1}{c}A_{t}+\frac{1}{pc}B_{t}$
$\displaystyle\overset{\tiny\eqref{eq:YoungPeterAndPaul}}{\lesssim}\hskip
3.00003pt$
$\displaystyle\frac{c^{\frac{p(1+\alpha)+2+\alpha}{\alpha}}}{p}\|Z\|^{\frac{p+2}{\alpha}}_{C_{t}\mathcal{C}^{\alpha}}+\frac{p+1}{cp}B_{t}+\frac{1}{c}A_{t}.$
(4.17)

Concerning the second term of (4.15), we have

$\displaystyle\frac{1}{p}|\langle Z_{t},w_{t}^{p+1}\rangle|$
$\displaystyle\overset{\tiny\eqref{eq:BesovDuality},\,\eqref{eq:Jensen1}}{\lesssim}\frac{1}{p}\|Z\|_{C_{t}\mathcal{C}^{\alpha}}B_{t}^{\frac{p+1}{p+2}}\overset{\tiny\eqref{eq:YoungPeterAndPaul}}{\lesssim}\,\frac{c^{p+1}}{p}\|Z\|^{p+2}_{C_{t}\mathcal{C}^{\alpha}}+\frac{1}{pc}B_{t}.$
(4.18)

For the third term of (4.15), by similar estimates and using that
$\mathfrak{m}\in\\{0,1\\}$ we have,

$\frac{1}{p}|\langle
Z_{t},w_{t}^{p}\mathfrak{m}\rangle|\overset{\tiny\eqref{eq:BesovDuality},\,\eqref{eq:Jensen1}}{\lesssim}\frac{1}{p}\|Z\|_{C_{t}\mathcal{C}^{\alpha}}\mathfrak{m}B^{\frac{p}{p+2}}\overset{\tiny\eqref{eq:YoungPeterAndPaul}}{\lesssim}\frac{c^{p}}{p^{2}}\|Z\|^{p+2}_{C_{t}\mathcal{C}^{\alpha}}+\frac{c^{p}}{p^{2}}\mathfrak{m}+\frac{1}{cp}B_{t}$
(4.19)

Regarding the eighth and final term of (4.6),

$\displaystyle\frac{1}{p-1}|\langle
Z_{t}^{2},\partial_{x}\rho_{w_{t}}\partial_{x}w_{t}^{p-1}\rangle|\hskip
7.20003pt\overset{\tiny\eqref{eq:BesovDuality},\,\eqref{eq:BesovProduct}}{\lesssim}\hskip
4.00002pt$
$\displaystyle\frac{1}{p}\|Z\|^{2}_{C_{t}\mathcal{C}^{\alpha}}\|\partial_{x}\rho_{w_{t}}\|_{L^{\infty}}\|\partial_{x}w^{p-1}_{t}\|_{L^{1}}$
$\displaystyle\overset{\tiny\eqref{eq:BesovGradElliptic2},\,\eqref{eq:GradientOfPowerBound}}{\lesssim}\hskip
3.00003pt$
$\displaystyle\|Z\|^{2}_{C_{t}\mathcal{C}^{\alpha}}B_{t}^{\frac{p}{2(p+2)}}A_{t}^{\frac{1}{2}}$
$\displaystyle\overset{\tiny\eqref{eq:YoungPeterAndPaul}}{\leq}\hskip 7.5pt$
$\displaystyle
c\|Z\|^{4}_{C_{t}\mathcal{C}^{\alpha}}B_{t}^{\frac{p}{p+2}}+\frac{1}{c}A_{t}$
$\displaystyle\overset{\tiny\eqref{eq:YoungPeterAndPaul}}{\lesssim}\hskip
7.5pt$
$\displaystyle\frac{c^{p+1}}{p}\|Z\|^{2(p+2)}_{C_{t}\mathcal{C}^{\alpha}}+\frac{1}{c}B_{t}+\frac{1}{c}A_{t}.$
(4.20)

So, since $p+1\leq\frac{p(1+\alpha)+2+\alpha}{\alpha}=:\beta$ and $c\geq 1$,
combining (4.17)–(4.20),

$\sum_{k=1}^{2}\binom{2}{k}\frac{1}{p+1-k}|\langle
Z_{t}^{k},\partial_{x}\rho_{w_{t}}\partial_{x}w_{t}^{p+1-k}\rangle|\lesssim\frac{c^{\beta}}{p}\|Z\|^{\frac{p+2}{\alpha}}_{C_{t}\mathcal{C}^{\alpha}}+\frac{c^{p}}{p^{2}}\mathfrak{m}+\frac{1}{c}B_{t}+\frac{1}{c}A_{t}.$
(4.21)

Since $\frac{1}{4(p-1)}\leq\frac{3/4}{p+1}$, combining (4.6) with (4.11),
(4.14), (4.21) and choosing $c\geq(p\vee\chi)C$ with $C>0$ the sufficiently
large implied constant, we obtain that

$\displaystyle\frac{1}{\chi
p(p-1)}\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}t}\|w_{t}\|^{p}_{L^{p}}\leq$
$\displaystyle\max\Big{\\{}\frac{(C(p\vee\chi))^{\beta}}{p}\|Z\|^{\frac{p+2}{\alpha}}_{C_{t}\mathcal{C}^{\alpha}},\,\frac{(C(p\vee\chi))^{p+1}\mathfrak{m}}{p^{2}}\Big{\\}}$
$\displaystyle-\frac{1}{4(p-1)}\|w^{p+2}_{t}\|_{L^{1}}.$

By Jensen’s
inequality,$(\|w_{t}\|^{p}_{L^{p}})^{\frac{p+2}{p}}\leq\|w^{p+2}_{t}\|_{L^{1}}$,
so that for $C>0$ sufficiently large

$\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}t}\|w_{t}\|^{p}_{L^{p}}+\frac{\chi
p}{4}\left(\|w_{t}\|^{p}_{L^{p}}\right)^{\frac{p+2}{p}}\leq\chi\max\Big{\\{}p(C(p\vee\chi))^{\beta}\|Z\|^{\frac{p+2}{\alpha}}_{C_{t}\mathcal{C}^{\alpha}},\,(C(p\vee\chi))^{p+1}\mathfrak{m}\Big{\\}}.$

Applying Lemma 4.2 with $f(t)=\|w_{t}\|^{p}_{L^{p}}$, $\lambda=\frac{p+2}{p}$
and possibly increasing $C>0$,

$\|w_{t}(\zeta)\|^{p}_{L^{p}}\leq\max\Big{\\{}(\chi
t/4)^{-\frac{p}{2}},(C(p\vee\chi))^{\beta\frac{p}{p+2}}\|Z\|^{\frac{p}{\alpha}}_{C_{t}\mathcal{C}^{\alpha}},\,(C(p\vee\chi))^{\frac{p(p+1)}{p+2}}\mathfrak{m}/p^{\frac{p}{p+2}}\,\Big{\\}},$

for every $t\in(0,T]$ and $\zeta\in\mathcal{C}^{\alpha_{0}}$. The stated
bound, (4.5), then follows after taking the $p^{\text{th}}$ root on both sides
and noticing that $\gamma=\frac{\beta}{p+2}\leq\frac{4+3\alpha}{4\alpha}$. ∎

######  Remark 4.5.

Observe that the bound (4.5) is trivial in the limit $p\rightarrow\infty$,
since $\gamma\geq 1$ and so $(p\vee\chi)^{\gamma}\rightarrow\infty$. This is
in contrast to [20], where an a priori bound of the same form in
$L^{\infty}(\mathbb{T})$ is shown directly for stochastic reaction-diffusions.

######  Remark 4.6.

In [20], the equivalent of (4.5), in the context of stochastic reaction-
diffusion equations, is established with
$\|Z\|^{1/\alpha}_{C_{T}\mathcal{C}^{\alpha}}$ replaced by
$\|Z\|^{1/(1+\alpha)}_{C_{T}\mathcal{C}^{\alpha}}$. This leads to lighter than
Gaussian tail bounds in the case of the reaction-diffusion equation, again
contrasted with the heavier than Gaussian tails that we are able to establish
for (1.1), see Theorem 1.4.

######  Remark 4.7.

While a mild generalisation of this analysis to $f(u)$, smooth and
asymptotically quadratic ($c_{1}u^{2}<f(u)<c_{2}u^{2}$), would likely be
possible, more natural generalisations such as $f(u)=|u|$ or $f(u)=u^{m-1}$
for $m\geq 5$ and odd, seem to present significant challenges. In the case of
$f(u)=|u|$, since the testing argument we follow only applies to the remainder
$w=u-Z$, the nonlinearity becomes

$\partial_{x}(|w_{t}+Z_{t}|\partial_{x}\rho_{w_{t}+Z_{t}})=\partial_{x}(|w_{t}+Z_{t}|)\partial_{x}\rho_{w_{t}+Z_{t}}-|w_{t}+Z_{t}|(w_{T}-\bar{w}+Z_{t}).$

When testing with $w_{t}^{p-1}$ this no longer produces the necessary damping
term $-\|w_{t}^{p+2}\|_{L^{1}}$ and so it is not directly clear how one could
proceed in this case. Regarding higher polynomial powers, it appears that
controlling the transport term causes a problem for $m>3$. In particular
estimating the equivalent of (4.15) in the same way, leads us to control
$|\langle\partial_{x}Z,w^{p+m-2}\partial_{x}\rho_{w}\rangle|$. Repeating
similar estimates as in (4.16) and (4.17) lead to the condition
$m<\frac{3-\alpha}{1-\alpha}$. For $\alpha\in(0,1/2)$ this restricts us to
$m<5$. Letting $\alpha\rightarrow 1$ would allow $m\rightarrow\infty$.
Informally speaking, integrability of the solution resulting from the
transport term appears to be linked to regularity of the noise - higher
regularity leads to higher integrability. This was also observed in the case
of stochastic reaction-diffusion equations in [20]. Note also that with
$\alpha\geq 1$ one could apply the testing argument directly to $u$, rather
than working with the remainder. In [17] it was in fact shown that for the
same model but with $\xi\equiv 0$, global existence and convergence to
equilibrium holds for all $m>0$. In the case of more regular noise $v\in
C_{T}\mathcal{C}^{1}(\mathbb{T})$, we would therefore expect at least an
analogue of Theorem 4.4 to hold directly for $u$ also for all $m>0$.

######  Proposition 4.8.

Let $\mathfrak{m}\in\\{0,1\\}$,
$\zeta\in\mathcal{C}^{\alpha_{0}}_{\mathfrak{m}}(\mathbb{T})$ and $Z\in
C_{T}\mathcal{C}^{\alpha}_{0}$. Then there exists a unique mild solution
$w(\zeta)\in C_{\eta;T}\mathcal{C}^{\alpha}$ to (3.1). Furthermore, the map,

$\displaystyle\mathscr{S}:\mathcal{C}^{\alpha_{0}}_{\mathfrak{m}}(\mathbb{T})\times
C_{T}\mathcal{C}^{\alpha}_{0}(\mathbb{T})$ $\displaystyle\rightarrow
C_{\eta;T}\mathcal{C}^{\alpha}_{\mathfrak{m}}(\mathbb{T})$ (4.22)
$\displaystyle(\zeta,Z)$ $\displaystyle\mapsto u(\zeta):=w(\zeta)+Z,$

is jointly, locally Lipschitz.

###### Proof.

We recall from Theorem 3.4 that there exists a $T_{*}\in(0,1)$, depending only
on $\|\zeta\|_{\mathcal{C}^{\alpha_{0}}}$ and
$\|Z\|_{C_{T}\mathcal{C}^{\alpha}}$, such that a mild solution $w\in
C_{\eta;T_{*}}\mathcal{C}^{\alpha}$ exists to $\eqref{eq:1dRepRemainder}$.
Without loss of generality let us assume $T>T_{*}$ and that we fix an even
$p\in(-\frac{1}{\alpha_{0}},\infty)$ so that
$L^{p}(\mathbb{T})\hookrightarrow\mathcal{C}^{\alpha_{0}}(\mathbb{T})$. In
this case, it is clear that we can extend the solution for a positive time of
existence so long as $\|w_{t}\|_{L^{p}}$ remains finite. However, by Theorem
4.4, $\|w_{t}\|_{L^{p}}$ is bounded above by a function of $t$ independent of
the initial data and so we may continue the solution to all of $[0,T]$. From
Corollary 3.7, for $u_{t}:=w_{t}+Z_{t}$, we have
$\bar{u}_{t}=\bar{\zeta}+\bar{Z}_{t}$, for all $t\in(0,T]$. Similarly, for all
$t>0$,
$\|u_{t}\|_{\mathcal{C}^{\alpha}}\leq\|w_{t}\|_{\mathcal{C}^{\alpha}}+\|Z_{t}\|_{\mathcal{C}^{\alpha}}$.
Hence the solution map (4.22) is well-defined and
$\|u\|_{C_{\eta,T}\mathcal{C}^{\alpha}}$ depends only on
$\|Z\|_{C_{T}\mathcal{C}^{\alpha}}$.

To continue the proof, we state the following

######  Lemma 4.9.

Consider $\mathfrak{R}>0$ and define the set

$D_{\mathfrak{R}}:=\\{(\zeta,Z)\in\mathcal{C}^{\alpha_{0}}_{\mathfrak{m}}\times
C_{T}\mathcal{C}^{\alpha}\,:\,\|\zeta\|_{\mathcal{C}^{\alpha_{0}}}+\|Z\|_{C_{T}\mathcal{C}^{\alpha}}<\mathfrak{R}\\}.$

Then for $T_{*}=T_{*}(\mathfrak{R})>0$ sufficiently small, $\mathscr{S}\colon
D_{\mathfrak{R}}\to C_{\eta}((0,2T_{*}],\mathcal{C}^{\alpha}_{\mathfrak{m}})$
is $K$-Lipschitz where $K=K(\mathfrak{R})>0$.

###### Proof.

Let $(\zeta,Z),(\tilde{\zeta},\tilde{Z})\in D_{\mathfrak{R}}$ and consider the
corresponding solutions

$\displaystyle u_{t}$
$\displaystyle=e^{t\Delta}\zeta+\int_{0}^{t}e^{(t-s)\Delta}\chi\partial_{x}(u^{2}_{s}\partial_{x}\rho_{u_{s}})\,\mathop{}\\!\mathrm{d}s+Z_{t},$
$\displaystyle\tilde{u}_{t}$
$\displaystyle=e^{t\Delta}\tilde{\zeta}+\int_{0}^{t}e^{(t-s)\Delta}\chi\partial_{x}(\tilde{u}^{2}_{s}\partial_{x}\rho_{\tilde{u}_{s}})\,\mathop{}\\!\mathrm{d}s+\tilde{Z}_{t}.$

From Theorem 3.4, there exists $T_{*}(\mathfrak{R})>0$ such that
$\|u\|_{C_{\eta;2T_{*}}\mathcal{C}^{\alpha}}\vee\,\|\tilde{u}\|_{C_{\eta;2T_{\star}}\mathcal{C}^{\alpha}}\leq
2$. For $t\in(0,2T_{*}]$, using Theorem A.6 and similar bounds as in the proof
of Theorem 3.4 we see that

$\displaystyle\|u_{t}-\tilde{u}_{t}\|_{\mathcal{C}^{\alpha}}$
$\displaystyle\lesssim
t^{-\frac{\alpha-\alpha_{0}}{2}}\|\zeta-\tilde{\zeta}\|_{\mathcal{C}^{\alpha_{0}}}+t^{\frac{1}{2}-3\eta}\|u-\tilde{u}\|_{C_{\eta;t}\mathcal{C}^{\alpha}}+\|Z_{t}-\tilde{Z}_{t}\|_{\mathcal{C}^{\alpha}},$

where the proportionality constant does not depend on
$\zeta,\tilde{\zeta},Z,\tilde{Z}$. So multiplying through by $t^{\eta}$ and
taking the supremum over $t\in(0,2T_{*}]$ we have that

$\|u-\tilde{u}\|_{C_{\eta;2T_{*}}\mathcal{C}^{\alpha}}\lesssim\|\zeta-\tilde{\zeta}\|_{\mathcal{C}^{\alpha_{0}}}+T_{*}^{\frac{1}{2}-2\eta}\|u-\tilde{u}\|_{C_{\eta;S}\mathcal{C}^{\alpha}}+\|Z-\tilde{Z}\|_{C_{T}\mathcal{C}^{\alpha}}.$

By lowering $T_{*}$ further, we obtain

$\|u-\tilde{u}\|_{C_{\eta;2T_{*}}\mathcal{C}^{\alpha}}\lesssim\|\zeta-\tilde{\zeta}\|_{\mathcal{C}^{\alpha_{0}}}+\|Z-\tilde{Z}\|_{C_{T}\mathcal{C}^{\alpha}}.\qed$

We now prove that $\mathscr{S}$ is jointly locally Lipschitz. Consider
$\mathfrak{R}>1$, $D_{\mathfrak{R}}$, and $T_{*}(\mathfrak{R})>0$ as in Lemma
4.9. Then $\mathscr{S}$ is $K(\mathfrak{R})$-Lipschitz from $D_{\mathfrak{R}}$
to $C_{\eta}((0,2T_{*}],\mathcal{C}^{\alpha})$. In particular, the map
$(\zeta,Z)\mapsto u|_{[T_{*},2T_{*}]}\in
C([T_{*},2T_{*}],\mathcal{C}^{\alpha})$ is $K(\mathfrak{R})$-Lipschitz. If
$T\leq 2T_{*}$, then we are done. Hence, suppose $T>2T_{*}$. We will prove
that, for $\bar{T}_{*}(\mathfrak{R})\in(0,T_{*}]$ sufficiently small,
$u|_{[T_{*}+\bar{T}_{*},T]}\in C([\bar{T}_{*},T],\mathcal{C}^{\alpha})$ is a
Lipschitz function of $(\zeta,Z)$, from which the conclusion will follow.

To this end, consider $(\zeta,Z)\in D_{\mathfrak{R}}$ and let $u$ be the
corresponding solution. By the a priori estimate, for $p\geq 2$ sufficiently
large
$\|u_{t}\|_{\mathcal{C}^{\alpha_{0}}}\lesssim_{\alpha_{0},p}\|u_{t}\|_{L^{p}}\lesssim
t^{-1/2}\vee\mathfrak{R}^{1/\alpha}$ for all $t>0$. Therefore, there exists
$\bar{\mathfrak{R}}(\mathfrak{R})>0$ such that
$\|u_{t}\|_{\mathcal{C}^{\alpha_{0}}}\leq\bar{\mathfrak{R}}$ for all
$t\in[T_{*},T]$.

Let $\bar{T}_{*}(\bar{\mathfrak{R}})>0$ be the constant from Lemma 4.9.
Without loss of generality, we can assume that $\bar{T}_{*}\leq T_{*}$ and
that $T=T_{*}+N\bar{T}_{*}$ for some integer $N\geq 2$. By the above remark,
for all $1\leq n<N$, $u|_{[T_{*}+n\bar{T}_{*},T_{*}+(n+1)\bar{T}_{*}]}\in
C([T_{*}+n\bar{T}_{*},T_{*}+(n+1)\bar{T}_{*}],\mathcal{C}^{\alpha})$ is a
$K(\mathfrak{R})$-Lipschitz function of
$(u_{T_{*}+(n-1)\bar{T}_{*}},Z)\in\mathcal{C}^{\alpha_{0}}\times
C_{T}\mathcal{C}^{\alpha}$. Combining these estimates together, it follows
that $u|_{[T_{*}+\bar{T}_{*},T]}\in C([\bar{T}_{*},T],\mathcal{C}^{\alpha})$
is a Lipschitz function of $(\zeta,Z)$. ∎

We conclude this section with the proof of Theorem 1.1.

###### Proof of Theorem 1.1.

From Theorem 2.3, for any $T>0$ and $\alpha<\frac{1}{2}$, $v_{0,\,\cdot\,}\in
C_{T}\mathcal{C}_{0}^{\alpha}$ (in fact
$v_{0,\,\cdot\,}\in\cap_{\kappa\in[0,1/2)}\mathcal{C}^{\kappa}_{T}\mathcal{C}_{0}^{\alpha-2\kappa}$).
Hence Proposition 4.8 implies that $\mathbb{P}$-a.s. there exists a unique
mild solution
$u(\zeta):=w(\zeta)+v_{0,\,\cdot\,}=\mathscr{S}(\zeta,v_{0,\,\cdot\,})$ to
(1.1) on $[0,T]$. ∎

##  5 Invariant Measure

Having established global well-posedness of the SPDE (1.1) we turn to study
its long time behaviour. In this section we prove Theorem 1.4 which implies
existence and uniqueness of the invariant measure and demonstrates exponential
convergence of the adjoint semi-group to the invariant measure in the topology
of total variation.

By Remark 4.3, appropriately adjusting the spatial mean $\mathfrak{m}$ of the
initial condition $\zeta$, it suffices to prove Theorem 1.4 in the case
$\chi=1$. We thus assume $\chi=1$ throughout this section.

######  Definition 5.1.

Let $v_{t_{0},\,\cdot\,}$ denote the mild solution to the stochastic heat
equation as in Definition 2.2 and denote $v_{t}:=v_{0,t}$. For
$\zeta\in\mathcal{C}^{\alpha_{0}}$, the process
$u(\zeta):=w(\zeta)+v_{\cdot}=\mathscr{S}(\zeta,v_{\cdot})$, with
$\mathscr{S}$ as defined in Proposition 4.8, is called the mild solution on
$[0,T]$ to (1.1) with initial condition $\zeta$.

We define the semi-group associated to (1.1) by setting, for all $t\geq 0$,

$P_{t}\Phi(\zeta):=\mathbb{E}\left[\Phi(u_{t}(\zeta))\right],\quad\text{ for
all }\Phi\in\mathcal{B}_{b}(\mathcal{C}^{\alpha_{0}}(\mathbb{T})).$

We first show that $(u_{t})_{t>0}$ is a Markov process with Feller semi-group
$(P_{t})_{t>0}$ and that for each
$\zeta\in\mathcal{C}^{\alpha_{0}}_{\mathfrak{m}}(\mathbb{T})$, there exists a
measure $\nu_{\zeta}\in\mathcal{P}(\mathcal{C}^{\alpha_{0}}_{\mathfrak{m}})$,
invariant for $(P_{t})_{t>0}$.

######  Lemma 5.2.

Let $u$ be a mild solution to (1.1). Then for every $t\in[0,T]$ and
$h\in(0,T-t)$ we have the identity

$u_{t+h}=\tilde{w}_{t,t+h}+v_{t,t+h},$ (5.1)

where $\tilde{w}_{t,t+h}$ solves

$\tilde{w}_{t,t+h}=e^{h\Delta}u_{t}+\int_{0}^{h}e^{(h-r)\Delta}\partial_{x}((\tilde{w}_{t,t+r}+v_{t,t+r})^{2}\partial_{x}\rho_{\tilde{w}_{t,t+r}+v_{t,t+r}})\,\mathop{}\\!\mathrm{d}r.$
(5.2)

###### Proof.

A simple consequence of the heat semi-group property. ∎

######  Theorem 5.3.

For $\mathfrak{m}\in\mathbb{R}$, and
$\zeta\in\mathcal{C}^{\alpha_{0}}_{\mathfrak{m}}(\mathbb{T})$, let
$u(\zeta)\in C_{\eta;T}\mathcal{C}^{\alpha}$ be the unique solution to (1.1)
as in Definition 5.1. Then for every $p\in[1,\infty)$, we have that

$\sup_{\zeta\in\mathcal{C}^{\alpha_{0}}_{\mathfrak{m}}(\mathbb{T})}\sup_{T>0}\sup_{t\in(0,T]}\left(t^{\frac{p}{2}}\wedge
1\right)\mathbb{E}\left[\|u_{t}(\zeta)\|^{p}_{L^{p}}\right]<\infty.$ (5.3)

###### Proof.

It suffices to consider $T>1$. Consider first $p\geq 2$ even. For $t\in(0,1]$,
applying (4.5) and Theorem 2.3,

$\mathbb{E}\left[\|u_{t}\|^{p}_{L^{p}}\right]\lesssim\mathbb{E}\left[\|w_{t}\|_{L^{p}}^{p}\right]+\mathbb{E}\left[\|v_{t}\|_{L^{p}}^{p}\right]\lesssim
t^{-\frac{p}{2}}+1.$ (5.4)

For $t\in(1,T]$ we employ the Markov decomposition (5.1) to give

$\mathbb{E}\left[\|u_{t}\|_{L^{p}}^{p}\right]\lesssim\mathbb{E}\left[\|\tilde{w}_{t-1,t}\|_{L^{p}}^{p}\right]+\mathbb{E}\left[\|v_{t-1,t}\|_{L^{p}}^{p}\right].$

Then, observing that $\tilde{w}_{t-1,t}$ solves (3.1) with initial condition
$u_{t-1}(\zeta)$ and driving path $Z_{t}=v_{t-1,t}$, by Theorem 4.4 we have

$\mathbb{E}\left[\|\tilde{w}_{t-1,t}\|_{L^{p}}^{p}\right]\leq
C_{p,\mathfrak{m}}\max\Big{\\{}\|v_{t-1,\,\cdot\,}\|^{\frac{p}{\alpha}}_{C_{[t-1,t]}\mathcal{C}^{\alpha}},\,1\,\Big{\\}}.$

So, since the law of $v_{t-1,t}$ does not depend on $t>1$, we have

$\sup_{\zeta\in\mathcal{C}^{\alpha_{0}}_{\mathfrak{m}}(\mathbb{T})}\sup_{T>0}\sup_{t\in(1,T]}\mathbb{E}\left[\|u_{t}(\zeta)\|_{L^{p}}^{p}\right]<\infty.$
(5.5)

Combining (5.4) with (5.5) proves (5.3) for even $p\geq 2$. The case of
general $p\geq 1$ follows by Jensen’s inequality. ∎

###  5.1 Existence of Invariant Measures

We use the decomposition (5.1) to show that $t\mapsto u_{t}$ defines a Markov
process.

######  Theorem 5.4.

Let $\zeta\in\mathcal{C}^{\alpha_{0}}(\mathbb{T})$ and $u_{t}(\zeta)$ as in
Definition 5.1. Then for any
$\Phi\in\mathcal{B}_{b}(\mathcal{C}^{\alpha_{0}}(\mathbb{T}))$

$\mathbb{E}\left[\Phi(u_{t+h}(\zeta))\,\big{|}\,\mathcal{F}_{t}\right]=P_{h}(\Phi(u_{t}(\zeta)).$

In particular, $t\mapsto u_{t}$ is a Markov process with associated Markov
semi-group $(P_{t})_{t>0}$.

###### Proof.

Using the Markov decomposition (5.1) we see that

$\Phi(u_{t+h}(\zeta))=\Phi(\tilde{w}_{t,t+h}(u_{t}(\zeta))+v_{t,t+h})=\Phi(u_{t,t+h}(u_{t}(\zeta))),$

where $\tilde{w}_{t,t+h}(u_{t}(\zeta))$ solves (5.2). We define

$\psi(u_{t}(\zeta),v_{t,t+h}):=\Phi(\tilde{w}_{t,t+h}(u_{t}(\zeta))+v_{t,t+h})$

and then since $u_{t}(\zeta)$ is $\mathcal{F}_{t}$ measurable and
$v_{t,t+h}\perp\mathcal{F}_{t}$, applying [6, Prop. 1.12], we have that

$\displaystyle\mathbb{E}\left[\Phi(u_{t+h}(\zeta))\,\big{|}\,\mathcal{F}_{t}\right]=\mathbb{E}[\Phi(u_{t,t+h}(u_{t}(\zeta)))]=P_{h}(\Phi(u_{t}(\zeta)))$

and so the claim is shown. ∎

The following lemma shows that $(P_{t})_{t>0}$ is a Feller semi-group in the
sense of [5, Sec. 3.1].

######  Lemma 5.5.

For any $\Phi\in\mathcal{C}_{b}(\mathcal{C}^{\alpha_{0}}(\mathbb{T}))$ and
$t>0$ we have that
$P_{t}\Phi\in\mathcal{C}_{b}(\mathcal{C}^{\alpha_{0}}(\mathbb{T}))$ and for
any $\zeta\in\mathcal{C}^{\alpha_{0}}$, $\lim_{t\rightarrow
0}|P_{t}\Phi(\zeta)-\Phi(\zeta)|=0$.

###### Proof.

By Proposition 4.8, $\mathcal{C}^{\alpha_{0}}(\mathbb{T})\ni\zeta\mapsto
u_{t}(\zeta)\in\mathcal{C}^{\alpha}(\mathbb{T})$ is $\mathbb{P}$-a.s.
continuous for every $t>0$. The fact that
$P_{t}\Phi\in\mathcal{C}_{b}(\mathcal{C}^{\alpha_{0}}(\mathbb{T}))$ thus
follows from the dominated convergence theorem. The fact that $\lim_{t\searrow
0}|P_{t}\Phi(\zeta)-\Phi(\zeta)|=0$ for all
$\zeta\in\mathcal{C}^{\alpha_{0}},\,\Phi\in\mathcal{C}_{b}(\mathcal{C}^{\alpha_{0}}(\mathbb{T}))$,
similarly follows the dominated convergence theorem and (3.9). ∎

We are now in a position to show the existence of an invariant measure,
$\nu_{\zeta}\in\mathcal{P}(\mathcal{C}^{\alpha_{0}}_{\mathfrak{m}})$ for every
$\zeta\in\mathcal{C}^{\alpha_{0}}_{\mathfrak{m}}(\mathbb{T})$.

######  Theorem 5.6.

Let $\mathfrak{m}\in\mathbb{R}$. Then for every
$\zeta\in\mathcal{C}^{\alpha_{0}}_{\mathfrak{m}}(\mathbb{T})$, there exists a
measure
$\nu_{\zeta}\in\mathcal{P}(\mathcal{C}^{\alpha_{0}}_{\mathfrak{m}}(\mathbb{T}))$
and an increasing sequence of times $t_{k}\nearrow\infty$ such that,

$\frac{1}{t_{k}}\int_{0}^{t_{k}}P^{*}_{r}\,\delta_{\zeta}\,\mathop{}\\!\mathrm{d}r\rightharpoonup\nu_{\zeta},\quad\text{
as }k\rightarrow\infty.$

Furthermore $\nu_{\zeta}$ is invariant for $(P_{t})_{t>0}$.

###### Proof.

By the compact embedding $L^{p}\hookrightarrow\mathcal{C}^{\alpha_{0}}$ for
sufficiently large $p\geq 1$, it follows from Theorem 5.3 that the family of
measures $\\{P^{*}_{t}\delta_{\zeta}\\}_{t\geq
1,\zeta\in\mathcal{C}^{\alpha_{0}}(\mathbb{T})}$ is tight. In particular, for
every sequence of times $(t_{k})_{k>0}$,
$\left(R_{t_{k}}^{*}\delta_{\zeta}\right)_{k>0}:=\left(\frac{1}{t_{k}}\int_{0}^{t_{k}}P^{*}_{r}\,\delta_{\zeta}\,\mathop{}\\!\mathrm{d}r\right)_{k>0}$
is a tight family of measures. Applying Prokhorov’s theorem gives the
existence of a sequence $(t_{k})_{k>0}$ for which the necessary weak
convergence holds. From [5, Thm. 3.1.1] we see that the limit measure is
necessarily invariant for $(P_{t})_{t>0}$. ∎

###  5.2 Strong Feller Property

We prove that $(P_{t})_{t>0}$ possesses the strong Feller property closely
following the method employed in [34]. The main step is to establish a
Bismut–Elworthy–Li (BEL) type formula, similar to that given in [34, Thm.
5.5].

For technical reasons in this section we replace (1.3) with the assumption
that $\alpha_{0}\in\left(-\frac{1}{3},0\right)$,
$\alpha\in\left(0,\alpha_{0}+\frac{1}{3}\right)$ and $\eta>0$ such that,

$\frac{\alpha-\alpha_{0}}{2}<\eta<\frac{1}{6}.$ (5.6)

We note that all previous results also hold for this more restrictive
parameter range and this restriction does not affect the main result as
stated, see the proof of Theorem 1.4 at the conclusion of Section 5.4. Let
$(e_{m})_{m\in\mathbb{Z}}$, $e_{m}(x)=e^{i2\pi mx}$, be the usual Fourier
basis elements of $L^{2}(\mathbb{T})$ and $(\Delta_{k})_{k\geq-1}$ be the
Littlewood–Paley projection operators, see Appendix A for more details. For
$\varepsilon\in(0,1)$, we define $\Pi_{\varepsilon}(L^{2}(\mathbb{T}))$ as the
space of real functions spanned by
$\left\\{(e_{m})_{|m|<\frac{1}{\varepsilon}}\right\\}$ and

$\hat{\Pi}_{\varepsilon}:=\sum_{-1\leq
k\leq-\log_{2}(9\varepsilon)}\Delta_{k}\colon
L^{2}(\mathbb{T})\to\Pi_{\varepsilon}(L^{2}(\mathbb{T}^{d})).$

Observe that there exists $\ell_{\varepsilon}\colon\mathbb{Z}\to[0,1]$ with
$\ell_{\varepsilon}(m)=0$ for $|m|\geq\frac{1}{\varepsilon}$ such that
$\mathcal{F}(\hat{\Pi}_{\varepsilon}f)(m)=\ell_{\varepsilon}(m)\mathcal{F}f(m)$
where $\mathcal{F}$ is the Fourier transform. Furthermore (see e.g. [34, p.
1213]),

  1. i)

$\displaystyle\|\hat{\Pi}_{\varepsilon}\|_{\text{Op}(\mathcal{C}^{\beta};\,\mathcal{C}^{\beta})}:=\sup_{\|u\|_{\mathcal{C}^{\beta}}\leq
1}\|\hat{\Pi}_{\varepsilon}u\|_{\mathcal{C}^{\beta}}\leq 1$ for all
$\varepsilon\in(0,1)$ and $\beta\in\mathbb{R}$,

  2. ii)

for every $\beta\in\mathbb{R}$ and $\delta>0$, there exists $C>0$ such that
for all $\varepsilon\in(0,1)$

$\|\hat{\Pi}_{\varepsilon}f-f\|_{\mathcal{C}^{\beta-\delta}}\leq
C\varepsilon^{\delta}\|f\|_{\mathcal{C}^{\beta}}.$

We fix for the rest of the subsection $\delta>0$

$\alpha+\delta<\frac{1}{2},\quad\eta-\frac{\alpha-\alpha_{0}+\delta}{2}>0,\quad\frac{1-\delta}{2}-3\eta>0.$
(5.7)

Such a $\delta$ exists due to (5.6).

Consider now $Z\in C_{T}\mathcal{C}^{\alpha+\delta}$ and let
$Z_{\varepsilon;t}:=\hat{\Pi}_{\varepsilon}Z_{t}$. Consider likewise
$\zeta\in\mathcal{C}^{\alpha_{0}}$ and let
$\zeta_{\varepsilon}:=\hat{\Pi}_{\varepsilon}\zeta$. We define a smooth
approximation to (1.1), with $\chi=1$ and deterministic noise, by

$u_{\varepsilon;t}=e^{t\Delta}\zeta_{\varepsilon}+\int_{0}^{t}e^{(t-s)\Delta}\partial_{x}\hat{\Pi}_{\varepsilon}(u^{2}_{\varepsilon;s}\partial_{x}\rho_{u_{\varepsilon;s}})\mathop{}\\!\mathrm{d}s+Z_{\varepsilon;t},$
(5.8)

where, as usual,

$-\partial_{xx}\rho_{u_{\varepsilon;t}}=u_{\varepsilon;t}-\bar{u}_{\varepsilon},\text{
on }\mathbb{T}.$

######  Remark 5.7.

We will later set, right before Theorem 5.13,
$Z_{\varepsilon}=\hat{\Pi}_{\varepsilon}v$, where $v$ is the SHE.

######  Theorem 5.8.

Let $T>0$, $\zeta\in\mathcal{C}^{\alpha_{0}}_{\mathfrak{m}}(\mathbb{T})$, and
$Z\in C_{T}\mathcal{C}^{\alpha+\delta}$. For every $\bar{\varepsilon}>0$,
there exists
$\varepsilon_{0}(\bar{\varepsilon},T,\|\zeta\|_{\mathcal{C}^{\alpha_{0}}}+\|Z\|_{C_{T}\mathcal{C}^{\alpha+\delta}})>0$
such that, for every $\varepsilon\in(0,\varepsilon_{0})$ there exists a unique
solution $u_{\varepsilon}\in C_{T}\Pi_{\varepsilon}(L^{2}(\mathbb{T}))$ to
(5.8). Furthermore

$\|u_{t}-u_{\varepsilon;t}\|_{C_{\eta;T}\mathcal{C}^{\alpha}}\leq\bar{\varepsilon}.$
(5.9)

###### Proof.

It follows from Properties i) and ii) of the projection and the same steps as
in the proof of Theorem 3.4 that for every $\varepsilon>0$ there exists
$T_{\varepsilon}(\|\zeta_{\varepsilon}\|_{\mathcal{C}^{\alpha_{0}}}+\|Z_{\varepsilon}\|_{C_{T}\mathcal{C}^{\alpha}})>0$,
such that $u_{\varepsilon}\in C_{\eta;T_{\varepsilon}}\mathcal{C}^{\alpha}$
solves (5.8) and satisfies
$\|u_{\varepsilon}\|_{C_{\eta;T_{\varepsilon}}\mathcal{C}^{\alpha}}\leq 1$.
The fact that $u_{\varepsilon}\in
C_{T_{\varepsilon}}\Pi_{\varepsilon}(L^{2}(\mathbb{T}))$ follows from the same
argument as the end of Theorem 3.4 since $\zeta_{\varepsilon}\in C^{\infty}$.

To continue, we state the following

######  Lemma 5.9.

Consider $\mathfrak{R}>0$ and define $D_{\mathfrak{R}}$ as in Lemma 4.9. There
exists $T_{*}(\mathfrak{R})>0$ such that, if
$(\zeta_{\varepsilon},Z_{\varepsilon}),(\tilde{\zeta},\tilde{Z})\in
D_{\mathfrak{R}}$, then

$\|\tilde{u}_{t}-u_{\varepsilon;t}\|_{C_{\eta;T_{*}}\mathcal{C}^{\alpha}}\leq\|\zeta_{\varepsilon}-\tilde{\zeta}\|_{\mathcal{C}^{\alpha_{0}-\delta}}+\varepsilon^{\delta}+\|\tilde{Z}-Z_{\varepsilon}\|_{C_{T}\mathcal{C}^{\alpha}},$

where $\tilde{u}=\mathscr{S}(\tilde{\zeta},\tilde{Z})$ with $\mathscr{S}$ as
in Proposition 4.8.

###### Proof.

There exists $T_{*}(\mathfrak{R})>0$ such that
$\|\tilde{u}\|_{C_{\eta;2T_{*}}\mathcal{C}^{\alpha}}+\|u_{\varepsilon}\|_{C_{\eta;2T_{\star}}\mathcal{C}^{\alpha}}\leq
2$. For $t\in(0,2T_{*}]$,

$\displaystyle\|\tilde{u}_{t}-u_{\varepsilon;t}\|_{\mathcal{C}^{\alpha}}$
$\displaystyle\leq\|e^{t\Delta}(\tilde{\zeta}-\zeta_{\varepsilon})\|_{\mathcal{C}^{\alpha}}+\int_{0}^{t}\|e^{(t-s)\Delta}\partial_{x}(\tilde{u}^{2}_{s}\partial_{x}\rho_{\bar{u}_{s}}-\hat{\Pi}_{\varepsilon}(u^{2}_{\varepsilon;s}\partial_{x}\rho_{u_{\varepsilon;s}}))\|_{\mathcal{C}^{\alpha}}\,\mathop{}\\!\mathrm{d}s$
$\displaystyle\quad+\|\tilde{Z}_{t}-Z_{\varepsilon;t}\|_{\mathcal{C}^{\alpha}}.$

Then

$\|e^{t\Delta}(\tilde{\zeta}-\zeta_{\varepsilon})\|_{\mathcal{C}^{\alpha}}\lesssim
t^{-\frac{\alpha-\alpha_{0}+\delta}{2}}\|\tilde{\zeta}-\zeta_{\varepsilon}\|_{\mathcal{C}^{\alpha_{0}-\delta}},$

and applying the triangle inequality and using Property ii) of the projection,

$\displaystyle\|\tilde{u}_{s}^{2}\partial_{x}\rho_{\bar{u}_{s}}-\hat{\Pi}_{\varepsilon}(u^{2}_{\varepsilon;s}\partial_{x}\rho_{u_{\varepsilon;s}})\|_{\mathcal{C}^{\alpha-\delta}}$
$\displaystyle\leq\|\tilde{u}_{s}^{2}\partial_{x}\rho_{\tilde{u}_{s}}-u^{2}_{\varepsilon;s}\partial_{x}\rho_{u_{\varepsilon;s}}\|_{\mathcal{C}^{\alpha}}$
$\displaystyle\qquad+\|u^{2}_{\varepsilon;s}\partial_{x}\rho_{u_{\varepsilon;s}}-\hat{\Pi}_{\varepsilon}(u^{2}_{\varepsilon;s}\partial_{x}\rho_{u_{\varepsilon;s}})\|_{\mathcal{C}^{\alpha-\delta}}$
$\displaystyle\lesssim
s^{-2\eta}\|\tilde{u}_{s}-u_{\varepsilon;s}\|_{\mathcal{C}^{\alpha}}+\varepsilon^{\delta}s^{-3\eta}$
$\displaystyle\leq
s^{-3\eta}\|\tilde{u}-u_{\varepsilon}\|_{C_{\eta;s}\mathcal{C}^{\alpha}}+\varepsilon^{\delta}s^{-3\eta}.$

Therefore, using that $\eta<\frac{1}{6}$, we may integrate
$\|e^{(t-s)\Delta}\partial_{x}(\tilde{u}^{2}_{s}\partial_{x}\rho_{\bar{u}_{s}}-\hat{\Pi}_{\varepsilon}(u^{2}_{\varepsilon;s}\partial_{x}\rho_{u_{\varepsilon;s}}))\|_{\mathcal{C}^{\alpha}}$
from $0$ to $t$ and take suprema to obtain

$\displaystyle\sup_{t\in(0,T_{*}]}t^{\eta}\|\tilde{u}_{s}-u_{\varepsilon;s}\|_{\mathcal{C}^{\alpha}}$
$\displaystyle\lesssim
T_{*}^{\eta-\frac{\alpha-\alpha_{0}-\delta}{2}}\|\tilde{\zeta}-\zeta_{\varepsilon}\|_{\mathcal{C}^{\alpha_{0}-\delta}}+T_{*}^{\frac{1-\delta}{2}-3\eta}\sup_{t\in(0,T_{*}]}t^{\eta}\|\tilde{u}_{t}-u_{\varepsilon;t}\|_{\mathcal{C}^{\alpha}}$
$\displaystyle\quad\quad+T_{*}^{\frac{1-\delta}{2}-2\eta}\varepsilon^{\delta}+T_{*}^{\eta}\|\tilde{Z}-Z_{\varepsilon}\|_{C_{T}\mathcal{C}^{\alpha}}.$

After potentially lowering $T_{*}(\mathfrak{R})>0$ further, we obtain

$\sup_{t\in(0,T_{*}]}t^{\eta}\|\tilde{u}_{t}-u_{\varepsilon;t}\|_{\mathcal{C}^{\alpha}}\leq\|\tilde{\zeta}-\zeta_{\varepsilon}\|_{\mathcal{C}^{\alpha_{0}-\delta}}+\varepsilon^{\delta}+\|\tilde{Z}-Z_{\varepsilon}\|_{C_{T}\mathcal{C}^{\alpha}}.\qed$

To conclude the proof of Theorem 5.8, observe that
$\|\zeta-\Pi_{\varepsilon}\zeta\|_{\mathcal{C}^{\alpha_{0}-\delta}}\lesssim\varepsilon^{\delta}\|\zeta\|_{\mathcal{C}^{\alpha_{0}}}$
and
$\|Z_{t}-Z_{\varepsilon;t}\|_{\mathcal{C}^{\alpha}}\lesssim\varepsilon^{\delta}\|Z_{t}\|_{\mathcal{C}^{\alpha+\delta}}$
by Property ii). By the a priori estimate, for every $\mathfrak{R}>0$ there
exists $T_{*}(\mathfrak{R})>0$ as in Lemma 5.9 and
$\bar{\mathfrak{R}}(\mathfrak{R})>0$ such that
$\|u_{t}\|_{\mathcal{C}^{\alpha_{0}}}\leq\bar{\mathfrak{R}}$ for all
$t\in[T_{*},T]$. Iteratively applying Lemma 5.9 as in the proof of Proposition
4.8, there exists $\varepsilon_{0}(\mathfrak{R})>0$ sufficiently small such
that $\|u_{\varepsilon;t}\|_{\mathcal{C}^{\alpha}}\leq 2\bar{\mathfrak{R}}$ on
$[\bar{T}_{*},T]$ for all $\varepsilon<\varepsilon_{0}$, from which (5.9)
follows after another iterative application of Lemma 5.9. ∎

By Theorem 5.8, for every $\mathfrak{R}>0$ and $T>0$, there exists
$\varepsilon_{0}(\mathfrak{R},T)>0$ sufficiently small such that for all
$\varepsilon\in(0,\varepsilon_{0})$ the solution map to (5.8)

$\displaystyle\mathscr{S}^{\varepsilon}\colon A_{\mathfrak{R}}$
$\displaystyle\to C_{T}\Pi_{\varepsilon}(L^{2}(\mathbb{T}))$ (5.10)
$\displaystyle\left(\zeta,Z\right)$ $\displaystyle\mapsto u_{\varepsilon},$

is well-defined and
$\|u_{\varepsilon}-u\|_{C_{\eta;T}\mathcal{C}^{\alpha}}\leq 1$, where
$u=\mathscr{S}(\zeta,Z)$ and
$A_{\mathfrak{R}}:=\\{(\zeta,Z\\}\in\mathcal{C}^{\alpha_{0}}\times
C_{T}\mathcal{C}^{\alpha+\delta}\,:\,\|\zeta\|_{\mathcal{C}^{\alpha_{0}}}+\|Z\|_{C_{T}\mathcal{C}^{\alpha+\delta}}<\mathfrak{R}\\}$.

We now fix for the remainder of the section
$\zeta\in\mathcal{C}^{\alpha_{0}}_{\mathfrak{m}}$, $T>0$, and $Z\in
C_{T}\mathcal{C}^{\alpha+\delta}$ with $Z_{0}=0$. We let $\varepsilon_{0}>0$
sufficiently small such that $\mathscr{S}^{\varepsilon}(\zeta,Z)$ is well-
defined and $\|u_{\varepsilon}-u\|_{C_{\eta;T}\mathcal{C}^{\alpha}}\leq 1$ for
all $\varepsilon\in(0,\varepsilon_{0})$. In the sequel we will always let
$\varepsilon\in(0,\varepsilon_{0})$.

We define
$N_{\varepsilon}:=\\{m\in\mathbb{Z}\,:\,|m|<\frac{1}{\varepsilon},\;m\neq
0\\}$ and further assume that
$Z_{\varepsilon;t}:=\hat{\Pi}_{\varepsilon}Z_{t}$ is of the form

$Z_{\varepsilon;t}=\sum_{m\in
N_{\varepsilon}}\int_{0}^{t}\ell_{\varepsilon}(m)e^{-4\pi^{2}|m|^{2}(t-s)}\mathop{}\\!\mathrm{d}\hat{B}_{m;s}\,e_{m},$
(5.11)

where $\\{\hat{B}_{m}\\}_{m\in\mathbb{Z},m\neq 0}$ are functions
$\hat{B}_{m}\in C([0,T],\mathbb{C})$ which satisfy the reality condition

$\hat{B}_{-m}=\overline{\hat{B}_{m}}.$ (5.12)

We denote

$\hat{B}_{\varepsilon}:=(\hat{B}_{m})_{m\in
N_{\varepsilon}}\in\bar{C}_{T}\mathbb{C}^{N_{\varepsilon}},$

where $\bar{C}_{T}\mathbb{C}^{N_{\varepsilon}}\subset
C_{T}\mathbb{C}^{N_{\varepsilon}}$ is the subspace of $(\hat{B}_{m})_{m\in
N_{\varepsilon}}$ which satisfies the reality condition (5.12).

######  Remark 5.10.

We will henceforth identify $\bar{C}_{T}\mathbb{C}^{N_{\varepsilon}}$ with a
subspace of $C_{T}C^{\infty}_{0}\subset C_{T}\mathcal{C}^{\alpha+\delta}_{0}$
by mapping $(\hat{B}_{m})_{m\in N_{\varepsilon}}$ to $Z_{\varepsilon}$ via
(5.11). Observe that the integral in (5.11) is well-defined as a
Riemann–Stieltjes integral for any
$\hat{B}\in\bar{C}_{T}\mathbb{C}^{N_{\varepsilon}}$. For fixed
$\zeta\in\mathcal{C}^{\alpha_{0}}$, we will treat in this way
$u_{\varepsilon}(\zeta,Z):=\mathscr{S}^{\varepsilon}(\zeta,Z)$ as a function
of $\hat{B}_{\varepsilon}$ whenever it is well-defined.

We now prove the differentiability of $\mathscr{S}^{\varepsilon}$ with respect
to both arguments separately, using $\mathcal{D}$ for derivatives with respect
to $\hat{B}_{\varepsilon}$ and $D$ for derivatives with respect to $\zeta$.
For $\mathfrak{R}\geq 1$, we recall the definition of $T_{*}(\mathfrak{R})$
given by (3.7).

######  Lemma 5.11 (Derivative in Noise).

There exists an open neighbourhood
$\mathcal{O}_{\hat{B}_{\varepsilon}}\subset\bar{C}_{T}\mathbb{C}^{N_{\varepsilon}}$
containing $\hat{B}_{\varepsilon}$ such that $u_{\varepsilon}(\zeta,\cdot)$ is
Fréchet differentiable as a mapping from $\mathcal{O}_{\hat{B}_{\varepsilon}}$
to $C_{T}\Pi_{\varepsilon}L^{2}(\mathbb{T})$. Furthermore, for any
$f\in\bar{C}_{T}\mathbb{C}^{N_{\varepsilon}}$, such that $f|_{t=0}=0$, the
directional derivative $\mathcal{D}_{f}u_{\varepsilon}$ satisfies the equation

$\displaystyle\mathcal{D}_{f}u_{\varepsilon;t}$
$\displaystyle=\int_{0}^{t}e^{(t-s)\Delta}\partial_{x}\hat{\Pi}_{\varepsilon}\left(2u_{\varepsilon;s}\mathcal{D}_{f}u_{\varepsilon;s}\partial_{x}\rho_{u_{\varepsilon;s}}+u^{2}_{\varepsilon;s}\partial_{x}\rho_{\mathcal{D}_{f}u_{\varepsilon;s}}\right)\,\mathop{}\\!\mathrm{d}s$
(5.13) $\displaystyle\quad+\sum_{m\in
N_{\varepsilon}}\int_{0}^{t}\ell_{\varepsilon}(m)e^{-4\pi^{2}|m|^{2}(t-s)}\mathop{}\\!\mathrm{d}f_{m;s}e_{m}.$

Finally, for $\varepsilon\in(0,\varepsilon_{0})$, there exists a
$C(T,\,\|\zeta\|_{\mathcal{C}^{\alpha_{0}}},\,\|Z_{\varepsilon}\|_{C_{T}\mathcal{C}^{\alpha}})>0$
such that,

$\|\mathcal{D}_{f}u_{\varepsilon}\|_{C_{T}\mathcal{C}^{\alpha}}\leq
C\|f\|_{C_{T}\mathbb{C}^{N_{\varepsilon}}}.$ (5.14)

###### Proof.

Integration by parts implies that, for any $m\in\mathbb{Z}$ and $f\in
C_{T}\mathbb{C}$ with $f_{0}=0$

$\int_{0}^{t}e^{-4\pi^{2}|m|^{2}(t-s)}\mathop{}\\!\mathrm{d}f_{s}=f_{t}-4\pi^{2}|m|^{2}\int_{0}^{t}e^{-4\pi^{2}|m|^{2}(t-s)}f_{s}\,\mathop{}\\!\mathrm{d}s.$

It follows that $Z_{\varepsilon}$ is a bounded, linear function of
$\hat{B}_{\varepsilon}$ with values in
$C_{T}\Pi_{\varepsilon}L^{2}_{0}(\mathbb{T})$, and so is continuously Fréchet
differentiable. Furthermore, for any
$f\in\bar{C}_{T}\mathbb{C}^{N_{\varepsilon}}$ with $f_{0}=0$

$\mathcal{D}_{f}Z_{\varepsilon;t}=\sum_{m\in
N_{\varepsilon}}\left(f_{m;t}-4\pi^{2}|m|^{2}\int_{0}^{t}e^{-4\pi^{2}|m|^{2}(t-s)}f_{m;s}\,\mathop{}\\!\mathrm{d}s\right)\ell_{\varepsilon}(m)e_{m},$

Regarding the approximate solution, $u_{\varepsilon;t}$, the mappings
$h\mapsto h^{2}$ and $h\mapsto\partial_{x}\rho_{h}$ are Fréchet differentiable
on $\Pi_{\varepsilon}(L^{2}(\mathbb{T}))$, so the map

$\Phi_{T}\colon(z,(f_{m})_{m\in
N_{\varepsilon}})\,\mapsto\,-z+e^{\,\cdot\,\Delta}\zeta_{\varepsilon}+\int_{0}^{\,\cdot\,}e^{(\,\cdot\,-s)\Delta}\partial_{x}\hat{\Pi}_{\varepsilon}\left(z_{s}^{2}\partial_{x}\rho_{z_{s}}\right)\,\mathop{}\\!\mathrm{d}s\\\
+\sum_{m\in
N_{\varepsilon}}\int_{0}^{\,\cdot\,}\ell_{\varepsilon}(m)e^{-4\pi^{2}|m|^{2}(t-s)}\mathop{}\\!\mathrm{d}f_{m;s}\,e_{m},$

is Fréchet differentiable as a mapping
$\Phi_{T}\colon(C_{T}\Pi_{\varepsilon}L^{2}(\mathbb{T}),\bar{C}_{T}\mathbb{C}^{N_{\varepsilon}})\to
C_{T}\Pi_{\varepsilon}L^{2}(\mathbb{T})$ and is such that
$\Phi_{T}(u_{\varepsilon},\hat{B}_{\varepsilon})=0$. Moreover, writing
$\mathscr{D}$ for the Fréchet derivative,

$(\mathscr{D}\Phi_{T_{*}})(u_{\varepsilon},\hat{B}_{\varepsilon})(\,\cdot\,,0)\colon
C_{T_{*}}\Pi_{\varepsilon}L^{2}(\mathbb{T})\to
C_{T_{*}}\Pi_{\varepsilon}L^{2}(\mathbb{T})$

is a Banach space isomorphism for
$T_{*}(\|u_{\varepsilon}\|_{C_{T}\mathcal{C}^{\alpha}})>0$ sufficiently small.
Applying the implicit function theorem for Banach spaces, [1, Thm. 2.3], we
obtain that $u_{\varepsilon}(\zeta,\cdot)|_{[0,T_{*}]}$ is Fréchet
differentiable in a neighbourhood of $\hat{B}_{\varepsilon}$. Patching
together a sufficient (but finite) number of intervals of length $T_{*}$ to
cover $[0,T]$, we obtain the first claim.

To derive (5.13), recall that the Fréchet and Gateaux derivatives of a Fréchet
differentiable map agree. Hence, for any
$f\in\bar{C}_{T}\mathbb{C}^{N_{\varepsilon}}$,
$\mathcal{D}_{f}u_{\varepsilon}=\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}\lambda}\mathscr{S}^{\varepsilon}(\zeta,\hat{B}_{\varepsilon}+\lambda
f_{\varepsilon})\big{|}_{\lambda=0}$ from which (5.13) follows now from the
approximate equation, (5.8). We finally show (5.14). By the triangle
inequality, the properties of $\hat{\Pi}_{\varepsilon}$, (A.13) and applying
(A.10), for any $t\in[0,T]$,

$\|\mathcal{D}_{f}u_{\varepsilon;t}\|_{\mathcal{C}^{\alpha}}\lesssim\|u_{\varepsilon}\|^{2}_{C_{\eta;t}\mathcal{C}^{\alpha}}\int_{0}^{t}s^{-\frac{1}{2}-2\eta}\|\mathcal{D}_{f}u_{\varepsilon;s}\|_{\mathcal{C}^{\alpha}}\,\mathop{}\\!\mathrm{d}s+\sup_{s\in[0,t]}|f_{s}|_{\mathbb{C}^{N_{\varepsilon}}}.$

Therefore, by Grönwall’s inequality, there exists $C>0$ such that

$\|\mathcal{D}_{f}u_{\varepsilon;t}\|_{\mathcal{C}^{\alpha}}\lesssim\|f\|_{C_{T}\mathbb{C}^{N_{\varepsilon}}}\exp\big{(}Ct^{\frac{1}{2}-2\eta}\|u_{\varepsilon}\|^{2}_{C_{\eta;t}\mathcal{C}^{\alpha}}\big{)}.$

Due to the global existence of $u_{\varepsilon}\in
C_{\eta;T}\mathcal{C}^{\alpha}$ (shown in Theorem 5.8 for
$\varepsilon\in(0,\varepsilon_{0})$ where $\varepsilon_{0}$ depends on
$\|\zeta\|_{\mathcal{C}^{\alpha_{0}}},\|Z_{\varepsilon}\|_{C_{T}\mathcal{C}^{\alpha}}$),
for a new constant
$C(T,\|\zeta\|_{\mathcal{C}^{\alpha_{0}}},\|Z\|_{C_{T}\mathcal{C}^{\alpha}})>0$,

$\|\mathcal{D}_{f}u_{\varepsilon}\|_{C_{T}\mathcal{C}^{\alpha}}\leq
C\|f\|_{C_{T}\mathbb{C}^{N_{\varepsilon}}}.$

∎

Regarding the derivative of $u_{\varepsilon}(\zeta)$ with respect to the
initial condition, for $g\in\mathcal{C}^{\alpha_{0}}$, we set
$g_{\varepsilon}:=\hat{\Pi}_{\varepsilon}g$ and then for any $0\leq s\leq
t\leq T$ we let $J_{s,t}^{\varepsilon}g$ solve the equation

$J^{\varepsilon}_{s,t}g=e^{(t-s)\Delta}g_{\varepsilon}+\int_{s}^{t}e^{(t-r)\Delta}\partial_{x}\hat{\Pi}_{\varepsilon}[2u_{\varepsilon;r}\partial_{x}\rho_{u_{\varepsilon;r}}J_{s,r}^{\varepsilon}g+u^{2}_{\varepsilon;r}\partial_{x}\rho_{J_{s,r}^{\varepsilon}g}]\,\mathop{}\\!\mathrm{d}r.$
(5.15)

We show below that for any $\zeta\in\mathcal{C}^{\alpha_{0}}(\mathbb{T})$ and
$t\in[0,T]$, $J^{\varepsilon}_{0,t}g=D_{g}u_{\varepsilon}(\zeta)$, the
directional derivative of $u_{\varepsilon}(\zeta)$ in $\zeta$. Note that
$J^{\varepsilon}$ satisfies the flow property, that is for $0\leq s\leq t\leq
T$ one has $J^{\varepsilon}_{0,t}=J^{\varepsilon}_{s,t}J^{\varepsilon}_{0,s}$.
In particular $J^{\varepsilon}_{s,s}=\mathrm{id}$.

######  Lemma 5.12.

There exists an open neighbourhood
$\mathcal{O}_{\zeta_{\varepsilon}}\subset\Pi_{\varepsilon}(L^{2}(\mathbb{T}))$
containing $\zeta_{\varepsilon}$ such that
$u_{\varepsilon}(\,\cdot\,,\hat{B}_{\varepsilon})$ is Fréchet differentiable
as a mapping from $\mathcal{O}_{\zeta_{\varepsilon}}$ to
$C_{T}\Pi_{\varepsilon}(L^{2}(\mathbb{T}))$. For any
$g\in\mathcal{C}^{\alpha_{0}}$, the derivative is given by
$D_{g}u_{\varepsilon;t}(\zeta)=J_{0,t}^{\varepsilon}g$. Furthermore, setting
$\mathfrak{R}=\|Z\|_{C_{T}\mathcal{C}^{\alpha}}+\|\zeta\|_{\mathcal{C}^{\alpha_{0}}}$,
there exists $C(\mathfrak{R})>0$ and $T_{*}(\mathfrak{R})>0$ such that for all
$t\in(0,T_{*}]$

$\sup_{s\in[0,t]}s^{\eta}\|J_{0,s}^{\varepsilon}g\|_{\mathcal{C}^{\alpha}}\leq
C\|g\|_{\mathcal{C}^{\alpha_{0}}}.$ (5.16)

###### Proof.

The same argument as in the proof of Lemma 5.11 shows that the map
$\Pi_{\varepsilon}L^{2}_{\mathfrak{m}}(\mathbb{T})\ni\zeta_{\varepsilon}\mapsto
u_{\varepsilon}(\zeta)\in
C_{T}\Pi_{\varepsilon}(L^{2}_{\mathfrak{m}}(\mathbb{T}))$ is Fréchet
differentiable in a neighbourhood of $\zeta_{\varepsilon}$. It is then readily
checked that on $\mathcal{O}_{\zeta_{\varepsilon}}$, for any
$g\in\mathcal{C}^{\alpha_{0}}$ the Fréchet derivative is equal to the map
$t\mapsto J^{\varepsilon}_{0,t}g$.

To prove (5.16), observe that

$\|J^{\varepsilon}_{0,s}g\|_{\mathcal{C}^{\alpha}}\lesssim
s^{-\frac{\alpha-\alpha_{0}}{2}}\|g\|_{\mathcal{C}^{\alpha_{0}}}+\|u_{\varepsilon}\|^{2}_{C_{\eta;s}\mathcal{C}^{\alpha}}\int_{0}^{s}(s-r)^{-\frac{1}{2}}r^{-2\eta}\|J^{\varepsilon}_{0,r}g\|_{\mathcal{C}^{\alpha}}\,\mathop{}\\!\mathrm{d}r.$

Therefore, for any $t\in(0,T]$,

$\sup_{s\in[0,t]}s^{\eta}\|J^{\varepsilon}_{0,s}g\|_{\mathcal{C}^{\alpha}}\lesssim
t^{\eta-\frac{\alpha-\alpha_{0}}{2}}\|g\|_{\mathcal{C}^{\alpha_{0}}}+t^{\frac{1}{2}-\eta}\|u_{\varepsilon}\|^{2}_{C_{\eta;t}\mathcal{C}^{\alpha}}\sup_{s\in[0,t]}s^{\eta}\|J^{\varepsilon}_{0,s}g\|_{\mathcal{C}^{\alpha}}.$

Since $\|u_{\varepsilon}-u\|_{C_{\eta;T}\mathcal{C}^{\alpha}}\leq 1$, by
Theorem 3.4 there exists a $T_{*}(\mathfrak{R})\in(0,1)$ such that
$\|u_{\varepsilon}\|_{C_{\eta;T_{*}}\mathcal{C}^{\alpha}}\leq 2$. Hence, for
all $t\in(0,T_{*}]$,

$\sup_{s\in[0,t]}s^{\eta}\|J^{\varepsilon}_{0,s}g\|_{\mathcal{C}^{\alpha}}\lesssim
t^{\eta-\frac{\alpha-\alpha_{0}}{2}}\|g\|_{\mathcal{C}^{\alpha_{0}}}+t^{\frac{1}{2}-\eta}\sup_{s\in[0,t]}s^{\eta}\|J^{\varepsilon}_{0,s}g\|_{\mathcal{C}^{\alpha}},$

so then choosing a sufficiently small time $t_{1}(\mathfrak{R})\in(0,t]$,

$\sup_{s\in[0,t_{1}]}s^{\eta}\|J^{\varepsilon}_{0,s}g\|_{\mathcal{C}^{\alpha}}\lesssim
t_{1}^{\eta-\frac{\alpha-\alpha_{0}}{2}}\|g\|_{\mathcal{C}^{\alpha_{0}}}.$

Repeating this procedure, we find a constant $C:=C(\mathfrak{R})>0$ such that

$\sup_{s\in[0,t]}s^{\eta}\|J^{\varepsilon}_{0,s}g\|_{\mathcal{C}^{\alpha}}\leq
C\|g\|_{\mathcal{C}^{\alpha_{0}}},\quad\forall\,t\in(0,T_{*}].$

∎

We finally reintroduce probability and consider in the remainder of the
section a finite dimensional approximation $B_{\varepsilon;t}$ of the white
noise defined by

$B_{\varepsilon;t}:=\sum_{m\in
N_{\varepsilon}}e_{m}\hat{B}_{t;m},\quad\hat{B}_{m;t}:=\xi(\mathds{1}_{[0,t]}\times
e_{m}),\quad\text{ for }m\in N_{\varepsilon}.$

Note that $(\hat{B}_{m})_{m\in N_{\varepsilon}}$ is a family of complex valued
Brownian motions which satisfy the reality condition (5.12). Our approximation
of the SHE corresponding to (5.11) is then

$v_{\varepsilon;s,t}:=\hat{\Pi}_{\varepsilon}v_{s,t}=\sum_{m\in
N_{\varepsilon}}\int_{s}^{t}\ell_{\varepsilon}(m)e^{-4\pi^{2}|m|^{2}(t-r)}\mathop{}\\!\mathrm{d}\hat{B}_{m;r}\,e_{m},$
(5.17)

By Remark 5.10, since
$\hat{B}_{\varepsilon}\in\bar{C}_{T}\mathbb{C}^{N_{\varepsilon}}$,
$(s,t)\mapsto v_{\varepsilon;s,t}$ is well defined pathwise. Furthermore, by
Property ii), there exists a $\mathbb{P}$-null set $\mathcal{N}\subset\Omega$
such that, fixing any realisation $\xi(\omega)$ for
$\omega\in\Omega\setminus\mathcal{N}$ gives realisations of
$v_{\varepsilon}(\omega),\,v(\omega)$ and for which
$v_{\varepsilon;0,\,\cdot\,}(\omega)\rightarrow v_{0,\,\cdot\,}(\omega)$ in
$\mathcal{C}^{\kappa}_{T}\mathcal{C}^{\alpha-2\kappa}$ for every
$\kappa\in[0,1/2)$. In the rest of the subsection, we will let
$v_{\varepsilon}$ denote the random path $t\mapsto v_{\varepsilon;0,t}$.

For each $\varepsilon>0$, the Cameron–Martin space of $\hat{B}_{\varepsilon}$
is

$\mathcal{C}\mathcal{M}^{\varepsilon}=\bar{W}^{1,2}_{0}\left([0,T];\mathbb{C}^{N_{\varepsilon}}\right):=\bigg{\\{}f\in
L^{2}([0,T];\,\mathbb{C}^{N_{\varepsilon}})\,:\,\partial_{t}f\in
L^{2}\left([0,T];\,\mathbb{C}^{N_{\varepsilon}}\right),\,\\\
f_{-m}=\bar{f}_{m},\,f|_{t=0}=0\bigg{\\}}.$

By the Sobolev embedding,
$W^{1,2}(\mathbb{R})\hookrightarrow\mathcal{C}^{1/2}(\mathbb{R})$, we see
$\mathcal{C}\mathcal{M}^{\varepsilon}\subset\bar{C}_{T}\mathbb{C}^{N_{\varepsilon}}$.
Therefore, Lemma 5.11 applies with $f\in\mathcal{C}\mathcal{M}^{\varepsilon}$.
We also choose a smooth, compactly supported, cut-off function
$\Theta:\mathbb{R}_{+}\rightarrow[0,1]$ such that $\Theta(z)=1$ for
$z<\frac{1}{2}$ and $\Theta(z)=0$ for $z\geq 1$.

We introduce the notion of right sided derivatives, $\mathcal{D}^{+}$, of
$\|Z\|_{C_{T}\mathcal{C}^{\alpha}}$, which, for
$f\in\bar{C}_{T}\mathcal{C}^{\alpha}$, is defined by

$\mathcal{D}^{+}_{f}\|Z\|_{C_{T}\mathcal{C}^{\alpha}}:=\lim_{\lambda\searrow
0}\frac{\|Z+\lambda
f\|_{C_{T}\mathcal{C}^{\alpha}}-\|Z\|_{C_{T}\mathcal{C}^{\alpha}}}{\lambda}.$

######  Theorem 5.13 (Bismut–Elworthy–Li Formula).

Let $t>0$, $\zeta\in\mathcal{C}^{\alpha_{0}}_{\mathfrak{m}}(\mathbb{T})$,
$\Phi\in\mathcal{C}^{1}_{b}(\mathcal{C}^{\alpha_{0}}(\mathbb{T}))$ and
$f^{\varepsilon}$ be a $\mathcal{C}\mathcal{M}^{\varepsilon}$ valued process
taking for which $\partial_{t}f^{\varepsilon}$ is adapted and such that
$\|\partial_{t}f^{\varepsilon}\|_{L^{2}([0,t];\mathbb{C}^{N_{\varepsilon}})}\leq
C$, $\mathbb{P}$-a.s. for a deterministic constant $C:=C(t)$. Then we have the
identity

$\displaystyle\mathbb{E}\big{[}\mathcal{D}_{f^{\varepsilon}}[\Phi(u_{\varepsilon;t}(\zeta,\hat{B}_{\varepsilon}))]\Theta(\|v_{\varepsilon}\|_{C_{t}\mathcal{C}^{\alpha}})\big{]}$
$\displaystyle=\mathbb{E}\Big{[}\Phi(u_{\varepsilon;t}(\zeta,\hat{B}_{\varepsilon}))\Theta(\|v_{\varepsilon}\|_{C_{t}\mathcal{C}^{\alpha}})\int_{0}^{t}\partial_{t}f^{\varepsilon}_{s}\cdot\,\mathop{}\\!\mathrm{d}\hat{B}_{\varepsilon;s}\Big{]}$
(5.18)
$\displaystyle\quad-\mathbb{E}\Big{[}\Phi(u_{\varepsilon;t}(\zeta,\hat{B}_{\varepsilon}))\Theta^{\prime}(\|v_{\varepsilon}\|_{C_{t}\mathcal{C}^{\alpha}})\mathcal{D}^{+}_{f^{\varepsilon}}\|v_{\varepsilon}\|_{C_{t}\mathcal{C}^{\alpha}}\Big{]},$

where we recall that $f^{\varepsilon}$ is identified with a function in
$C_{t}C^{\infty}_{0}$ as in Remark 5.10.

###### Proof.

For any $f^{\varepsilon}\in\mathcal{C}\mathcal{M}^{\varepsilon}$ and
$\lambda>0$, define the shift operator
$\mathcal{T}^{\lambda}_{f^{\varepsilon}}\hat{B}_{\varepsilon}:=\hat{B}_{\varepsilon}+\lambda
f^{\varepsilon}$. Define further

$\displaystyle u^{\lambda;f}_{\varepsilon}(\zeta)$
$\displaystyle:=\mathscr{S}^{\varepsilon}\left(\zeta,\mathcal{T}^{\lambda}_{f^{\varepsilon}}\hat{B}^{\varepsilon}\right),$
$\displaystyle\mathcal{T}^{\lambda}_{f^{\varepsilon}}v_{\varepsilon;0,t}$
$\displaystyle:=\sum_{m\in
N_{\varepsilon}}\int_{0}^{t}\ell_{\varepsilon}(m)e^{-4\pi^{2}|m|^{2}(t-r)}(\mathop{}\\!\mathrm{d}\hat{B}^{m}_{r}+\lambda\partial_{t}f^{\varepsilon}_{m;r}\,\mathop{}\\!\mathrm{d}r)\,e_{m}.$

We now look for a family of measures $\mathbb{P}_{\lambda}$ such that the law
of $\mathcal{T}^{\lambda}_{f^{\varepsilon}}\hat{B}^{\varepsilon}$ under
$\mathbb{P}_{\lambda}$ is equal to the law of $\hat{B}^{\varepsilon}$ under
$\mathbb{P}$ for each $\lambda$. The assumptions on $f^{\varepsilon}$ imply
that Novikov’s condition [30, Ch. 8, Prop. 1.15] is satisfied by
$\lambda\partial_{t}f^{\varepsilon}$. Hence, by Girsanov’s theorem [30, Ch. 4,
Thm. 1.4], the desired measure $\mathbb{P}_{\lambda}$ is given by

$\frac{\mathop{}\\!\mathrm{d}\mathbb{P}_{\lambda}}{\mathop{}\\!\mathrm{d}\mathbb{P}}:=A^{\lambda}_{t},\quad
A_{t}^{\lambda}:=\exp\left(-\lambda\int_{0}^{t}\partial_{t}f^{\varepsilon}_{s}\,\cdot\mathop{}\\!\mathrm{d}\hat{B}^{\varepsilon}_{s}-\frac{\lambda^{2}}{2}\int_{0}^{t}|\partial_{t}f^{\varepsilon}_{s}|^{2}\,\mathop{}\\!\mathrm{d}s\right).$

As a consequence, and using
$\mathbb{E}_{\mathbb{P}_{\lambda}}\left[\,\cdot\,\right]=\mathbb{E}\left[\,\cdot\,A^{\lambda}_{t}\right]$,
we have

$\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}\lambda}\mathbb{E}\left[\Phi(u^{\lambda;f^{\varepsilon}}_{\varepsilon;t}(\zeta))\Theta(\|\mathcal{T}^{\lambda}_{f^{\varepsilon}}v_{\varepsilon}\|_{C_{t}\mathcal{C}^{\alpha}})A^{\lambda}_{t}\right]\bigg{|}_{\lambda=0+}=0.$

We now show that we may exchange differentiation and integration in the above
expression, after which we will derive the BEL formula, (5.18). Let us define
the difference operator $\Delta_{\lambda}F:=F(\lambda)-F(0)$ so that, for
example,

$\Delta_{\lambda}\Phi(u^{\,\cdot\,;f^{\varepsilon}}_{\varepsilon;t}(\zeta))=\Phi(u^{\lambda;f^{\varepsilon}}_{\varepsilon;t}(\zeta))-\Phi(u_{\varepsilon;t}(\zeta)).$

We demonstrate that the family
$\\{\frac{1}{\lambda}(\Delta_{\lambda}(\Phi(u^{\,\cdot\,;f^{\varepsilon}}_{\varepsilon;t}(x))\Theta(\|\mathcal{T}^{\,\cdot\,}_{f^{\varepsilon}}v_{\varepsilon;0,\,\cdot\,}\|_{C_{T}\mathcal{C}^{\alpha}})A^{\,\cdot\,}_{t}))\\}_{\lambda\in(0,1]}$
is $\mathbb{P}$-uniformly integrable. For any $\lambda\in(0,1]$ we observe
that,

$\displaystyle\Delta_{\lambda}(\Phi(u^{\,\cdot\,;f^{\varepsilon}}_{\varepsilon;t}(\zeta))\Theta(\|\mathcal{T}^{\,\cdot\,}_{f^{\varepsilon}}v_{\varepsilon}\|_{C_{t}\mathcal{C}^{\alpha}})A^{\,\cdot\,}_{t})$
$\displaystyle=\Delta_{\lambda}\Phi(u^{\,\cdot\,;f^{\varepsilon}}_{\varepsilon;t}(\zeta))\Theta(\|\mathcal{T}^{\lambda}_{f^{\varepsilon}}v_{\varepsilon}\|_{C_{t}\mathcal{C}^{\alpha}})A^{\lambda}_{t}$
$\displaystyle\quad+\Phi(u_{\varepsilon;t}(\zeta))\Delta_{\lambda}\Theta(\|\mathcal{T}^{\,\cdot\,}_{f^{\varepsilon}}v_{\varepsilon}\|_{C_{t}\mathcal{C}^{\alpha}})A^{\lambda}_{t}$
$\displaystyle\quad+\Phi(u_{\varepsilon;t}(\zeta))\Theta(\|v_{\varepsilon}\|_{C_{t}\mathcal{C}^{\alpha}})\Delta_{\lambda}A^{\,\cdot\,}_{t}$

For the first term, by the assumption that
$\Phi\in\mathcal{C}^{1}_{b}(\mathcal{C}^{\alpha_{0}}(\mathbb{T}))$,

$\displaystyle|\Delta_{\lambda}(\Phi(u^{\,\cdot\,;f^{\varepsilon}}_{\varepsilon;t}(\zeta))|$
$\displaystyle\leq\|\Phi\|_{\mathcal{C}^{1}_{b}}\|u^{\lambda;f^{\varepsilon}}_{\varepsilon;t}(\zeta)-u_{\varepsilon;t}(\zeta)\|_{\mathcal{C}^{\alpha_{0}}}$
$\displaystyle\leq\|\Phi\|_{\mathcal{C}^{1}_{b}}\int_{0}^{\lambda}\left\|\mathcal{D}_{f}u^{\tilde{\lambda};f}_{\varepsilon;t}(\zeta)\right\|_{\mathcal{C}^{\alpha}}\mathop{}\\!\mathrm{d}\tilde{\lambda}.$

If
$\|\mathcal{T}^{{\lambda}}_{f^{\varepsilon}}v_{\varepsilon}\|_{C_{t}\mathcal{C}^{\alpha}}>1$,
then the first term vanishes, so suppose that
$\|\mathcal{T}^{{\lambda}}_{f^{\varepsilon}}v_{\varepsilon}\|_{C_{t}\mathcal{C}^{\alpha}}\leq
1$. Hence, due to the bound on $f^{\varepsilon}$, there exists a deterministic
$C>0$ such that
$\|\mathcal{T}^{\tilde{\lambda}}_{f^{\varepsilon}}v_{\varepsilon}\|_{C_{t}\mathcal{C}^{\alpha}}\leq
C$ for all $\tilde{\lambda}\in[0,\lambda]$. Therefore, applying (5.14)
uniformly for $\tilde{\lambda}\in(0,\lambda]$, we have the bound

$|\Delta_{\lambda}\Phi(u^{\,\cdot\,;f^{\varepsilon}}_{\varepsilon;t}(\zeta))\Theta(\|\mathcal{T}^{\lambda}_{f^{\varepsilon}}v_{\varepsilon}\|_{C_{t}\mathcal{C}^{\alpha}})A^{\lambda}_{t}|\,\lesssim_{f,\Theta,t}\lambda\|\Phi\|_{\mathcal{C}^{1}_{b}}A^{\lambda}_{t},$

where the implicit constant is independent of the realisation of $v$ and
$\lambda\in(0,1]$. For the second term, arguing similarly and using our
assumptions on $\Theta$ we obtain the similar bound,

$|\Phi(u_{\varepsilon;t}(\zeta))\Delta_{\lambda}\Theta(\|\mathcal{T}^{\,\cdot\,}_{f^{\varepsilon}}v_{\varepsilon}\|_{C_{T}\mathcal{C}^{\alpha}})A^{\lambda}_{t}|\,\lesssim_{f,\Theta,t}\lambda\|\Phi\|_{\infty}A^{\lambda}_{t}.$

Finally, for the third term,

$|\Phi(u_{\varepsilon;t}(\zeta))\Theta(\|v_{\varepsilon}\|_{C_{t}\mathcal{C}^{\alpha}})\Delta_{\lambda}A^{\,\cdot\,}_{t}|\lesssim_{\Theta}\|\Phi\|_{\infty}\int_{0}^{\lambda}\left|\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}\lambda}A^{\tilde{\lambda}}_{t}\right|\mathop{}\\!\mathrm{d}\tilde{\lambda}.$

Therefore there exists a constant
$C(f,\Theta,t,\|\Phi\|_{\mathcal{C}^{1}_{b}})>0$ such that for all
$\lambda\in(0,1]$ and $p\geq 1$

$\mathbb{E}\left[\left|\frac{\Delta_{\lambda}(\Phi(u^{\,\cdot\,;f^{\varepsilon}}_{\varepsilon;t}(\zeta))\Theta(\|\mathcal{T}^{\,\cdot\,}_{f^{\varepsilon}}v_{\varepsilon}\|_{C_{t}\mathcal{C}^{\alpha}})A^{\,\cdot\,}_{t})}{\lambda}\right|^{p}\right]\leq
C\left(\sup_{\lambda\in(0,1]}\mathbb{E}[(A^{\lambda}_{t})^{p}]+\sup_{\lambda\in(0,1]}\mathbb{E}\left[\left|\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}\lambda}A^{\lambda}_{t}\right|^{p}\right]\right),$

where we applied Jensen’s inequality to obtain
$(\int_{0}^{\lambda}|\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}\lambda}A^{\tilde{\lambda}}_{t}|\mathop{}\\!\mathrm{d}\tilde{\lambda})^{p}\leq\lambda^{p-1}\int_{0}^{\lambda}|\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}\lambda}A^{\tilde{\lambda}}_{t}|^{p}\mathop{}\\!\mathrm{d}\tilde{\lambda}$.
For the first term we have
$(A^{\lambda}_{t})^{p}=A^{p\lambda}_{t}\exp\left(\frac{p^{2}-p}{2}\lambda^{2}\int_{0}^{t}|\partial_{t}f^{\varepsilon}_{s}|^{2}\mathop{}\\!\mathrm{d}s\right)$
where $A^{p\lambda}$ is also a strictly positive martingale with expectation
$1$ and
$\exp\left(\frac{p^{2}-p}{2}\lambda^{2}\int_{0}^{t}|\partial_{t}f^{\varepsilon}_{s}|^{2}\mathop{}\\!\mathrm{d}s\right)$
is uniformly bounded in $L^{\infty}(\Omega;\mathbb{R})$ across
$\lambda\in(0,1]$ due to the assumptions on $f^{\varepsilon}$. Regarding the
second term, using the explicit form,

$\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}\lambda}A^{\lambda}_{t}=-A^{\lambda}_{t}\left(\int_{0}^{t}\partial_{t}f^{\varepsilon}_{s}\cdot\mathop{}\\!\mathrm{d}\hat{B}_{\varepsilon;s}+\lambda\int_{0}^{t}|\partial_{t}f^{\varepsilon}_{s}|^{2}\mathop{}\\!\mathrm{d}s\right)$
(5.19)

and Cauchy–Schwarz, we see that

$\displaystyle\mathbb{E}\left[\left|\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}\lambda}A^{\tilde{\lambda}}_{t}\right|^{p}\right]^{2}\leq\mathbb{E}\left[(A^{\tilde{\lambda}}_{t})^{2p}\right]\mathbb{E}\left[\left(\int_{0}^{t}\partial_{t}f^{\varepsilon}_{s}\cdot\mathop{}\\!\mathrm{d}\hat{B}_{s}^{\varepsilon}+\tilde{\lambda}\int_{0}^{t}|\partial_{t}f^{\varepsilon}_{s}|^{2}\mathop{}\\!\mathrm{d}s\right)^{2p}\right].$

The first factor is uniformly bounded for all $\tilde{\lambda}\in(0,1]$ by the
same arguments as we applied for $(A^{\lambda}_{t})^{p}$. The second factor
can be controlled using the Burkholder–Davis–Gundy inequality and again our
assumption of uniform boundedness on
$\|\partial_{t}f^{\varepsilon}\|_{L^{2}([0,t])}$. Hence

$\sup_{\lambda\in(0,1]}\mathbb{E}\left[\,\left|\frac{\Delta_{\lambda}(\Phi(u^{\,\cdot\,;f^{\varepsilon}}_{\varepsilon;t}(x))\Theta(\|\mathcal{T}^{\,\cdot\,}_{f^{\varepsilon}}v_{\varepsilon}\|_{C_{t}\mathcal{C}^{\alpha}})A^{\,\cdot\,}_{t})}{\lambda}\right|^{p}\,\right]<\infty,$

and we have uniform integrability. Therefore, by Vitali’s convergence theorem,
we may exchange differentiation and expectation to give that

$\mathbb{E}\left[\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}\lambda}\left(\Phi(u^{\lambda;f^{\varepsilon}}_{\varepsilon;t}(\zeta))\Theta(\|\mathcal{T}^{\lambda}_{f^{\varepsilon}}v_{\varepsilon}\|_{C_{t}\mathcal{C}^{\alpha}})A^{\lambda}_{t}\right)\bigg{|}_{\lambda=0+}\right]=0.$

In order to conclude the proof it only remains to show that the identities

$\displaystyle\mathcal{D}^{+}_{f^{\varepsilon}}\Phi\left(u_{\varepsilon;t}(\zeta)\right)$
$\displaystyle=\mathcal{D}_{f^{\varepsilon}}\Phi\left(u_{\varepsilon;t}(\zeta)\right),$
$\displaystyle\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}\lambda}A^{\lambda}_{t}\bigg{|}_{\lambda=0+}$
$\displaystyle=-\int_{0}^{t}\partial_{t}f^{\varepsilon}_{s}\cdot\mathop{}\\!\mathrm{d}\hat{B}_{\varepsilon;s},$
$\displaystyle\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}\lambda}\Theta\left(\|\mathcal{T}^{\lambda}_{f^{\varepsilon}}v_{\varepsilon}\|_{C_{t}\mathcal{C}^{\alpha}}\right)\bigg{|}_{\lambda=0+}A^{0}_{t}$
$\displaystyle=\Theta^{\prime}(\|v_{\varepsilon}\|_{C_{t}\mathcal{C}^{\alpha}})\mathcal{D}^{+}_{f^{\varepsilon}}\|v_{\varepsilon}\|_{C_{t}\mathcal{C}^{\alpha}}.$

The first follows directly from Lemma 5.11, since one sided derivatives and
full derivatives agree for any Fréchet differentiable map. The second identity
follows after setting $\lambda=0$ in (5.19) and observing that
$A^{0}_{t}\equiv 1$. The final identity follows in the same way from our
assumptions on $\Theta$ and the chain rule. ∎

We can apply the identity (5.18) to obtain an initial bound on the difference
of the semi-group acting on elements of
$\mathcal{C}^{1}_{b}(\mathcal{C}^{\alpha_{0}})$, from two initial points. The
following result and its proof is close to that of [34, Prop. 5.8].

######  Proposition 5.14.

Let $\zeta,\,\tilde{\zeta}\in\mathcal{C}_{\mathfrak{m}}^{\alpha_{0}}$ with
$\tilde{\zeta}\in B_{1}(\zeta)$, set
$\mathfrak{R}:=1+2\|\zeta\|_{\mathcal{C}^{\alpha_{0}}}$ and let
$T_{*}(\mathfrak{R})>0$ be defined according to (3.7). Then there exists a
constant $C:=C(\mathfrak{R},\alpha,\alpha_{0},\eta)>0$ and exponent
$\theta:=\theta(\eta)>0$ such that for any
$\Phi\in\mathcal{C}^{1}_{b}(\mathcal{C}^{\alpha_{0}}_{\mathfrak{m}})$ and
$t\in(0,T_{*}]$,

$|P_{t}\Phi(\zeta)-P_{t}\Phi(\tilde{\zeta})|\leq
C\frac{1}{t^{\theta}}\|\Phi\|_{L^{\infty}}\|\zeta-\tilde{\zeta}\|_{\mathcal{C}^{\alpha_{0}}}+2\|\Phi\|_{L^{\infty}}\mathbb{P}(\|v\|_{C_{t}\mathcal{C}^{\alpha}}>1).$
(5.20)

###### Proof.

First, defining the semi-group for the approximate equation and applying the
triangle inequality, for every
$\Phi\in\mathcal{C}^{1}_{b}(\mathcal{C}^{\alpha_{0}})$ one obtains

$\displaystyle|P^{\varepsilon}_{t}\Phi(\zeta)-P^{\varepsilon}_{t}\Phi(\tilde{\zeta})|$
$\displaystyle\leq\left|\mathbb{E}\left[(\Phi(u_{\varepsilon;t}(\zeta))-\Phi(u_{\varepsilon;t}(\tilde{\zeta})))\Theta(\|v_{\varepsilon}\|_{C_{t}\mathcal{C}^{\alpha}})\right]\right|$
$\displaystyle\quad+\left|\mathbb{E}\left[(\Phi(u_{\varepsilon;t}(\zeta))-\Phi(u_{\varepsilon;t}(\zeta)))\left(1-\Theta(\|v_{\varepsilon}\|_{C_{t}\mathcal{C}^{\alpha}})\right)\right]\right|$
$\displaystyle=:I_{1}(t)+I_{2}(t).$ (5.21)

For the second term,

$I_{2}(t)\leq
2\|\Phi\|_{L^{\infty}}\mathbb{P}\left(\|v_{\varepsilon}\|_{C_{t}\mathcal{C}^{\alpha}}>1\right),$
(5.22)

and using the fact that $v_{\varepsilon}\rightarrow v$ in
$C_{t}\mathcal{C}^{\alpha}$ in probability we obtain the second term on the
right hand side of (5.20). Regarding the first term, we use the fundamental
theorem of calculus along with Fubini to write,

$I_{1}(t)=\left|\int_{0}^{1}\mathbb{E}\left[D_{\zeta-\tilde{\zeta}}[\Phi(u_{t}(\zeta+\lambda(\zeta-\tilde{\zeta}))]\Theta(\|v_{\varepsilon}\|_{C_{t}\mathcal{C}^{\alpha}})\right]\mathop{}\\!\mathrm{d}\lambda\,\right|.$

We now note that for any
$f^{\varepsilon}\in\mathcal{C}\mathcal{M}^{\varepsilon}$, it follows from the
mild forms of $\mathcal{D}_{f^{\varepsilon}}u_{\varepsilon;t}$ and
$J^{\varepsilon}_{s,t}\partial_{t}f^{\varepsilon}_{m;s}$ given in (5.13) and
(5.15) respectively, that,

$\mathcal{D}_{f^{\varepsilon}}u_{\varepsilon;t}(\zeta)=\sum_{m\in
N_{\varepsilon}}\int_{0}^{t}J^{\varepsilon}_{s,t}\partial_{t}f^{\varepsilon}_{m;s}e_{m}\,\mathop{}\\!\mathrm{d}s.$

Denote
$z_{\lambda}:=\zeta-\lambda(\zeta-\tilde{\zeta})\in\mathcal{C}^{\alpha_{0}}$
and let $f^{\varepsilon}\in\mathcal{C}\mathcal{M}^{\varepsilon}$ such that
$\partial_{t}f^{\varepsilon}_{m;s}=\frac{1}{t}\langle
J^{\varepsilon}_{0,s}(\zeta-\tilde{\zeta}),e_{m}\rangle$ for every $m\in
N_{\varepsilon}$, where
$J^{\varepsilon}_{0,s}(\zeta-\tilde{\zeta})=D_{\zeta-\tilde{\zeta}}u_{\varepsilon;s}(z_{\lambda})$
and for notational ease, we have suppressed the dependence on $z_{\lambda}$ in
$J^{\varepsilon}_{0,s}(\zeta-\tilde{\zeta})$. Then

$\mathcal{D}_{f^{\varepsilon}}u_{\varepsilon;t}(z_{\lambda})=J_{0,t}^{\varepsilon}(\zeta-\tilde{\zeta})=D_{\zeta-\tilde{\zeta}}u_{\varepsilon;t}(z_{\lambda}).$
(5.23)

Furthermore, for $f^{\varepsilon}$ defined in this way,
$\partial_{t}f^{\varepsilon}\in
L^{\infty}(\Omega;L^{2}([0,t];\mathbb{C}^{N_{\varepsilon}}))$. Note also that
$\|z_{\lambda}\|_{\mathcal{C}^{\alpha_{0}}}\leq\mathfrak{R}$ for all
$\lambda\in(0,1)$, so the local bounds of Lemmas 5.11 and 5.12 both hold
uniformly in $z_{\lambda}$. Using (5.23) and (5.18) we obtain the bound

$\left|\mathbb{E}\left[D_{\zeta-\tilde{\zeta}}[\Phi(u_{\varepsilon;t}(z_{\lambda}))]\Theta(\|v_{\varepsilon}\|_{C_{t}\mathcal{C}^{\alpha}})\right]\right|\\\
\leq\frac{1}{t}\|\Phi\|_{L^{\infty}}\Big{(}\mathbb{E}\Big{[}\,\Big{|}\int_{0}^{t}\langle
J^{\varepsilon}_{0,s}(\zeta-\tilde{\zeta}),\mathop{}\\!\mathrm{d}W_{\varepsilon;s}\rangle\,\Theta(\|v_{\varepsilon}\|_{C_{t}\mathcal{C}^{\alpha}})\Big{|}\,\,\Big{]}\\\
+\mathbb{E}\left[\,\left|\,\Theta^{\prime}(\|v_{\varepsilon}\|_{C_{t}\mathcal{C}^{\alpha}})\mathcal{D}_{f^{\varepsilon}}^{+}\|v_{\varepsilon}\|_{C_{t}\mathcal{C}^{\alpha}}\,\right|\,\right]\Big{)}.$

For the first term, one can apply Cauchy–Schwarz, Itô’s isometry and the
results of Lemma 5.12 to obtain

$\displaystyle\mathbb{E}\left[\,\left|\int_{0}^{t}\langle
J^{\varepsilon}_{0,s}(\zeta-\tilde{\zeta}),\mathop{}\\!\mathrm{d}W_{\varepsilon;s}\rangle\,\Theta(\|v_{\varepsilon;0,\,\cdot\,}\|_{C_{t}\mathcal{C}^{\alpha}})\right|\right]$
$\displaystyle\leq\mathbb{E}\left[\int_{0}^{t}\|J^{\varepsilon}_{0,s}(\zeta-\tilde{\zeta})\|^{2}_{L^{2}}\mathop{}\\!\mathrm{d}s\right]^{1/2}$
$\displaystyle\lesssim_{\mathfrak{R},\alpha,\alpha_{0},\eta}t^{\frac{1}{2}-\eta}\|\zeta-\tilde{\zeta}\|_{\mathcal{C}^{\alpha_{0}}}.$
(5.24)

For the second term, using the explicit form of the directional derivative and
Lemma 5.12 again applied to our choice of $f^{\varepsilon}$, we have

$\displaystyle\mathbb{E}\left[\,\left|\,\Theta^{\prime}(\|v_{\varepsilon}\|_{C_{t}\mathcal{C}^{\alpha}})\mathcal{D}_{f^{\varepsilon}}^{+}\|v_{\varepsilon}\|_{C_{t}\mathcal{C}^{\alpha}}\,\right|\,\right]$
$\displaystyle\leq
t^{-\eta}\mathbb{E}\left[\sup_{s\in[0,t]}s^{\eta}\|J^{\varepsilon}_{0,s}(\zeta-\tilde{\zeta})\|_{\mathcal{C}^{\alpha}}|\Theta^{\prime}(\|v_{\varepsilon}\|_{C_{t}\mathcal{C}^{\alpha}})|\right]$
$\displaystyle\leq
Ct^{-\eta}\|\zeta-\tilde{\zeta}\|_{\mathcal{C}^{\alpha_{0}}},$ (5.25)

where in the last line we also used that $\Theta^{\prime}$ is bounded.

Combining (5.22), (5.24) and (5.25) yields the stated inequality, (5.20), for
the approximate semi-group $P^{\varepsilon}$. Taking the limit
$\varepsilon\rightarrow 0$ and applying the dominated convergence theorem
concludes the proof. ∎

We conclude this section by obtaining a local Hölder bound for the dual semi-
group. It follows from this bound that, for all $t>0$,
$P_{t}\colon\mathcal{B}_{b}(\mathcal{C}_{\mathfrak{m}}^{\alpha_{0}})\rightarrow\mathcal{C}_{b}(\mathcal{C}_{\mathfrak{m}}^{\alpha_{0}})$,
which concludes our proof of the strong Feller property for $(P_{t})_{t>0}$.

######  Theorem 5.15.

Let $\zeta,\,\tilde{\zeta}\in\mathcal{C}_{\mathfrak{m}}^{\alpha_{0}}$ with
$\tilde{\zeta}\in B_{1}(\zeta)$. Then there exists $C>0$, $\theta\in(0,1)$ and
$\sigma>0$ such that, for every $t\geq 1$,

$\|P^{*}_{t}\delta_{\zeta}-P^{*}_{t}\delta_{\tilde{\zeta}}\|_{\mathrm{TV}}\,\,\leq
C(1+\|\zeta\|_{\mathcal{C}^{\alpha_{0}}})^{\sigma}\|\zeta-\tilde{\zeta}\|^{\theta}_{\mathcal{C}^{\alpha_{0}}}.$

###### Sketch of Proof.

It follows from [5, Lem. 7.1.5] that (5.20) is equivalent to the statement,

$\|P^{*}_{t}\delta_{\zeta}-P^{*}_{t}\delta_{\tilde{\zeta}}\|_{\mathrm{TV}}\,\,\leq
C\frac{1}{t^{\theta}}\|\zeta-\tilde{\zeta}\|_{\alpha_{0}}+2\mathbb{P}(\|v\|_{C_{t}\mathcal{C}^{\alpha}}>1).$

From this bound, one may follow exactly the steps of [34, Thm. 5.10]. The key
idea is to use the Gaussian nature of $v$ to obtain the control
$\mathbb{P}(\|v\|_{C_{t}\mathcal{C}^{\alpha}}>1)\lesssim t^{\theta_{2}}$ for
some $\theta_{2}\in(0,1)$. Then we use the explicit form of
$T_{\ast}(\mathfrak{R})$ defined in (3.7) and monotonicity of the map
$t\mapsto\|P^{*}_{t}\delta_{\zeta}-P^{*}_{t}\delta_{\tilde{\zeta}}\|_{\mathrm{TV}}$
to obtain the final result. ∎

###  5.3 Full Support

We demonstrate that $u_{T}(\zeta)$, and thus any invariant measure
$\nu_{\zeta}$ for $(P_{t})_{t>0}$ as in Theorem 5.6, has full support in
$\mathcal{C}^{1/2-\delta}_{\bar{\zeta}}(\mathbb{T})$ for any
$\delta\in(0,1/2)$. In this subsection we are not concerned with the behaviour
of the solution near zero, so until the start of Section 5.4 we just consider
$\alpha\in(0,1/2)$ and $\mathfrak{m}\in\mathbb{R}$.

Let $L^{2}_{0}(\mathbb{R}_{+}\times\mathbb{T})$ be the space of square
integrable functions on $\mathbb{R}_{+}\times\mathbb{T}$ such that for any
$t\geq 0$, $\bar{f}_{t}=0$. Then for any $T\geq 0$ we define the
Cameron–Martin space of $v:=v_{0,\,\cdot\,}$,

$\mathscr{H}_{T}:=\left\\{h\colon[0,T]\times\mathbb{T}\to\mathbb{R}\,:\,h_{t}=\int_{0}^{t}e^{(t-s)\Delta}f_{s}\,\mathop{}\\!\mathrm{d}s,\,f\in
L^{2}_{0}(\mathbb{R}_{+}\times\mathbb{T})\right\\},$

Note that by Theorems A.2 and A.6, $\mathscr{H}_{T}$ is continuously and
densely embedded in $\\{h\in C_{T}\mathcal{C}^{\alpha}_{0}\,:\,h(0)=0\\}$. The
following is a direct consequence of the Cameron–Martin theorem.

######  Lemma 5.16 (Theorem 3.6.1 of [3]).

Let $T>0$ and
$\mathcal{L}(v)=(v)\\#\mathbb{P}\in\mathcal{P}(C_{T}\mathcal{C}^{\alpha})$ be
the law of $v$. Then
$\mathrm{supp}(\mathcal{L}(v))=\overline{\mathscr{H}_{T}}^{\|\,\cdot\,\|_{C_{T}\mathcal{C}^{\alpha}}}$.

In the following theorem, we treat $\mathcal{L}(u_{T}(\zeta))$ as a
probability measure on $\mathcal{C}^{\alpha}_{\mathfrak{m}}(\mathbb{T})$.

######  Theorem 5.17.

Let $T>0$, $\zeta\in\mathcal{C}^{\alpha}_{\mathfrak{m}}(\mathbb{T})$. Then

$\mathrm{supp}(\mathcal{L}(u_{T}(\zeta)))=\mathcal{C}^{\alpha}_{\mathfrak{m}}(\mathbb{T}).$

###### Proof.

We first show

$\overline{\left\\{\mathscr{S}_{T}(\zeta,h)\,:\,h\in\mathscr{H}_{T}\right\\}}^{\|\,\cdot\,\|_{\mathcal{C}^{\alpha}}}\subseteq\mathrm{supp}(\mathcal{L}(u_{T}(\zeta))).$
(5.26)

Recall that the map $\mathscr{S}_{T}(\zeta,\,\cdot\,)\colon
C_{T}\mathcal{C}^{\alpha}_{0}(\mathbb{T})\rightarrow\mathcal{C}^{\alpha}_{\mathfrak{m}}$
is continuous and $\mathscr{H}_{T}\subset
C_{T}\mathcal{C}^{\alpha}_{0}(\mathbb{T})$. Consider now
$h\in\mathscr{H}_{T}$. Then for any $\varepsilon>0$, there exists a $\delta>0$
such that

$\|v-h\|_{C_{T}\mathcal{C}^{\alpha}_{0}}<\delta\,\Rightarrow\,\|u_{T}-\mathscr{S}_{T}(\zeta,h)\|_{\mathcal{C}^{\alpha}}<\varepsilon,$

and therefore

$\mathbb{P}(\|u_{T}-\mathscr{S}_{T}(\zeta,h)\|_{\mathcal{C}^{\alpha}}<\varepsilon)\geq\mathbb{P}(\|v-h\|_{C_{T}\mathcal{C}^{\alpha}_{0}}<\delta)>0,$

where the last inequality follows from Lemma 5.16 and this shows (5.26). It
now suffices to show that,

$\overline{\left\\{\mathscr{S}_{T}(\zeta,h)\,:\,h\in\mathscr{H}_{T}\right\\}}^{\|\,\cdot\,\|_{\mathcal{C}^{\alpha}}}=\mathcal{C}_{\mathfrak{m}}^{\alpha}(\mathbb{T}).$

Let $y\in C_{\mathfrak{m}}^{\infty}(\mathbb{T})$. Then for $t\in[0,T]$ define

$u^{y}_{t}:=e^{t\Delta}\zeta+\frac{t}{T}(y-e^{T\Delta}\zeta),$ (5.27)

and

$h^{y}_{t}:=-\int_{0}^{t}e^{(t-s)\Delta}\partial_{x}((u^{y}_{s})^{2}\partial_{x}\rho_{u^{y}_{s}})\mathop{}\\!\mathrm{d}s+\frac{t}{T}(y-e^{T\Delta}\zeta).$

Since $\zeta\in\mathcal{C}^{\alpha}_{\mathfrak{m}}$, we have $u^{y}\in
C_{T}\mathcal{C}_{\mathfrak{m}}^{\alpha}$ and it also follows that $h^{y}\in
C_{T}\mathcal{C}_{0}^{\alpha}$ with $h^{y}(0)=0$. Furthermore, by
construction,

$\mathscr{S}_{T}(\zeta,h^{y})=u_{T}^{y}=y.$ (5.28)

Approximating $h^{y}$ by functions in
$C^{\infty}_{0}([0,T]\times\mathbb{T})\cap\mathscr{H}_{T}$ and using the
density of $C^{\infty}(\mathbb{T})$ in $\mathcal{C}^{\alpha}(\mathbb{T})$
concludes the proof. ∎

###  5.4 Exponential Mixing

In Theorem 5.18 and Corollary 5.19 below, we keep $(\alpha_{0},\,\alpha)$
satisfying (5.6), the more restrictive parameter set from the start of
Subsection 5.2.

######  Theorem 5.18.

There exists $\lambda\in(0,1)$ such that for every
$\zeta,\,\tilde{\zeta}\in\mathcal{C}_{\mathfrak{m}}^{\alpha_{0}}$ and $t\geq
1$,

$\|P^{*}_{t}\delta_{\zeta}-P^{*}_{t}\delta_{\tilde{\zeta}}\|_{\mathrm{TV}}\,\,\leq\,\lambda.$
(5.29)

###### Sketch of Proof.

It suffices to essentially repeat the proof of [34, Thm. 6.5]. Given an
arbitrary fixed time interval, the idea is to divide the interval into three
sub-intervals and repeatedly evolve the solutions,
$u(\zeta),\,u(\tilde{\zeta})$, for the duration of each sub-interval into an
open bounded set, couple the semi-groups started from these points and
successively apply the results of Theorems 5.3, 5.15 and 5.17. ∎

######  Corollary 5.19.

For any $\mathfrak{m}\in\mathbb{R}$, there exists a unique measure
$\nu\in\mathcal{P}(\mathcal{C}^{\alpha_{0}}_{\mathfrak{m}})$ which is
invariant for the semi-group $(P_{t})_{t\geq 0}$. Furthermore, for any
$\delta\in(0,1/2)$, $\nu(\mathcal{C}^{1/2-\delta}_{\mathfrak{m}})=1$ and $\nu$
has full support in $\mathcal{C}^{1/2-\delta}_{\mathfrak{m}}$. Finally, there
exists $\lambda\in(0,1)$ such that for every
$\mu\in\mathcal{P}(\mathcal{C}^{\alpha_{0}}_{\mathfrak{m}})$ and $t\geq 1$

$\|P^{*}_{t}\mu-\nu\|_{\mathrm{TV}}\,\leq\lambda^{\lfloor
t\rfloor}\|\mu-\nu\|_{\mathrm{TV}(\mathcal{C}^{\alpha_{0}}_{\mathfrak{m}})}.$
(5.30)

###### Proof.

Using the existence of invariant measures, shown in Theorem 5.6, the proof of
uniqueness follows exactly as that of [34, Cor. 6.6]. ∎

We now complete the proof of Theorem 1.4.

###### Proof of Theorem 1.4.

Let $(\alpha_{0},\,\alpha,\,\eta)$ satisfy the criteria of (1.3), the less
restrictive set of parameters, and
$(\alpha^{\prime},\,\alpha_{0}^{\prime},\,\eta^{\prime})$ satisfy the more
restrictive set, (5.6), and be such that $\alpha_{0}^{\prime}>\alpha_{0}$.
Then applying Corollary 5.19 with
$(\alpha_{0}^{\prime},\,\alpha^{\prime},\,\eta^{\prime})$ we find that there
exists a unique invariant measure
$\nu\in\mathcal{P}(\mathcal{C}^{\alpha}_{\mathfrak{m}})$ of $(P_{t})_{t\geq
0}$ and that $\nu$ has full support in $\mathcal{C}^{\alpha}_{\mathfrak{m}}$.
From Theorem 1.1, given $\zeta\in\mathcal{C}^{\alpha_{0}}(\mathbb{T})$, for
any $t>0$,
$u_{t}(\zeta)\in\mathcal{C}^{\alpha}_{\mathfrak{m}}(\mathbb{T})\subset\mathcal{C}^{\alpha_{0}^{\prime}}_{\mathfrak{m}}(\mathbb{T})$.
Therefore, setting $\mu=\mathcal{L}(u_{t}(\zeta))$, in Corollary 5.19, we
have, for some $\lambda\in(0,1)$,

$\displaystyle\|P_{2t}^{*}\delta_{\zeta}-\nu\|_{\mathrm{TV}(\mathcal{C}^{\alpha})}$
$\displaystyle=\|P^{*}_{t}\mathcal{L}\left(u_{t}(\zeta)\right)-\nu\|_{\mathrm{TV}(\mathcal{C}^{\alpha_{0}})}$
$\displaystyle\leq\lambda^{\lfloor
t\rfloor}\|\mathcal{L}\left(u_{t}(\zeta)\right)-\nu\|_{\mathrm{TV}(\mathcal{C}^{\alpha^{\prime}_{0}})}$
$\displaystyle\leq\lambda^{\lfloor t\rfloor},$

where in the penultimate line we used that
$\mathcal{C}_{b}(\mathcal{C}^{\alpha_{0}}(\mathbb{T}))\subset\mathcal{C}_{b}(\mathcal{C}^{\alpha_{0}^{\prime}}(\mathbb{T}))$
and in the last line that the total variation distance is bounded by $1$.
Therefore (1.4) holds for $c>0$ sufficiently small.

Consider now $p\geq 1$. By Fernique’s theorem ([6, Thm. 2.6]), there exists
$\Lambda>0$ such that
$\mathbb{E}[\exp(\Lambda\|v\|_{C_{1}\mathcal{C}^{\alpha}}^{2})]<\infty$. Since
$\mathcal{C}^{\alpha}\hookrightarrow L^{p}$ and
$\|u_{1}(\zeta)\|_{L^{p}}\lesssim
1+\|v\|_{C_{1}\mathcal{C}^{\alpha}}^{1/\alpha}$ by Theorem 4.4, there exists
$\Lambda>0$ such that

$\sup_{\zeta\in\mathcal{C}^{\alpha_{0}}}\mathbb{E}\left[\exp\left(\Lambda\|u_{1}(\zeta)\|^{2\alpha}_{L^{p}}\right)\right]<\infty\;.$
(5.31)

Hence, taking $\alpha$ arbitrarily close to $\frac{1}{2}$ (after possibly
restarting the equation and choosing a new $\alpha_{0}$ close to $0$), we
obtain (1.5). ∎

##  Appendix A Hölder-Besov Spaces on $\mathbb{T}$

For $\alpha\in\mathbb{R}$ and $p,\,q\in[1,\infty]$ we define the non-
homogeneous Besov norm for $f\in C^{\infty}(\mathbb{T})$ by the expression

$\|f\|_{\mathcal{B}^{\alpha}_{p,\,q}(\mathbb{T})}:=\left\|\left(2^{\alpha
k}\|\Delta_{k}f\|_{L^{p}(\mathbb{T})}\right)_{k}\right\|_{l^{q}(\mathbb{Z})},$
(A.1)

where $\Delta_{k}:L^{1}(\mathbb{T})\rightarrow L^{1}(\mathbb{T})$ are the
Littlewood–Paley projection operators on the periodic, integrable functions.
See for example [2, Ch. 2] for a construction on $\mathbb{R}^{d}$ and [21,
App. A] for a discussion of the same construction in the periodic setting. We
recall here that there exist kernels $h_{k}\in C^{\infty}(\mathbb{T})$ such
that, for $f\in L^{1}(\mathbb{T})$ and $k\geq-1$, $\Delta_{k}f=h_{k}\ast f$ \-
see [2, Sec. 2.2]. Furthermore $\mathcal{F}h_{k}$, the Fourier transform of
$h_{k}$, takes values in $[0,1]$ and is supported in the ball $B(0,2)$ for
$k=-1$ and in the annulus $B(0,9\cdot 2^{k})\setminus B(0,2^{k})$ for $k\geq
0$.

We denote by $\mathcal{B}^{\alpha}_{p,\,q}(\mathbb{T})$ the completion of
$C^{\infty}(\mathbb{T})$ with respect to (A.1), which ensures these spaces are
separable.

######  Theorem A.1 (Duality Pairing - [21, Prop. A.1]).

For $\alpha\in\mathbb{R}$, $p,\,q\in[1,\infty]$ and
$p^{\prime},q^{\prime}\in[1,\infty]$ such that
$\frac{1}{p}+\frac{1}{p^{\prime}}=\frac{1}{q}+\frac{1}{q^{\prime}}=1$, there
exists a constant $C:=C(\alpha,p,q)>0$ such that

$|\langle f,g\rangle|\,\leq
C\|f\|_{\mathcal{B}^{\alpha}_{p,q}}\|g\|_{\mathcal{B}^{-\alpha}_{p^{\prime},q^{\prime}}}.$
(A.2)

######  Theorem A.2 (Besov Embeddings - [21, Prop. A.2, Rem. A.3]).

Let $\alpha\leq\beta\in\mathbb{R}$, $q_{1},\,q_{2}\in[1,\infty]$, $p\geq
r\in[1,\infty]$ be such that
$\beta=\alpha+\left(\frac{1}{r}-\frac{1}{p}\right)$ and $q_{1}\geq q_{2}$.
Then there exists a constant $C:=C(\alpha,p,r,q_{1},q_{2})>0$ such that

$\|f\|_{\mathcal{B}^{\alpha}_{p,q_{1}}}\leq
C\|f\|_{\mathcal{B}^{\beta}_{r,q_{1}}},\quad\text{and}\quad\|f\|_{\mathcal{B}^{\alpha}_{p,q_{1}}}\leq
C\|f\|_{\mathcal{B}^{\alpha}_{p,q_{2}}}.$ (A.3)

Furthermore, for any $\beta>\alpha$ and $p,q\in[1,\infty]$ the embedding
$\mathcal{B}^{\beta}_{p,q}\hookrightarrow\mathcal{B}^{\alpha}_{p,q}$ is
compact. Finally, there exists $C:=C(p)>0$ such that

$C^{-1}\|f\|_{\mathcal{B}^{0}_{p,\infty}}\leq\|f\|_{L^{p}}\leq\|f\|_{\mathcal{B}^{0}_{p,1}}\;.$
(A.4)

######  Theorem A.3 (Effect of Derivatives - [21, Prop. A.5]).

Let $\alpha\in\mathbb{R}$, $p,\,q\in[1,\infty]$ and $k\in\mathbb{N}_{>0}$.
Then there exists a constant $C:=C(k)>0$ such that for any
$\sigma\in\mathbb{N}$

$\|D^{\sigma}f\|_{\mathcal{B}^{\alpha-\sigma}_{p,q}}\leq
C\|f\|_{\mathcal{B}^{\alpha}_{p,q}}.$ (A.5)

######  Theorem A.4 (Bounds in Terms of the Derivative - [21, Prop. A.6]).

Let $\alpha\in(0,1]$ and $p,\,q\in[1,\infty]$, where if $\alpha=1$ we impose
$q=\infty$. Then there exists a constant $C:=C(\alpha,p,q)>0$ such that

$\|f\|_{\mathcal{B}^{\alpha}_{p,q}}\leq
C\left(\|f\|_{L^{p}}^{1-\alpha}\|\partial_{x}f\|^{\alpha}_{L^{p}}+\|f\|_{L^{p}}\right)$
(A.6)

######  Theorem A.5 (Product Bounds - [21, Prop. A.7]).

Let $p,q\in[1,\infty]$, $\alpha,\,\beta\in\mathbb{R}$ be such that
$\alpha+\beta>0$ with $\beta>0$. Then for $\alpha>0$ and $\nu\in[0,1]$, there
exists a constant $C:=C(\alpha,\beta,p,q)>0$ such that

$\|fg\|_{\mathcal{B}^{\alpha}_{p,q}}\leq
C\|f\|_{\mathcal{B}^{\alpha}_{\frac{p}{\nu},q}}\|g\|_{\mathcal{B}^{\beta}_{\frac{p}{1-\nu},q}}\leq
C\|f\|_{\mathcal{B}^{\alpha+(1-\nu)\frac{d}{p}}_{p,q}}\|g\|_{\mathcal{B}^{\beta+\nu\frac{d}{p}}_{p,q}}.$
(A.7)

Furthermore, if $\alpha>0$ and $p_{1},p_{2},p_{3},p_{4}\in[1,\infty]$ are such
that
$\frac{1}{p_{1}}+\frac{1}{p_{2}}=\frac{1}{p_{3}}+\frac{1}{p_{4}}=\frac{1}{p}$,
then there exists a $C:=C(\alpha,p,q,p_{1},p_{2},p_{3},p_{4})>0$ such that

$\|fg\|_{\mathcal{B}^{\alpha}_{p,q}}\leq
C\left(\|f\|_{L^{p_{1}}}\|g\|_{\mathcal{B}^{\alpha}_{p_{2},q}}+\|f\|_{\mathcal{B}^{\alpha}_{p_{3},q}}\|g\|_{L^{p_{4}}}\right).$
(A.8)

###  A.1 Parabolic and Elliptic Regularity Estimates

We define the action of the heat semi-group on $f\in L^{1}(\mathbb{T})$ by,

$e^{t\Delta}f:=\mathcal{F}^{-1}\left(e^{-4\pi^{2}|\,\cdot\,|^{2}t}\hat{f}_{\,\cdot\,}\right)=\mathcal{H}_{t}\ast
f,\quad t>0,$

where $\mathcal{F}^{-1}$ denotes the inverse Fourier transform on
$L^{1}(\mathbb{T})$ and

$\mathcal{H}_{t}(x):=\frac{1}{\sqrt{4\pi
t}}\sum_{n\in\mathbb{Z}}e^{-\frac{|x-n|^{2}}{4t}}\mathds{1}_{(0,\infty)}(t)=\sum_{m\in\mathbb{Z}}e^{-4\pi^{2}|m|^{2}t}e_{m}(x)\mathds{1}_{(0,\infty)}(t),$
(A.9)

defines the heat kernel on $(0,\infty)\times\mathbb{T}$. We refer to [22,
Prop. 5 & 6] for a proof of the following theorem.

######  Theorem A.6 (Regularising Effect of the Heat Flow - [22, Prop. 5 &
6]).

Let $\alpha,\,\beta\in\mathbb{R}$, $p\geq r\in[1,\infty]$ and
$q\in[1,\infty]$. Then, if $\beta\leq\alpha\leq\beta+2$, there exists a