# Multiple and weak Markov properties in Hilbert spaces with applications to
fractional stochastic evolution equations

Kristin Kirchner and Joshua Willems  Delft Institute of Applied Mathematics
Delft University of Technology
P.O. Box 5031
2600 GA Delft
The Netherlands<EMAIL_ADDRESS>Delft Institute of Applied Mathematics
Delft University of Technology
P.O. Box 5031
2600 GA Delft
The Netherlands<EMAIL_ADDRESS>
###### Abstract.

We define various higher-order Markov properties for stochastic processes
$(X(t))_{t\in\mathbb{T}}$, indexed by an interval
$\mathbb{T}\subseteq\mathbb{R}$ and taking values in a real and separable
Hilbert space $U$. We furthermore investigate the relations between them. In
particular, for solutions to the stochastic evolution equation
$\mathcal{L}X=\dot{W}^{Q}\\!$, where $\mathcal{L}$ is a linear operator acting
on functions mapping from $\mathbb{T}$ to $U$ and
$(\dot{W}^{Q}(t))_{t\in\mathbb{T}}$ is the formal derivative of a $U$-valued
(cylindrical) $Q$-Wiener process, we prove necessary and sufficient conditions
for the weakest Markov property via locality of the precision operator
$\mathcal{L}^{*}\\!\mathcal{L}$.

As an application, we consider the space–time fractional parabolic operator
$\mathcal{L}={(\partial_{t}+A)^{\gamma}}$ of order ${\gamma\in(1/2,\infty)}$,
where $-A$ is a linear operator generating a $C_{0}$-semigroup on $U$. We
prove that the resulting solution process satisfies an $N$th order Markov
property if $\gamma=N\in\mathbb{N}$ and show that a necessary condition for
the weakest Markov property is generally not satisfied if
$\gamma\notin\mathbb{N}$. The relevance of this class of processes is twofold:
Firstly, it can be seen as a spatiotemporal generalization of Whittle–Matérn
Gaussian random fields if $U=L^{2}(\mathcal{D})$ for a spatial domain
$\mathcal{D}\subseteq\mathbb{R}^{d}\\!$. Secondly, we show that a $U$-valued
analog to the fractional Brownian motion with Hurst parameter $H\in(0,1)$ can
be obtained as the limiting case of
$\mathcal{L}=(\partial_{t}+\varepsilon\,\mathrm{Id}_{U})^{H+\frac{1}{2}}$ for
$\varepsilon\downarrow 0$.

###### Key words and phrases:

Higher-order Markov property, infinite-dimensional fractional Wiener process,
Matérn covariance, spatiotemporal Gaussian processes.

###### 2020 Mathematics Subject Classification:

Primary: 60J25, 60G15; secondary: 60G22, 60H15.

##  1\. Introduction

###  1.1. Background and motivation

Gaussian Markov random fields play an important role for various applications,
such as the analysis of time series or longitudinal data, image processing and
spatial statistics, see e.g. [42, Section 1.3]. The latter focuses on the
statistical modeling of spatial or spatiotemporal dependence in data collected
from phenomena encountered in disciplines such as climatology [1],
epidemiology [26] and neuroimaging [34]. The popularity of Gaussian Markov
random fields among the larger class of Gaussian random fields is a
consequence of their additional conditional independence properties, which
entail a sparse precision structure and facilitate efficient computational
methods for statistical inference. In particular, hierarchical models based on
Gaussian Markov random fields allow for efficient Bayesian inference using
Markov chain Monte Carlo methods, see e.g. [42, Section 4.1].

Since a Gaussian process is fully characterized by its second-order structure,
i.e., the mean and covariance function, a natural way to specify its
distribution is to choose a suitable second-order structure. Alternatively,
the dynamics of Gaussian random fields defined on a Euclidean domain
$\mathcal{D}\subseteq\mathbb{R}^{d}$ can be specified by means of stochastic
partial differential equations (SPDEs), such as the white noise
$(\mathcal{W}(x))_{x\in\mathcal{D}}$ driven equation

$LX(x)=\mathcal{W}(x),\quad x\in\mathcal{D}.$ (1.1)

Here, $L$ is a linear operator acting on real-valued functions defined on
$\mathcal{D}$. A spatial Gaussian random field $(X(x))_{x\in\mathcal{D}}$ is
said to have the Markov property if the subcollections
$(X(x))_{x\in\mathcal{D}_{1}}$ and $(X(x))_{x\in\mathcal{D}_{2}}$
corresponding to pairs of disjoint subdomains
$\mathcal{D}_{1},\mathcal{D}_{2}\subseteq\mathcal{D}$ are independent
conditional on $(X(x))_{x\in\mathcal{D}^{\prime}}$ for some non-trivial
‘splitting’ set $\mathcal{D}^{\prime}\subseteq\mathcal{D}$ separating the two.
The precise specification of these sets, which respectively carry the
intuitive interpretations of _past_ , _future_ and _present_ , leads to
various definitions of the Markov property. According to the theory of Rozanov
[41], a real-valued Gaussian random field satisfying (1.1) has such a Markov
property if and only if its precision operator $L^{*}\\!L$ is local, where
$L^{*}$ denotes the adjoint of $L$.

An important example in spatial statistics is the choice of a fractional-order
differential operator $L:=\tau(\kappa^{2}-\Delta)^{\beta}$ in (1.1), where
$\Delta$ is the Laplacian, $\mathcal{W}$ is Gaussian white noise and
$\tau,\kappa,\beta\in(0,\infty)$. Whittle [47] observed that the covariance
function of the stationary solution $(X(x))_{x\in\mathcal{D}}$ to (1.1) with
$\mathcal{D}=\mathbb{R}^{d}$ then belongs to the widely used _Matérn
covariance class_ [33]. This observation motivated the _SPDE approach_ for
spatial statistical modeling proposed by Lindgren, Rue and Lindström [29].
Here, one considers (1.1) on a bounded Euclidean domain
$\mathcal{D}\subsetneq\mathbb{R}^{d}$, augmented with boundary conditions, and
approximates the resulting _Whittle–Matérn fields_ by means of efficient
numerical methods available for (S)PDEs. Owing to its ease of generalization
and its computational efficiency as compared to covariance-based techniques,
this approach has gained widespread popularity, see e.g. [5, 6, 7, 17, 28,
43]. Since in this case the precision operator is given by
$L^{*}\\!L=\tau^{2}(\kappa^{2}-\Delta)^{2\beta}\\!$, we find that
Whittle–Matérn fields are Gaussian _Markov_ random fields in the sense of
Rozanov [41] precisely when $2\beta\in\mathbb{N}$.

Recently, extensions of the SPDE approach incorporating time dependence have
been discussed. A class of space–time equations which has been proposed in
this context is

$(\partial_{t}+L)^{\gamma}X(t,x)=\dot{\mathcal{W}}^{Q}(t,x),\quad(t,x)\in\mathbb{T}\times\mathcal{D},$
(1.2)

where $\mathbb{T}\subseteq\mathbb{R}$ represents a time interval and
$\dot{\mathcal{W}}^{Q}$ is spatiotemporal Gaussian noise, which is spatially
colored by an operator $Q$, see [24, 27]. In particular, it has been shown in
[24] that equation (1.2) extends the Matérn model in terms of spatial marginal
covariance, and that the interplay of its parameters governs smoothness in
space and time as well as the degree of separability.

Spatiotemporal random fields can be viewed as $U$-valued stochastic processes
by letting a Hilbert space $U$ encode the spatial variable, so that (1.2)
corresponds to a stochastic fractional evolution equation of the form

$(\partial_{t}+A)^{\gamma}X(t)=\dot{W}^{Q}(t),\quad t\in\mathbb{T}.$ (1.3)

The (temporal) Markov property of solutions to (1.2) is then equivalent to
that of the $U$-valued solution process $(X(t))_{t\in\mathbb{T}}$, where the
Markov behavior is considered with respect to the index set $\mathbb{T}$.
Moreover, viewing (1.3) as a special case of

$\mathcal{L}X(t)=\dot{W}^{Q}(t),\quad t\in\mathbb{T},$ (1.4)

where $\mathcal{L}$ is now a linear operator acting on functions from
$\mathbb{T}$ to $U$, the theory of Rozanov [41] suggests that locality of the
precision operator $\mathcal{L}^{*}\\!\mathcal{L}$, also acting on functions
$f\colon\mathbb{T}\to U$, can be used to characterize temporal Markov behavior
of the solution $X$.

###  1.2. Contributions

In this work we define _simple_ , _multiple_ (_$N$ -ple_ for $N\in\mathbb{N}$)
and _weak_ Markov properties for stochastic processes which take values in a
Hilbert space $U$. These definitions generalize those appearing for instance
in [18, 40, 41] for real-valued processes to infinite dimensions, see
Definitions 3.1, 3.3 and 3.5, respectively. Besides gathering them in once
place, we establish their interrelations, see Proposition 3.6 and Remark 3.7.
The main results are Theorems 3.8 and 3.10, which give necessary and
sufficient conditions, in terms of the precision operator
$\mathcal{L}^{*}\\!\mathcal{L}$, for the weakest notion of Markovianity for a
$U$-valued Gaussian process defined via (1.4). These results are proven by a
non-trivial extension of the theory by Rozanov [41, Chapters 2 and 3] from the
real-valued to the $U$-valued setting.

In order to consider the more concrete class of processes defined via (1.3),
we construct a stochastic integral for deterministic operator-valued
integrands defined on the whole of $\mathbb{R}$ with respect to a two-sided
(cylindrical) $Q$-Wiener process $(W^{Q}(t))_{t\in\mathbb{R}}$, see Subsection
2.2. We employ this stochastic integral to define the mild solution process
$Z_{\gamma}=(Z_{\gamma}(t))_{t\in\mathbb{R}}$ to (1.3) on
$\mathbb{T}=\mathbb{R}$, see Definition 4.5. Our rigorous definition of the
fractional space–time operator $(\partial_{t}+A)^{\gamma}$ for
$\gamma\in\mathbb{R}$, see Definition 4.3, extends the Weyl fractional
calculus in the sense that one recovers the Weyl fractional derivatives and
integrals defined in [22, Section 2.3] upon specializing to $U=\mathbb{R}$ and
$A=0$.

We show that the mild solution $Z_{\gamma}$ to (1.3) satisfies the $N$-ple
Markov property if $\gamma=N\in\mathbb{N}$, see Theorem 4.9. Conversely, we
use Theorem 3.8 to show that $Z_{\gamma}$ is in general not weakly Markov for
$\gamma\not\in\mathbb{N}$. This complements [14, Theorem 2.7], where the
authors show that any time-homogeneous $U$-valued Gaussian simple Markov
process is the solution to a first-order stochastic evolution equation.

Finally, we discuss another interesting aspect of the SPDE (1.3): A fractional
$Q$-Wiener process $(W_{H}^{Q}(t))_{t\in\mathbb{R}}$ with Hurst parameter
$H\in(0,1)$, as defined for instance in [12], can be obtained as a limiting
case of (1.3) with $\gamma=H+\frac{1}{2}$ and
$A=\varepsilon\operatorname{Id}_{U}$ as $\varepsilon\downarrow 0$, see
Proposition 5.3. The proof is based on a Mandelbrot–Van Ness [31] type
integral representation of $W^{Q}_{H}$, again using the two-sided stochastic
integral from Subsection 2.2, see Proposition 5.2. The case $H=\frac{1}{2}$
corresponds to a (non-fractional) $Q$-Wiener process and is thus Markov.
Conversely, although the results of Theorems 3.8 and 3.10 do not apply
directly, the above observation provides evidence that $W^{Q}_{H}$ does not
satisfy a weak Markov property for $H\in(0,1)\setminus\\{\frac{1}{2}\\}$.

###  1.3. Outline

In Section 2 we begin by establishing the necessary notation, see Subsection
2.1, followed by the construction of the stochastic integral with respect to a
two-sided (cylindrical) $Q$-Wiener process in Subsection 2.2. Section 3 is
devoted to defining, relating and (for solutions to (1.4)) characterizing
various notions of Markov behavior for $U$-valued stochastic processes. The
goal of Section 4 is to define and analyze the mild solution to (1.3) on
$\mathbb{T}=\mathbb{R}$. To this end, we first describe the setting and define
$(\partial_{t}+A)^{\gamma}$ with $\gamma\in\mathbb{R}$ in Subsections 4.1 and
4.2, respectively, and subsequently investigate for which values of $\gamma$
the process exhibits Markov behavior in Subsection 4.4. In Section 5 we recall
the definition from [12] of a $Q$-fractional Wiener process and prove a
Mandelbrot–Van Ness type integral representation, which allows us to exhibit
it as a limiting case of (1.3).

This article is supplemented by three appendices: Appendix A collects some
definitions and facts about spaces of functions with values in a Hilbert or
Banach space. It is followed by Appendix B, which contains auxiliary results
relating to specific results from the main text whose statements and proofs
were postponed for readability; subjects include conditional independence,
filtrations indexed by $\mathbb{R}$ and the mean-square differentiability of
stochastic convolutions. Appendix C is a short overview of results regarding
fractional powers of linear operators and their relation to the fractional
parabolic operator $(\partial_{t}+A)^{\gamma}$.

##  2\. Preliminaries

###  2.1. Notation

The sets $\mathbb{N}:=\\{1,2,3,\dots\\}$ and
$\mathbb{N}_{0}:=\mathbb{N}\cup\\{0\\}$ denote the positive and non-negative
integers, respectively. We write $s\wedge t$ (or $s\vee t$) for the minimum
(or maximum) of two real numbers $s,t\in\mathbb{R}$. Empty sums and products
are assigned the values zero and one, respectively. Given two arbitrary sets
$D$ and $E$, the preimage of $E_{0}\subseteq E$ under the function $f\colon
D\to E$ is defined by ${f^{-1}[E_{0}]=\\{f\in E_{0}\\}=\\{x\in D:f(x)\in
E_{0}\\}}$. The identity on $D$ and the indicator function of $D_{0}\subseteq
D$ are respectively denoted by $\operatorname{Id}_{D}\colon D\to D$ and
$\mathbf{1}_{D_{0}}\colon D\to\\{0,1\\}$.

Let $(U,\langle\,\cdot\,,\,\cdot\,\rangle_{U})$ and
$(\widetilde{U},\langle\,\cdot\,,\,\cdot\,\rangle_{\widetilde{U}})$ denote
Hilbert spaces with induced norms ${\|\cdot\|_{U}}$ and
$\|\cdot\|_{\widetilde{U}}$. Banach spaces are usually denoted by
$(E,\|\cdot\|_{E})$ and $(F,\|\cdot\|_{F})$. These spaces are all assumed to
be real and separable.

We write ${\mathscr{L}(E;F)}$ for the Banach space of bounded linear operators
equipped with the usual operator norm. The adjoint operator of
$T\in\mathscr{L}(E;F)$ is denoted by ${T^{*}\in\mathscr{L}(F^{*};E^{*})}$,
where $E^{*},F^{*}$ denote the dual spaces; if
$T\in\mathscr{L}({U};\widetilde{U})$, then we can identify $U^{*}=U$ and
$\widetilde{U}^{*}=\widetilde{U}$ via the Riesz maps, so that
$T^{*}\in\mathscr{L}(\widetilde{U};{U})$. We abbreviate
$\mathscr{L}(E):=\mathscr{L}(E;E)$, and similarly for the other spaces of
linear operators.

We write $T\in\mathscr{L}^{+}(U)$ to indicate that $T\in\mathscr{L}(U)$ is
self-adjoint positive definite, meaning that $T^{*}=T$ and there exists
$\theta\in(0,\infty)$ such that $\langle
Tx,x\rangle_{U}\geq\theta\|x\|_{U}^{2}$ for all $x\in U$. Let
$\mathscr{L}_{2}(U;\widetilde{U})\subseteq\mathscr{L}(U;\widetilde{U})$ denote
the Hilbert space of all Hilbert–Schmidt operators equipped with the inner
product $\langle
T,S\rangle_{\mathscr{L}_{2}(U;\widetilde{U})}:=\sum_{j\in\mathbb{N}}\langle
Te_{j},Se_{j}\rangle_{\widetilde{U}}$, where $(e_{j})_{j\in\mathbb{N}}$ is any
orthonormal basis for $U$. The trace of $T\in\mathscr{L}^{+}(U)$ is defined by
$\operatorname{tr}T:=\sum_{j\in\mathbb{N}}\langle Te_{j},e_{j}\rangle_{U}.$ We
say that $T\in\mathscr{L}(U)$ belongs to the Banach space $\mathscr{L}_{1}(U)$
of trace class operators if
$\|T\|_{\mathscr{L}_{1}(U)}:=\operatorname{tr}|T|<\infty$ for
$|T|:=(T^{*}T)^{\frac{1}{2}}$, using the operator square root (see e.g. [45,
Proposition 8.27]) and set
$\mathscr{L}_{1}^{+}(U):=\mathscr{L}^{+}(U)\cap\mathscr{L}_{1}(U)$.

A linear operator $A$ on $E$ with domain $\mathsf{D}(A)$ is denoted by
${A\colon\mathsf{D}(A)\subseteq E\to E}$; for its range we write
$\mathsf{R}(A)$. We call $A$ closed if its graph
$\mathsf{G}(A):={\\{(x,Ax):x\in\mathsf{D}(A)\\}}$ is closed with respect to
the norm
$\|(x,Ax)\|_{\mathsf{G}(A)}=\|x\|_{\mathsf{D}(A)}=\|x\|_{E}+\|Ax\|_{E}$, and
densely defined if $\mathsf{D}(A)$ is dense in $E$. If the closure
$\overline{\mathsf{G}(A)}$ is the graph of a linear operator $\overline{A}$,
then we call $\overline{A}$ the closure of $A$. We write
$A\subseteq\widetilde{A}$ if
$\mathsf{G}(A)\subseteq\mathsf{G}(\widetilde{A})$.

Throughout this work, we assume that a complete probability space
$(\Omega,\mathcal{F},\mathbb{P})$ is given, meaning that $\mathcal{F}$
contains the collection $\mathcal{N}_{\mathbb{P}}$ of $\mathbb{P}$-null sets.
We abbreviate the phrase “$\mathbb{P}$-almost surely” by “$\mathbb{P}$-a.s.”
In what follows, we call a function $Z\colon\Omega\to E$ an $E$-valued random
variable if it is strongly $\mathbb{P}$-measurable, see Subsection A.1 in
Appendix A. We write $Z\sim N(m,Q)$ if $Z$ is a $U$-valued Gaussian random
variable with mean $m\in U$ and covariance operator
$Q\in\mathscr{L}^{+}_{1}(U)$; its existence is guaranteed by [4, Theorem
2.3.1]. Two stochastic processes $(X(t))_{t\in\mathbb{T}}$ and
$(\widetilde{X}(t))_{t\in\mathbb{T}}$ are said to be modifications of each
other if $\mathbb{P}(X(t)=\widetilde{X}(t))=1$ for all $t\in\mathbb{T}$, where
$\mathbb{T}=[0,\infty)$, $\mathbb{T}=[0,T]$ for some $T\in(0,\infty)$ or
$\mathbb{T}=\mathbb{R}$.

Let $\mathcal{G}_{1},\mathcal{H},\mathcal{G}_{2}\subseteq\mathcal{F}$ be
sub-$\sigma$-algebras of $\mathcal{F}$. The join of two $\sigma$-algebras is
denoted by
$\mathcal{G}_{1}\vee\mathcal{G}_{2}:=\sigma(\mathcal{G}_{1}\cup\mathcal{G}_{2})$.
We write $\mathcal{G}_{1}\perp\\!\\!\\!\perp\mathcal{G}_{2}$ to indicate that
$\mathcal{G}_{1}$ and $\mathcal{G}_{2}$ are independent. The expression
$\mathbb{E}[Z\mid\mathcal{H}]$ denotes the conditional expectation of a random
variable $Z$ given $\mathcal{H}$, and the conditional probability of
$A\in\mathcal{F}$ given $\mathcal{H}$ is defined by
$\mathbb{P}(A\mid\mathcal{H}):=\mathbb{E}[1_{A}\mid\mathcal{H}]$,
$\mathbb{P}$-a.s. The notation
$\mathcal{G}_{1}\perp\\!\\!\\!\perp_{\mathcal{H}}\mathcal{G}_{2}$ indicates
that $\mathcal{G}_{1}$ and $\mathcal{G}_{2}$ are conditionally independent
given $\mathcal{H}$, i.e.,

$\forall
G_{1}\in\mathcal{G}_{1},\,G_{2}\in\mathcal{G}_{2}:\quad\mathbb{P}(G_{1}\cap
G_{2}\mid\mathcal{H})=\mathbb{P}(G_{1}\mid\mathcal{H})\mathbb{P}(G_{2}\mid\mathcal{H}),\quad\mathbb{P}\text{-a.s.},$
(2.1)

When conditioning on the natural $\sigma$-algebra
$\sigma(Y)=\\{Y^{-1}[B]:B\in\mathcal{B}(E)\\}$ generated by an $E$-valued
random variable $Y$, we may simply write $Y$ instead of $\sigma(Y)$; e.g.,
$\mathbb{E}[Z\mid Y]$ or
$\mathcal{G}_{1}\perp\\!\\!\\!\perp_{Y}\mathcal{G}_{2}$.

###  2.2. Stochastic integration with respect to a two-sided Wiener process

Let $(W_{1}^{Q}(t))_{t\geq 0}$ and $(W_{2}^{Q}(t))_{t\geq 0}$ be independent
$U$-valued standard $Q$-Wiener processes for a given
$Q\in\mathscr{L}_{1}^{+}(U)$, see for instance [30, Section 2.1], and define

$W^{Q}(t):=\begin{cases}W_{1}^{Q}(t),&\quad t\in[0,\infty);\\\
W_{2}^{Q}(-t),&\quad t\in(-\infty,0).\end{cases}$

Then the _two-sided $Q$-Wiener process_ $W^{Q}:=(W^{Q}(t))_{t\in\mathbb{R}}$
satisfies the following:

  1. (WP1)

$W^{Q}(t)$ has mean zero and $W^{Q}(t)-W^{Q}(s)\sim N(0,(t-s)Q)$ for $t\geq
s$;

  2. (WP2)

$W^{Q}$ has continuous sample paths;

  3. (WP3)

$W^{Q}(t_{4})-W^{Q}(t_{3})\perp\\!\\!\\!\perp W^{Q}(t_{2})-W^{Q}(t_{1})$ for
$t_{1}<t_{2}\leq t_{3}<t_{4}$.

One can define a stochastic integral with respect to such a process using a
construction analogous to the one-sided case, as presented for instance in
[30, Section 2.3]. Restricting ourselves to deterministic integrands
$\Phi\colon\mathbb{R}\to\mathscr{L}(U;\widetilde{U})$, this procedure yields a
square-integrable stochastic integral
$\int_{\mathbb{R}}\Phi(t)\mathop{}\\!\mathrm{d}W^{Q}(t)\in
L^{2}(\Omega;\widetilde{U})$ which exists if and only if
$\Phi(\,\cdot\,)Q^{\frac{1}{2}}\in
L^{2}(\mathbb{R};\mathscr{L}_{2}(U;\widetilde{U}))$; see Subsection A.2 in
Appendix A for the definitions of these (Bochner) spaces. In this case, it
satisfies the following Itô isometry:

$\left\|\int_{\mathbb{R}}\Phi(t)\mathop{}\\!\mathrm{d}W^{Q}(t)\right\|_{L^{2}(\Omega;\widetilde{U})}^{2}=\int_{\mathbb{R}}\|\Phi(t)Q^{\frac{1}{2}}\|_{\mathscr{L}_{2}(U;\widetilde{U})}^{2}\mathop{}\\!\mathrm{d}t.$
(2.2)

As in the one-sided case, we can extend the definition of the stochastic
integral to allow for an operator
$Q\in\mathscr{L}^{+}(U)\setminus\mathscr{L}_{1}^{+}(U)$ which is not trace-
class. To this end, one defines a _cylindrical $Q$-Wiener process_
$(W^{Q}(t))_{t\in\mathbb{R}}$ on $U$, which can be identified with a
$\widetilde{Q}$-Wiener process on a larger Hilbert space $\mathscr{U}\supseteq
U$ with Hilbert–Schmidt embedding ${\iota\in\mathscr{L}_{2}(U;\mathscr{U})}$,
where $\widetilde{Q}=\iota Q\iota^{*}\in\mathscr{L}_{1}^{+}(\mathscr{U})$. We
can then define the stochastic integral with respect to the associated
cylindrical $Q$-Wiener process, which still satisfies (2.2). Such an embedding
$\iota\in\mathscr{L}_{2}(U;\mathscr{U})$ always exists, and the resulting
integral does not depend on $\iota$, see [30, Section 2.5].

Now we turn to the matter of $\mathbb{R}$-indexed filtrations on
$(\Omega,\mathcal{F},\mathbb{P})$ associated to $(W^{Q}(t))_{t\in\mathbb{R}}$.
First we recall the situation in the one-sided case. Here, the integral
process
$\bigl{(}\int_{0}^{t}\Phi(r)\mathop{}\\!\mathrm{d}W_{1}^{Q}(r)\bigr{)}_{t\geq
0}:=\bigl{(}\int_{0}^{\infty}\mathbf{1}_{[0,t]}(r)\Phi(r)\mathop{}\\!\mathrm{d}W_{1}^{Q}(r)\bigr{)}_{t\geq
0}$ is adapted to the natural filtration

$\mathcal{F}_{t}^{W_{1}^{Q}}:=\sigma\bigl{(}W_{1}^{Q}(s):0\leq s\leq
t\bigr{)}\vee\sigma(\mathcal{N}_{\mathbb{P}}),\quad t\in[0,\infty),$

whenever $\Phi(\,\cdot\,)Q^{\frac{1}{2}}\in
L^{2}(0,t;\mathscr{L}_{2}(U;\widetilde{U}))$ for all $t\in[0,\infty)$, which
is immediate from the definition of the stochastic integral. Since property
(WP3) for $(W_{1}^{Q}(t))_{t\geq 0}$ implies that
$W_{1}^{Q}(t^{\prime})-W_{1}^{Q}(s^{\prime})\perp\\!\\!\\!\perp\mathcal{F}_{s}^{W_{1}^{Q}}$
for all $0\leq s\leq s^{\prime}<t^{\prime}\leq t$, the stochastic integral
$\int_{s}^{t}\Phi(r)\mathop{}\\!\mathrm{d}W_{1}^{Q}(r)$ is also independent of
$\mathcal{F}_{s}^{W_{1}^{Q}}$. The combination of the previous two facts
implies that
$\bigl{(}\int_{0}^{t}\Phi(r)\mathop{}\\!\mathrm{d}W_{1}^{Q}(r)\bigr{)}_{t\geq
0}$ is a martingale with respect to
$\bigl{(}\mathcal{F}^{W_{1}^{Q}}_{t}\bigr{)}_{t\geq 0}$.

In the two-sided case, we instead work with the _(completed) filtration
$(\mathcal{F}_{t}^{\delta W^{Q}})_{t\in\mathbb{R}}$ generated by the
increments of $W^{Q}$_, defined by

$\mathcal{F}_{t}^{\delta W^{Q}}:=\sigma(W^{Q}(u)-W^{Q}(s):s<u\leq
t)\vee\sigma(\mathcal{N}_{\mathbb{P}}),\quad t\in\mathbb{R}.$ (2.3)

Note that we have $\mathcal{F}^{\delta
W^{Q}}_{t}\subseteq\mathcal{F}^{W^{Q}}_{t}$ for all $t\in\mathbb{R}$ and
$\mathcal{F}^{W^{Q}}_{t}=\mathcal{F}^{\delta W^{Q}}_{t}$ for $t\in[0,\infty)$,
where $\mathcal{F}^{W^{Q}}_{t}$ is generated by $(W^{Q}(s))_{s\in(-\infty,t]}$
for each $t\in\mathbb{R}$. We point out that $(\mathcal{F}_{t}^{\delta
W^{Q}})_{t\in\mathbb{R}}$ is normal: It is complete by definition and right-
continuous by Proposition B.6 in Appendix B. By property (WP3), our two-sided
Wiener process $(W^{Q}(t))_{t\in\mathbb{R}}$ now satisfies

$\forall s\leq s^{\prime}<t:\quad
W^{Q}(t)-W^{Q}(s^{\prime})\perp\\!\\!\\!\perp\mathcal{F}_{s}^{\delta
W^{Q}}\\!,$ (WP3′)

so that by an argument, analogous to the one-sided case,
$\bigl{(}\int_{-\infty}^{t}\Phi(r)\mathop{}\\!\mathrm{d}W^{Q}(r)\bigr{)}_{t\in\mathbb{R}}$
is a martingale with respect to $(\mathcal{F}_{t}^{\delta
W^{Q}})_{t\in\mathbb{R}}$ for every $\Phi(\,\cdot\,)Q^{\frac{1}{2}}\in
L^{2}(\mathbb{R};\mathscr{L}_{2}(U,\widetilde{U}))$. Unlike
$(W_{1}^{Q}(t))_{t\geq 0}$, however, the process $(W^{Q}(t))_{t\in\mathbb{R}}$
itself will not be a martingale with respect to any filtration, see
Proposition B.7 in Appendix B. We refer the reader to [2, 3] for more details
on the subject of real-valued martingale type processes indexed by
$\mathbb{R}$ and stochastic integration with respect to such processes.

We finally record a sufficient condition for changing the order of integration
between a deterministic Lebesgue integral over a $\sigma$-finite measure space
and a stochastic integral over $\mathbb{R}$ with respect to
$(W^{Q}(t))_{t\in\mathbb{R}}$. See [46] for a version of this result in the
one-sided case with stochastic integrands. For deterministic integrands, the
two-sided generalization admits the following short proof.

######  Theorem 2.1 (Stochastic Fubini theorem).

Let $(S,\mathscr{A},\mu)$ be a $\sigma$-finite measure space and suppose
$\Psi\colon S\times\mathbb{R}\to\mathscr{L}(U,\widetilde{U})$ satisfies $\Psi
Q^{\frac{1}{2}}\in
L^{1,2}(S\times\mathbb{R};\mathscr{L}_{2}(U;\widetilde{U}))$ (see Subsection
A.2 of Appendix A). Then $\Psi Q^{\frac{1}{2}}\in L^{2,1}(\mathbb{R}\times
S;\mathscr{L}_{2}(U;\widetilde{U}))$ and

$\int_{S}\int_{\mathbb{R}}\Psi(s,r)\mathop{}\\!\mathrm{d}W^{Q}(r)\mathop{}\\!\mathrm{d}\mu(s)=\int_{\mathbb{R}}\int_{S}\Psi(s,r)\mathop{}\\!\mathrm{d}\mu(s)\mathop{}\\!\mathrm{d}W^{Q}(r),\quad\mathbb{P}\text{-a.s.}$
(2.4)

###### Proof.

Let $(\Psi_{n})_{n\in\mathbb{N}}$ be a sequence of functions $\Psi_{n}\colon
S\times\mathbb{R}\to\mathscr{L}(U;\widetilde{U})$ which are linear
combinations of functions of the form
$(\mathbf{1}_{B}\otimes\mathbf{1}_{(a,b]})\otimes T$ for $a<b$,
$B\in\mathscr{A}$ with $\mu(B)<\infty$ and $T\in\mathscr{L}(U;\widetilde{U})$
such that

$\lim_{n\to\infty}\int_{S}\biggl{[}\int_{\mathbb{R}}\|[\Psi(s,r)-\Psi_{n}(s,r)]Q^{\frac{1}{2}}\|^{2}_{\mathscr{L}_{2}(U;\widetilde{U})}\mathop{}\\!\mathrm{d}r\biggr{]}^{\frac{1}{2}}\mathop{}\\!\mathrm{d}\mu(s)=0.$
(2.5)

An application of the Minkowski inequality for integrals [44, Section A.1] to
the function
$g_{n}(s,r):=\|[\Psi(s,r)-\Psi_{n}(s,r)]Q^{\frac{1}{2}}\|_{\mathscr{L}_{2}(U;\widetilde{U})}$
for each $n\in\mathbb{N}$ then yields

$\biggl{[}\int_{\mathbb{R}}\biggl{|}\int_{S}g_{n}(s,r)\mathop{}\\!\mathrm{d}\mu(s)\biggr{|}^{2}\mathop{}\\!\mathrm{d}r\biggr{]}^{\frac{1}{2}}\leq\int_{S}\biggl{[}\int_{\mathbb{R}}|g_{n}(s,r)|^{2}\mathop{}\\!\mathrm{d}r\biggr{]}^{\frac{1}{2}}\mathop{}\\!\mathrm{d}\mu(s),$

which together with (2.5) shows that

$\lim_{n\to\infty}\int_{\mathbb{R}}\biggl{|}\int_{S}\|[\Psi(s,r)-\Psi_{n}(s,r)]Q^{\frac{1}{2}}\|_{\mathscr{L}_{2}(U;\widetilde{U})}\mathop{}\\!\mathrm{d}\mu(s)\biggr{|}^{2}\mathop{}\\!\mathrm{d}r=0.$
(2.6)

By the respective definitions of the (deterministic and stochastic) integrals
involved, (2.5) and (2.6) imply the first and last steps of

$\displaystyle\int_{S}\int_{\mathbb{R}}\Psi(s,r)\mathop{}\\!\mathrm{d}W^{Q}(r)\mathop{}\\!\mathrm{d}\mu(s)=\lim_{n\to\infty}\int_{S}\int_{\mathbb{R}}\Psi_{n}(s,r)\mathop{}\\!\mathrm{d}W^{Q}(r)\mathop{}\\!\mathrm{d}\mu(s)$
$\displaystyle\>=\lim_{n\to\infty}\int_{\mathbb{R}}\int_{S}\Psi_{n}(s,r)\mathop{}\\!\mathrm{d}\mu(s)\mathop{}\\!\mathrm{d}W^{Q}(r)=\int_{\mathbb{R}}\int_{S}\Psi(s,r)\mathop{}\\!\mathrm{d}\mu(s)\mathop{}\\!\mathrm{d}W^{Q}(r)\text{
in }L^{2}(\Omega;\widetilde{U}),$

the second identity can be verified by direct computation for simple
functions. ∎

##  3\. Markov properties for Hilbert space valued stochastic processes

Let $X=(X(t))_{t\in\mathbb{T}}$ be a $U$-valued stochastic process indexed by
$\mathbb{T}$, see Subsection 2.1. Intuitively, $X$ is said to be a Markov
process if, at any instant, its past and future states are independent
conditional on the present. Varying the amount of information from the present
gives rise to different Markov properties, which we will list in decreasing
order of strength.

###  3.1. Simple Markov property

The following definition is often just referred to as the Markov property, see
also [9, p. 77] or [11, Equation (6.2), p. 81]. We use the adjective _simple_
to differentiate it from the weaker notions of Markov behavior which will be
given below.

######  Definition 3.1.

An $(\mathcal{F}_{t})_{t\in\mathbb{T}}$-adapted $U$-valued stochastic process
$(X(t))_{t\in\mathbb{T}}$ is said to have the _simple Markov property_ if for
all $s\leq t$ and $B\in\mathcal{B}(U)$, we have

$\mathbb{P}(X(t)\in B\mid\mathcal{F}_{s})=\mathbb{P}(X(t)\in B\mid
X(s)),\quad\mathbb{P}\text{-a.s.}$ (3.1)

Note that (3.1) in the above definition can equivalently be replaced by

$\mathbb{E}[\varphi(X(t))\mid\mathcal{F}_{s}]=\mathbb{E}[\varphi(X(t))\mid
X(s)],\quad\mathbb{P}\text{-a.s.},$ (3.2)

for all $s\leq t$ and $\varphi$ belonging to the Banach space $B_{b}(U)$ of
bounded measurable functions from $U$ to $\mathbb{R}$ equipped with the norm
$\|\varphi\|_{B_{b}(U)}:=\sup_{x\in U}|f(x)|$; this can be shown using
approximation by simple functions.

In particular, this viewpoint suggests the following characterization of the
simple Markov property by means of _transition operators_ , which is sometimes
taken as the definition, for instance in [40, Chapter III, Definition 1.3].

######  Proposition 3.2.

An $(\mathcal{F}_{t})_{t\in\mathbb{T}}$-adapted $U$-valued stochastic process
$(X(t))_{t\in\mathbb{T}}$ is simple Markov if and only if there exists a
family $(T_{s,t})_{s\leq t\in\mathbb{T}}$ of linear operators on $B_{b}(U)$
satisfying, for all $s\leq t$ in $\mathbb{T}$ and $\varphi\in B_{b}(U)$,

$\mathbb{E}[\varphi(X(t))\mid\mathcal{F}_{s}]=T_{s,t}\varphi(X(s)),\quad\mathbb{P}\text{-a.s.}$
(3.3)

In this case, the _transition operators_ $(T_{s,t})_{s\leq t\in\mathbb{T}}$
have the following properties:

  1. (TO1)

$T_{s,t}\varphi(x)\geq 0$ for all $x\in U$ if $\varphi\in B_{b}(U)$ is non-
negative,

  2. (TO2)

$T_{s,t}\mathbf{1}_{U}=\mathbf{1}_{U}$,

  3. (TO3)

$T_{s,u}\varphi(X(s))=T_{s,t}T_{t,u}\varphi(X(s))$, $\mathbb{P}$-a.s., for all
$\varphi\in B_{b}(U)$ and $s\leq t\leq u$.

###### Proof.

The sufficiency part of the assertion can be verified by applying
$\mathbb{E}[\;\cdot\mid X(s)]$ on both sides of (3.3) and using elementary
properties of conditional expectations to conclude that
$\mathbb{E}[\varphi(X(t))\mid X(s)]=T_{s,t}\varphi(X(s))$ holds
$\mathbb{P}$-a.s. for any $\varphi\in B_{b}(U)$. This implies (3.2), hence $X$
has the simple Markov property.

Conversely, suppose that $X$ is simple Markov. Since $U$ is separable and thus
a Polish space, there exists a regular conditional distribution of $X(t)$
given $X(s)$ for each $s\leq t$ [21, Theorem 8.5], i.e., a mapping
$\pi_{s,t}\colon U\times\mathcal{B}(U)\to[0,1]$ such that for any $x\in U$ and
$B\in\mathcal{B}(U)$, $\pi_{s,t}(x,\,\cdot\,)$ is a probability measure on
$(U,\mathcal{B}(U))$, $\pi_{s,t}(\,\cdot\,,B)\colon U\to[0,1]$ is Borel
measurable and

$\pi_{s,t}(X(s),B)=\mathbb{P}(X(t)\in B\mid X(s))=\mathbb{P}(X(t)\in
B\mid\mathcal{F}_{s}),\quad\mathbb{P}\text{-a.s.}$

We use this to define the transition operator

$T_{s,t}\varphi(x):=\int_{U}\varphi(y)\,\pi_{s,t}(x,\mathrm{d}y),\quad\varphi\in
B_{b}(U),\;x\in U.$

Properties (TO1) and (TO2) are consequences of the fact that
$\pi_{s,t}(x,\,\cdot\,)$ is a probability measure on $\mathcal{B}(U)$ for each
$x\in U$. Equation (3.3) can be verified using approximation by simple
functions as before; in conjunction with the tower property of conditional
expectations, we find (TO3). ∎

Property (TO3) is also known as the _Chapman–Kolmogorov relation_. If
$T_{s,t}$ only depends on the difference $t-s$, then $X$ is said to be _time-
homogeneous_ and we write $T_{t}:=T_{0,t}$, so that (TO3) reduces to the
semigroup law $T_{t+s}=T_{t}T_{s}$.

Lastly, we can also characterize the simple Markov property in terms of
conditional independence: By Theorem B.1 in Appendix B, the simple Markov
property is equivalent to

$\forall s\leq t:\quad\mathcal{F}_{s}\perp\\!\\!\\!\perp_{X(s)}\sigma(X(t)).$

In fact, according to [21, Lemma 11.1], this is in turn equivalent to

$\forall
s\in\mathbb{T}:\quad\mathcal{F}_{s}\perp\\!\\!\\!\perp_{X(s)}\sigma(X(t):t\geq
s).$ (3.4)

###  3.2. Multiple Markov property

The following weaker notion of Markov behavior dates back to Doob, who
introduced it in the context of stationary real-valued Gaussian processes [10,
pp. 271–272]. We generalize it to square-integrable $U$-valued processes with
some mean-square differentiability, i.e., $(X(t))_{t\in\mathbb{T}}\subseteq
L^{2}(\Omega;U)$ such that $t\mapsto X(t)$ is classically differentiable from
$\mathbb{T}$ to $L^{2}(\Omega;U)$. In works such as [18, 41], which treat the
real-valued setting, this is called the _multiple Markov property (in the
restricted sense)_ , where “restricted” refers to the required
differentiability; we will omit this descriptor.

######  Definition 3.3.

Let $X=(X(t))_{t\in\mathbb{T}}\subseteq L^{2}(\Omega;U)$ be an
$(\mathcal{F}_{t})_{t\in\mathbb{T}}$-adapted $U$-valued stochastic process and
suppose that $N\in\mathbb{N}$. Then $X$ has the _$N$ -ple Markov property_ if
it has $N-1$ mean square derivatives and, for all $s\leq t$ in $\mathbb{T}$
and $B\in\mathcal{B}(U)$,

$\mathbb{P}(X(t)\in B\mid\mathcal{F}_{s})=\mathbb{P}(X(t)\in B\mid
X(s),X^{\prime}(s),\dots,X^{(N-1)}(s)),\quad\mathbb{P}\text{-a.s.}$ (3.5)

Setting $\mathbf{X}(t):=(X^{(k)}(t))_{k=0}^{N-1}$, one defines a process
taking values in the direct product Hilbert space
$(U^{N},\langle\,\cdot\,,\,\cdot\,\rangle_{U^{N}}\\!)$, where the inner
product

$\langle\mathbf{x},\mathbf{y}\rangle_{U^{N}}:=\sum_{j=1}^{N}\langle
x_{j},y_{j}\rangle_{U},\qquad\mathbf{x}=(x_{j})_{j=1}^{N},\,\mathbf{y}=(y_{j})_{j=1}^{N}\in
U^{N}\\!,$ (3.6)

induces the product topology on the set $U^{N}$. In particular, the Borel
$\sigma$-algebra of $U^{N}$ satisfies
$\mathcal{B}(U^{N})=\otimes^{N}\mathcal{B}(U)$ by [21, Lemma 1.2]. Theorem B.1
in Appendix B again yields an equivalent formulation of the $N$-ple Markov
property in terms of conditional independence:

$\forall
s\in\mathbb{T}:\quad\mathcal{F}_{s}\perp\\!\\!\\!\perp_{\mathbf{X}(s)}\sigma(X(t):t\geq
s).$ (3.7)

Note that $\sigma(\mathbf{X}(s))\vee\mathcal{F}_{s}=\mathcal{F}_{s}$ since the
mean-square derivatives of $X$ can be replaced by left derivatives, see the
proof of Proposition 3.6 below. Arguing as in [21, Lemma 11.1], one can show
that this is in turn equivalent to

$\forall
s\in\mathbb{T}:\quad\mathcal{F}_{s}\perp\\!\\!\\!\perp_{\mathbf{X}(s)}\sigma(\mathbf{X}(t):t\geq
s),$ (3.8)

which is nothing more than the simple Markov property for $\mathbf{X}$. Thus,
we can apply Proposition 3.2 to derive the following characterization.

######  Corollary 3.4.

An $(\mathcal{F}_{t})_{t\in\mathbb{T}}$-adapted and square-integrable
$U$-valued stochastic process $X=(X(t))_{t\in\mathbb{T}}$ with $N-1$ mean-
square derivatives is $N$-ple Markov if and only if there exists a family
$(T_{s,t})_{s\leq t\in\mathbb{T}}$ of linear operators on $B_{b}(U^{N})$
satisfying, for all $s\leq t\leq u$ in $\mathbb{T}$ and $\varphi\in
B_{b}(U^{N})$,

$\mathbb{E}[\varphi(\mathbf{X}(t))\mid\mathcal{F}_{s}]=T_{s,t}\varphi(\mathbf{X}(s)),\quad\mathbb{P}\text{-a.s.}$
(3.9)

In this case, $(T_{s,t})_{s\leq t\in\mathbb{T}}$ satisfies properties
(TO1)–(TO3).

###  3.3. Weak Markov properties; relations between concepts

We now define two Markov properties for which the “present” at time
$s\in\mathbb{T}$ is represented by information from neighborhoords around $s$.
As we will prove in Proposition 3.6 below, these two notions are equivalent.
They appear in, e.g., [41, p. 62] and [18, Equation (5.87), p. 115].

######  Definition 3.5.

An $(\mathcal{F}_{t})_{t\in\mathbb{T}}$-adapted $U$-valued stochastic process
$(X(t))_{t\in\mathbb{T}}$ has

  1. (i)

the _weak Markov property_ if, for every $s\in\mathbb{T}$,

$\exists\delta>0:\forall\varepsilon\in(0,\delta):\quad\mathcal{F}_{s}\perp\\!\\!\\!\perp_{\mathscr{A}_{\varepsilon}(s)}\sigma(X(t):t\geq
s),$ (3.10)

where
$\mathscr{A}_{\varepsilon}(s):=\sigma(X(u):u\in(s-\varepsilon,s+\varepsilon)\cap\mathbb{T})$;

  2. (ii)

the _$\sigma$ -Markov property_ if, for every $s\in\mathbb{T}$,

$\mathcal{F}_{s}\perp\\!\\!\\!\perp_{\partial\mathscr{A}(s)}\sigma(X(t):t\geq
s),$ (3.11)

where
$\partial\mathscr{A}(s):=\bigcap_{\varepsilon>0}\mathscr{A}_{\varepsilon}(s)$.

######  Proposition 3.6.

Let $X=(X(t))_{t\in\mathbb{T}}$ be an
$(\mathcal{F}_{t})_{t\in\mathbb{T}}$-adapted $U$-valued stochastic process. We
have the following relations between Markov properties:

$\text{simple Markov}\implies\text{$\sigma$-Markov}\iff\text{weak Markov}.$

If $N,M\in\mathbb{N}$ are such that $N\geq M$ and $X$ has $N-1$ mean-square
derivatives, then we moreover have

$\text{$M$-ple Markov}\implies\text{$N$-ple Markov}\implies\text{weak
Markov}.$

###### Proof.

If $X$ has the weak Markov property, then by definition we have the following
identity for fixed $s\in\mathbb{T}$, $B_{-}\in\mathcal{F}_{s}$ and
$B_{+}\in\sigma(X(t):t\geq s)$:

$\mathbb{P}(B_{-}\mid\mathscr{A}_{\nicefrac{{1}}{{n}}}(s))\mathbb{P}(B_{+}\mid\mathscr{A}_{\nicefrac{{1}}{{n}}}(s))=\mathbb{P}(B_{-}\cap
B_{+}\mid\mathscr{A}_{\nicefrac{{1}}{{n}}}(s)),\quad\mathbb{P}\text{-a.s.},$
(3.12)

whenever $n\in\mathbb{N}$ is large enough. Note that
$(\mathscr{G}_{n})_{n\in\mathbb{N}}:=(\mathscr{A}_{\nicefrac{{1}}{{n}}}(s))_{n\in\mathbb{N}}$
is a non-increasing sequence of sub-$\sigma$-algebras of $\mathcal{F}$, i.e.,
a backward filtration on $(\Omega,\mathcal{F},\mathbb{P})$. Therefore,
$(\mathbb{P}(B\mid\mathscr{G}_{n}))_{n\in\mathbb{N}}$ is a backward martingale
with respect to $(\mathscr{G}_{n})_{n\in\mathbb{N}}$ for any
$B\in\mathcal{F}$. Combined with the fact that
$\bigcap_{n\in\mathbb{N}}\mathscr{G}_{n}=\partial\mathscr{A}(s)$, the backward
martingale convergence theorem [15, Section 12.7, Theorem 4] implies that we
may take the $\mathbb{P}$-a.s. limit as $n\to\infty$ in (3.12) to find that
$X$ is $\sigma$-Markov.

Now let $N,M\in\mathbb{N}$ with $N\geq M$ be such that $X$ has the $M$-ple
Markov property and $N-1$ mean-square derivatives. When considering
$\sigma(X^{\prime}(s))$ at $s\in\mathbb{T}$, we can restrict ourselves to
mean-square left derivatives, i.e., we consider the sequence

$(\Delta_{n}(s))_{n\in\mathbb{N}}:=(n[X(s)-X(s-n^{-1})])_{n\in\mathbb{N}}$
(3.13)

converging to $X^{\prime}(s)$ in the $L^{2}(\Omega;U)$-norm as $n\to\infty$.
Consequently, there exists a subsequence
$(\Delta_{n_{k}}(s))_{k\in\mathbb{N}}$ such that $\Delta_{n_{k}}(s)\to
X^{\prime}(s)$, $\mathbb{P}$-a.s., as $k\to\infty$. Since $\Delta_{n_{k}}(s)$
is $\mathcal{F}_{s}$-measurable for each $k\in\mathbb{N}$, we conclude that
$X^{\prime}(s)$ is $\mathcal{F}_{s}$-measurable and thus
$\sigma(X^{\prime}(s))\subseteq\mathcal{F}_{s}$. By induction, this extends to

$\sigma(X(s),X^{\prime}(s),\dots,X^{(M-1)}(s))\subseteq\sigma(X(s),X^{\prime}(s),\dots,X^{(N-1)}(s))\subseteq\mathcal{F}_{s},$

so that Lemma B.2b yields the $N$-ple Markov property as formulated in (3.7).

It remains to show that the $N$-ple Markov property for $N\in\mathbb{N}$ and
the $\sigma$-Markov property imply the weak Markovianity of $X$. Fixing
$s\in\mathbb{T}$ and $\varepsilon>0$, we can set

$\displaystyle\mathcal{H}_{1}^{\prime}:=\sigma(X(u):u\in(s-\varepsilon,s]\cap\mathbb{T})\subseteq\mathcal{F}_{s},$
(3.14)
$\displaystyle\mathcal{H}_{2}^{\prime}:=\sigma(X(u):u\in[s,s+\varepsilon)\cap\mathbb{T})\subseteq\sigma(X(t):t\in[s,\infty)\cap\mathbb{T}),$
$\displaystyle\mathcal{H}^{\prime}:=\mathcal{H}_{1}^{\prime}\vee\mathcal{H}_{2}^{\prime}=\mathscr{A}_{\varepsilon}(s).$

Since
$\sigma(X(s))\subseteq\partial\mathscr{A}(s)\subseteq\mathscr{A}_{\varepsilon}(s)$,
by Lemma B.2c the simple (i.e., 1-ple) Markov or $\sigma$-Markov property of
$X$ would imply

$\mathcal{F}_{s}\perp\\!\\!\\!\perp_{\mathscr{A}_{\varepsilon}(s)}\sigma(X(t):t\in[s,\infty)\cap\mathbb{T}),$
(3.15)

and thus the weak Markov property since $\varepsilon>0$ was arbitrary. It
remains to show that (3.15) also holds if $X$ is $N$-ple Markov. Choosing
$K\in\mathbb{N}$ so large that $n_{k}>\varepsilon^{-1}$ for all $k\geq K$, we
find that $(\Delta_{n_{k}}(s))_{k\geq K}$ (see (3.13)) is a sequence of
$\mathscr{A}_{\varepsilon}(s)$-measurable random variables converging
$\mathbb{P}$-a.s. to $X^{\prime}(s)$. As before, this procedure can be
repeated inductively to yield

$\mathcal{H}:=\sigma(X(s),X^{\prime}(s),\dots,X^{(N-1)}(s))\subseteq\mathscr{A}_{\varepsilon}(s).$

This justifies the use of Lemma B.2c to establish (3.15) for the remaining
case, and the desired conclusion follows. ∎

######  Remark 3.7.

An analog to Definition 3.5 for _generalized_ $U$-valued stochastic processes
$(X(\phi))_{\phi\in C_{c}^{\infty}(\mathbb{T})}$ is obtained by replacing
$\sigma(X(u):u\in J)$ with the natural $\sigma$-algebra
$\sigma(X(\phi):\phi\in
C_{c}^{\infty}(\mathbb{T}),\,\operatorname{supp}\phi\subseteq J)$ generated by
$X$ on an open set $J\subseteq\mathbb{T}$. Since pointwise evaluation is not
meaningful for such processes, there is no analog to the simple Markov
property. Furthermore, although the proof of the implication

$\text{weak Markov}\implies\text{$\sigma$-Markov}$

carries over, its converse now fails: The distributional derivative of white
noise is a generalized process which is $\sigma$-Markov but not weak Markov,
see [41, p. 62].

###  3.4. Characterization of weakly Markov Gaussian processes

A $U$-valued stochastic process $X=(X(t))_{t\in\mathbb{T}}$ is said to be
Gaussian if, for any $n\in\mathbb{N}$ and
$\\{t_{i}\\}_{i=1}^{n}\subseteq\mathbb{T}$, the $U^{n}$-valued random variable
$(X(t_{1}),X(t_{2}),\dots,X(t_{n}))$ is Gaussian. For such processes, we shall
characterize the weak Markov property of Definition 3.5 by extending the
theory of Rozanov [41] from real-valued to $U$-valued processes.

We consider the case of a mean-square continuous Gaussian process $X$ which is
the solution of a stochastic evolution equation of the form
$\mathcal{L}X=\dot{W}$ for some linear operator
$\mathcal{L}\colon\mathsf{D}(\mathcal{L})\subseteq L^{2}(\mathbb{T};U)\to
L^{2}(\mathbb{T};U)$; here, $\dot{W}$ denotes spatiotemporal Gaussian white
noise, cf. (1.1) and (1.3). More precisely, we assume that $\mathcal{L}$ has a
bounded inverse $\mathcal{L}^{-1}$ which _colors_ $X$, meaning

$\langle
X,\phi\rangle_{L^{2}(\mathbb{T};U)}\overset{d}{=}\mathscr{W}(\mathcal{L}^{-*}\phi),\quad\forall\phi\in
C_{c}^{\infty}(\mathbb{T};U),$ (3.16)

where $\mathcal{L}^{-*}\coloneqq(\mathcal{L}^{-1})^{*}=(\mathcal{L}^{*})^{-1}$
and $\overset{d}{=}$ indicates equality in distribution. Here,
$(\mathscr{W}(f))_{f\in L^{2}(\mathbb{T};U)}\subseteq L^{2}(\Omega)$ is an
$L^{2}(\mathbb{T};U)$-isonormal Gaussian process, i.e., a family of mean-zero
and real-valued Gaussian random variables satisfying

$\langle\mathscr{W}(f),\mathscr{W}(g)\rangle_{L^{2}(\Omega)}=\langle
f,g\rangle_{L^{2}(\mathbb{T};U)},\quad\forall f,g\in L^{2}(\mathbb{T};U).$
(3.17)

The following theorem then states that the locality of the _precision
operator_ $\mathcal{L}^{*}\mathcal{L}$ is necessary for $X$ to possess the
weak Markov property.

######  Theorem 3.8.

Let $\mathcal{L}\colon\mathsf{D}(\mathcal{L})\subseteq L^{2}(\mathbb{T};U)\to
L^{2}(\mathbb{T};U)$ be a boundedly invertible linear operator, and suppose
that $X=(X(t))_{t\in\mathbb{T}}$ is a mean-square continuous Gaussian
$U$-valued process colored by $\mathcal{L}^{-1}$. Let $F$ be a dense subset of
$U$ for which $C_{c}^{\infty}(\mathbb{T};F)\subseteq\mathsf{D}(\mathcal{L})$.
Furthermore, suppose that $C_{c}^{\infty}(\mathbb{T};F)$ and its image under
$\mathcal{L}$ are dense subsets of $L^{2}(\mathbb{T};U)$.

If $X$ has the weak Markov property from Definition 3.5 with respect to its
natural filtration $(\mathcal{F}_{t}^{X})_{t\in\mathbb{T}}$, then

$\forall
J\in\mathscr{I}:\quad\langle\mathcal{L}\phi,\mathcal{L}\psi\rangle_{L^{2}(\mathbb{T};U)}=0,\quad\forall\phi\in
C_{c}^{\infty}(J;F),\psi\in
C_{c}^{\infty}(\mathbb{T}\setminus\overline{J};F),$ (3.18)

where $\mathscr{I}$ denotes the set of all open intervals
$J\subseteq\mathbb{T}$.

###### Proof.

For all $J\in\mathscr{I}$ we define a closed subspace $\mathfrak{H}(J)$ of
$L^{2}(\Omega)$ by

$\mathfrak{H}(J):=\overline{\\{\langle
X,\phi\rangle_{L^{2}(\mathbb{T};U)}:\phi\in
C_{c}^{\infty}(J;F)\\}}^{L^{2}(\Omega)}.$ (3.19)

Then the family $(\mathfrak{H}(J))_{J\in\mathscr{I}}$ is a _Gaussian random
field_ in the sense of [41, Chapter 2, Section 3.1], and we can connect it to
the present setting by showing that

$\sigma(X(t):t\in J)=\sigma(\mathfrak{H}(J)).$ (3.20)

Indeed, we have $\sigma(\mathfrak{H}(J))\subseteq\sigma(X(t):t\in J)$ since
$\langle X,\phi\rangle_{L^{2}(\mathbb{T};U)}$ is measurable with respect to
the latter $\sigma$-algebra for all $\phi\in C_{c}^{\infty}(J;F)$ with
$\operatorname{supp}\phi\subseteq J$. In order to establish the converse
inclusion, it suffices to verify the claim that $X(t)$ is
$\sigma(\mathfrak{H}(J))$-measurable for each $t\in J$. Let
$(e_{j})_{j\in\mathbb{N}}$ be an orthonormal basis of $U$ and write

$X(t)=\sum_{j=1}^{\infty}\langle X(t),e_{j}\rangle_{U}\,e_{j}\quad\text{in
}L^{2}(\Omega;U).$ (3.21)

Now we will show that $\langle X(t),e_{j}\rangle_{U}$ is
$\sigma(\mathfrak{H}(J))$-measurable for every $j\in\mathbb{N}$. In fact, by
the density of $F\subseteq U$ it suffices to consider $\langle
X(t),x\rangle_{U}$ for $x\in F$. Let $(\phi_{n})_{n\in\mathbb{N}}\subseteq
C_{c}^{\infty}(J)$ be a sequence of bump functions concentrating around $t$,
i.e., we have
$\lim_{n\to\infty}\int_{\mathbb{T}}f(s)\phi_{n}(s)\mathop{}\\!\mathrm{d}s=f(t)$
in $E$ for any $f\in C(\mathbb{T};E)$, where $E$ is an arbitrary Banach space.
It follows from the mean-square continuity of $X$ that $f:=\langle
X(\,\cdot\,),x\rangle_{U}\in C(\mathbb{T};L^{2}(\Omega))$, thus with
$E:=L^{2}(\Omega)$ we obtain

$\langle X(t),x\rangle_{U}=\lim_{n\to\infty}\int_{\mathbb{T}}\langle
X(s),x\rangle_{U}\phi_{n}(s)\mathop{}\\!\mathrm{d}s=\lim_{n\to\infty}\int_{\mathbb{T}}\langle
X(s),\phi_{n}(s)x\rangle_{U}\mathop{}\\!\mathrm{d}s\quad\text{in
}L^{2}(\Omega).$

Passing to a $\mathbb{P}$-a.s.-convergent subsequence in the rightmost
expression, we find that $\langle X(t),x\rangle_{U}$ is a limit of
$\sigma(\mathfrak{H}(J))$-measurable random variables. Thus, each summand in
(3.21) is $\sigma(\mathfrak{H}(J))$-measurable, and passing to a
$\mathbb{P}$-a.s.-convergent subsequence of $(\sum_{j=1}^{N}\langle
X(t),e_{j}\rangle_{U}\,e_{j})_{N\in\mathbb{N}}$ proves the claim.

The theory of [41, Chapter 2, Section 3.1], now implies that $X$ has the weak
Markov property from Definition 3.5 if and only if
$(\mathfrak{H}(J))_{J\in\mathscr{I}}$ is Markov in the sense of [41, p. 97].
For a general $B\subseteq\mathbb{T}$, we define

$\mathfrak{H}_{+}(B):=\bigcap\nolimits_{\varepsilon>0}\mathfrak{H}(B^{\varepsilon}),$
(3.22)

where
$B^{\varepsilon}:=\\{t\in\mathbb{T}:\operatorname{dist}(t,B)<\varepsilon\\}$
denotes an open $\varepsilon$-neighborhood of $B$. Using the definition (3.22)
for $B\in\\{\partial J,J,\mathbb{T}\setminus J\\}$, the Markov property for
$(\mathfrak{H}(J))_{J\in\mathscr{I}}$ implies that [41, Equations (3.14), p.
97] are satisfied for every $J\in\mathscr{I}$:

$\mathfrak{H}_{+}(\partial
J)=\mathfrak{H}_{+}(J)\cap\mathfrak{H}_{+}(\mathbb{T}\setminus
J)\quad\text{and}\quad\mathfrak{H}_{+}(J)^{\perp}\perp\mathfrak{H}_{+}(\mathbb{T}\setminus
J)^{\perp},$ (3.23)

where we take $L^{2}(\Omega)$-orthogonal complements in
$\mathfrak{H}(\mathbb{T})$.

Next we define $\langle X^{*},\,\cdot\,\rangle\colon
C_{c}^{\infty}(\mathbb{T};F)\to L^{2}(\Omega)$ by

$\langle
X^{*},\phi\rangle:=\mathscr{W}(\mathcal{L}\phi),\quad\mathbb{P}\text{-a.s.},\quad\phi\in
C_{c}^{\infty}(\mathbb{T};F),$ (3.24)

to which we associate the spaces

$\mathfrak{H}^{*}(J):=\overline{\\{\langle X^{*},\phi\rangle:\phi\in
C_{c}^{\infty}(J;F)\\}}^{L^{2}(\Omega)},\quad J\in\mathscr{I}.$ (3.25)

Then $(\langle X^{*},\phi\rangle)_{\phi\in C_{c}^{\infty}(\mathbb{T};F)}$ is
dual to $(\langle X,\psi\rangle_{L^{2}(\mathbb{T};U)})_{\psi\in
C_{c}^{\infty}(\mathbb{T};F)}$ in the sense that

$\displaystyle\mathbb{E}[\langle X,\phi\rangle_{L^{2}(\mathbb{T};U)}\langle
X^{*},\psi\rangle]=\mathbb{E}[\mathscr{W}(\mathcal{L}^{-*}\phi)\mathscr{W}(\mathcal{L}\psi)]=\langle\phi,\psi\rangle_{L^{2}(\mathbb{T};U)}$
(3.26)

for $\phi,\psi\in C_{c}^{\infty}(\mathbb{T};F)$. Next we will prove

$\mathfrak{H}(\mathbb{T})=\mathfrak{H}^{*}(\mathbb{T}).$ (3.27)

by showing that both of these sets are equal to

$\mathscr{Z}:=\overline{\\{\mathscr{W}(f):f\in
L^{2}(\mathbb{T};U)\\}}^{L^{2}(\Omega)}.$

First, we note that $\mathfrak{H}(\mathbb{T})$ and
$\mathfrak{H}^{*}(\mathbb{T})$ are clearly contained in $\mathscr{Z}$. Now let
$Z\in\mathscr{Z}$ and $\varepsilon>0$ be arbitrary, and choose $f\in
L^{2}(\mathbb{T};U)$ such that
$\|\mathscr{W}(f)-Z\|_{L^{2}(\Omega)}<\frac{1}{3}\varepsilon$. Since the image
of $C_{c}^{\infty}(\mathbb{T};F)$ under $\mathcal{L}$ is assumed to be dense
in $L^{2}(\mathbb{T};U)$, we may furthermore choose $\phi\in
C_{c}^{\infty}(\mathbb{T};F)$ such that
$\|\mathcal{L}\phi-f\|_{L^{2}(\Omega;U)}<\frac{2}{3}\varepsilon$. It follows
that

$\displaystyle\|Z-\langle X^{*},\phi\rangle\|_{L^{2}(\Omega)}$
$\displaystyle\leq\|Z-\mathscr{W}(f)\|_{L^{2}(\Omega)}+\|\mathscr{W}(\mathcal{L}\phi-f)\|_{L^{2}(\Omega)}$
$\displaystyle=\|Z-\mathscr{W}(f)\|_{L^{2}(\Omega)}+\|\mathcal{L}\phi-f\|_{L^{2}(\mathbb{T};U)}<\varepsilon,$

which shows $Z\in\mathfrak{H}^{*}(\mathbb{T})$ since $\varepsilon>0$ was
arbitrary. On the other hand, since $\mathcal{L}$ is densely defined and has a
bounded inverse, it is in particular closed, hence $\mathcal{L}^{*}$ exists
and is also densely defined by [45, Proposition 10.22]. It thus follows that
$\mathsf{R}(\mathcal{L}^{-*})=\mathsf{R}((\mathcal{L}^{*})^{-1})=\mathsf{D}(\mathcal{L}^{*})$
is dense in $L^{2}(\mathbb{T};U)$, so that there exists $g\in
L^{2}(\mathbb{T};U)$ satisfying
$\|f-\mathcal{L}^{-*}g\|_{L^{2}(\mathbb{T};U)}<\frac{1}{3}\varepsilon$.
Finally, we choose $\psi\in C_{c}^{\infty}(\mathbb{T};F)$ such that
$\|\psi-g\|_{L^{2}(\mathbb{T};U)}<\|\mathcal{L}^{-*}\|_{\mathscr{L}(L^{2}(\mathbb{T};U))}^{-1}\frac{1}{3}\varepsilon$
so that

$\displaystyle\|Z-\langle
X,\psi\rangle_{L^{2}(\mathbb{T};U)}\|_{L^{2}(\Omega)}$
$\displaystyle<\tfrac{1}{3}\varepsilon+\|\mathscr{W}(f-\mathcal{L}^{-*}g)\|_{L^{2}(\Omega)}+\|\mathscr{W}(\mathcal{L}^{-*}g-\mathcal{L}^{-*}\psi)\|_{L^{2}(\Omega)}$
$\displaystyle=\tfrac{1}{3}\varepsilon+\|f-\mathcal{L}^{-*}g\|_{L^{2}(\mathbb{T};U)}+\|\mathcal{L}^{-*}(g-\psi)\|_{L^{2}(\mathbb{T};U)}<\varepsilon,$

hence also $Z\in\mathfrak{H}(\mathbb{T})$. We conclude that (3.27) holds.

The necessity of (3.18) for the weak Markov property of $X$ will follow from

$\mathfrak{H}^{*}(J)\subseteq\mathfrak{H}_{+}(\mathbb{T}\setminus
J)^{\perp},\quad\forall J\in\mathscr{I}.$ (3.28)

Indeed, if $X$ is weakly Markov, then (3.28) would imply that the random
variables

$\displaystyle\xi:=\langle X^{*},\phi\rangle$
$\displaystyle\in\mathfrak{H}^{*}(J)\subseteq\mathfrak{H}_{+}(\mathbb{T}\setminus
J)^{\perp},$ $\displaystyle\eta:=\langle X^{*},\psi\rangle$
$\displaystyle\in\mathfrak{H}^{*}(\mathbb{T}\setminus
J)\subseteq\mathfrak{H}_{+}(J)^{\perp},$

where $\phi\in C_{c}^{\infty}(J;F)$ and $\psi\in
C_{c}^{\infty}(\mathbb{T}\setminus J;F)$, are orthogonal by (3.23). Therefore,

$0=\langle\xi,\eta\rangle_{L^{2}(\Omega)}=\mathbb{E}[\langle
X^{*},\phi\rangle\langle
X^{*},\psi\rangle]=\langle\mathcal{L}\phi,\mathcal{L}\psi\rangle_{L^{2}(\mathbb{T};U)},$

which shows (3.18). Note that by definition (3.25) and density, the
orthogonality extends to all $\xi\in\mathfrak{H}^{*}(J)$ and
$\eta\in\mathfrak{H}^{*}(\mathbb{T}\setminus J)$.

In order to verify (3.28), we again take $\xi=\langle X^{*},\phi\rangle$ with
$\phi$ as above. By the compact support of the latter, there exists
$\varepsilon>0$ such that $\langle\phi,\psi\rangle_{L^{2}(\mathbb{T};U)}=0$
for all $\psi\in C_{c}^{\infty}((\mathbb{T}\setminus J)^{\varepsilon};F)$.
Hence, for $\eta=\langle
X,\psi\rangle_{L^{2}(\mathbb{T};U)}\in\mathfrak{H}((\mathbb{T}\setminus
J)^{\varepsilon})$ we have $\xi\perp\eta$ by (3.26), from which we can deduce
$\xi\perp\mathfrak{H}_{+}(\mathbb{T}\setminus J)$, and thus (3.28). ∎

In order to state and prove sufficient conditions for weak Markovianity of $X$
in terms of the locality of its precision operator, we first need to collect
some definitions which are based on objects encountered in the proof of
Theorem 3.8. Namely, we will define spaces
$(\mathcal{H}(J))_{J\in\mathscr{I}}$ such that $\mathcal{H}(\mathbb{T})$ is
unitarily isomorphic to $\mathfrak{H}(\mathbb{T})$ and there exists a dense
injection $\iota\colon
C_{c}^{\infty}(\mathbb{T};F)\to\mathcal{H}(\mathbb{T})$.

Associating, to each $\eta\in\mathfrak{H}(\mathbb{T})$, a mapping
$I^{-1}\eta\colon C_{c}^{\infty}(\mathbb{T};F)\to\mathbb{R}$ given by

$I^{-1}\eta(\phi):=\mathbb{E}[\langle
X,\phi\rangle_{L^{2}(\mathbb{T};U)}\eta],\quad\phi\in
C_{c}^{\infty}(\mathbb{T};F),$ (3.29)

sets up a linear map
$I^{-1}\colon\mathfrak{H}(\mathbb{T})\to\mathcal{H}(\mathbb{T})$, where
$\mathcal{H}(\mathbb{T})$ is defined as the range of $I^{-1}$. It is also
injective since $I^{-1}\eta(\phi)=0$ for all $\phi\in
C_{c}^{\infty}(\mathbb{T};F)$ means $\eta\perp C_{c}^{\infty}(\mathbb{T};F)$
in $\mathfrak{H}(\mathbb{T})$, and thus $\eta=0$ by (3.19). Equipping
$\mathcal{H}(\mathbb{T})$ with the inner product

$\langle v_{1},v_{2}\rangle_{\mathcal{H}(\mathbb{T})}:=\langle
Iv_{1},Iv_{2}\rangle_{L^{2}(\Omega)},\quad
v_{1},v_{2}\in\mathfrak{H}(\mathbb{T}),$

renders $I\colon\mathcal{H}(\mathbb{T})\to\mathfrak{H}(\mathbb{T})$ a unitary
isomorphism. For $J\in\mathscr{I}$ we can then define

$\mathcal{H}(J):=\bigvee\nolimits_{\\!\\!\varepsilon>0}\,\\{v\in\mathcal{H}(\mathbb{T}):v(\phi)=0\text{
for all }\phi\in C_{c}^{\infty}((\mathbb{T}\setminus J)^{\varepsilon};F)\\},$
(3.30)

where $\bigvee$ denotes the closed linear span.

A dense injection $\iota\colon
C_{c}^{\infty}(\mathbb{T};F)\to\mathcal{H}(\mathbb{T})$ is obtained by
defining ${\iota v\colon C_{c}^{\infty}(\mathbb{T};F)\to\mathbb{R}}$ in the
following way, for any $v\in C_{c}^{\infty}(\mathbb{T};F)$:

$\iota v(\phi):=\langle v,\phi\rangle_{L^{2}(\mathbb{T};U)},\quad\phi\in
C_{c}^{\infty}(\mathbb{T};F).$

Indeed, we find $\iota v\in\mathcal{H}(\mathbb{T})$ since the duality
relations (3.26) and (3.27) between $X$ and $X^{*}$ imply that $\langle
X^{*},v\rangle\in\mathfrak{H}^{*}(\mathbb{T})=\mathfrak{H}(\mathbb{T})$
satisfies

$\iota v(\phi)=\mathbb{E}[\langle X,\phi\rangle_{L^{2}(\mathbb{T};U)}\langle
X^{*},v\rangle]=I^{-1}\langle X^{*},v\rangle(\phi),\quad\forall\phi\in
C_{c}^{\infty}(\mathbb{T};F).$

Moreover, the injectivity follows in the same way as for $I^{-1}$. To show
density of the range, fix an arbitrary $v\in\mathcal{H}(\mathbb{T})$. Then
${Iv\in\mathfrak{H}(\mathbb{T})=\mathfrak{H}^{*}(\mathbb{T})}$ and thus there
exists a sequence $(\psi_{n})_{n\in\mathbb{N}}\subseteq
C_{c}^{\infty}(\mathbb{T};F)$ such that $\langle X^{*},\psi_{n}\rangle\to Iv$
in $L^{2}(\Omega)$. Consequently, we have $\iota\psi_{n}=I^{-1}\langle
X^{*},\psi_{n}\rangle\to v$ in $\mathcal{H}(\mathbb{T})$.

######  Remark 3.9.

For centered, real-valued Gaussian random fields $(Z(x))_{x\in\mathcal{X}}$
which are indexed by a compact metric space $(\mathcal{X},d_{\mathcal{X}})$
and moreover mean-square continuous, a unitary isomorphism can be established
between the $L^{2}(\Omega;\mathbb{R})$-closure of all linear combinations of
point evaluations and the dual of the Cameron–Martin space for its associated
Gaussian measure on the space $L^{2}(\mathcal{X};\mathbb{R})$, see [23, Lemma
4.1(iii)]. We point out its analogy to the unitary isomorphism
$I\colon\mathcal{H}(\mathbb{T})\to\mathfrak{H}(\mathbb{T})$ defined above.

######  Theorem 3.10.

Let the linear operator $\mathcal{L}\colon\mathsf{D}(\mathcal{L})\subseteq
L^{2}(\mathbb{T};U)\to L^{2}(\mathbb{T};U)$, the $U$-valued process
$X=(X(t))_{t\in\mathbb{T}}$ and the subset $F\subseteq U$ be as in Theorem
3.8. Recall that $\mathcal{H}(\mathbb{T})$ is the range of the linear mapping
$I^{-1}$ defined by (3.29) and $\mathcal{H}(J)$ is given by (3.30) for all
$J\in\mathscr{I}$. If

$\mathcal{H}(J)=\overline{\iota
C_{c}^{\infty}(J;F)}^{\mathcal{H}(\mathbb{T})},\quad\forall J\in\mathscr{I},$
(3.31)

then (3.18) implies that $X$ has the weak Markov property from Definition 3.5.

###### Proof.

We will show that, under the additional assumption (3.31),

$\mathfrak{H}^{*}(J)=\mathfrak{H}_{+}(\mathbb{T}\setminus
J)^{\perp},\quad\forall J\in\mathscr{I}.$ (3.32)

Note that inclusion (3.28) also holds without this assumption, see the proof
of Theorem 3.8. Identities (3.27) and (3.32) express that
$(\mathfrak{H}(J))_{J\in\mathscr{I}}$ and
$(\mathfrak{H}^{*}(J))_{J\in\mathscr{I}}$ are _dual_ in the sense of [41,
Chapter 2, Section 3.5]. In this situation, the theorem on [41, p. 100] yields
that $X$ is weakly Markov if and only if
$(\mathfrak{H}^{*}(J))_{J\in\mathscr{I}}$ is _orthogonal_ , meaning that

$\mathfrak{H}^{*}(J)\perp\mathfrak{H}^{*}(\mathbb{T}\setminus\overline{J}),\quad\forall
J\in\mathscr{I},$ (3.33)

which is equivalent to (3.18) by the respective definitions of $X^{*}$ and
$(\mathfrak{H}^{*}(J))_{J\in\mathscr{I}}$.

To verify (3.32), note that for each $\varepsilon>0$,

$\displaystyle\mathfrak{H}((\mathbb{T}\setminus J)^{\varepsilon})^{\perp}$
$\displaystyle=\\{\eta\in\mathfrak{H}(\mathbb{T}):\langle\eta,\xi\rangle_{L^{2}(\Omega)}=0\text{
for all }\xi\in\mathfrak{H}((\mathbb{T}\setminus J)^{\varepsilon})\\}$
$\displaystyle=\\{\eta\in\mathfrak{H}(\mathbb{T}):U^{-1}\eta(\phi)=0\text{ for
all }\phi\in C_{c}^{\infty}((\mathbb{T}\setminus J)^{\varepsilon};F)\\}$
$\displaystyle\cong\\{v\in\mathcal{H}(\mathbb{T}):v(\phi)=0\text{ for all
}\phi\in C_{c}^{\infty}((\mathbb{T}\setminus J)^{\varepsilon};F)\\},$

and it follows that

$\mathcal{H}(J)\cong\mathfrak{H}_{+}(\mathbb{T}\setminus J)^{\perp}.$ (3.34)

On the other hand, definition (3.25) implies

$\mathcal{H}(J)\cong\mathfrak{H}^{*}(J),$

so together we indeed find (3.32). ∎

######  Remark 3.11.

In order for locality of the precision operator $\mathcal{L}^{*}\mathcal{L}$
to imply the weak Markovianity of $X$, one needs to verify the additional
condition (3.31). In the real-valued case, two examples of sufficient
conditions on $\mathcal{L}$ for (3.31) to hold are [41, Lemmas 1 and 2, pp.
108–111], which are expressed in terms of boundedness of multipication and
translation operators, respectively, with respect to the norms
$\|\mathcal{L}^{-*}\,\cdot\,\|_{L^{2}(\mathbb{T})}$ and/or
$\|\mathcal{L}\,\cdot\,\|_{L^{2}(\mathbb{T})}$. In [41, Chapter 3, Section
3.2], these results are applied to a class of differential operators with
sufficiently regular coefficients.

Although it is expected that analogous results can be derived in the
$U$-valued setting, this subject is out of scope for the processes considered
in the remainder of this article, since we establish Markovianity using direct
methods instead of Theorem 3.10, see Subsection 4.4. However, we do use
Theorem 3.8 to show when the process lacks Markov behavior in Subsection
4.4.3.

##  4\. Fractional stochastic abstract Cauchy problem on $\mathbb{R}$

The aim of this section is to define a Hilbert space valued stochastic process
$(Z_{\gamma}(t))_{t\in\mathbb{R}}$ which can be interpreted as a _mild
solution_ to an equation of the form

$(\partial_{t}+A)^{\gamma}X(t)=\dot{W}^{Q}(t),\quad t\in\mathbb{R}.$ (4.1)

In Section 4.1 we specify the setting in which equation (4.1) will be
considered. The fractional parabolic differential operator
$(\partial_{t}+A)^{\gamma}$ is defined in Section 4.2, whereas the noise term
$\dot{W}^{Q}$ in (4.1) is the formal time derivative of the two-sided
$Q$-Wiener process defined in Section 2.2. In Section 4.3 we combine these two
notions to give a rigorous definition of the process
$(Z_{\gamma}(t))_{t\in\mathbb{R}}$, and we indicate its relation to the
fractional $Q$-Wiener process defined in Section 5. Lastly, in Section 4.4 we
prove that $(Z_{\gamma}(t))_{t\in\mathbb{R}}$ is $N$-ple Markov if
$\gamma=N\in\mathbb{N}$, but does in general not satisfy the weak Markov
property when $\gamma\not\in\mathbb{N}$.

###  4.1. Setting

The standing assumption throughout this section on the Hilbert space $U$ and
the linear operator $A$ is as follows.

######  Assumption 4.1.

The linear operator $-A\colon\mathsf{D}(A)\subseteq U\to U$ on the separable
real Hilbert space $U$ generates an _exponentially stable_ $C_{0}$-semigroup
$(S(t))_{t\geq 0}$ of bounded linear operators on $U$, i.e.,

$\exists M_{0}\in[1,\infty),w\in(0,\infty):\forall
t\in[0,\infty):\quad\|S(t)\|_{\mathscr{L}(U)}\leq M_{0}e^{-wt}.$ (4.2)

In addition, we may assume one or both of the following conditions on the
fractional power $\gamma$ and the linear operator $Q$:

######  Assumption 4.2.

  1. (i)

There exist $\gamma_{0}\in(\nicefrac{{1}}{{2}},\infty)$ and
$Q\in\mathscr{L}^{+}(U)$ such that

$\int_{0}^{\infty}\|t^{\gamma_{0}-1}S(t)Q^{\frac{1}{2}}\|_{\mathscr{L}_{2}(U)}^{2}\mathop{}\\!\mathrm{d}t<\infty.$

  2. (ii)

The $C_{0}$-semigroup $(S(t))_{t\geq 0}$ is analytic.

For a general account of the theory of $C_{0}$-semigroups, the reader is
referred to [13, 37]. Note that the results in these works, while stated for
complex Hilbert spaces, can be applied to the real setting by employing
_complexifications_ of (linear operators on) $U$, see e.g. Subsection B.2.1 of
[24, Appendix B]. In particular, by Assumption 4.2(ii) we mean that the
complexification of $(S(t))_{t\geq 0}$ is analytic in the sense of [37,
Section 2.5]. We will not emphasize this matter in what follows.

Note that $(\nicefrac{{1}}{{2}},\infty)$ is the maximal range on which
Assumption 4.2(i) can hold. Moreover, if Assumption 4.2(i) holds for some
$\gamma_{0}\in(\nicefrac{{1}}{{2}},\infty)$ then the same is true for all
$\gamma^{\prime}\in[\gamma_{0},\infty)$. These facts are proven in Subsection
B.2 of Appendix B.

Under Assumption 4.2(ii), we have the classical derivative
$\frac{\mathrm{d}}{\mathrm{d}t}A^{j}S(t)=-A^{j+1}S(t)$ from $(0,\infty)$ to
$\mathscr{L}(U)$ for all $j\in\mathbb{N}$; moreover, by [37, Chapter 2,
Theorem 6.13(c)],

$\exists M_{j}\in[1,\infty)\colon\>\forall
t\in(0,\infty):\quad\|A^{j}S(t)\|_{\mathscr{L}(U)}\leq M_{j}t^{-j}e^{-wt}.$
(4.3)

In what follows, we will simply write
$(\mathcal{F}_{t})_{t\in\mathbb{R}}:=(\mathcal{F}^{\delta
W^{Q}}_{t})_{t\in\mathbb{R}}$ for the increment filtration of $W^{Q}$ defined
by (2.3).

###  4.2. Fractional parabolic calculus and the deterministic problem

In this section we first consider the following deterministic counterpart to
equation (4.1):

$(\partial_{t}+A)^{\gamma}u(t)=f(t),\quad t\in\mathbb{R},$ (4.4)

where $f\in L^{2}(\mathbb{R};U)$. In order to define its mild solution, we
introduce the operations of fractional parabolic integration and
differentiation, generalizing the scalar-valued setting with $A=0$ which is
treated in [22, Chapter 2].

######  Definition 4.3.

Let Assumption 4.1 hold. Given a function $f\colon\mathbb{R}\to U$, we define
its _Weyl type fractional parabolic integral_
$\mathfrak{I}^{\gamma}f\colon\mathbb{R}\to U$ of order $\gamma\in(0,\infty)$
by

$\mathfrak{I}^{\gamma}f(t):=\frac{1}{\Gamma(\gamma)}\int_{-\infty}^{t}(t-s)^{\gamma-1}S(t-s)f(s)\mathop{}\\!\mathrm{d}s=\frac{1}{\Gamma(\gamma)}\int_{0}^{\infty}r^{\gamma-1}S(r)f(t-r)\mathop{}\\!\mathrm{d}r$
(4.5)

if these Bochner integrals, which are equivalent by the change of variables
${r:=t-s}$, exist for almost all $t\in\mathbb{R}$.

Viewing $\mathfrak{I}^{\gamma}$ as a linear operator, it turns out that

$\forall
p\in(1,\infty):\quad\mathfrak{I}^{\gamma}\in\mathscr{L}(L^{p}(\mathbb{R};U))\quad\text{with}\quad\|\mathfrak{I}^{\gamma}\|_{\mathscr{L}(L^{p}(\mathbb{R};U))}\leq\frac{M_{0}}{w^{\gamma}},$

which follows by combining estimate (4.2) with Minkowski’s integral inequality
[44, Section A.1]. Setting $\mathfrak{I}^{0}:=\operatorname{Id}_{U}$, the
family $(\mathfrak{I}^{\gamma})_{\gamma\geq 0}$ is a semigroup of bounded
operators on $L^{2}(\mathbb{R};U)$. Indeed, the semigroup law

$\forall\gamma_{1},\gamma_{2}\in[0,\infty)\colon\quad\mathfrak{I}^{\gamma_{1}+\gamma_{2}}=\mathfrak{I}^{\gamma_{1}}\mathfrak{I}^{\gamma_{2}},$
(4.6)

follows from an argument involving Fubini’s theorem and [36, Equation
(5.12.1)]. The adjoint operator $\mathfrak{I}^{\gamma*}$ of
$\mathfrak{I}^{\gamma}$ satisfies the following formula for all
$t\in\mathbb{R}$:

$\mathfrak{I}^{\gamma*}f(t)=\frac{1}{\Gamma(\gamma)}\int_{t}^{\infty}(s-t)^{\gamma-1}[S(s-t)]^{*}f(s)\mathop{}\\!\mathrm{d}s.$
(4.7)

Given a linear operator $T\colon\mathsf{D}(T)\subseteq U\to U$ and a measure
space $(S,\mathscr{A},\mu)$, let us define
$\mathcal{T}_{S}\colon\mathsf{D}(\mathcal{T}_{S})\subseteq L^{2}(S;U)\to
L^{2}(S;U)$ by

$\mathsf{D}(\mathcal{T}_{S})=L^{2}(S;\mathsf{D}(T))\quad\text{and}\quad[\mathcal{T}_{S}f](s):=Tf(s),\;\text{a.a.
}s\in S,\;f\in\mathsf{D}(\mathcal{T}_{S}).$ (4.8)

Using this notion, we have

$(\partial_{t}+\mathcal{A}_{\mathbb{R}})f=\partial_{t}f+\mathcal{A}_{\mathbb{R}}f,\quad
f\in\mathsf{D}(\partial_{t}+\mathcal{A}_{\mathbb{R}})=H^{1}(\mathbb{R};U)\cap
L^{2}(\mathbb{R};\mathsf{D}(A)),$ (4.9)

where $\partial_{t}$ denotes the Bochner–Sobolev weak derivative from
Subsection A.2 in Appendix A with
$\mathsf{D}(\partial_{t})=H^{1}(\mathbb{R};U)\subset L^{2}(\mathbb{R};U)$.

Now define the _Weyl type fractional parabolic derivative_ of order
$\gamma\in[0,\infty)$ as the linear operator
$\mathfrak{D}^{\gamma}:=(\partial_{t}+\mathcal{A}_{\mathbb{R}})^{\lceil\gamma\rceil}\mathfrak{I}^{\lceil\gamma\rceil-\gamma}$
on $L^{2}(\mathbb{R};U)$ on its natural domain

$\mathsf{D}(\mathfrak{D}^{\gamma})=\Bigl{\\{}f\in
L^{2}(\mathbb{R};U):\mathfrak{I}^{\lceil\gamma\rceil-\gamma}f\in\mathsf{D}\bigl{(}{(\partial_{t}+\mathcal{A}_{\mathbb{R}})^{\lceil\gamma\rceil}}\bigr{)}\Bigr{\\}}.$

The operator $\mathfrak{D}^{\gamma}$ is to be interpreted as
$(\partial_{t}+A)^{\gamma}$ in (4.4). This raises the question of how this
definition relates to the concept of fractional powers, such as the one
defined for $C_{0}$-semigroup generators in Appendix C. In general, we cannot
define fractional powers of the operator
$\partial_{t}+\mathcal{A}_{\mathbb{R}}$ itself since it may fail to be closed,
which would exclude the applicability of both Definition C.1 (since generators
of $C_{0}$-semigroups are closed [37, Chapter 1, Corollary 2.5]) and more
general definitions (see the introduction of [32]). In fact, by the operator
sum approach to maximal $L^{p}$-regularity, see [25, Discussion 1.18],
$\partial_{t}+\mathcal{A}_{\mathbb{R}}$ being closed is equivalent $A$ having
maximal $L^{p}$-regularity. In the Hilbert space setting, this is equivalent
to $-A$ generating a bounded analytic semigroup (i.e., Assumptions 4.1 and
4.2(ii)), see [25, Corollary 1.7].

However, Proposition C.4 shows that the closure
$\mathcal{B}:=\overline{\partial_{t}+\mathcal{A}_{\mathbb{R}}}$ exists and
does admit fractional powers $\mathcal{B}^{\gamma}$ in the sense of Definition
C.1 for all $\gamma\in\mathbb{R}$. In particular, $\mathcal{B}$ is boundedly
invertible and in fact we have $\mathcal{B}^{-\gamma}=\mathfrak{I}^{\gamma}$
for $\gamma\in[0,\infty)$. By [16, Proposition 3.2.1(b)], it holds that
$\mathcal{B}^{\gamma_{1}}\mathcal{B}^{\gamma_{2}}\subseteq\mathcal{B}^{\gamma_{1}+\gamma_{2}}$
for all $\gamma_{1},\gamma_{2}\in\mathbb{R}$ and
$\mathsf{D}(\mathcal{B}^{\gamma_{1}}\mathcal{B}^{\gamma_{2}})=\mathsf{D}(\mathcal{B}^{\gamma_{2}})\cap\mathsf{D}(\mathcal{B}^{\gamma_{1}+\gamma_{2}})$.
We thus recover (4.6) and find

$\forall\gamma\in[0,\infty):\quad\mathfrak{D}^{\gamma}=(\partial_{t}+\mathcal{A}_{\mathbb{R}})^{\lceil\gamma\rceil}\mathfrak{I}^{\lceil\gamma\rceil-\gamma}\subseteq\mathcal{B}^{\lceil\gamma\rceil}\mathfrak{I}^{\lceil\gamma\rceil-\gamma}=\mathcal{B}^{\lceil\gamma\rceil}\mathcal{B}^{\gamma-\lceil\gamma\rceil}=\mathcal{B}^{\gamma}.$

It follows that $\mathfrak{D}^{\gamma}$ and $\mathfrak{I}^{\gamma}$ are
inverses of each other in the sense that
$\mathfrak{D}^{\gamma}\mathfrak{I}^{\gamma}f=f$ and
$\mathfrak{I}^{\gamma}\mathfrak{D}^{\gamma}f=f$ for all $f\in
L^{2}(\mathbb{R};U)$ such that the respective left-hand sides exist.
Therefore, $u:=\mathfrak{I}^{\gamma}f$ is the unique solution to the equation
$\mathfrak{D}^{\gamma}u=f$ whenever it belongs to
$\mathsf{D}(\mathfrak{D}^{\gamma})$ (which may fail if
$\partial_{t}+\mathcal{A}_{\mathbb{R}}$ is not closed). Relaxing this
assumption, we shall call $u\in C(\mathbb{R};U)$ a _mild solution_ to (4.4) if
it is given by the formula

$\forall t\in\mathbb{R}:\quad
u(t)=\mathfrak{I}^{\gamma}f(t)=\frac{1}{\Gamma(\gamma)}\int_{-\infty}^{t}(t-s)^{\gamma-1}S(t-s)f(s)\mathop{}\\!\mathrm{d}s.$
(4.10)

###  4.3. Mild solution process

Combining the spatiotemporal fractional integration theory from Subsection 4.2
with the stochastic integral defined in Subsection 2.2, we can give a rigorous
definition for the mild solution to (4.1). We first introduce the notion of
predictability for a stochastic process indexed by $\mathbb{R}$.

######  Definition 4.4.

An $(\mathcal{F}_{t})_{t\in\mathbb{R}}$-adapted $U$-valued process
$(X(t))_{t\in\mathbb{R}}$ is _predictable_ with respect to
$(\mathcal{F}_{t})_{t\in\mathbb{R}}$ if the mapping $(t,\omega)\mapsto
X(t,\omega)$ is strongly measurable with respect to the _predictable
$\sigma$-algebra_ $\mathcal{P}_{\mathbb{R}\times\Omega}:=\sigma((s,t]\times
F_{s}:s<t,\,F_{s}\in\mathcal{F}_{s})\vee\sigma(\mathcal{N}_{\mathbb{P}}).$

######  Definition 4.5.

Let Assumption 4.1 be satisfied and let
$\gamma\in(\nicefrac{{1}}{{2}},\infty)$ be such that Assumption 4.2(i) holds
with $\gamma_{0}=\gamma$. Define the process
$Z_{\gamma}=(Z_{\gamma}(t))_{t\in\mathbb{R}}$ by

$Z_{\gamma}(t):=\frac{1}{\Gamma(\gamma)}\int_{-\infty}^{t}(t-s)^{\gamma-1}S(t-s)\mathop{}\\!\mathrm{d}W^{Q}(s),\quad
t\in\mathbb{R}.$ (4.11)

A $(\mathcal{F}_{t}^{\delta W^{Q}})_{t\in\mathbb{R}}$-predictable modification
of $Z_{\gamma}$ is said to be a _mild solution_ to (4.1).

The first part of the following result asserts that $Z_{\gamma}$ is mean-
square continuous. Since $Z_{\gamma}$ is adapted to
$\bigl{(}\mathcal{F}_{t}^{\delta W^{Q}}\bigr{)}_{t\in\mathbb{R}}$, there does
indeed exist a modification of $Z_{\gamma}$ which is predictable with respect
to this filtration, cf. [9, Proposition 3.7(ii)], the proof of which can be
generalized to unbounded index sets.

######  Proposition 4.6.

Let Assumption 4.1 be satisfied. Suppose that $t_{0}\in[-\infty,\infty)$,
$\gamma\in(\nicefrac{{1}}{{2}},\infty)$ are given. Define
$\mathbb{T}:=[t_{0},\infty)$ if $t_{0}\in\mathbb{R}$ or
$\mathbb{T}:=\mathbb{R}$ if $t_{0}=-\infty$ and let Assumption 4.2(i) hold for
$\gamma_{0}=\gamma$. Then the process $(Z_{\gamma}(t\mid
t_{0}))_{t\in\mathbb{T}}$ defined by

$Z_{\gamma}(t\mid
t_{0}):=\frac{1}{\Gamma(\gamma)}\int_{t_{0}}^{t}(t-s)^{\gamma-1}S(t-s)\mathop{}\\!\mathrm{d}W^{Q}(s),\quad
t\in\mathbb{T},$ (4.12)

where $Z_{\gamma}(\,\cdot\,\mid-\infty):=Z_{\gamma}$, is mean-square
continuous on $\mathbb{T}$.

If in addition Assumption 4.2(ii) is satisfied and there exists
$N\in\mathbb{N}$ such that Assumptions 4.2(i) holds for $\gamma_{0}=\gamma-N$,
then $(Z_{\gamma}(t\mid t_{0}))_{t\in\mathbb{T}}$ has $N$ mean square
derivatives and

$\forall
t\in[t_{0},\infty),\,n\in\\{0,\dots,N-1\\}\colon\>\left(\tfrac{\mathrm{d}}{\mathrm{d}t}+A\right)\tfrac{\mathrm{d}^{n}}{\mathrm{d}t^{n}}Z_{\gamma}(t\mid
t_{0})=\tfrac{\mathrm{d}^{n}}{\mathrm{d}t^{n}}Z_{\gamma-1}(t\mid t_{0}).$
(4.13)

###### Proof.

The mean-square continuity follows directly by Lemma B.8 in Appendix B, so we
turn to the mean-square differentiability. We introduce the auxiliary
processes

$Z_{\beta,j}(t):=\frac{1}{\Gamma(\beta)}\int_{t_{0}}^{t}(t-s)^{\beta-1}A^{j}S(t-s)\mathop{}\\!\mathrm{d}W^{Q}(s),\quad
t\in[t_{0},\infty),$

for $j\in\mathbb{N}_{0}$ and $\beta\in(\nicefrac{{1}}{{2}},\infty)$ such the
right-hand side exists.

We claim that, under Assumptions 4.2(i)–(ii) with $\gamma_{0}=\gamma-N$, the
function ${t\mapsto t^{\beta-1}A^{j}S(t)Q^{\frac{1}{2}}}$ belongs to
$H^{1}_{0,\\{0\\}}(0,\infty;\mathscr{L}_{2}(U))$ if
$\beta-j-\gamma+N\in[1,\infty)$. To this end, we first note that the product
rule for the (classical) derivative yields

$\tfrac{\mathrm{d}}{\mathrm{d}t}t^{\beta-1}A^{j}S(t)Q^{\frac{1}{2}}=(\beta-1)t^{\beta-2}A^{j}S(t)Q^{\frac{1}{2}}-t^{\beta-1}A^{j+1}S(t)Q^{\frac{1}{2}}$

with values in $\mathscr{L}(U)$ for all $t\in(0,\infty)$. Combining (4.3) with
an argument involving a change of variables and the semigroup property (cf.
the proof of Lemma B.4 in Appendix B), one can show that the
$L^{2}(0,\infty;\mathscr{L}_{2}(U))$-norms of the two functions on the right-
hand side can be estimated by that of $t\mapsto
t^{\beta-j-2}S(t)Q^{\frac{1}{2}}$, which is finite since
$\beta-j-1\geq\gamma_{0}$. Again by (4.3), we have

$\|t^{\beta-1}A^{j}S(t)Q^{\frac{1}{2}}\|_{\mathscr{L}(U)}\leq
M_{j}t^{\beta-j-1}\|Q^{\frac{1}{2}}\|_{\mathscr{L}(U)}\to 0\quad\text{as
}t\downarrow 0$

since $\beta-j-1\geq\gamma_{0}>0$. Noting that
$\mathscr{L}_{2}(U)\hookrightarrow\mathscr{L}(U)$ and using [24, Lemma A.9]
then proves the claim.

Thus, we may apply Lemma B.8 from Appendix B, write the result as two separate
integrals, and pull the closed operator $A$ out of the stochastic integral
defining $Z_{\beta,j+1}$ (cf. [9, Proposition 4.30]) to find

$\displaystyle Z_{\beta,j}^{\prime}(t)$
$\displaystyle=Z_{\beta-1,j}(t)-Z_{\beta,j+1}(t).$ (4.14)
$\displaystyle=Z_{\beta-1,j}(t)-AZ_{\beta,j}(t).$ (4.15)

Rearranging equation (4.15) for $\beta=\gamma$ and $j=0$ implies (4.13) in the
case $n=0$. Applying (4.14) iteratively, we find that $Z_{\beta,j}$ has the
$n$th mean-square derivative

$Z_{\beta,j}^{(n)}(t)=\sum_{m=0}^{n}(-1)^{m}{n\choose m}Z_{\beta-n+m,j+m}(t),$
(4.16)

provided that $\beta-j-\gamma+N\in[n,\infty)$. Now we again let $\beta=\gamma$
and $j=0$ and apply (4.15) with $\beta^{\prime}=\gamma-n+m$ and $j^{\prime}=m$
to each term on the right-hand side to derive (4.13) for the remaining values
of $n$. ∎

The statistical relevance of the process $Z_{\gamma}$ is motivated by the
following fact, which demonstrates a situation in which it has a marginal
temporal covariance structure of Matérn type. Let
$(Q_{Z_{\gamma}}(s,t))_{s,t\in\mathbb{R}}\subseteq\mathscr{L}_{1}^{+}(U)$
denote the covariance operators of $Z_{\gamma}$, defined in general via the
relation

$\langle Q_{Z_{\gamma}}(s,t)x,y\rangle_{U}=\mathbb{E}\left[\left\langle
Z_{\gamma}(s)-\mathbb{E}\bigl{[}Z_{\gamma}(s)\bigr{]},x\right\rangle_{U}\left\langle
Z_{\gamma}(t)-\mathbb{E}[Z_{\gamma}(t)],y\right\rangle_{U}\right]$ (4.17)

for all $s,t\in\mathbb{R}$ and $x,y\in U$. Note that
$\mathbb{E}\bigl{[}Z_{\gamma}(\,\cdot\,)\bigr{]}\equiv 0$ in this case.

######  Proposition 4.7.

Let $\gamma\in(\nicefrac{{1}}{{2}},\infty)$, $A:=\kappa\operatorname{Id}_{U}$
with ${\kappa\in(0,\infty)}$ and suppose that Assumption 4.2(i) is satisfied
for $\gamma_{0}=\gamma$. Then the covariance of $Z_{\gamma}$ is separable and
its temporal part is of Matérn type, i.e.,

$\forall s,t\in\mathbb{R},\,s\neq t:\quad
Q_{Z_{\gamma}}(s,t)=\frac{2^{\frac{1}{2}-\gamma}\kappa^{1-2\gamma}}{\sqrt{\pi}\,\Gamma(\gamma)}(\kappa|t-s|)^{\gamma-\frac{1}{2}}K_{\gamma-\frac{1}{2}}(\kappa|t-s|)\,Q.$
(4.18)

###### Proof.

For $A=\kappa\operatorname{Id}_{U}$, Assumption 4.1 is trivially satisfied and
the definition of $Z_{\gamma}$ takes on the following form for all
$t\in\mathbb{R}$:

$Z_{\gamma}(t)=\frac{1}{\Gamma(\gamma)}\int_{-\infty}^{t}(t-r)^{\gamma-1}e^{-\kappa(t-r)}\mathop{}\\!\mathrm{d}W^{Q}(r)=\int_{\mathbb{R}}k_{\gamma,\kappa}(t-r)\mathop{}\\!\mathrm{d}W^{Q}(r),$
(4.19)

with real-valued convolution kernel
$k_{\gamma,\kappa}(t):=\frac{1}{\Gamma(\gamma)}t^{\gamma-1}_{+}e^{-\kappa t}$,
where $t_{+}^{\gamma-1}:=t^{\gamma-1}$ if $t\in[0,\infty)$ and
$t_{+}^{\gamma-1}:=0$ otherwise. Define the linear operator
$\widetilde{k}(s,r;x)\in\mathscr{L}(U;\mathbb{R})$ for $s,r\in\mathbb{R}$ and
$x\in U$ by

$\widetilde{k}(s,r;x)h:=k_{\gamma,\kappa}(s-r)\langle x,h\rangle_{U},\quad
h\in U.$

Then combining the Itô isometry (2.2) and the polarization identity yields

$\displaystyle\mathbb{E}[\langle Z_{\gamma}(s),x\rangle_{U}\,\langle
Z_{\gamma}(t),y\rangle_{U}]=\mathbb{E}\biggl{[}\int_{\mathbb{R}}\widetilde{k}(s,r;x)\mathop{}\\!\mathrm{d}W^{Q}(r)\int_{\mathbb{R}}\widetilde{k}(t,r;y)\mathop{}\\!\mathrm{d}W^{Q}(r)\biggr{]}$
$\displaystyle=\int_{\mathbb{R}}\langle\widetilde{k}(s,r;x)Q^{\frac{1}{2}},\widetilde{k}(t,r;y)Q^{\frac{1}{2}}\rangle_{\mathscr{L}_{2}(U;\mathbb{R})}\mathop{}\\!\mathrm{d}r=\biggl{\langle}\int_{\mathbb{R}}k_{\gamma,\kappa}(s-r)k_{\gamma,\kappa}(t-r)\mathop{}\\!\mathrm{d}r\,Qx,y\biggr{\rangle}_{U}.$

Since $x,y\in U$ were arbitrary, we find for all
$h\in\mathbb{R}\setminus\\{0\\}$ the covariance operators

$Q_{Z_{\gamma}}(t+h,t)=Q_{Z_{\gamma}}(t,t+h)=\int_{\mathbb{R}}k_{\gamma,\kappa}(t-r)k_{\gamma,\kappa}(t+h-r)\mathop{}\\!\mathrm{d}r\,Q.$
(4.20)

Using the change of variables $u(r):=h+2(t-r)$ in the integral, we obtain

$\displaystyle\int_{\mathbb{R}}$ $\displaystyle
k_{\gamma,\kappa}(t-r)k_{\gamma,\kappa}(t+h-r)\mathop{}\\!\mathrm{d}r=\frac{1}{2}\int_{\mathbb{R}}k_{\gamma,\kappa}(\tfrac{u-h}{2})k_{\gamma,\kappa}(\tfrac{u+h}{2})\mathop{}\\!\mathrm{d}u$
(4.21)
$\displaystyle=\frac{2^{1-2\gamma}}{[\Gamma(\gamma)]^{2}}\int_{|h|}^{\infty}(u^{2}-h^{2})^{\gamma-1}e^{-\kappa
u}\mathop{}\\!\mathrm{d}u=\frac{2^{\frac{1}{2}-\gamma}\kappa^{1-2\gamma}}{\sqrt{\pi}\,\Gamma(\gamma)}(\kappa|h|)^{\gamma-\frac{1}{2}}K_{\gamma-\frac{1}{2}}(\kappa|h|),$

where the last identity follows by [35, Part I, Equation (3.13)]. ∎

###  4.4. Markov behavior

In this section we consider the Markov behavior of the process $Z_{\gamma}$
defined in Section 4.3. Namely, we will show that $Z_{N}$ is $N$-ple Markov
for $N\in\mathbb{N}$, whereas $Z_{\gamma}$ is in general not weak Markov if
$\gamma\not\in\mathbb{N}$, see Theorem 4.9 and Example 4.15, respectively.

####  4.4.1. Integer case; main results

We first introduce the necessary notation and intermediate results leading up
to the main theorem asserting the $N$-ple Markov property of $Z_{N}$. The
proofs are postponed to Subsection 4.4.2.

If $\gamma\in(\nicefrac{{1}}{{2}},\infty)$ is such that Assumptions 4.1 and
4.2(i) hold with $\gamma_{0}=\gamma$, then we define for $t_{0}\in\mathbb{R}$
the truncated integral process $(\widetilde{Z}_{\gamma}(t\mid
t_{0}))_{t\in\mathbb{R}}$ by

$\widetilde{Z}_{\gamma}(t\mid
t_{0}):=\frac{1}{\Gamma(\gamma)}\int_{-\infty}^{t\wedge
t_{0}}(t-s)^{\gamma-1}S(t-s)\mathop{}\\!\mathrm{d}W^{Q}(s),$ (4.22)

so that $\widetilde{Z}_{\gamma}(\,\cdot\mid t_{0})=Z_{\gamma}$ on
$(-\infty,t_{0}]$ and $Z_{\gamma}=\widetilde{Z}_{\gamma}(\,\cdot\mid
t_{0})+Z_{\gamma}(\,\cdot\,\mid t_{0})$ on $(t_{0},\infty)$, where we recall
the process $(Z_{\gamma}(t\mid t_{0}))_{t\in[t_{0},\infty)}$ from (4.12). It
immediately follows that $\widetilde{Z}_{\gamma}(t\mid t_{0})$ has the same
temporal regularity at $t\in\mathbb{R}\setminus\\{t_{0}\\}$ as $Z_{\gamma}(t)$
(and ${Z_{\gamma}(t\mid t_{0})}$ if $t\in(t_{0},\infty)$). In the case
$\gamma=N\in\mathbb{N}$, both have $N-1$ mean-square derivatives by
Proposition 4.6 if Assumption 4.2(i) holds for $\gamma_{0}=\gamma-(N-1)=1$.
The same holds at the critical point $t=t_{0}$ since the first $N-1$ mean-
square (right) derivatives of $Z_{\gamma}(\,\cdot\mid t_{0})$ vanish there,
see (4.16) in the proof of Proposition 4.6.

Under Assumption 4.2(ii), we have $A^{j}S(t)\in\mathscr{L}(U)$ for all
$j\in\mathbb{N}_{0}$ and $t\in(0,\infty)$, see (4.3). Therefore, we can define
the function
$\overline{\Gamma}(n,(\,\cdot\,)A)\colon[0,\infty)\to\mathscr{L}(U)$ by

$\displaystyle\overline{\Gamma}(n,tA):=\begin{cases}\sum_{j=0}^{n-1}\frac{t^{j}}{j!}A^{j}S(t),\quad&t\in(0,\infty);\\\
\operatorname{Id}_{U},\quad&t=0,\end{cases}$ (4.23)

for $n\in\mathbb{N}$. We point out the analogy with integer-order scalar-
valued normalized upper incomplete gamma functions [36, Equations (8.4.10) and
(8.4.11)]. We will use these functions to derive an expression for
$(\widetilde{Z}_{N}(t\mid t_{0}))_{t\in[t_{0},\infty)}$ in terms of $Z_{N}$
and its mean-square derivatives at $t_{0}$. Recall that
$\mathbf{Z}_{N}=(Z_{N}^{(n)})_{n=0}^{N-1}$ indicates the $U^{N}$-valued
process consisting of $Z_{N}$ and its first $N-1$ mean-square derivatives.

######  Proposition 4.8.

Let Assumptions 4.1 and 4.2(ii) be satisfied and suppose that Assumption
4.2(i) holds for $\gamma_{0}=1$. Then for all $N\in\mathbb{N}$,
$t_{0}\in\mathbb{R}$ and $t\in[t_{0},\infty)$,

$\widetilde{Z}_{N}(t\mid t_{0})=\zeta_{N}(t\mid
t_{0})\mathbf{Z}_{N}(t_{0}),\quad\mathbb{P}\text{-a.s.},$ (4.24)

where we define, for any $\bm{\xi}\in
L^{2}(\Omega,\mathcal{F}_{t_{0}},\mathbb{P};U^{N})$,

$\zeta_{N}(t\mid
t_{0})\bm{\xi}:=\sum_{k=0}^{N-1}\frac{(t-t_{0})^{k}}{k!}\overline{\Gamma}(N-k,(t-t_{0})A)\,\xi_{k},$
(4.25)

using the incomplete gamma functions defined in (4.23).

In particular, adding $Z_{N}(t\mid t_{0})$ on both sides of equation (4.24)
yields

$\forall t\in[t_{0},\infty):\quad Z_{N}(t)=Z_{N}(t\mid
t_{0},\mathbf{Z}_{N}(t_{0})),\quad\mathbb{P}\text{-a.s.},$ (4.26)

where the process $(Z_{N}(t\mid t_{0},\bm{\xi}))_{t\in[t_{0},\infty)}$ is
defined by

$Z_{N}(t\mid t_{0},\bm{\xi}):=\zeta_{N}(t\mid t_{0})\bm{\xi}+Z_{N}(t\mid
t_{0}),\quad t\in[t_{0},\infty).$ (4.27)

By (4.26), it suffices to show that $(Z_{N}(t\mid
t_{0},\bm{\xi}))_{t\in[t_{0},\infty)}$ has the $N$-ple Markov property in the
sense of Definition 3.3 for arbitrary $t_{0}\in\mathbb{R}$ and $\bm{\xi}\in
L^{2}(\Omega,\mathcal{F}_{t_{0}},\mathbb{P};U^{N})$. In fact, we will show
that it is $N$-ple Markov using Corollary 3.4; this is the subject of the
following result, which is the main theorem of this section.

######  Theorem 4.9.

Let $N\in\mathbb{N}$, $t_{0}\in\mathbb{R}$ and
$\bm{\xi}=(\xi_{k})_{k=0}^{N-1}\in
L^{2}(\Omega,\mathcal{F}_{t_{0}},\mathbb{P};U^{N+1})$ be given. Let
Assumptions 4.1 and 4.2(ii) hold and suppose that Assumption 4.2(i) is
satisfied for $\gamma_{0}=1$. Then the process $(Z_{N}(t\mid
t_{0},\bm{\xi}))_{t\in[t_{0},\infty)}$ from (4.27) has the $N$-ple Markov
property in the sense of Definition 3.3 with respect to the transition
operators $(T_{s,t})_{t_{0}\leq s\leq t}$ on $B_{b}(U^{N})$ defined by

$T_{s,t}\varphi(\mathbf{x}):=\mathbb{E}[\varphi(\mathbf{Z}_{N}(t\mid
s,\mathbf{x}))],\quad\varphi\in B_{b}(U^{N}),\;\mathbf{x}\in U^{N},$ (4.28)

and the increment filtration
$(\mathcal{F}_{t})_{t\in[t_{0},\infty)}:=(\mathcal{F}_{t}^{\delta
W^{Q}})_{t\in[t_{0},\infty)}$ from (2.3).

The process $(Z_{N}(t))_{t\in\mathbb{R}}$ from (4.11) has $N-1$ mean-square
derivatives and is $N$-ple Markov in the sense of Definition 3.3 with respect
to $(\mathcal{F}_{t})_{t\in\mathbb{R}}$.

The statements and proofs of Proposition 4.8 and Theorem 4.9 use the following
result regarding the mean-square differentiability of $(\zeta_{N}(t\mid
t_{0})\bm{\xi})_{t\in[t_{0},\infty)}$, which is similar to Proposition 4.6.

######  Proposition 4.10.

Let $N\in\\{2,3,\dots\\}$, $t_{0}\in\mathbb{R}$ and $\bm{\xi}\in
L^{2}(\Omega,\mathcal{F}_{t_{0}},\mathbb{P};U^{N})$ be given, where
$\xi_{k}\in L^{2}(\Omega;\mathsf{D}(A))$ for $k\in\\{0,\dots,N-2\\}$. Suppose
that Assumptions 4.1 and 4.2(ii) hold. Then the process $(\zeta_{N}(t\mid
t_{0})\bm{\xi})_{t\in[t_{0},\infty)}$ from (4.25) is infinitely mean-square
differentiable at any $t\in(t_{0},\infty)$ and, for $n\in\\{0,\dots,N-2\\}$,

$\left(\tfrac{\mathrm{d}}{\mathrm{d}t}+A\right)\tfrac{\mathrm{d}^{n}}{\mathrm{d}t^{n}}\zeta_{N}(t\mid
t_{0})\bm{\xi}=\tfrac{\mathrm{d}^{n}}{\mathrm{d}t^{n}}\zeta_{N-1}(t\bigm{|}t_{0})(\xi_{k+1}+A\xi_{k})_{k=0}^{N-2},\quad\mathbb{P}\text{-a.s.}$
(4.29)

Moreover, we have
$\left(\tfrac{\mathrm{d}}{\mathrm{d}t}+A\right)\tfrac{\mathrm{d}^{n}}{\mathrm{d}t^{n}}\zeta_{1}(t\mid
t_{0})\xi=0$, $\mathbb{P}$-a.s., for $\xi\in
L^{2}(\Omega,\mathcal{F}_{t_{0}},\mathbb{P};U)$.

Combining Propositions 4.6, 4.8 and 4.10 yields:

######  Corollary 4.11.

Let $N\in\\{2,3,\dots\\}$, $t_{0}\in\mathbb{R}$ and $\bm{\xi}\in
L^{2}(\Omega,\mathcal{F}_{t_{0}},\mathbb{P};U^{N})$ be given, where
$\xi_{k}\in L^{2}(\Omega;\mathsf{D}(A))$ for $k\in\\{0,\dots,N-2\\}$. Let
Assumptions 4.1 and 4.2(ii) hold and suppose that Assumption 4.2(i) is
satisfied for $\gamma_{0}=1$. Then the process $(Z_{N}(t\mid
t_{0},\bm{\xi}))_{t\in[t_{0},\infty)}$ from (4.27) is $N-1$ times mean-square
differentiable at any $t\in(t_{0},\infty)$ and satisfies, for
$n\in\\{0,\dots,N-2\\}$,

$\left(\tfrac{\mathrm{d}}{\mathrm{d}t}+A\right)\tfrac{\mathrm{d}^{n}}{\mathrm{d}t^{n}}Z_{N}(t\mid
t_{0},\bm{\xi})=\tfrac{\mathrm{d}^{n}}{\mathrm{d}t^{n}}Z_{N-1}\bigl{(}t\bigm{|}t_{0},(\xi_{k+1}+A\xi_{k})_{k=0}^{N-2}\,\bigr{)},\quad\mathbb{P}\text{-a.s.}$
(4.30)

In particular, it holds for all $t\in\mathbb{R}$ and $n\in\\{0,\dots,N-2\\}$
that

$\left(\tfrac{\mathrm{d}}{\mathrm{d}t}+A\right)\tfrac{\mathrm{d}^{n}}{\mathrm{d}t^{n}}Z_{N}(t)=\tfrac{\mathrm{d}^{n}}{\mathrm{d}t^{n}}Z_{N-1}(t),\quad\mathbb{P}\text{-a.s.}$
(4.31)

######  Remark 4.12.

Corollary 4.11 can be interpreted as saying that $(Z_{N}(t\mid
t_{0},\bm{\xi}))_{t\in[t_{0},\infty)}$ for $N\in\\{2,3,\dots\\}$ solves the
$L^{2}(\Omega,\mathcal{F},\mathbb{P};U)$-valued initial value problem

$\left\\{\begin{aligned}
\left(\tfrac{\mathrm{d}}{\mathrm{d}t}+A\right)X(t)&=Z_{N-1}(t\mid
t_{0},(\xi_{k+1}+A\xi_{k})_{k=0}^{N-2}),\quad\forall t\in(t_{0},\infty),\\\
X(t_{0})&=\xi_{0},\end{aligned}\right.$ (4.32)

whenever $\xi_{k}\in
L^{2}(\Omega,\mathcal{F}_{t_{0}},\mathbb{P};\mathsf{D}(A))$ for
$k\in\\{0,\dots,N-2\\}$. This observation is the key to the proofs of
Propositions 4.8 and 4.13 below. It is also of interest for computational
methods, as it implies that the computation of $Z_{N}(t\mid t_{0},\bm{\xi})$
amounts to solving a first-order problem $N-1$ times. In fact, inductively
applying this result and the fact that
$(\tfrac{\mathrm{d}}{\mathrm{d}t}+A)\zeta_{1}(t\mid t_{0})\eta=0$ for any
$\eta\in L^{2}(\Omega,\mathcal{F}_{t_{0}},\mathbb{P};U)$, we see that for
$N\in\mathbb{N}$ we may interpret $(Z_{N}(t\mid
t_{0},\bm{\xi}))_{t\in[t_{0},\infty)}$ as the mild solution to the $N$th order
initial value problem

$\left\\{\begin{aligned}
\left(\tfrac{\mathrm{d}}{\mathrm{d}t}+A\right)^{N}X(t)&=\dot{W}^{Q}(t),&&\forall
t\in(t_{0},\infty),\\\
\tfrac{\mathrm{d}^{k}}{\mathrm{d}t^{k}}X(t_{0})&=\xi_{k},&&\forall
k\in\\{0,\dots,N-1\\}.\end{aligned}\right.$ (4.33)

Another key step in the proof of in Theorem 4.9 is given by the following
result, which essentially amounts to the fact that $(T_{s,t})_{s\leq t}$
satisfies (TO3).

######  Proposition 4.13.

Let $N\in\mathbb{N}$, $t_{0}\in\mathbb{R}$ and $\bm{\xi}\in
L^{2}(\Omega,\mathcal{F}_{t_{0}},\mathbb{P};U^{N})$ be given. Let Assumptions
4.1 and 4.2(ii) hold and suppose that Assumption 4.2(i) is satisfied for
$\gamma_{0}=1$. The stochastic process $(Z_{N}(t\mid
t_{0},\bm{\xi}))_{t\in[t_{0},\infty)}$ from (4.27) has the _$N$ -ple
Chapman–Kolmogorov property_, i.e., for all $t_{0}\leq s\leq t$ we have

$\mathbf{Z}_{N}(t\mid t_{0},\bm{\xi})=\mathbf{Z}_{N}(t\mid
s,\mathbf{Z}_{N}(s\mid t_{0},\bm{\xi})),\quad\mathbb{P}\text{-a.s.}$ (4.34)

####  4.4.2. Integer case; proofs

As indicated in Subsection 4.4.1 above, the statements and proofs of
Proposition 4.8 and Theorem 4.9 rely on Proposition 4.10, which we prove
first.

###### Proof of Proposition 4.10.

We first make some general remarks regarding the operators
$\overline{\Gamma}(n,tA)$ from (4.23). Under Assumption 4.2(ii), estimate
(4.3) implies that the set
$\\{t^{j}A^{j}S(t):t\in(0,\infty)\\}\subseteq\mathscr{L}(U)$ is uniformly
bounded. It follows that $t\mapsto\overline{\Gamma}(n,tA)$ is a strongly
continuous function from $[0,\infty)$ to $\mathscr{L}(U)$ for any
$n\in\mathbb{N}$, which at $t\in(0,\infty)$ admits a classical derivative
satisfying the recurrence relation

$\left(\tfrac{\mathrm{d}}{\mathrm{d}t}+A\right)\overline{\Gamma}(n,tA)=\begin{cases}0,&n=1;\\\
A\overline{\Gamma}(n-1,tA),&n\in\\{2,3,\dots\\}.\end{cases}$ (4.35)

To prove the proposition, we may assume $t_{0}=0$, so fix $t\in(0,\infty)$.
For arbitrary $M\in\mathbb{N}$, $j\in\mathbb{N}_{0}$ and $\bm{\eta}\in
L^{2}(\Omega;U^{M})$, we define $\zeta_{M,j}(t)\bm{\eta}:=A^{j}\zeta_{M}(t\mid
0)\bm{\eta}.$ Combining the product rule in the form
$(\frac{\mathrm{d}}{\mathrm{d}t}+A)(uv)=u^{\prime}v+u[(\frac{\mathrm{d}}{\mathrm{d}t}+A)v]$
with the above recurrence relation yields for $M\in\mathbb{N}$

$\displaystyle(\tfrac{\mathrm{d}}{\mathrm{d}t}$
$\displaystyle+A)\zeta_{M,j}(t)\bm{\eta}$ (4.36)
$\displaystyle=\sum_{k=1}^{M-1}\frac{t^{k-1}}{(k-1)!}A^{j}\overline{\Gamma}(M-k,tA)\eta_{k}+\sum_{k=0}^{M-2}\frac{t^{k}}{k!}A^{j+1}\overline{\Gamma}(M-1-k,tA)\eta_{k}$
$\displaystyle=\zeta_{(M-1),j}(t)(\eta_{k+1})_{k=0}^{M-1}+\zeta_{(M-1),(j+1)}(t)(\eta_{k})_{k=0}^{M-1}.$
(4.37)

This shows in particular that (4.29) holds for integers $N\geq 2$ and $n=0$,
by applying (4.37) with $M=N$, $\bm{\eta}=\bm{\xi}$ and $j=0$. Moreover, note
that

$\left(\tfrac{\mathrm{d}}{\mathrm{d}t}+A\right)\zeta_{1,j}(t)\,\eta=\left(\tfrac{\mathrm{d}}{\mathrm{d}t}+A\right)A^{j}S(t)\,\eta=0.$

Iteratively applying (4.37) and the latter identity then yields that
$\zeta_{M,j}\bm{\eta}$ is $M-1$ times (mean-square) differentiable with an
$n$th derivative of the form

$\zeta^{(n)}_{M,j}(t)\bm{\eta}=\sum_{\ell=0}^{n}\sum_{m=0}^{\ell}C_{\ell,m}\zeta_{(M-\ell),(j+n-m)}(t)B_{\ell,m}\bm{\eta},$
(4.38)

where $C_{\ell,m}\in\mathbb{R}$, $B_{\ell,m}\in\mathscr{L}(U^{M};U^{M-\ell})$
and $\zeta_{(M-\ell),(j+n-m)}:=0$ if $M-\ell<1$. In particular,
$\zeta_{N}(\,\cdot\mid t_{0})\bm{\xi}$ is $N-1$ times (mean-square)
differentiable as claimed. In order to deduce that (4.29) also holds for
$n\in\\{1,\dots,N-2\\}$, we need to justify taking the $n$th derivative on
both sides and commuting it with $A$. Since $A$ is closed, it suffices to
verify that $\zeta_{N}^{\prime}(\,\cdot\mid t_{0})\bm{\xi}$,
$A^{j}\zeta_{N}(\,\cdot\mid t_{0})\bm{\xi}$, $\zeta_{N-1}(\,\cdot\mid
t_{0})(\xi_{j+1})_{j=0}^{N-1}$ and $A^{j}\zeta_{N-1}(\,\cdot\mid
t_{0})(\xi_{j})_{j=0}^{N-1}$ admit $n$th derivatives for $j\in\\{0,1\\}$.
Indeed, these assertions follow from (4.38). ∎

We can now prove Propositions 4.8 and 4.13. Note that for the proof of the
latter we may use Corollary 4.11, since it combines Propositions 4.6, 4.8 and
4.10.

###### Proof of Proposition 4.8.

We use induction on $N\in\mathbb{N}$. For $N=1$ and $t\in(t_{0},\infty)$,

$\displaystyle\widetilde{Z}_{1}(t\mid
t_{0})=S(t-t_{0})\int_{-\infty}^{t_{0}}S(t_{0}-s)\mathop{}\\!\mathrm{d}W^{Q}(s)=\zeta_{1}(t\mid
t_{0})Z_{1}(t_{0}),\quad\mathbb{P}\text{-a.s.}$ (4.39)

Now suppose that the statement is true for a given $N\in\mathbb{N}$. By
Proposition 4.6 and the discussion below (4.22), $Z_{N+1}$ and
$\widetilde{Z}_{N+1}(\,\cdot\mid t_{0})$ have $N$ mean-square derivatives
which satisfy

$\left(\tfrac{\mathrm{d}}{\mathrm{d}t}+A\right)Z_{N+1}^{(k)}(t)=Z_{N}^{(k)}(t)\quad\text{and}\quad\left(\tfrac{\mathrm{d}}{\mathrm{d}t}+A\right)\widetilde{Z}_{N+1}(t\mid
t_{0})=\widetilde{Z}_{N}(t\mid t_{0}),\quad\mathbb{P}\text{-a.s.},$ (4.40)

for all $k\in\\{0,\dots,N-1\\}$ and $t\in(t_{0},\infty)$. Combined with
Proposition 4.10 and the induction hypothesis, we find

$\displaystyle\bigl{(}\tfrac{\mathrm{d}}{\mathrm{d}t}$
$\displaystyle+A\bigr{)}\zeta_{N+1}(t\mid
t_{0})\mathbf{Z}_{N+1}(t_{0})=\zeta_{N}(t\mid
t_{0})\bigl{[}\bigl{(}\tfrac{\mathrm{d}}{\mathrm{d}t}+A\bigr{)}Z_{N+1}^{(k)}(t_{0})\bigr{]}_{k=0}^{N-1}$
(4.41) $\displaystyle=\zeta_{N}(t\mid
t_{0})\mathbf{Z}_{N}(t_{0})=\widetilde{Z}_{N}(t\mid
t_{0})=\left(\tfrac{\mathrm{d}}{\mathrm{d}t}+A\right)\widetilde{Z}_{N+1}(t\mid
t_{0}).$ (4.42)

Since $\widetilde{Z}_{N+1}(t_{0}\mid
t_{0})=Z_{N+1}(t_{0})=\zeta_{N+1}(t_{0}\mid t_{0})\mathbf{Z}_{N+1}(t_{0})$, we
conclude that (4.24) with $N+1$ holds on $[t_{0},\infty)$ by the uniqueness of
solutions to $L^{2}(\Omega,\mathcal{F},\mathbb{P};U)$-valued abstract Cauchy
problems, see [37, Chapter 4, Theorem 1.3]. ∎

###### Proof of Proposition 4.13.

Let $t_{0}\leq s\leq t$. We use induction on $N\in\mathbb{N}$. For the base
case $N=1$ we have

$\displaystyle Z_{1}(t\mid s,Z_{1}(s\mid t_{0},\xi))=S(t-s)Z_{1}(s\mid
t_{0},\xi)+\int_{s}^{t}S(t-r)\mathop{}\\!\mathrm{d}W^{Q}(r)$
$\displaystyle=S(t-s)S(s-t_{0})\xi+S(t-s)\int_{t_{0}}^{s}S(s-r)\mathop{}\\!\mathrm{d}W^{Q}(r)+\int_{s}^{t}S(t-r)\mathop{}\\!\mathrm{d}W^{Q}(r)$
$\displaystyle=S(t-t_{0})\xi+\int_{t_{0}}^{s}S(t-r)\mathop{}\\!\mathrm{d}W^{Q}(r)+\int_{s}^{t}S(t-r)\mathop{}\\!\mathrm{d}W^{Q}(r)=Z_{1}(t\mid
t_{0},\xi),$

$\mathbb{P}$-a.s., for $\xi\in
L^{2}(\Omega,\mathcal{F}_{t_{0}},\mathbb{P};\mathsf{D}(A))$. Now suppose that
the result holds for $N\in\mathbb{N}$ and let $\bm{\xi}\in
L^{2}(\Omega,\mathcal{F}_{t_{0}},\mathbb{P};\mathsf{D}(A)^{N})$. Then for any
$t\in(s,\infty)$ we have

$\displaystyle\bigl{(}\tfrac{\mathrm{d}}{\mathrm{d}t}$
$\displaystyle+A\bigr{)}Z_{N+1}(t\mid s,\mathbf{Z}_{N+1}(s\mid
t_{0},\bm{\xi}))=Z_{N}\left(t\,\middle|\,s,\bigl{[}\bigl{(}\tfrac{\mathrm{d}}{\mathrm{d}t}+A\bigr{)}Z_{N+1}^{(k)}(s\mid
t_{0},\bm{\xi})\bigr{]}_{k=0}^{N-1}\right)$
$\displaystyle=Z_{N}\bigl{(}t\bigm{|}s,{\mathbf{Z}}_{N}(s\mid
t_{0},[\xi_{k+1}+A\xi_{k}]_{k=0}^{N-1})\bigr{)}=Z_{N}(t\mid
t_{0},[\xi_{k+1}+A\xi_{k}]_{k=0}^{N-1})$
$\displaystyle=\bigl{(}\tfrac{\mathrm{d}}{\mathrm{d}t}+A\bigr{)}Z_{N+1}(t\mid
t_{0},\bm{\xi}),\quad\text{$\mathbb{P}$-a.s.,}$

where we applied Corollary 4.11 in every identity except the third, which uses
the induction hypothesis. Moreover, $Z_{N+1}(s\mid s,\mathbf{Z}_{N+1}(s\mid
t_{0},\bm{\xi}))=Z_{N+1}(s\mid t_{0},\bm{\xi})$ is evident from the
definitions. Together, these facts imply that the difference process
$Y:=Z_{N+1}(\,\cdot\mid s,\mathbf{Z}_{N+1}(s\mid
t_{0},\bm{\xi}))-Z_{N+1}(\,\cdot\mid t_{0},\bm{\xi})$ solves the abstract
Cauchy problem

$\left\\{\begin{aligned}
\left(\tfrac{\mathrm{d}}{\mathrm{d}t}+\mathcal{A}_{\Omega}\right)Y(t)&=0,\quad\forall
t\in(s,\infty);\\\ Y(s)&=0,\end{aligned}\right.$ (4.43)

where $\mathcal{A}_{\Omega}\colon L^{2}(\Omega;\mathsf{D}(A))\subseteq
L^{2}(\Omega;U)\to L^{2}(\Omega;U)$ is as in (4.8). Since
$-\mathcal{A}_{\Omega}$ is the generator of a $C_{0}$-semigroup on
$L^{2}(\Omega;U)$, see Lemma C.2(c) in Appendix C, the uniqueness result [37,
Chapter 4, Theorem 1.3] shows that $Y\equiv 0$ on $[s,\infty)$, meaning that
for all $t\in[s,\infty)$ we have

$Z_{N+1}(t\mid t_{0},\bm{\xi})=Z_{N+1}(t\mid s,\mathbf{Z}_{N+1}(s\mid
t_{0},\bm{\xi})),\quad\mathbb{P}\text{-a.s.}$ (4.44)

Taking $n$th mean-square derivatives on both sides for $n\in\\{0,\dots,N\\}$,
which is justified by Corollary 4.11, we find (4.34). In order to establish
this identity for general $\bm{\xi}\in
L^{2}(\Omega,\mathcal{F}_{t_{0}},\mathbb{P};U^{N})$, we employ the density of
$\mathsf{D}(A)$ in $U$ [37, Chapter 1, Corollary 2.5], which implies the
density of $L^{2}(\Omega;\mathsf{D}(A)^{N})$ in $L^{2}(\Omega;U^{N})$, so that
it suffices to argue that the mapping $\bm{\eta}\mapsto\mathbf{Z}_{N+1}(t\mid
t_{0},\bm{\eta})$ from $L^{2}(\Omega;U^{N})$ to itself is continuous for any
fixed $t\in[t_{0},\infty)$. The continuity of
$\bm{\eta}\mapsto\zeta_{N+1}(t\mid t_{0})\bm{\eta}$ follows from the fact that
$\eta\mapsto\overline{\Gamma}(N+1-k,(t-t_{0}))\eta$ is bounded on
$L^{2}(\Omega;U)$ for any $k\in\\{0,\dots,N\\}$. The same holds for the
derivatives of $\zeta_{N+1}(\,\cdot\mid t_{0})\bm{\eta}$ since they are of the
same form by Proposition 4.10. Together, the conclusion follows. ∎

With these intermediate results in place, we are ready to prove the main
theorem asserting the $N$-ple Markovianity of $Z_{N}$. Its proof is a
generalization of [9, Theorem 9.14] and [38, Theorem 9.30], which concern
simple Markovianity for $N=1$. We divide it in two parts: In the first part,
we verify that $(T_{s,t})_{t_{0}\leq s\leq t}$ is a well-defined family of
transition operators on $B_{b}(U^{N})$. In the second part, we show that
$Z_{N}(\,\cdot\mid t_{0},\bm{\xi})$ is $N$-ple Markov with respect to
$(T_{s,t})_{t_{0}\leq s\leq t}$, see Corollary 3.4

###### Proof of Theorem 4.9.

Step 1: Well-definedness of $(T_{s,t})_{t_{0}\leq s\leq t}$. We have to show
that $T_{s,t}\varphi=\mathbb{E}[\varphi(\mathbf{Z}_{N}(t\,|\,s,\,\cdot\,))]$
is measurable for $\varphi\in B_{b}(U^{N})$. For a monotone class argument, we
introduce the linear space $\mathscr{H}$ of bounded functions $\varphi\colon
U^{N}\to\mathbb{R}$ such that
$\mathbf{x}\mapsto\mathbb{E}[\varphi(\mathbf{Z}_{N}(t\mid s,\mathbf{x}))]$ is
measurable. Arguing similarly to the end of the proof of Proposition 4.13, we
find that ${\mathbf{x}\mapsto[\mathbf{Z}_{N}(t\mid s,\mathbf{x})](\omega)}$ is
continuous on $U^{N}$ for $\mathbb{P}$-a.e. $\omega\in\Omega$. Thus, for
$\varphi\in\mathscr{C}:=C_{b}(U^{N})$, the dominated convergence theorem
implies that $\mathbf{x}\mapsto\mathbb{E}[\varphi(\mathbf{Z}_{N}(t\mid
s,\mathbf{x}))]$ is also continuous, hence (Borel) measurable. It follows that
$\mathscr{C}\subseteq\mathscr{H}$. Moreover, $\mathscr{H}$ contains all
constant functions, and given a sequence
$(\varphi_{n})_{n\in\mathbb{N}}\subseteq\mathscr{H}$ such that
${0\leq\varphi_{n}\uparrow\varphi}$ pointwise for some bounded limit function
$\varphi$, we find $\varphi\in\mathscr{H}$ by monotone convergence. Since
$\mathscr{C}$ is closed under pointwise multiplication, we conclude that
$B_{b}(U^{N}\\!,\sigma(\mathscr{C}))=B_{b}(U^{N})\subseteq\mathscr{H}$ by the
monotone class theorem [40, Chapter 0, Theorem 2.2].

Step 2: $N$-ple Markovianity. For $t_{0}\leq s\leq t$ and $\varphi\in
B_{b}\left(U^{N}\right)$, we show

$\mathbb{E}[\varphi(\mathbf{Z}_{N}(t\mid
t_{0},\bm{\xi}))\mid\mathcal{F}_{s}]=T_{s,t}\varphi(\mathbf{Z}_{N}(s\mid
t_{0},\bm{\xi})),\quad\mathbb{P}\text{-a.s.},$ (4.45)

for all $\bm{\xi}\in L^{2}(\Omega,\mathcal{F}_{s},\mathbb{P};U^{N})$. By
Proposition 4.13, it suffices to verify that

$\mathbb{E}[\varphi(\mathbf{Z}_{N}(t\mid
s,\bm{\xi}))\mid\mathcal{F}_{s}]=T_{s,t}\varphi(\bm{\xi}),\quad\mathbb{P}\text{-a.s.}$
(4.46)

By a monotone class argument similar to that of Step 1, it suffices to
consider $\varphi\in C_{b}(U^{N})$. If
$\bm{\xi}=\sum_{j=1}^{n}\mathbf{x}_{j}\mathbf{1}_{A_{j}}$ with
$n\in\mathbb{N}$, $\\{\mathbf{x}_{1},\dots,\mathbf{x}_{n}\\}\subseteq U^{N}$
and disjoint events $\\{A_{1},\dots,A_{n}\\}\subseteq\mathcal{F}_{s}$ covering
$\Omega$, then

$\mathbf{Z}_{N}(t\mid s,\bm{\xi})=\sum\nolimits_{j=1}^{n}\mathbf{Z}_{N}(t\mid
s,\mathbf{x}_{j})\mathbf{1}_{A_{j}},\quad\mathbb{P}\text{-a.s.}$

For every $j\in\\{1,\dots,n\\}$, $\zeta_{N}(t\,|\,s)\mathbf{x}_{j}$ is
deterministic, whereas $Z_{N}(t\,|\,s,\bm{0})$ is independent of
$\mathcal{F}_{s}$ by (WP3′) and the definition of the stochastic integral,
thus $Z_{N}(t\mid s,\mathbf{x}_{k})\perp\\!\\!\\!\perp\mathcal{F}_{s}$. Since
the mean-square derivatives of ${Z_{N}(\,\cdot\mid s,\mathbf{x}_{k})}$ have
the same form (see Proposition 4.10), we deduce $\mathbf{Z}_{N}(t\mid
s,\mathbf{x}_{k})\perp\\!\\!\\!\perp\mathcal{F}_{s}$, so that

$\displaystyle\mathbb{E}$ $\displaystyle[\varphi(\mathbf{Z}_{N}(t\mid
s,\bm{\xi}))\mid\mathcal{F}_{s}]=\sum\nolimits_{k=1}^{n}\mathbb{E}[\varphi(\mathbf{Z}_{N}(t\mid
s,\mathbf{x}_{k}))\mathbf{1}_{A_{k}}\mid\mathcal{F}_{s}]$ (4.47)
$\displaystyle=\sum\nolimits_{k=1}^{n}\mathbb{E}[\varphi(\mathbf{Z}_{N}(t\mid
s,\mathbf{x}_{k}))]\mathbf{1}_{A_{k}}=\mathbb{E}[\varphi(\mathbf{Z}_{N}(t\mid
s,\bm{\xi}))]=[T_{s,t}\varphi](\bm{\xi}),\quad\mathbb{P}\text{-a.s.}$

This shows the desired property for simple $\bm{\xi}$. For a general
$\bm{\xi}\in L^{2}(\Omega,\mathcal{F}_{s},\mathbb{P};U^{N})$, we can find a
sequence $(\bm{\xi}_{n})_{n\in\mathbb{N}}$ of $\mathcal{F}_{s}$-measurable,
$U^{N}$-valued simple random variables such that $\bm{\xi}_{n}\to\bm{\xi}$ in
$L^{2}(\Omega;U^{N})$. By the continuity of the mapping
$\bm{\xi}\mapsto\mathbf{Z}_{N}(t\mid s,\bm{\xi})$ on $L^{2}(\Omega;U^{N})$ and
of $\varphi$, we have ${\varphi(\mathbf{Z}_{N}(t\mid
s,\bm{\xi}_{n}))\to\varphi(\mathbf{Z}_{N}(t\mid s,\bm{\xi}))}$ in
$L^{2}(\Omega;\mathbb{R})$. From this, one can derive that

$\mathbb{E}[\varphi(\mathbf{Z}_{N}(t\mid
s,\bm{\xi}_{n}))\mid\mathcal{F}_{s}](\omega)\to\mathbb{E}[\varphi(\mathbf{Z}_{N}(t\mid
s,\bm{\xi}))\mid\mathcal{F}_{s}](\omega),$

$\mathbb{P}$-a.s., passing to a subsequence if needed. On the other hand, we
can pass to a further subsequence in order to assume that
$\bm{\xi}_{n}(\omega)\to\bm{\xi}(\omega)$ in $U^{N}$, $\mathbb{P}$-a.s., and
deduce that
$[T_{s,t}\varphi](\bm{\xi}_{n}(\omega))\to[T_{s,t}\varphi](\bm{\xi}(\omega))$
in $\mathbb{R}$, finishing the density argument.

Finally, the statement regarding $(Z_{N}(t))_{t\in\mathbb{R}}$ follows from
Proposition 4.8. ∎

####  4.4.3. Non-Markovianity in the fractional case

We conclude this section by showing how Theorem 3.8 can be used to deduce that
$Z_{\gamma}$ is not weakly Markov (see Definition 3.5) if
$\gamma\not\in\mathbb{N}$. To this end, we determine the coloring operator of
$Z_{\gamma}$.

######  Proposition 4.14.

Let Assumption 4.1 be satisfied and suppose that
$\gamma\in(\nicefrac{{1}}{{2}},\infty)$ is such that Assumption 4.2(i) holds
for $\gamma_{0}=\gamma$. Then the coloring operator of $Z_{\gamma}$, see
(3.16), is given by
$\mathcal{L}^{-1}_{\gamma}=\mathfrak{I}^{\gamma}\mathcal{Q}_{\mathbb{R}}^{\frac{1}{2}}\in\mathscr{L}(L^{2}(\mathbb{R};U))$,
where $\mathfrak{I}^{\gamma}$ is as in (4.5).

###### Proof.

Note that for each $f\in C_{c}^{\infty}(\mathbb{R};U)$, we have

$\displaystyle\langle Z_{\gamma},f\rangle_{L^{2}(\mathbb{R};U)}$
$\displaystyle=\frac{1}{\Gamma(\gamma)}\int_{\mathbb{R}}\int_{-\infty}^{t}\langle(t-s)^{\gamma-1}S(t-s)\mathop{}\\!\mathrm{d}W^{Q}(s),f(t)\rangle_{U}\mathop{}\\!\mathrm{d}t,$
(4.48)
$\displaystyle=\int_{\mathbb{R}}\int_{\mathbb{R}}\widetilde{\Psi}_{f}(t,s)\mathop{}\\!\mathrm{d}W^{Q}(s)\mathop{}\\!\mathrm{d}t,\quad\mathbb{P}\text{-a.s.},$
(4.49)

where $\widetilde{\Psi}_{f}\colon\mathbb{R}^{2}\to\mathscr{L}(U;\mathbb{R})$
is given by

$\widetilde{\Psi}_{f}(t,s)u:=\frac{1}{\Gamma(\gamma)}\langle(t-s)^{\gamma-1}S(t-s)u,f(t)\rangle_{U}\,\mathbf{1}_{(-\infty,t)}(s),\quad
t,s\in\mathbb{R},\,u\in U.$ (4.50)

Indeed, the latter integral is well-defined since

$\displaystyle\int_{\mathbb{R}}$
$\displaystyle\biggl{[}\int_{\mathbb{R}}\left\|\widetilde{\Psi}_{f}(t,s)Q^{\frac{1}{2}}\right\|_{\mathscr{L}_{2}(U;\mathbb{R})}^{2}\mathop{}\\!\mathrm{d}s\biggr{]}^{\frac{1}{2}}\mathop{}\\!\mathrm{d}t$
$\displaystyle=\frac{1}{\Gamma(\gamma)}\int_{\mathbb{R}}\biggl{[}\int_{-\infty}^{t}\|(t-s)^{\gamma-1}Q^{\frac{1}{2}}[S(t-s)]^{*}f(t)\|_{U}^{2}\mathop{}\\!\mathrm{d}s\biggr{]}^{\frac{1}{2}}\mathop{}\\!\mathrm{d}t$
$\displaystyle\leq\frac{1}{\Gamma(\gamma)}\biggl{[}\int_{0}^{\infty}\|s^{\gamma-1}Q^{\frac{1}{2}}[S(s)]^{*}\|_{\mathscr{L}_{2}(U)}^{2}\mathop{}\\!\mathrm{d}s\biggr{]}^{\frac{1}{2}}\biggl{[}\int_{\mathbb{R}}\|f(t)\|_{U}\mathop{}\\!\mathrm{d}t\biggr{]}<\infty;$

note that the first integral on the last line is finite by Assumption 4.2(i)
and the second by the compact support of $f$. This also shows that Theorem 2.1
is applicable, hence

$\displaystyle\langle Z_{\gamma},f\rangle_{L^{2}(\mathbb{R};U)}$
$\displaystyle=\int_{\mathbb{R}}\int_{\mathbb{R}}\widetilde{\Psi}_{f}(t,s)\mathop{}\\!\mathrm{d}t\mathop{}\\!\mathrm{d}W^{Q}(s)=\int_{\mathbb{R}}\widetilde{\Phi}_{f}(s)\mathop{}\\!\mathrm{d}W^{Q}(s),\quad\mathbb{P}\text{-a.s.},$
(4.51)

where $\widetilde{\Phi}_{f}\colon\mathbb{R}\to\mathscr{L}(U;\mathbb{R})$ is
given by

$\displaystyle\widetilde{\Phi}_{f}(s)u:=\int_{\mathbb{R}}\widetilde{\Psi}_{f}(t,s)u\mathop{}\\!\mathrm{d}t$
$\displaystyle=\frac{1}{\Gamma(\gamma)}\int_{s}^{\infty}\langle(t-s)^{\gamma-1}S(t-s)u,f(t)\rangle_{U}\mathop{}\\!\mathrm{d}t$
(4.52) $\displaystyle=\langle
u,\mathfrak{I}^{\gamma*}f(s)\rangle_{U},\quad\forall s\in\mathbb{R},\,u\in U,$
(4.53)

where we recall equation (4.7) for the adjoint of $\mathfrak{I}^{\gamma}$.
Consequently, the Itô isometry (2.2) yields

$\|\langle
Z_{\gamma},f\rangle_{L^{2}(\mathbb{R};U)}\|_{L^{2}(\Omega)}^{2}=\int_{\mathbb{R}}\|\widetilde{\Phi}_{f}(t)Q^{\frac{1}{2}}\|_{\mathscr{L}_{2}(U;\mathbb{R})}^{2}\mathop{}\\!\mathrm{d}t=\|\mathcal{Q}_{\mathbb{R}}^{\frac{1}{2}}\mathfrak{I}^{\gamma*}f\|_{L^{2}(\mathbb{R};U)}^{2}.$
(4.54)

An application of the polarization identity then shows the coloring property
(3.16) with
$\mathcal{L}^{-*}_{\gamma}=\mathcal{Q}_{\mathbb{R}}^{\frac{1}{2}}\mathfrak{I}^{\gamma*}$.
∎

######  Example 4.15.

Let Assumptions 4.1 and 4.2(ii) be satisfied and suppose that
$\gamma\in(\nicefrac{{1}}{{2}},\infty)$ is such that Assumption 4.2(i) holds
for $\gamma_{0}=\gamma$. The latter implies that
$\partial_{t}+\mathcal{A}_{\mathbb{R}}=\mathcal{B}$, and we always have
$\mathcal{B}^{-\gamma}=\mathfrak{I}^{\gamma}$, see Subsection 4.2. Thus,

$\mathcal{L}_{\gamma}^{*}\mathcal{L}_{\gamma}=\bigl{(}\mathcal{Q}_{\mathbb{R}}^{-\frac{1}{2}}\mathcal{B}^{\gamma}\bigr{)}^{*}\mathcal{Q}_{\mathbb{R}}^{-\frac{1}{2}}\mathcal{B}^{\gamma}=\mathcal{B}^{\gamma*}\mathcal{Q}_{\mathbb{R}}^{-1}\mathcal{B}^{\gamma}=(\partial_{t}+\mathcal{A}_{\mathbb{R}})^{\gamma*}\mathcal{Q}_{\mathbb{R}}^{-1}(\partial_{t}+\mathcal{A}_{\mathbb{R}})^{\gamma}.$

Moreover, this assumption implies that $\mathsf{D}(A^{n})$ is dense in $U$ for
all $n\in\mathbb{N}$ by [37, Chapter 2, Theorem 6.8(c)]; choosing $n$ large
enough, we also find that $C_{c}^{\infty}(\mathbb{T};\mathsf{D}(A^{n}))$ is
dense in $\mathsf{D}(\mathcal{B}^{\gamma})$, so we can take
$F=\mathsf{D}(A^{n})$ in Theorem 3.8.

Although $Q^{-1}$ may be a nonlocal spatial operator,
$\mathcal{Q}_{\mathbb{R}}^{-1}$ is always local in time. Thus for
$\gamma\in\mathbb{N}$, the precision operator is local as a composition of
three local operators, which is in accordance with the Markovianity shown in
Subsection 4.4.

For $\gamma\not\in\mathbb{N}$, we will show that the precision operator is not
local in general. Suppose that $A$ has an eigenvector $v\in U$ with
corresponding eigenvalue $\lambda\in\mathbb{R}$. Such eigenpairs exists for
example if $A=(\kappa^{2}-\Delta)^{\beta}$ with $\kappa,\beta\in(0,\infty)$
and $\Delta$ the Dirichlet Laplacian on a bounded Euclidean domain
$\mathcal{D}\subsetneq\mathbb{R}^{d}$. If we moreover assume that
$v\in\mathsf{D}(Q^{-\frac{1}{2}})$, then we find

$\forall\phi\in
C_{c}^{\infty}(\mathbb{R}):\quad\mathcal{Q}_{\mathbb{R}}^{-\frac{1}{2}}\mathcal{B}^{\gamma}(\phi\otimes
v)=[(\partial_{t}+\lambda)^{\gamma}\phi]\otimes Q^{-\frac{1}{2}}v$ (4.55)

since the spectral mapping theorem implies $S(t)v=e^{-\lambda t}v$ for all
$t\in[0,\infty)$. It thus suffices to consider the case
$A=\lambda\in\mathbb{R}$, i.e., we wish to find disjointly supported
$\phi,\psi\in C_{c}^{\infty}(\mathbb{R})$ such that

$F_{\gamma}(\phi,\psi):=|\langle(\partial_{t}+\lambda)^{\gamma}\phi,(\partial_{t}+\lambda)^{\gamma}\psi\rangle_{L^{2}(\mathbb{R})}|\neq
0.$ (4.56)

We will discuss this by means of a numerical experiment for the case
$\lambda=1$, using the following smooth function $\phi\in
C_{c}^{\infty}(\mathbb{R})$ supported on $[-1,1]$:

$\phi(t):=\begin{cases}\exp\left(-\frac{1}{1-x^{2}}\right)\\!,&x\in(-1,1),\\\
0,&x\in\mathbb{R}\setminus(-1,1),\end{cases}$ (4.57)

and taking $\psi:=\phi(\,\cdot\,-2-\delta)$ for some $\delta\in(0,\infty)$. In
Figure 1, we see that the parabolic derivatives of $\phi$ consists of
(positive or negative) peaks. For $\gamma\not\in\mathbb{N}$, the support of
the last of these peaks appears to include the whole of $[1,\infty)$, with its
absolute value taking rapidly decaying yet nonzero values there. Therefore,
the idea is to take $\delta$ small enough, making the right-hand side tail of
$\phi$ overlap with the first peak of $\psi$ to obtain a non-zero
$L^{2}(\mathbb{R})$-inner product. Table 1 shows the approximate outcomes of
this process for various values of $\gamma$ and $\delta$, using symbolic
differentiation and numerical integration.

$\delta$ $\gamma$ | 0.25 | 0.50 | 0.75 | 1 | 1.25 | 1.50 | 1.75 | 2 | 2.25 | 2.50 | 2.75 | 3
---|---|---|---|---|---|---|---|---|---|---|---|---
$10^{-1}$ | 0.004 | 0.007 | 0.007 | 0 | 0.016 | 0.042 | 0.059 | 0 | 0.298 | 1.078 | 2.111 | 0
$10^{-2}$ | 0.005 | 0.009 | 0.009 | 0 | 0.024 | 0.065 | 0.098 | 0 | 0.622 | 2.568 | 5.829 | 0
$10^{-3}$ | 0.005 | 0.009 | 0.009 | 0 | 0.025 | 0.068 | 0.104 | 0 | 0.678 | 2.850 | 6.601 | 0
Table 1.  Numerically approximated values of $F_{\gamma}(\phi,\psi)$, see
(4.56), with $\phi$ from (4.57) and $\psi:=\phi(\,\cdot\,-2-\delta)$ for
certain values of $\gamma$ and $\delta$.

Figure 1. Graphs of the fractional parabolic derivative
$(\partial_{t}+1)^{\gamma}\phi$ with $\phi$ from (4.57) for certain values of
$\gamma\in[0,2]$. Note the different scales on the $y$-axes.

Note the contrast with the merely spatial Matérn case, where the self-
adjointness of the shifted Laplacian $\kappa^{2}-\Delta$ causes
$L_{\beta}^{*}L_{\beta}=\tau^{2}(\kappa^{2}-\Delta)^{2\beta}$, thus we find a
weak Markov property also for half-integer values of $\beta\in(0,\infty)$.

##  5\. Fractional $Q$-Wiener process

In this section we consider the _fractional $Q$-Wiener process_, which is a
$U$-valued analog to fractional Brownian motion; we define it as in [12,
Definition 2.1].

######  Definition 5.1.

Let $Q\in\mathscr{L}_{1}^{+}(U)$. A $U$-valued Gaussian process
$(W^{Q}_{H}(t))_{t\in\mathbb{R}}$ is called a _fractional $Q$-Wiener process
with Hurst parameter $H\in(0,1)$_ if

  1. (f-WP1)

$\mathbb{E}[W^{Q}_{H}(t)]=0$ for all $t\in\mathbb{R}$;

  2. (f-WP2)

$Q_{H}(s,t)=\frac{1}{2}(|t|^{2H}+|s|^{2H}-|t-s|^{2H})Q$ for all
$s,t\in\mathbb{R}$;

  3. (f-WP3)

$W^{Q}_{H}$ has continuous sample paths.

Here, $(Q_{H}(s,t))_{s,t\in\mathbb{R}}\subseteq\mathscr{L}_{1}^{+}(U)$ are the
covariance operators of $W^{Q}_{H}$, cf. (4.17).

Note that for $H=\frac{1}{2}$, the above definition reduces to a
characterization of a standard (non-fractional) $Q$-Wiener process when
restricted to $[0,\infty)$.

In the real-valued setting, the fractional Brownian motion can also be
represented as a stochastic integral over $\mathbb{R}$ [31, Definition 2.1].
Proposition 5.2 below shows that such a representation is also available in
the $U$-valued setting using the stochastic integral over $\mathbb{R}$ from
Section 2.2. This representation can be leveraged to prove that a fractional
$Q$-Wiener process can be viewed as a limiting case of the class of processes
discussed in Section 4, see Proposition 5.3. We conclude this section by
making some remarks about the Markov behavior of $W^{Q}_{H}$, see Subsection
5.2.

###  5.1. Integral representation and relation to $Z_{\gamma}$

Let $Q\in\mathscr{L}_{1}^{+}(U)$ and let $(W^{Q}(t))_{t\in\mathbb{R}}$ be a
two-sided $Q$-Wiener process, see Subsection 2.2. We define the process
$(\widehat{W}^{Q}_{H}(t))_{t\in\mathbb{R}}$ via the following Mandelbrot–Van
Ness [31] type integral:

$\widehat{W}_{H}^{Q}(t):=\int_{\mathbb{R}}K_{H}(t,r)\mathop{}\\!\mathrm{d}W^{Q}(r),\quad
t\in\mathbb{R},$ (5.1)

where the integration kernel $K_{H}\colon\mathbb{R}^{2}\to\mathbb{R}$ given by

$K_{H}(t,r):=\frac{(t-r)^{H-\frac{1}{2}}_{+}-(-r)_{+}^{H-\frac{1}{2}}}{C_{H}},\quad(t,r)\in\mathbb{R}^{2}.$
(5.2)

The normalizing constant $C_{H}$ is chosen as

$C_{H}:=\int_{\mathbb{R}}\Bigl{|}(1-r)_{+}^{H-\frac{1}{2}}-(-r)^{H-\frac{1}{2}}_{+}\Bigr{|}^{2}\mathop{}\\!\mathrm{d}r=\frac{3-2H}{4H}\mathrm{B}(2-2H,H+\tfrac{1}{2}),$

where $\mathrm{B}$ denotes the beta function [39, Theorem B.1], so that
$\widehat{Q}_{H}(1,1)=Q$, where $(\widehat{Q}_{H}(t,s))_{t,s\in\mathbb{R}}$
denotes the family of covariance operators associated to
$\widehat{W}^{Q}_{H}$. From this definition it follows that

$\widehat{W}^{Q}_{H}(t)=\frac{1}{C_{H}}\int_{-\infty}^{0}[(t-r)^{H-\frac{1}{2}}-(-r)^{H-\frac{1}{2}}]\mathop{}\\!\mathrm{d}W^{Q}(r)+\frac{1}{C_{H}}\int_{0}^{t}(t-r)^{H-\frac{1}{2}}\mathop{}\\!\mathrm{d}W^{Q}(r)$
(5.3)

for $t\in[0,\infty)$, whereas for $t\in(-\infty,0)$ we have

$\widehat{W}^{Q}_{H}(t)=\frac{1}{C_{H}}\int_{-\infty}^{t}[(t-r)^{H-\frac{1}{2}}-(-r)^{H-\frac{1}{2}}]\mathop{}\\!\mathrm{d}W^{Q}(r)+\frac{1}{C_{H}}\int_{t}^{0}(-r)^{H-\frac{1}{2}}\mathop{}\\!\mathrm{d}W^{Q}(r).$
(5.4)

In particular, for $H=\frac{1}{2}$ and $t\in[0,\infty)$ we find
$\widehat{W}^{Q}_{\nicefrac{{1}}{{2}}}(t)=W^{Q}(t)$, $\mathbb{P}$-a.s.

We then have the following relation to the fractional $Q$-Wiener process:

######  Proposition 5.2.

Formula (5.1) yields a well-defined square-integrable random variable
$\widehat{W}^{Q}_{H}(t)\in L^{2}(\Omega,\mathcal{F}_{t}^{\delta
W^{Q}},\mathbb{P};U)$ for each $t\in\mathbb{R}$, and there exists a
modification of the process $(\widehat{W}^{Q}_{H}(t))_{t\in\mathbb{R}}$ which
is a fractional $Q$-Wiener process in the sense of Definition 5.1.

###### Proof.

By the Itô isometry (2.2) it suffices to note that

$\mathbb{E}\Bigl{[}\bigl{\|}\widehat{W}^{Q}_{H}(t)\bigr{\|}_{U}^{2}\Bigr{]}=\int_{\mathbb{R}}\|K_{H}(t,r)Q^{\frac{1}{2}}\|_{\mathscr{L}_{2}(U)}^{2}\mathop{}\\!\mathrm{d}r=\operatorname{tr}Q\int_{\mathbb{R}}|K_{H}(t,r)|^{2}\mathop{}\\!\mathrm{d}r<\infty$

to see that $\widehat{W}^{Q}_{H}(t)\in L^{2}(\Omega;U)$ is well-defined. Since
$K_{H}$ is a deterministic kernel integrated with respect to a mean-zero
Gaussian process $W^{Q}$, it readily follows that $\widehat{W}^{Q}_{H}$ is
also mean-zero Gaussian. For the covariance operators
$(\widehat{Q}_{H}(t,s))_{t,s\in\mathbb{R}}$ of $\widehat{W}^{Q}_{H}$, we can
argue as in the proof of Proposition 4.7 to find

$\forall
t,s\in\mathbb{R}:\quad\widehat{Q}_{H}(t,s)=\int_{\mathbb{R}}K_{H}(s,r)K_{H}(t,r)\mathop{}\\!\mathrm{d}r\,Q=\mathbb{E}[B_{H}(t)B_{H}(s)]\,Q,$
(5.5)

where $B_{H}=(B_{H}(t))_{t\in\mathbb{R}}$ denotes (real-valued) fractional
Brownian motion. Hence (f-WP2) holds by the properties of $B_{H}$.

Lastly, we shall establish the existence of a continuous modification of
(5.1). To this end, we first remark that $\widehat{W}^{Q}_{H}$ is self-similar
with exponent $H$ and that its increments are stationary; this means for any
$t\in\mathbb{R}$ we have, respectively,

$\displaystyle\forall\alpha\in(0,\infty)$
$\displaystyle:\quad\widehat{W}^{Q}_{H}(\alpha
t)\overset{d}{=}\alpha^{H}\widehat{W}^{Q}_{H}(t),$ (5.6) $\displaystyle\forall
h\in\mathbb{R}$
$\displaystyle:\quad\widehat{W}^{Q}_{H}(t+h)-\widehat{W}^{Q}_{H}(t)\overset{d}{=}\widehat{W}^{Q}_{H}(h).$
(5.7)

The proofs of these statements are analogous to the real-valued case; they
involve changing variables in the integral (5.1) and using the facts that
$\alpha^{-\frac{1}{2}}W^{Q}(\alpha\,\cdot\,)$ and $W^{Q}(\,\cdot\,+h)$,
respectively, are also two-sided $Q$-Wiener processes. Thus,

$\displaystyle\bigl{\|}\widehat{W}^{Q}_{H}(t+h)-\widehat{W}^{Q}_{H}(t)\bigr{\|}_{L^{p}(\Omega;U)}=\bigl{\|}\widehat{W}^{Q}_{H}(|h|)\bigr{\|}_{L^{p}(\Omega;U)}=|h|^{H}\bigl{\|}\widehat{W}^{Q}_{H}(1)\bigr{\|}_{L^{p}(\Omega;U)}$
(5.8)

for any $p\in(1,\infty)$. By the Kahane–Khintchine inequalities (see, e.g.,
[20, Theorem 6.2.6]), there exists a constant $C_{p}\in(0,\infty)$ such that

$\bigl{\|}\widehat{W}^{Q}_{H}(1)\bigr{\|}_{L^{p}(\Omega;U)}\leq
C_{p}\bigl{\|}\widehat{W}^{Q}_{H}(1)\bigr{\|}_{L^{2}(\Omega;U)}=C_{p}\operatorname{tr}Q.$

We have thus shown that $t\mapsto\widehat{W}^{Q}_{H}(t)$ is $H$-Hölder
continuous from $\mathbb{R}$ to $L^{p}(\Omega;U)$. Since $p\in(1,\infty)$ was
arbitrary, we can apply the Kolmogorov–Chentsov theorem [8, Corollary 3.10]
with $p\in(\nicefrac{{1}}{{H}},\infty)$ to conclude that $\widehat{W}^{Q}_{H}$
has a modification with continuous sample paths. ∎

Now we consider the relation between fractional $Q$-Wiener process and the
process $Z_{\gamma}$ considered in Section 4. For $\varepsilon\in(0,\infty)$,
let $Z_{\gamma}^{\varepsilon}$ denote the mild solution to (4.1) with
$A=\varepsilon\operatorname{Id}_{U}$ and define the process
$\bigl{(}\overline{Z}_{\gamma}^{\varepsilon}(t)\bigr{)}_{t\in\mathbb{R}}$ by

$\overline{Z}_{\gamma}^{\varepsilon}(t):=\frac{\Gamma(\gamma)}{C_{\gamma-\frac{1}{2}}}(Z_{\gamma}^{\varepsilon}(t)-Z_{\gamma}^{\varepsilon}(0)),\quad
t\in\mathbb{R}.$ (5.9)

Note that $\widehat{W}^{Q}_{H}(t)$ can formally be written as a “convergent
difference of divergent integrals,” cf. [31, Footnote 3]:

$\frac{1}{C_{H}}\biggl{[}\int_{-\infty}^{t}(t-s)^{H-\frac{1}{2}}\mathop{}\\!\mathrm{d}W^{Q}(s)-\int_{-\infty}^{0}(-s)^{H-\frac{1}{2}}\mathop{}\\!\mathrm{d}W^{Q}(s)\biggr{]}.$

The above expression would correspond to taking $\varepsilon=0$ in (5.9),
which is ill-defined since Assumption 4.2(i) cannot be satisfied. However, the
next result shows that a fractional $Q$-Wiener process can be viewed as a
limiting case of $\overline{Z}_{\gamma}^{\varepsilon}$ as
$\varepsilon\downarrow 0$.

######  Proposition 5.3.

Let $Q\in\mathscr{L}_{1}^{+}(U)$ and
$\gamma\in(\nicefrac{{1}}{{2}},\nicefrac{{3}}{{2}})$. The family of stochastic
processes $(\overline{Z}_{\gamma}^{\varepsilon})_{\varepsilon\in(0,\infty)}$
defined by (5.9) converges uniformly on compact subsets of $\mathbb{R}$ in
mean-square sense to the fractional $Q$-Wiener process $\widehat{W}^{Q}_{H}$
in (5.1) with Hurst parameter $H=\gamma-\frac{1}{2}$ as $\varepsilon\downarrow
0$:

$\forall\,T\in(0,\infty):\quad\lim_{\varepsilon\downarrow
0}\sup_{t\in[-T,T]}\left\|\widehat{W}^{Q}_{\gamma-\frac{1}{2}}(t)-\overline{Z}_{\gamma}^{\varepsilon}(t)\right\|_{L^{2}(\Omega;U)}=0.$
(5.10)

###### Proof.

For $t\in[0,\infty)$, we can write

$\displaystyle\widehat{W}^{Q}_{\gamma-\frac{1}{2}}(t)$
$\displaystyle-\overline{Z}_{\gamma}^{\varepsilon}(t)=\frac{1}{C_{\gamma-\frac{1}{2}}}\int_{0}^{t}(t-s)^{\gamma-1}\bigl{(}1-e^{-\varepsilon(t-s)}\bigr{)}\mathop{}\\!\mathrm{d}W^{Q}(s)$
$\displaystyle+\frac{1}{C_{\gamma-\frac{1}{2}}}\int_{-\infty}^{0}\left[(t-s)^{\gamma-1}\bigl{(}1-e^{-\varepsilon(t-s)}\bigr{)}-(-s)^{\gamma-1}\bigl{(}1-e^{\varepsilon
s}\bigr{)}\right]\mathop{}\\!\mathrm{d}W^{Q}(s).$

After applying the Itô isometry to each of these integrals and using the
respective changes of variables $s^{\prime}:=-s$ and $s^{\prime}:=t-s$, we
obtain

$\displaystyle\biggl{\|}\widehat{W}^{Q}_{\gamma-\frac{1}{2}}(t)$
$\displaystyle-\overline{Z}_{\gamma}^{\varepsilon}(t)\biggr{\|}_{L^{2}(\Omega;U)}^{2}=\frac{I_{1}(t)+I_{2}(t)}{C_{\gamma-\frac{1}{2}}}(\operatorname{tr}Q)^{2},$
(5.11) $\displaystyle I_{1}(t)$
$\displaystyle:=\int_{0}^{|t|}s^{2\gamma-2}\bigl{(}1-e^{-\varepsilon
s}\bigr{)}^{2}\mathop{}\\!\mathrm{d}s,$ (5.12) $\displaystyle I_{2}(t)$
$\displaystyle:=\int_{0}^{\infty}\left|(|t|+s)^{\gamma-1}\bigl{(}1-e^{-\varepsilon(|t|+s)}\bigr{)}-s^{\gamma-1}\bigl{(}1-e^{-\varepsilon
s}\bigr{)}\right|^{2}\mathop{}\\!\mathrm{d}s.$ (5.13)

For $t\in(-\infty,0)$ we find (5.11) by instead splitting into integrals over
$(-\infty,t)$ and $(t,0)$ and changing variables $s^{\prime}:=t-s$ and
$s^{\prime}:=-s$, respectively.

For $t\in[-T,T]$, the estimate $1-e^{-x}\leq x$ for $x\in[0,\infty)$ yields
for the first term:

$\displaystyle I_{1}(t)\leq\varepsilon^{2}\int_{0}^{\lvert
t\rvert}s^{2\gamma}\mathop{}\\!\mathrm{d}s=\frac{\varepsilon^{2}\lvert
t\rvert^{2\gamma+1}}{2\gamma+1}\leq\frac{\varepsilon^{2}T^{2\gamma+1}}{2\gamma+1}.$

For $I_{2}$, we apply the fundamental theorem of calculus to the function

$u\mapsto(u+s)^{\gamma-1}(1-e^{-\varepsilon(u+s)}),$

followed by Minkowski’s integral inequality [44, Section A.1] to find

$\displaystyle I_{2}$
$\displaystyle(t)=\int_{0}^{\infty}\biggl{|}\int_{0}^{\lvert
t\rvert}[(\gamma-1)(u+s)^{\gamma-2}(1-e^{-\varepsilon(u+s)})+\varepsilon(u+s)^{\gamma-1}e^{-\varepsilon(u+s)}]\mathop{}\\!\mathrm{d}u\biggr{|}^{2}\mathop{}\\!\mathrm{d}s$
$\displaystyle\leq\Biggl{(}\int_{0}^{T}\biggl{[}\int_{0}^{\infty}\Bigl{|}(\gamma-1)(u+s)^{\gamma-2}(1-e^{-\varepsilon(u+s)})+\varepsilon(u+s)^{\gamma-1}e^{-\varepsilon(u+s)}\Bigr{|}^{2}\mathop{}\\!\mathrm{d}s\biggr{]}^{\frac{1}{2}}\mathop{}\\!\mathrm{d}u\Biggr{)}^{2}$
$\displaystyle=\varepsilon^{3-2\gamma}\Biggl{(}\int_{0}^{T}\biggl{[}\int_{\varepsilon
u}^{\infty}\Bigl{|}(\gamma-1)v^{\gamma-2}(1-e^{-v})+v^{\gamma-1}e^{-v}\Bigr{|}^{2}\mathop{}\\!\mathrm{d}v\biggr{]}^{\frac{1}{2}}\mathop{}\\!\mathrm{d}u\Biggr{)}^{2}$
$\displaystyle\leq\varepsilon^{3-2\gamma}\,T^{2}\int_{0}^{\infty}\Bigl{|}(\gamma-1)v^{\gamma-2}(1-e^{-v})+v^{\gamma-1}e^{-v}\Bigr{|}^{2}\mathop{}\\!\mathrm{d}v,$

where we performed the change of variables $v(s):=\varepsilon(u+s)$ on the
third line.

The improper integral on the last line converges: As $v\downarrow 0$, the
squares of both terms are of order $\mathcal{O}(v^{2\gamma-2})$, where we
again use $1-e^{-v}\leq v$ for the first term, and we have
$2\gamma-2\in(-1,1)$; the square of the first term is of order
$\mathcal{O}(v^{2\gamma-4})$ as $v\to\infty$, with $2\gamma-4\in(-3,-1)$,
whereas the second term decays exponentially.

The convergence thus follows by letting $\varepsilon\downarrow 0$ in the
previous two displays. ∎

###  5.2. Remarks on Markov behavior

Now we consider the Markov behavior of fractional $Q$-Wiener processes with
Hurst parameter $H\in(0,1)$. Since the case $H=\frac{1}{2}$ corresponds to a
standard $Q$-Wiener process, we find that $W^{Q}_{\nicefrac{{1}}{{2}}}$ is
simple Markov, whereas we can expect that $W^{Q}_{H}$ is not weakly Markov for
$H\neq\frac{1}{2}$.

In the real-valued case, the first published proof of non-Markovianity appears
to be [19], which shows that $B_{H}$ is not simple Markov for
$H\neq\frac{1}{2}$ using a characterization in terms of its covariance
function which is valid for Gaussian processes. This result can be improved by
applying the theory of [18, Chapter V] for Gaussian $N$-ple Markov processes
to the Mandelbrot–Van Ness representation of $B_{H}$. Namely, according to
[18, Theorem 5.1], any real-valued Gaussian process of the form

$\biggl{(}\int_{-\infty}^{t}K(t,s)\mathop{}\\!\mathrm{d}B(s)\biggr{)}_{t\in\mathbb{R}}$

can only be $N$-ple Markov for $N\in\mathbb{N}$ if its integration kernel $F$
can be written as

$K(s,t)=\sum_{j=1}^{N}f_{j}(s)g_{j}(t)$

for some functions $(f_{j})_{j=1}^{N},(g_{j})_{j=1}^{N}$. This condition is
not satisfied for $K_{H}$ from (5.2).

In order to establish that $W^{Q}_{H}$ (or $B_{H}$) does not have the weak
Markov property for $H\neq\frac{1}{2}$, one could attempt to associate a
nonlocal precision operator to the process and apply the necessary condition
from Theorem 3.8. Formally, the coloring operator $\mathcal{L}^{-1}_{H}$ of
$W^{Q}_{H}$ acts on functions $f\colon\mathbb{R}\to U$ as

$\mathcal{L}^{-1}_{H}f(t)=\frac{1}{C_{H}}\int_{\mathbb{R}}K_{H}(t,s)f(s)\mathop{}\\!\mathrm{d}s,\quad\forall
t\in\mathbb{R}.$

For certain ranges of $H$, see for instance [39, Equation (31)], an explicit
formula of its inverse $\mathcal{L}_{H}$ can also be determined. The operator
$\mathcal{L}^{-1}_{H}$ is bounded on some weighted Hölder space by [39,
Theorem 6], but there is no reason to expect that it is bounded on a Hilbert
space such as $L^{2}(\mathbb{R};U)$. Therefore, Theorem 3.8 is not directly
applicable, as it would need to be extended to the Banach space setting, which
is beyond the scope of this work.

## Acknowledgments

The authors acknowledge fruitful discussions with Richard Kraaij which led to
Proposition B.7 and the choice of the two-sided $Q$-Wiener process to define
the stochastic integral in Subsection 2.2. Moreover, the authors thank Jan van
Neerven for carefully reading the manuscript and providing valuable comments.

## Funding

K.K. acknowledges support of the research project _Efficient spatiotemporal
statistical modelling with stochastic PDEs_ (with project number
VI.Veni.212.021) by the talent programme _Veni_ which is financed by the Dutch
Research Council (NWO).

##  Appendix A Spaces of vector-valued functions

###  A.1. Spaces of measurable, continuous and differentiable functions

Let $(S,\mathscr{A},\mu)$ be a measure space. We abbreviate the phrases
“almost everywhere” and “almost all” by “a.e.” and “a.a.”, respectively.

Let $\mathcal{B}(E)$ denote the Borel $\sigma$-algebra of $E$. Given $x\in E$
and $f\colon S\to\mathbb{R}$, we define $f\otimes x\colon S\to E$ by
$[f\otimes x](\,\cdot\,):=f(\,\cdot\,)x$. A function
$f\colon(S,\mathscr{A},\mu)\to(E,\mathcal{B}(E))$ is said to be $\mu$-strongly
measurable if it is the $\mu$-a.e. limit of linear combinations of
$\mathbf{1}_{A}\otimes x$, where $x\in E$ and $A\in\mathscr{A}$ with
$\mu(A)<\infty$. We denote by $B_{b}(S,\mathscr{A})$ (resp. $C_{b}(S)$) the
Banach space of $\mathscr{A}$-measurable (resp. continuous) and bounded
functions $f\colon S\to\mathbb{R}$ endowed with the norm
$\|f\|_{B_{b}(S,\mathscr{A})}=\|f\|_{C_{b}(S)}=\sup_{s\in S}|f(s)|$. If the
$\sigma$-algebra is clear from context, we may simply write $B_{b}(S)$. The
set $C_{c}^{\infty}(J;E)$ consists of infinitely differentiable functions from
$J$ to $E$ with compact support.

###  A.2. Bochner and Sobolev spaces

For $p\in[1,\infty)$, let $L^{p}(S,\mathscr{A}\\!,\mu;E)$ denote the Bochner
space of (equivalence classes of) strongly measurable, $p$-integrable
functions equipped with the norm
$\|f\|_{L^{p}(S,\mathscr{A}\\!,\mu;E)}:=\left(\int_{S}\|f(s)\|_{E}^{p}\mathop{}\\!\mathrm{d}\mu(s)\right)^{\\!\nicefrac{{1}}{{p}}}\\!\\!.$
If the $\sigma$-algebra and measure are clear from context, we may suppress
them from our notation and simply write $L^{p}(S;E)$. The space
$L^{2}(S,\mathscr{A}\\!,\mu;U)$ is a Hilbert space when equipped with the
inner product $\langle
f,g\rangle_{L^{2}(S,\mathscr{A}\\!,\mu;U)}:=\int_{S}\langle
f(s),g(s)\rangle_{U}\mathop{}\\!\mathrm{d}\mu(s)$.

If $(S,\mathscr{A}\\!,\mu)$ and $(T,\mathscr{B},\nu)$ are $\sigma$-finite
measure spaces, then the measurable space $(S\times
T,\mathscr{A}\otimes\mathscr{B})$, equipped with the product $\sigma$-algebra

$\mathscr{A}\otimes\mathscr{B}:=\sigma(A\times
B:A\in\mathscr{A},B\in\mathscr{B}),$

admits a unique $\sigma$-finite _product measure $\mu\times\nu$_ satisfying

$(\mu\times\nu)(A\times B)=\mu(A)\nu(B)\quad\forall
A\in\mathscr{A},\,B\in\mathscr{B}.$

For $p,q\in[1,\infty)$, we define the _mixed-exponent Bochner space_
$L^{p,q}(S\times T;E)$ as the Banach space of (equivalence classes of)
strongly $(\mu\times\nu)$-measurable functions $f\colon S\times T\to E$ such
that

$\|f\|_{L^{p,q}(S\times
T;E)}:=\biggl{[}\int_{S}\biggl{[}\int_{T}\|f(s,t)\|_{E}^{q}\mathop{}\\!\mathrm{d}\nu(t)\biggr{]}^{\frac{p}{q}}\mathop{}\\!\mathrm{d}\mu(s)\biggr{]}^{\frac{1}{p}}<\infty.$
(A.1)

Intervals $J\subseteq\mathbb{R}$ will be equipped with the Lebesgue measure
$\lambda$ and Lebesgue $\sigma$-algebra
$\mathcal{B}(J)\vee\sigma(\mathcal{N}_{\lambda})$. A function $f\in
L^{p}(J;E)$ is said to be weakly differentiable, with weak derivative
$\partial_{t}f$, if

$\int_{J}\phi^{\prime}(t)f(t)\mathop{}\\!\mathrm{d}t=-\int_{J}\phi(t)[\partial_{t}f](t)\mathop{}\\!\mathrm{d}t\quad\forall\phi\in
C^{\infty}_{c}(J;\mathbb{R}).$

More generally, we denote its $n$th weak derivative by $\partial_{t}^{n}f$. If
$\partial_{t}^{\,j}f\in L^{p}(J;E)$ for all $j\in\\{0,\dots,n\\}$, then $f$
belongs to the Bochner–Sobolev space $W^{n,p}(J;E)$, which is a Banach space
using the norm
$\|f\|_{W^{n,p}(J;E)}:=(\sum_{j=0}^{n}\|\partial_{t}^{j}f\|_{L^{p}(J;E)}^{p})^{\nicefrac{{1}}{{p}}}$.
If $p=2$ and $E=U$, then $H^{n}(J;U):=W^{n,2}(J;U)$ is a Hilbert space whose
norm is induced by the inner product $\langle
f,g\rangle_{H^{n}(J;U)}:=\sum_{j=0}^{n}\langle\partial_{t}^{\,j}f,\partial_{t}^{\,j}g\rangle_{L^{2}(J;U)}$.

##  Appendix B Auxiliary results

This appendix collects some auxiliary results which are needed in the main
text but have been postponed for the sake of readability.

###  B.1. Facts regarding conditional independence

Let $\mathcal{G}_{1},\mathcal{H},\mathcal{G}_{2}\subseteq\mathcal{F}$ be
$\sigma$-algebras on $(\Omega,\mathcal{F},\mathbb{P})$. We recall a
characterization of conditional independence, see [21, Theorem 8.9], from
which we derive a lemma which is useful for establishing relations between
various (equivalent formulations of) Markov properties defined in Section 3.

######  Theorem B.1 (Doob’s conditional independence property).

We have $\mathcal{G}_{1}\perp\\!\\!\\!\perp_{\mathcal{H}}\mathcal{G}_{2}$ if
and only if

$\mathbb{P}(G_{2}\mid\mathcal{G}_{1}\vee\mathcal{H})=\mathbb{P}(G_{2}\mid\mathcal{H}),\quad\mathbb{P}\text{-a.s.},\quad\forall
G_{2}\in\mathcal{G}_{2}.$ (B.1)

######  Lemma B.2.

If $\mathcal{G}_{1}\perp\\!\\!\\!\perp_{\mathcal{H}}\mathcal{G}_{2}$, then

  1. (a)

$\mathcal{G}_{1}\vee\mathcal{H}\perp\\!\\!\\!\perp_{\mathcal{H}}\mathcal{G}_{2}$;

  2. (b)