# Cooling down stochastic differential equations:
almost sure convergence

Steffen Dereich  Steffen Dereich
Institut für Mathematische Stochastik
Fachbereich 10: Mathematik und Informatik
Westfälische Wilhelms-Universität Münster
Orléans-Ring 10
48149 Münster
Germany<EMAIL_ADDRESS>and  Sebastian Kassing  Sebastian Kassing
Institut für Mathematische Stochastik
Fachbereich 10: Mathematik und Informatik
Westfälische Wilhelms-Universität Münster
Orléans-Ring 10
48149 Münster
Germany<EMAIL_ADDRESS>
###### Abstract.

We consider almost sure convergence of the SDE
$dX_{t}=\alpha_{t}\,dt+\beta_{t}\,dW_{t}$ under the existence of a
$C^{2}$-Lyapunov function $F:{\mathbb{R}}^{d}\to{\mathbb{R}}$. More
explicitly, we show that on the event that the process stays local we have
almost sure convergence in the Lyapunov function $(F(X_{t}))_{t\geq 0}$ as
well as $\nabla F(X_{t})\to 0$, if $|\beta_{t}|=\mathcal{O}(t^{-\beta})$ for a
$\beta>1/2$. If, additionally, one assumes that $F$ is a Łojasiewicz-function,
we get almost sure convergence of the process itself, given that
$|\beta_{t}|=\mathcal{O}(t^{-\beta})$ for a $\beta>1$. The assumptions are
shown to be optimal in the sense that there is a divergent counterexample
where $|\beta_{t}|$ is of order $t^{-1}$.

###### Key words and phrases:

Stochastic gradient flow; Brownian particle; cooling down; Łojasiewicz-
inequality; almost sure convergence

###### 2020 Mathematics Subject Classification:

Primary 60J60; Secondary 60H10, 65C35

##  1\. Introduction

Let $(\Omega,(\mathcal{F}_{t})_{t\geq 0},\mathcal{F},{\mathbb{P}})$ be a
filtered probability space satisfying the usual conditions and suppose that
$(X_{t})_{t\geq 0}$ is a continuous semimartingale satisfying the equation

(1) $\displaystyle
X_{t}-X_{0}=\int_{0}^{t}\alpha_{s}\,ds+\int_{0}^{t}\beta_{s}\,dW_{s},$

for all $t\geq 0$, where

  * •

$(\alpha_{t})_{t\geq 0}$ is a progressive, ${\mathbb{R}}^{d}$-valued process,

  * •

$(\beta_{t})_{t\geq 0}$ is a progressive ${\mathbb{R}}^{d\times
d^{\prime}}$-valued process and

  * •

$(W_{t})_{t\geq 0}$ is an ${\mathbb{R}}^{d^{\prime}}$-valued
$(\mathcal{F}_{t})_{t\geq 0}$-Brownian motion with initial value $W_{0}=0$.

In this article, we analyse convergence of $(X_{t})_{t\geq 0}$ under the
assumption that the drift term satisfies a Lyapunov kind of condition and the
diffusivity converges to zero sufficiently fast. More explicitly, we denote by
$F:{\mathbb{R}}^{d}\to{\mathbb{R}}$ a $C^{2}$-function, by $f:=\nabla F$ its
gradient and by $H:=\text{Hess}\,F$ the Hessian of $F$ and consider the
following event

$\displaystyle{\mathbb{C}}:=\Bigl{\\{}\liminf_{t\to\infty}$
$\displaystyle\frac{\langle
f(X_{t}),-\alpha_{t}\rangle}{|f(X_{t})|^{2}}>0,\,\liminf_{t\to\infty}\frac{|f(X_{t})|}{|\alpha_{t}|}>0,$
$\displaystyle\quad\int_{0}^{\infty}|\beta_{s}|_{F}^{2}\,ds<\infty\,\text{
and}\ \limsup\limits_{t\to\infty}|\beta_{t}|<\infty\Bigr{\\}},$

where we interpret $\frac{0}{0}$ as $\infty$ and, for a matrix
$A\in{\mathbb{R}}^{d\times d^{\prime}}$, $|A|_{F}$ (resp. $|A|$) denotes the
Frobenius norm (resp. spectral norm). Intuitively, on ${\mathbb{C}}$ the drift
term $(\alpha_{t})_{t\geq 0}$ is comparable in size and direction to the
negative gradient of the Lyapunov function, i.e., $(-\nabla F(X_{t}))_{t\geq
0}$, and the size of the diffusivity $(\beta_{t})_{t\geq 0}$ can be controlled
at late times. The first and second condition in the definition of the event
${\mathbb{C}}$ are obviously satisfied if we consider the solution to the SDE

(2) $\displaystyle dX_{t}=-\nabla F(X_{t})\,dt+\beta_{t}\,dW_{t},$

which represents the canonical stochastic version of the gradient flow ODE

$\dot{x}_{t}=-\nabla F(x_{t}).$

However, the current framework allows more flexibility in the drift term. We
will restrict attention to certain events: set

$\mathbb{B}:=\Biggl{\\{}\liminf_{t\to\infty}F(X_{t})>-\infty,\,\limsup_{t\to\infty}|\alpha_{t}|<\infty\text{
and }\limsup_{\varepsilon\to 0}\limsup_{\begin{subarray}{c}t\to\infty\\\ y\in
B(X_{t},\varepsilon)\end{subarray}}|H(y)|<\infty\Biggr{\\}}$

and

${\mathbb{L}}:=\Bigl{\\{}\limsup_{t\to\infty}|X_{t}|<\infty\Bigr{\\}}.$

We separately show almost sure convergence of $(F(X_{t}))_{t\geq
0},(f(X_{t}))_{t\geq 0}$ and $(X_{t})_{t\geq 0}$ under mild assumptions on the
diffusivity and the underlying Lyapunov function which are shown to be optimal
(c.f. Chapter 4). The SDE (2) is the continuous-time counterpart for the
stochastic gradient descent scheme (when $(\beta_{t})_{t\geq 0}$ is chosen
properly, see [LTE17]) and, thus, a helpful tool for gaining intuition and
insight on the behaviour of SGD. It also appears naturally in applications
from Physics and Chemistry. For example, (2) can be viewed as the pathwise
solution to the Fokker–Planck equation (see [Gar85]) in a physical system,
where $F$ models the energy and $(\beta_{t})_{t\geq 0}$ an external source of
perturbation111$(2/\beta_{t}^{2})_{t\geq 0}$ is sometimes referred to as the
_inverse temperature_ , e.g. [CHS87, JKO98], or as the chemical Langevin
equation describing a chemical reaction evolving in time ([vK92], [Gil00],
[SSG14]). In our analysis, we allow general non-convex energy landscapes and
focus on the case where the perturbation fades within time.

There are several asymptotic results for the solution $(X_{t})_{t\geq 0}$ of
(2) when choosing a deterministic diffusivity $(\beta_{t})_{t\geq 0}$.
Inspired by simulated annealing, [GH86] and [CHS87] proved that for
$(\beta_{t})_{t\geq 0}=(\sqrt{2c/\log(t+1)})_{t\geq 0}$ with sufficiently
large $c>0$ and under additional assumptions on the target function $F$, the
solution of (2) converges weakly to a distribution $\pi$ concentrated on the
global minima of $F$ independent of the initial value
$X_{0}=x\in{\mathbb{R}}^{d}$, even in the case of non-convex target functions
$F$. More precisely, $\pi$ is the weak limit of the Gibbs densities
$\pi_{t}(x)\propto\exp(-2F(x)/\beta^{2}_{t})$. In this regime, [HKS89] showed
that if the diffusion process is defined on a compact manifold, then
$(F(X_{t}))_{t\geq 0}$ converges to the optimal value in probability if and
only if $c>c^{*}$, where $c^{*}$ denotes the minimal potential energy
necessary to connect every point with a global minimum using a continuous
path. Recently, this result has been generalised by [FT21] to diffusions on
${\mathbb{R}}^{d}$ in the case where $F\in C^{\infty}$ is assumed to satisfy
$\lim_{|x|\to\infty}F(x)=\infty$ and
$\int_{{\mathbb{R}}^{d}}e^{-\alpha_{0}F(x)}\,dx<\infty$ for some
$\alpha_{0}>0$. In that case one has $\liminf_{t\to\infty}|X_{t}|<\infty$,
almost surely.

However, in this particular scenario we cannot hope for the individual paths
of the solution $(X_{t})_{t\geq 0}$ to converge. Conversely, the diffusivity
$(\beta_{t})_{t\geq 0}$ considered here converges significantly faster to
zero. Therefore, we cannot guarantee that the process $(X_{t})_{t\geq 0}$
started in the basin of attraction of a bad local minimum is able to overcome
a barrier in the landscape in order to converge to a global minimum (see
Section 3.3) and the distribution of the limit points of $(X_{t})_{t\geq 0}$
depends heavily on the distribution of the initial value $X_{0}$. As we will
indicate below, the gradient flow converges if $F$ is a Łojasiewicz-landscape,
in the sense of Definition 1.4, and the gradient flow does not escape to
infinity. The main contribution of this article is the analysis in how far
this is still the case when additionally incorporating a stochastic
perturbation. We stress that we analyse the asymptotic behaviour of the system
as time tends to infinite which is significantly different from the analysis
of families of SDEs where the diffusivity decays with a family index
$\varepsilon>0$, see e.g. [FW12].

First, we will provide a short proof for almost sure convergence of
$(F(X_{t}))_{t\geq 0}$ and $(f(X_{t}))_{t\geq 0}$ with
$\lim_{t\to\infty}f(X_{t})=0$ on the event ${\mathbb{C}}\cap\mathbb{B}$. We
conclude that we have a.s. convergence of $(X_{t})_{t\geq 0}$ on the event
${\mathbb{C}}\cap{\mathbb{L}}$ in the case where where the set of critical
points $\mathcal{C}:=\\{x\in{\mathbb{R}}^{d}:f(x)=0\\}$ is at most countable.

######  Theorem 1.1.

On ${\mathbb{C}}\cap\mathbb{B}$, almost surely, $(F(X_{t}))_{t\geq 0}$
converges to a finite value and $\lim_{t\to\infty}f(X_{t})=0$ as well as
$\limsup_{t_{0}\to\infty}\int_{t_{0}}^{\infty}\langle
f(X_{s}),-\alpha_{s}\rangle\,ds<\infty$. If the set of critical points of $F$
is at most countable, then, on ${\mathbb{C}}\cap{\mathbb{L}}$, almost surely,
$(X_{t})_{t\geq 0}$ converges.

Theorem 1.1 is an immediate consequence of the more general Proposition 2.1
below.

######  Remark 1.2.

Note that as $F$ is $C^{2}$ we clearly have
${\mathbb{C}}\cap{\mathbb{L}}\subset{\mathbb{C}}\cap\mathbb{B}$ so that the
first statement of Theorem 1.1 also holds on ${\mathbb{C}}\cap{\mathbb{L}}$.
Moreover, for a Lyapunov function $F$ with $\liminf_{|x|\to\infty}|f(x)|>0$
the first statement implies that

${\mathbb{C}}\cap\mathbb{B}={\mathbb{C}}\cap{\mathbb{L}}.$

######  Remark 1.3.

A standard approach for proving convergence statements similar to those given
in Theorem 1.1 is the theory of asymptotic pseudotrajectories (see [Ben99],
especially Corollary 6.7). While these methods may lead to different and
typically weaker assumptions on the perturbation $(\beta_{t})_{t\geq 0}$,
Theorem 1.1 will mainly be used to prove Theorem 1.6 below. In order to be
self-contained we stated Theorem 1.1 in such a way that it can easily be
applied in Theorem 1.6 and has a direct and brief proof.

In the case where the set of critical points of $F$ contains a continuum of
elements the situation is more subtle. Even in the case without random
perturbations convergence of the solution $(x_{t})_{t\geq 0}$ to the ODE

(3) $\displaystyle\dot{x}_{t}=-\nabla F(x_{t})$

is a non-trivial issue: one can find $C^{\infty}$-functions $F$ together with
solutions $(x_{t})_{t\geq 0}$ that stay on compact sets but do not converge,
see example 3 on page 14 of [PdM82]. Counterexamples of this structure have
been known for a long time (see e.g. [Cur44]) and include the famous Mexican
hat function [AMA05]. To guarantee convergence (at least in the case where the
solution stays on a compact set), one needs to impose additional assumptions.
An appropriate assumption is the validity of a Łojasiewicz-inequality, see
Definition 1.4 below. This assumption has the appeal that it is satisfied by
analytic functions, see [Łoj63, Łoj65]. With the help of the inequality (4)
Łojasiewicz showed in [Łoj84] that any integral curve of the gradient flow
equation (3) for an analytic function $F$ that has an accumulation point has a
unique limit. A proof of that assertion can also be found in [Har12] or
[CMI14].

In this article, we device a probabilistic counterpart to the classical
analytic approach. We will derive moment estimates of $F(X_{t})$ for
Łojasiewicz-functions $F:{\mathbb{R}}^{d}\to{\mathbb{R}}$ on
${\mathbb{C}}\cap{\mathbb{L}}$ under the additional assumption that we can
asymptotically bound the diffusivity $(\beta_{t})_{t\geq 0}$ by a locally-
bounded deterministic function $\sigma:[0,\infty)\to[0,\infty)$ to deduce
almost sure convergence of $(\int_{0}^{t}\alpha_{s}\,ds)_{t\geq 0}$ and thus,
almost sure convergence of $(X_{t})_{t\geq 0}$.

######  Definition 1.4.

We call the function $F:{\mathbb{R}}^{d}\to{\mathbb{R}}$ _Łojasiewicz-
function_ , if for every $x\in\mathcal{C}:=f^{-1}(\\{0\\})$, the Łojasiewicz-
inequality is true on a neighbourhood $U_{x}$ of $x$ with parameters $\L>0$
and $\theta\in[\frac{1}{2},1)$, i.e., for all $y\in U_{x}$

(4) $\displaystyle|f(y)|\geq\L\,|F(y)-F(x)|^{\theta}.$

######  Remark 1.5.

Note that for all $x\notin\mathcal{C}$ and $\theta\in[\frac{1}{2},1)$ there
trivially exist a neighbourhood $U_{x}$ and constant $\L>0$ such that the
Łojasiewicz-inequality with parameter $\L$ and $\theta$ hold on $U_{x}$.

The main contribution of this article concerns convergence of the SDE in the
case where the Lyapunov function is a Łojasiewicz-function.

######  Theorem 1.6.

Let $F$ be a Łojasiewicz-function and $(\sigma_{t})_{t\geq
0}=((t+1)^{-\sigma})_{t\geq 0}$ with $\sigma>1$. Then, on
${\mathbb{C}}\cap{\mathbb{L}}\cap\bigl{\\{}\limsup_{t\to\infty}\frac{|\beta_{t}|}{\sigma_{t}}<\infty\bigr{\\}}$,
the process $(X_{t})_{t\geq 0}$ satisfying (1) converges, almost surely, to a
critical point of $F$.

The proof of the theorem is accomplished in Section 3. The theorem is optimal
in the sense that its conclusion is not true for the choice
$(\sigma_{t})_{t\geq 0}=((t+1)^{-1})_{t\geq 0}$. More explicitly, we provide
an SDE for which
${\mathbb{C}}\cap{\mathbb{L}}\cap\bigl{\\{}\limsup_{t\to\infty}(t+1)|\beta_{t}|<\infty\bigr{\\}}$
is an almost sure set and for which $(X_{t})_{t\geq 0}$ diverges, almost
surely, in Section 4.

######  Remark 1.7.

Choose $(\alpha_{t})_{t\geq 0}=(-f(X_{t}))_{t\geq 0}$ and consider the
solution to the SDE (2). In that case we have for $(\sigma_{t})_{t\geq 0}$ as
in Theorem 1.6 that
$\bigl{\\{}\limsup_{t\to\infty}\frac{|\beta_{t}|}{\sigma_{t}}<\infty\bigr{\\}}\subset{\mathbb{C}}.$
Now, assume that $F\in C^{2}$ satisfies $\lim_{|x|\to\infty}F(x)=\infty$. In
Appendix A we derive sufficient conditions that guarantee locality of the
process $(X_{t})_{t\geq 0}$. In particular, we show that if the Hessian of $F$
is uniformly bounded (i.e. $\sup_{x\in{\mathbb{R}}^{d}}|H(x)|<\infty$) then
$\bigl{\\{}\int_{0}^{\infty}|\beta_{s}|_{F}^{2}\,ds<\infty\bigr{\\}}\subset{\mathbb{L}}$.
Also, we get for any compact set $K\subset{\mathbb{R}}^{d}$

$\displaystyle\Bigl{\\{}$ $\displaystyle 1\mathrm{l}_{\\{X_{t}\notin
K\\}}(-|f(X_{t})|^{2}+\frac{1}{2}\mathrm{tr}((\beta_{t})^{\dagger}H(X_{t})\beta_{t}))\leq
0\text{ for all large }t$ $\displaystyle\text{ and
}\int_{0}^{\infty}1\mathrm{l}_{\\{X_{s}\in
K\\}}|\beta_{s}|_{F}^{2}\,ds<\infty\Bigr{\\}}\subset{\mathbb{L}}.$

The proof of Theorem 1.6 is based on moment estimates for $F(X_{t})$
restricted to particular events, see Proposition 3.1 below. In the particular
case where we approach a local minimum these estimates entail the following
moment estimate.

######  Theorem 1.8.

Let $K\subset{\mathbb{R}}^{d}$ be a compact set, $\L>0$, $\theta\in(1/2,1)$,
$\ell\in{\mathbb{R}}$, $C_{\beta}>0$ and $\sigma\in[0,\infty)$. Suppose that
for all $y\in K$

$F(y)\geq\ell\ \text{and }\ |f(y)|\geq\L(F(y)-\ell)^{\theta}.$

Consider the stopping time

$T:=\inf\Bigl{\\{}t\geq 0:X_{t}\notin K,\frac{\langle
f(X_{t}),-\alpha_{t}\rangle}{|f(X_{t})|^{2}}\leq\rho\ ,\ |\beta_{t}|\geq
C_{\beta}(t+1)^{-\sigma}\Bigr{\\}}.$

Then there exists $C\geq 0$ so that for all $t\geq 0$

${\mathbb{E}}[1\mathrm{l}_{\\{T>t\\}}(F(X_{t})-\ell)]\leq
C(t+1)^{-(\frac{\sigma}{\theta}\wedge\frac{1}{2\theta-1})}.$

Theorem 1.8 will be a consequence of Proposition 3.1 and Remark 3.2 below.
Note that if $\sigma\geq\theta/(2\theta-1)$ we essentially get the same order
of convergence as in the case without random perturbations. If the drift terms
vanishes at a slower rate, i.e. $\sigma<\theta/(2\theta-1)$, this effects the
order of decay of the expected function value $(F(X_{t})-\ell)_{t\geq 0}$.

######  Notation.

For a vector $v\in{\mathbb{R}}^{d}$ we denote by $|v|$ the Euclidean norm
induced by the scalar product, i.e. $|v|^{2}=\langle v,v\rangle$. For a matrix
$A\in{\mathbb{R}}^{d\times d^{\prime}}$ we denote by $|A|$ the spectral norm
and by $|A|_{F}$ the Frobenius norm, i.e.

$|A|:=\max\limits_{|x|\neq 0}\frac{|Ax|}{|x|}\quad\text{ and
}\quad|A|_{F}:=\sqrt{\mathrm{tr}(A^{\dagger}A)}.$

For $t_{0}\geq 0$ and a real-valued continuous semimartingale $(Y_{s})_{s\geq
t_{0}}$ we denote by $\langle Y\rangle=(\langle Y\rangle_{s})_{s\geq t_{0}}$
its quadratic variation process.

##  2\. Proof of Theorem 1.1

In this section we prove a slight generalisation of Theorem 1.1.

######  Proposition 2.1.

Consider

$\displaystyle{\mathbb{C}}_{1}:=\Bigl{\\{}$
$\displaystyle\liminf_{t\to\infty}F(X_{t})>-\infty,\
\limsup_{t\to\infty}|\beta_{t}|<\infty,\ \liminf_{t\to\infty}\frac{\langle
f(X_{t}),-\alpha_{t}\rangle}{|f(X_{t})|^{2}}>0\,\text{ and }$
$\displaystyle\Bigl{(}\int_{0}^{t}\mathrm{tr}(\beta_{s}^{\dagger}H(X_{s})\beta_{s})\,ds\Bigr{)}_{t\geq
0}\text{ converges to a finite value}\Bigr{\\}}$
${\mathbb{C}}_{2}:={\mathbb{C}}_{1}\cap\Bigl{\\{}\limsup_{t\to\infty}|\alpha_{t}|<\infty,\,\limsup_{\varepsilon\to
0}\limsup_{\begin{subarray}{c}t\to\infty\\\ y\in
B(X_{t},\varepsilon)\end{subarray}}|H(y)|<\infty,\int_{0}^{\infty}|\beta_{s}|_{F}^{2}<\infty\Bigr{\\}},$

and

${\mathbb{C}}_{3}:={\mathbb{C}}_{2}\cap\Bigl{\\{}\liminf_{t\to\infty}|X_{t}|<\infty\Bigr{\\}},$

where we interpret $\frac{0}{0}$ as $\infty$. On ${\mathbb{C}}_{1}$, almost
surely, $(F(X_{t}))_{t\geq 0}$ converges to a finite value and
$\limsup_{t_{0}\to\infty}\int_{t_{0}}^{\infty}\langle
f(X_{s}),-\alpha_{s}\rangle\,ds<\infty$. On ${\mathbb{C}}_{2}$, almost surely,
$\lim_{t\to\infty}f(X_{t})=0$. If the set of critical points of $F$ is at most
countable, then, on ${\mathbb{C}}_{3}$, almost surely, $(X_{t})_{t\geq 0}$
converges.222Note that ${\mathbb{C}}_{3}$ entails ${\mathbb{C}}_{1}$ and
${\mathbb{C}}_{2}$.

Note that ${\mathbb{C}}\cap\mathbb{B}\subset{\mathbb{C}}_{2}$ since
$\int_{0}^{\infty}|\beta_{s}|_{F}^{2}\,ds<\infty$ and
$\limsup_{t\to\infty}|H(X_{t})|<\infty$ imply convergence of

$\Bigl{(}\int_{0}^{t}\mathrm{tr}(\beta_{s}^{\dagger}H(X_{s})\beta_{s})\,ds\Bigr{)}_{t\geq
0}.$

Therefore, Theorem 1.1 directly follows from Proposition 2.1.

###### Proof of Proposition 2.1.

Let $t_{0},C_{1},C_{2}\geq 0$ and set

$\displaystyle T:=T^{(t_{0},C_{1},C_{2})}:=\inf\Bigl{\\{}t\geq t_{0}:\ $
$\displaystyle
F(X_{t})<-C_{1}\,\vee\,\int_{t_{0}}^{t}\mathrm{tr}(\beta_{t}^{\dagger}H(X_{t})\beta_{t})\,ds>C_{1}\,\vee\,|\beta_{t}|>C_{1},$
$\displaystyle\text{ \ or \ }\langle
f(X_{t}),\alpha_{t}\rangle>-C_{2}|f(X_{t})|^{2}\Bigr{\\}}$

By Itô’s formula one has

(5) $\displaystyle dF(X_{t})$ $\displaystyle=\langle
f(X_{t}),\alpha_{t}\rangle\,dt+\langle
f(X_{t})),\beta\,dW_{t}\rangle+\frac{1}{2}\mathrm{tr}(\beta_{t}^{\dagger}H(X_{t})\beta_{t})\,dt.$

We let

$(M_{t})_{t\geq t_{0}}:=\Bigl{(}\int_{t_{0}}^{t\wedge T}\langle
f(X_{t}),\beta\,dW_{t}\rangle\Bigr{)}_{t\geq t_{0}}$

and note that its variation process satisfies

$\langle M\rangle_{t}=\int_{t_{0}}^{t\wedge
T}|\beta^{\dagger}_{s}f(X_{s})|^{2}\,ds\leq C_{1}\int_{t_{0}}^{t\wedge
T}|f(X_{s})|^{2}\,ds.$

On the other hand,

$\int_{t_{0}}^{t\wedge T}\langle f(X_{s}),\alpha_{s}\rangle\,ds\leq-
C_{2}\int_{t_{0}}^{t\wedge T}|f(X_{s})|^{2}\,ds.$

Consequently,

$\int_{t_{0}}^{t\wedge T}\langle f(X_{s}),\alpha_{s}\rangle\,ds+M_{t}\leq
M_{t}-\frac{C_{2}}{C_{1}}\langle M\rangle_{t}$

We conclude that for $t\geq t_{0}$

$\displaystyle\underbrace{F(X_{t\wedge T})}_{\geq-C_{1}}-F(X_{t_{0}})$
$\displaystyle=\underbrace{\int_{t_{0}}^{t\wedge T}\langle
f(X_{s}),\alpha_{s}\rangle\,ds+M_{t}}_{\leq M_{t}-\frac{C_{2}}{C_{1}}\langle
M\rangle_{t}}+\frac{1}{2}\underbrace{\int_{t_{0}}^{t\wedge
T}\mathrm{tr}(\beta_{s}^{\dagger}H(X_{s})\beta_{s})\,ds}_{\leq C_{1}}.$

Note that on $\\{\langle M\rangle_{\infty}=\infty\\}$, almost surely,
$M_{t}-\frac{C_{2}}{C_{1}}\langle M\rangle_{t}\to-\infty$. Since the left-hand
side is bounded from below we get that $\langle M\rangle_{\infty}<\infty$,
a.s., so that the martingale $(M_{t})_{t\geq t_{0}}$ converges almost surely
to a finite value. Consequently,

$-\int_{t_{0}}^{t\wedge T}\langle f(X_{s}),\alpha_{s}\rangle\,ds\leq
F(X_{t_{0}})+M_{t}+\frac{3}{2}C_{1}$

and the left-hand side is increasing in $t$ with a finite limit. Consequently,
on

$\Big{\\{}\Big{(}\int_{0}^{t}\mathrm{tr}(\beta_{s}^{\dagger}H(X_{s})\beta_{s})\,ds\Big{)}_{t\geq
0}\text{ converges}\Big{\\}},$

we have that $(F(X_{t\wedge T}))_{t\geq 0}$ converges and
$-\int_{t_{0}}^{T}\langle f(X_{s}),\alpha_{s}\rangle\,ds<\infty$, almost
surely. We note that

${\mathbb{C}}_{1}=\Big{\\{}\Big{(}\int_{0}^{t}\mathrm{tr}(\beta_{s}^{\dagger}H(X_{s})\beta_{s})\,ds\Big{)}_{t\geq
0}\text{
converges}\Big{\\}}\cap\bigcup_{n\in{\mathbb{N}}}\\{T^{(n,n,1/n)}=\infty\\}$

which implies the first statement.

To show the second statement we set

$\tilde{T}:=\tilde{T}^{(t_{0},C_{1},C_{2})}:=T^{(t_{0},C_{1},C_{2})}\wedge\inf\Bigl{\\{}t\geq
t_{0}:|\alpha_{t}|\vee\sup_{y\in
B(X_{t},C_{2})}|H(y)|\vee\int_{t_{0}}^{t}|\beta_{s}|^{2}_{F}\,ds>C_{1}\Bigr{\\}}$

and note that it satisfies to show that $f(X_{t})\to 0$, almost surely, on
$\\{\tilde{T}^{(t_{0},C_{1},C_{2})}=\infty\\}$, since

${\mathbb{C}}_{2}=\bigcup_{n\in{\mathbb{N}}}\\{\tilde{T}^{(n,n,1/n)}=\infty\\}.$

Consider

$(\tilde{M}_{t})_{t\geq
t_{0}}:=\Bigl{(}\int_{t_{0}}^{t\wedge\tilde{T}}\beta_{s}\,dW_{s}\Bigr{)}_{t\geq
t_{0}}\text{ \ and \ }(B_{t})_{t\geq
t_{0}}:=\Bigl{(}\int_{t_{0}}^{t\wedge\tilde{T}}\langle
f(X_{s}),\alpha_{s}\rangle\,ds\Bigr{)}_{t\geq t_{0}}$

Note that $(\tilde{M}_{t})_{t\geq t_{0}}$ is a multivariate martingale with

$\langle\tilde{M}\rangle_{t}=\int_{0}^{t\wedge\tilde{T}}|\beta_{s}|_{F}^{2}\,ds\leq
C_{1}.$

Hence, $(\tilde{M}_{t})_{t\geq t_{0}}$ converges almost surely and together
with the proof of the first statement we get that

$\Omega_{0}:=\bigl{\\{}(\tilde{M}_{t})_{t\geq t_{0}}\text{ and }(B_{t})_{t\geq
t_{0}}\text{ converge}\bigr{\\}}$

is an almost sure event. Suppose now that there exists
$\omega\in\Omega_{0}\cap\\{\tilde{T}=\infty\\}$ for which
$(f(X_{t}(\omega)))_{t\geq 0}$ does not converge to zero. Then there exist
$\delta>0$ and an increasing $[t_{0},\infty)$-valued sequence
$(t_{k})_{k\in{\mathbb{N}}}$ converging to infinity with

$|f(X_{t_{k}}(\omega))|>\delta$

for all $k\in{\mathbb{N}}$. Pick $\varepsilon\in(0,C_{2})$ with
$C_{1}\varepsilon\leq\delta/2$. By thinning the original sequence
$(t_{k})_{k\in{\mathbb{N}}}$ we can ensure that
$([t_{k},t_{k}+\varepsilon/(2C_{1})]:k\in{\mathbb{N}})$ are disjoint intervals
and that for all $k\in{\mathbb{N}}$

$\sup\limits_{t\geq
t_{k}}|\tilde{M}_{t}(\omega)-\tilde{M}_{t_{k}}(\omega)|<\varepsilon/2.$

For $k\in{\mathbb{N}}$ and $t\in[t_{k},t_{k}+\varepsilon/(2C_{1})]$ one has

$\displaystyle|X_{t}(\omega)-X_{t_{k}}(\omega)|$
$\displaystyle\leq\int_{t_{k}}^{t}|\alpha_{s}(\omega)|\,ds+|\tilde{M}_{t}(\omega)-\tilde{M}_{t_{k}}(\omega)|$
$\displaystyle\leq C_{1}(t-t_{k})+\varepsilon/2\leq\varepsilon$

Moreover, since $\varepsilon<C_{2}$ we conclude for $x=X_{t}(\omega)$ and
$y=X_{t_{k}}(\omega)$ that

$\displaystyle|f(y)-f(x)|\leq|H(\xi)|\,|y-x|\leq C_{1}|y-x|,$

where $\xi$ lies on the segment joining $x$ and $y$. Hence,

$|f(X_{t}(\omega))|\geq|f(X_{t_{k}}(\omega))|-\underbrace{|f(X_{t})-f(X_{t_{k}}(\omega))|}_{\leq\delta/2}\geq\delta/2,$

so that

$\int_{t_{k}}^{t_{k}+\varepsilon/(2C_{1})}|f(X_{t}(\omega))|^{2}\,dt\geq\Bigl{(}\frac{\delta}{2}\Bigr{)}^{2}\frac{\varepsilon}{2C_{1}}.$

This entails that $\int_{t_{0}}^{\infty}|f(X_{t})|^{2}\,dt=\infty$ and, hence,

$-B_{t}(\omega)=\int_{t_{0}}^{t}\langle
f(X_{s}(\omega)),-\alpha_{s}(\omega)\rangle\,ds\geq
C_{2}\int_{t_{0}}^{t}|f(X_{s}(\omega)|^{2}\,ds\to\infty.$

This contradicts that $(B_{t}(\omega))_{t\geq t_{0}}$ converges to a finite
value, by choice of $\omega$.

It remains to prove almost sure convergence of $(X_{t})_{t\geq 0}$ on
${\mathbb{C}}_{3}$ in the case where the set
$\mathcal{C}:=\\{x\in{\mathbb{R}}^{d}:f(x)=0\\}$ is at most countable. Take
$\omega\in\Omega$ for which $\lim_{t\to\infty}f(X_{t}(\omega))=0$ and set
$r:=\limsup_{t\to\infty}|X_{t}(\omega)|<\infty$. We show by contradiction that
$(X_{t}(\omega))_{t\geq 0}$ converges. Suppose that $(X_{t}(\omega))_{t\geq
0}$ would not converge. Then there would exist a coordinate
$i\in\\{1,\dots,d\\}$ and reals $a<b$ such that the set of accumulation points

$\mathcal{K}:=\bigcap_{n\in{\mathbb{N}}}\bigl{\\{}X^{(i)}_{t}(\omega):t\geq
n,X_{t}(\omega)\in\overline{B(0,r+1)}\bigr{\\}}$

is a compact set containing $[a,b]$. By applying the principle of nested
intervals we can produce for every $u\in[a,b]$ an accumulation point $\eta(u)$
of $(X_{t}(\omega))_{t\geq 0}$ with $i$th coordinate equal to $u$. Recall that
$f(X_{t})\to 0$ so that $\eta(u)$ is a critical point of $F$. We thus observe
that there is an injective mapping taking $[a,b]$ to the set of critical
points of $F$ which contradicts the assumption.

∎

##  3\. Proof of Theorem 1.6

The proof of Theorem 1.6 is arranged as follows. In Section 3.1, we provide
moment estimates for the Lyapunov function seen by the SDE in a neighbourhood
$U$ of a point $y\in{\mathbb{R}}^{d}$ so that the Łojasiewicz-inequality is
satisfied on $U$ with parameters $\L>0$ and $\theta\in(\frac{1}{2},1)$. To
that purpose we fix a particular critical level and “take out” realisations
that undershoot the critical level by a certain amount (_lower dropout_). In
the case where no perturbation is present ($\beta_{t}\equiv 0$), we recover
the convergence rate for the respective ODE (3) derived in [Łoj84], i.e.

$|F(x_{t})-F(y)|\leq C(t+1)^{-\frac{1}{2\theta-1}}.$

We show that the convergence rate is maintained in the SDE setting as long as
$|\beta_{t}|\leq\tilde{C}(t+1)^{-\frac{\theta}{2\theta-1}}$ for sufficiently
large $t$. In Section 3.2, we provide estimates for the drift term
$(\int_{0}^{t}\alpha_{s}\,ds)_{t\geq 0}$ in the setting of Section 3.1 on the
event that no dropout occurs. In Section 3.3, we provide estimates that later
allow us to show that when a dropout occurs for a particular critical level it
is very likely that the limit of the Lyapunov function lies strictly below
that critical level. Finally, in Section 3.4 we combine the results of the
previous subsections to achieve the proof of Theorem 1.6.

###  3.1. Convergence rate for the target value in the case without lower
dropout

######  Proposition 3.1.

Suppose that the following assumptions are satisfied:

Assumptions on the error term. Let $\theta\in(\frac{1}{2},1)$, $(v_{t})_{t\geq
0}$ be a continuously differentiable, decaying, positive function and
$(\sigma_{t})_{t\geq 0}$ be a right-continuous and locally integrable
function, so that there exist constants $\kappa_{1},\kappa_{2}>0$ such that
for all $t\geq 0$

(6)
$\displaystyle-\frac{\dot{v}_{t}}{v_{t}}\leq\kappa_{1}v_{t}^{2\theta-1}\quad\text{
and }\quad\kappa_{2}v_{t}^{2\theta}\geq\sigma^{2}_{t}.$

No lower dropout/Łojasiewicz-inequality. Let $\ell\in{\mathbb{R}}$, $\L>0$ and
$(w_{t})_{t\geq 0}=(C_{w}v_{t})_{t\geq 0}$ for a $C_{w}\geq 0$. Let $T$ be a
stopping time, such that for all $t\geq 0$, on $\\{T>t\\}$,

$\displaystyle|f(X_{t})|\geq\L|F(X_{t})-\ell|^{\theta}\ ,\ F(X_{t})-\ell\geq-
w_{t}$

as well as

$\displaystyle-\langle f(X_{t}),\alpha_{t}\rangle\geq\rho|f(X_{t})|^{2},\ \
\mbox{$\frac{1}{2}$}\mathrm{tr}(\beta_{t}^{\dagger}H(X_{t})\beta_{t})\leq\sigma^{2}_{t}$

for a $\rho>0$. Additionally, suppose that for every $t>0$

(7) $\displaystyle{\mathbb{E}}\Bigl{[}\int_{0}^{t\wedge
T}|\beta_{s}^{\dagger}f(X_{s})|^{2}\,ds\Bigr{]}<\infty.$

Result. Suppose that the above assumptions hold and

${\mathbb{E}}[1\mathrm{l}_{\\{T>0\\}}(F(X_{0})-\ell)]\leq R.$

Then there exist $\alpha,\eta>0$ such that, for all $t\geq 0$, one has

(8)
$\displaystyle{\mathbb{E}}\bigl{[}1\mathrm{l}_{\\{T>t\\}}(F(X_{t})-\ell+w_{t})\bigr{]}\leq\Phi^{(R)}_{t}+\alpha
v_{t},$

where

$\Phi^{(R)}_{t}=R\Bigl{(}\big{(}(2\theta-1)\eta
R^{2\theta-1}\big{)}t+1\Bigr{)}^{-1/(2\theta-1)}.$

###### Proof.

We assume without loss of generality that $\ell=0$. By Itô’s formula

(9) $\displaystyle dF(X_{t})=\langle f(X_{t}),\alpha_{t}\rangle\,dt+\langle
f(X_{t}),\beta_{t}\,dW_{t}\rangle+\frac{1}{2}\mathrm{tr}(\beta_{t}^{\dagger}H(X_{t})\beta_{t})\,dt.$

Let $(Z_{t})_{t\geq 0}:=(1\mathrm{l}_{\\{T>t\\}}(F(X_{t})+w_{t}))_{t\geq 0}$.
By choice of $T$, $(Z_{t})_{t\geq 0}$ is a non-negative process and we have

(10) $\displaystyle dZ_{t}=1\mathrm{l}_{\\{T>t\\}}\bigl{(}\langle
f(X_{t}),\alpha_{t}\rangle+\frac{1}{2}\mathrm{tr}(\beta_{t}^{\dagger}H(X_{t})\beta_{t})+\dot{w}_{t}\bigr{)}\,dt+dM_{t}-d\xi_{t},$

where $(M_{t})_{t\geq 0}$ denotes the $L^{2}$-martingale

$(M_{t})_{t\geq 0}:=\Bigl{(}\int_{0}^{T\wedge t}\langle
f(X_{s})),\beta_{s}\,dW_{s}\rangle\Bigr{)}_{t\geq 0},$

see assumption (7), and $(\xi_{t})_{t\geq 0}$ is an increasing process given
by

$\xi_{t}:=\begin{cases}0,&\text{ if }t<T\text{ or }T=0,\\\
F(X_{T})+w_{T},&\text{ else}.\end{cases}$

For all $t\geq 0$, we let $\zeta_{t}:={\mathbb{E}}[Z_{t}]$. In the first step
we show that $(\zeta_{t})_{t\geq 0}$ is right-continuous and has no upward
jumps. By definition, $(Z_{t})_{t\geq 0}$ is right-continuous and it is
straight-forward to verify the continuity theorem for integration (Note that
$Z_{t}$ is non-negative and $\langle f(X_{t}),\alpha_{t}\rangle\leq 0$ on
$\\{t<T\\}$). To verify that $(\zeta_{t})_{t\geq 0}$ has no upward jump at
$t>0$ we note that for $\varepsilon\in(0,t)$

$\zeta_{t}-\zeta_{t-\varepsilon}={\mathbb{E}}[Z_{t}-Z_{t-\varepsilon}]\leq{\mathbb{E}}\Bigl{[}\int_{t-\varepsilon}^{t}1\mathrm{l}_{\\{s<T\\}}\frac{1}{2}\mathrm{tr}(\beta_{s}^{\dagger}H(X_{s})\beta_{s})\,ds\Bigr{]}\to
0$

as $\varepsilon\downarrow 0$.

Take $\alpha\geq C_{w}$ and $\eta>0$ such that for all $a,b>0$

(11)
$\displaystyle\rho\L^{2}2^{-(2\theta-1)}(a+b)^{2\theta}>\eta\,a^{2\theta}+\frac{\rho\L^{2}C_{w}^{2\theta}+\kappa_{2}+\kappa_{1}\alpha}{\alpha^{2\theta}}\,b^{2\theta}.$

We will now show that $\zeta_{t}\leq\bar{\zeta}_{t}:=\Phi^{(R)}_{t}+\alpha
v_{t}$ for all $t\geq 0$. Note that $\zeta_{0}\leq
R+C_{w}v_{0}\leq\Phi_{0}^{(R)}+\alpha v_{0}=\bar{\zeta}_{0}$ since $\alpha\geq
C_{w}$ so that the inequality is at least true for $t=0$. We prove the
statement by contradiction. Suppose the time

$\tau:=\inf\\{s\geq 0:\zeta_{s}>\bar{\zeta}_{s}\\}$

is finite. Since $\zeta-\bar{\zeta}$ has no upwards jumps and starts in a non-
positive value we have $\zeta_{\tau}=\bar{\zeta}_{\tau}$. We consider the
slope of the right secant of $\zeta$ with supporting points $\tau$ and
$\tau+\varepsilon$ for $\varepsilon>0$. Using the Łojasiewicz-inequality we
get that on $\\{T>t\\}$

$\displaystyle\langle f(X_{t}),\alpha_{t}\rangle$
$\displaystyle\leq-\rho|f(X_{t})|^{2}\leq-\rho\L^{2}|F(X_{t})|^{2\theta}$
$\displaystyle\leq-\rho\L^{2}\bigl{(}2^{-(2\theta-1)}|F(X_{t})+w_{t}|^{2\theta}-w_{t}^{2\theta}\bigr{)}$

so that with (10) and
$\frac{1}{2}\mathrm{tr}(\beta^{\dagger}_{t}H(X_{t})\beta_{t})\leq\sigma^{2}(t)$

$\frac{\zeta_{\tau+\varepsilon}-\zeta_{\tau}}{\varepsilon}\leq{\mathbb{E}}\Bigl{[}\frac{1}{\varepsilon}\int_{\tau}^{\tau+\varepsilon}1\mathrm{l}_{\\{T>t\\}}\bigl{(}-\rho\L^{2}\bigl{(}2^{-(2\theta-1)}|Z_{t}|^{2\theta}-w_{t}^{2\theta}\bigr{)}+\sigma_{t}^{2}\bigr{)}\,dt\Bigr{]}.$

By non-negativity and right-continuity of $(Z_{t})_{t\geq 0}$ and convexity of
$|\cdot|^{2\theta}$ one has

$\liminf_{\varepsilon\downarrow
0}{\mathbb{E}}\Bigl{[}\frac{1}{\varepsilon}\int_{\tau}^{\tau+\varepsilon}1\mathrm{l}_{\\{T>t\\}}|Z_{t}|^{2\theta}\,dt\Bigr{]}\geq{\mathbb{E}}[|Z_{\tau}|^{2\theta}]\geq|{\mathbb{E}}[Z_{\tau}]|^{2\theta}=\bar{\zeta}_{\tau}^{2\theta}.$

Consequently,

(12) $\displaystyle\limsup_{\varepsilon\downarrow
0}\frac{\zeta_{\tau+\varepsilon}-\zeta_{\tau}}{\varepsilon}\leq-\rho\L^{2}(2^{-(2\theta-1)}\bar{\zeta}_{\tau}^{2\theta}-w_{\tau}^{2\theta})+\sigma_{\tau}^{2}$

Conversely, $(\Phi^{(R)}_{t})_{t\geq 0}$ is the solution of the differential
equation

$\displaystyle\dot{\Phi}^{(R)}_{t}=-\eta(\Phi^{(R)}_{t})^{2\theta},\Phi^{(R)}_{0}=R$

so that

$\dot{\bar{\zeta}}_{\tau}=-\eta(\Phi^{(R)}_{\tau})^{2\theta}+\alpha\dot{v}_{\tau}.$

Note that if the right-hand side of (12) is strictly smaller than
$\dot{\bar{\zeta}}_{\tau}$, i.e., if

(13)
$\displaystyle-\rho\L^{2}(2^{-(2\theta-1)}\bar{\zeta}_{\tau}^{2\theta}-w_{\tau}^{2\theta})+\sigma_{\tau}^{2}<-\eta(\Phi^{(R)}_{\tau})^{2\theta}+\alpha\dot{v}_{\tau},$

then there exists an interval $[\tau,\tau+\varepsilon]$ on which
$\zeta_{t}\leq\bar{\zeta}_{t}$ which entails that $\zeta$ cannot overtake
$\bar{\zeta}$ at time $\tau$ which is a contradiction to the choice of $\tau$.
Using that $w_{\tau}=C_{w}v_{\tau}$,
$\dot{v}_{\tau}\geq-\kappa_{1}v_{\tau}^{2\theta}$ and
$\sigma_{\tau}^{2}\leq\kappa_{2}v_{\tau}^{2\theta}$ we conclude that (13) is
satisfied, if

$\rho\L^{2}2^{-(2\theta-1)}(\underbrace{\Phi_{\tau}^{(R)}+\alpha
v_{\tau}}_{=\bar{\zeta}_{\tau}})^{2\theta}>\eta(\Phi_{\tau}^{(R)})^{2\theta}+\frac{\rho\L^{2}C_{w}^{2\theta}+\kappa_{2}+\kappa_{1}\alpha}{\alpha^{2\theta}}(\alpha
v_{\tau})^{2\theta}.$

By choice of $\alpha$ and $\eta$ the latter inequality holds so that we
produced a contradiction. This finishes the proof. ∎

######  Remark 3.2.

We discuss Proposition 3.1 in the case where the critical level $\ell$ is a
local minimum and choose $C_{w}=0$ in the proposition. For given
$\theta\in(\frac{1}{2},1)$, we note that for $v_{t}=(t+1)^{-1/(2\theta-1)}$
the condition on the left-hand side of (6) is satisfied for an appropriate
constant $\kappa_{1}$ and if
$\sigma_{t}=\mathcal{O}((t+1)^{-\theta/(2\theta-1)})$, then also the second
condition is satisfied for an appropriate $\kappa_{2}$. In this case,
$(v_{t})_{t\geq 0}$ is of the same order as $(\Phi_{t}^{(R)})_{t\geq 0}$ so
that

${\mathbb{E}}[1\mathrm{l}_{\\{T>t\\}}(F(X_{t})-\ell)]=\mathcal{O}\bigl{(}(t+1)^{-1/(2\theta-1)}\bigr{)}.$

This agrees with the corresponding estimates for the ODE convergence. So if
$(\sigma_{t})_{t\geq 0}$ is of order $\mathcal{O}((t+1)^{-\theta/(2\theta-1)}$
we essentially get the same order of convergence as in the deterministic
setting.

Now suppose that $\sigma_{t}\approx(t+1)^{-\sigma}$ with
$0\leq\sigma<\theta/(2\theta-1)$. Then one can choose
$v_{t}=(t+1)^{-\sigma/\theta}$ and constants $\kappa_{1}$ and $\kappa_{2}$
appropriately so that (6) is satisfied. So in that case

${\mathbb{E}}[1\mathrm{l}_{\\{T>t\\}}(F(X_{t})-\ell)]=\mathcal{O}\bigl{(}(t+1)^{-\sigma/\theta}\bigr{)}.$

Altogether, we thus get for the choice $\sigma_{t}\approx(t+1)^{-\sigma}$ with
$\sigma\in[0,\infty)$ that

${\mathbb{E}}[1\mathrm{l}_{\\{T>t\\}}(F(X_{t})-\ell)]=\mathcal{O}\bigl{(}(t+1)^{-(\frac{\sigma}{\theta}\wedge\frac{1}{2\theta-1})}\bigr{)}.$

Observing that on a compact set $F$, $f$ and the Hessian are uniformly bounded
it is straight-forward to infer the statement of Theorem 1.8.

###  3.2. Bounding the drift term in the case where no dropout occurs

######  Proposition 3.3.

Suppose that $T$ is such that on $\\{T\geq t\\}$, $\langle
f(X_{t}),-\alpha_{t}\rangle\geq 0$ and $(F(X_{t}))_{t\geq 0}$ converges on
$\\{T=\infty\\}$, almost surely. Let $\Phi:[0,\infty)\to[0,\infty)$ be a
decreasing, continuously differentiable function so that
$t\mapsto-\dot{\Phi}_{t}$ is decreasing and

$\lim_{t\to\infty}\frac{\Phi_{t}}{\sqrt{-\dot{\Phi}_{t}}}=0.$

If for every $t\geq 0$,

${\mathbb{E}}[1\mathrm{l}_{\\{T>t\\}}(F(X_{t})-F(X_{T}))]+\frac{1}{2}{\mathbb{E}}\Bigl{[}\int_{t}^{\infty}1\mathrm{l}_{\\{s\leq
T\\}}\mathrm{tr}(\beta_{s}^{\dagger}H(X_{s})\beta_{s})\,ds\Bigr{]}\leq\Phi_{t}$

and $(M_{t})_{t\geq 0}$ given by

$M_{t}:=\int_{0}^{t\wedge T}\langle f(X_{s}),\beta_{s}\,dW_{s}\rangle$

is a regular martingale, then

${\mathbb{E}}\Bigl{[}\int_{0}^{\infty}1\mathrm{l}_{\\{s<T\\}}\sqrt{\langle
f(X_{s}),-\alpha_{s}\rangle}\,ds\Bigr{]}\leq\int_{0}^{\infty}\sqrt{-\dot{\Phi}_{s}}\,ds.$

###### Proof.

Since $(M_{t})_{t\geq 0}$ is a regular martingale we get with Itô’s formula,
see (9), that

$\displaystyle\Psi(t):=$ $\displaystyle\
{\mathbb{E}}\Bigl{[}\int_{t}^{\infty}1\mathrm{l}_{\\{s<T\\}}\langle
f(X_{s}),-\alpha_{s}\rangle\,ds\Bigr{]}$ $\displaystyle=$ $\displaystyle\
{\mathbb{E}}\bigl{[}1\mathrm{l}_{\\{T>t\\}}(F(X_{t})-F(X_{T}))\bigr{]}+\frac{1}{2}{\mathbb{E}}\Bigl{[}\int_{t}^{\infty}1\mathrm{l}_{\\{T>s\\}}\mathrm{tr}(\beta_{s}^{\dagger}H(X_{s})\beta_{s})\,ds\Bigr{]}\leq\Phi_{t}.$

The function $\Psi$ is right-continuous and monotonically decreasing and,
hence, càdlàg. We let $\varphi_{s}:=(-\dot{\Phi}_{s})^{-1/2}$ and note that by
Cauchy-Schwarz

$\displaystyle{\mathbb{E}}\Bigl{[}\int_{0}^{\infty}1\mathrm{l}_{\\{s<T\\}}\sqrt{\langle
f(X_{s}),-\alpha_{s}\rangle}\,ds\Bigr{]}\leq\Bigl{(}\int_{0}^{\infty}\varphi_{s}{\mathbb{E}}\bigl{[}1\mathrm{l}_{\\{s<T\\}}\langle
f(X_{s}),-\alpha_{s}\rangle\bigr{]}\,ds\Bigr{)}^{1/2}\Bigl{(}\int_{0}^{\infty}\varphi_{s}^{-1}\,ds\Bigr{)}^{1/2}.$

Moreover, using partial integration we get that

$\displaystyle\int_{0}^{\infty}\varphi_{s}{\mathbb{E}}\bigl{[}1\mathrm{l}_{\\{s<T\\}}\langle
f(X_{s}),-\alpha_{s}\rangle\bigr{]}\,ds$
$\displaystyle=-\int_{0}^{\infty}\varphi_{s}\,d\Psi_{s}$
$\displaystyle={-}[\varphi\Psi]_{0}^{\infty}+\int_{0}^{\infty}\Psi_{s}\,d\varphi_{s}.$

$(\varphi_{s})_{s\geq 0}$ is increasing by assumption and
$\varphi_{s}\Phi_{s}\to 0$ as $s\to\infty$ so that

$\displaystyle{-}[\varphi\Psi]_{0}^{\infty}+\int_{0}^{\infty}\Psi_{s}\,d\varphi_{s}$
$\displaystyle\leq{-}\varphi_{0}(\Phi_{0}-\Psi_{0}){-}[\varphi\Phi]_{0}^{\infty}+\int_{0}^{\infty}\Phi_{s}\,d\varphi_{s}$
$\displaystyle={-}\varphi_{0}(\Phi_{0}-\Psi_{0})+\int_{0}^{\infty}\underbrace{\varphi_{s}(-\dot{\Phi}_{s})}_{=\sqrt{-\dot{\Phi}_{s}}}\,ds.$

Altogether we get that

${\mathbb{E}}\Bigl{[}\int_{0}^{\infty}1\mathrm{l}_{\\{s<T\\}}\sqrt{\langle
f(X_{s}),-\alpha_{s}\rangle}\,ds\Bigr{]}\leq\int_{0}^{\infty}\sqrt{-\dot{\Phi}_{s}}\,ds.$

∎

###  3.3. Technical analysis of lower dropouts

Roughly speaking, the following two lemmas will later be used to show that for
a certain critical level of the target function, a lower dropout (in the sense
of the previous two propositions) entails that the Lyapunov function converges
to a value strictly below the respective critical level with high probability.

######  Lemma 3.4.

Let $(M_{t})_{t\geq 0}$ be a continuous local martingale started in zero. Then
for every $\kappa>0$

${\mathbb{P}}\Bigl{(}\sup_{t\geq 0}(M_{t}-\langle
M\rangle_{t})\geq\kappa\Bigr{)}\leq\frac{1}{\kappa^{2}}+\sum_{n\in{\mathbb{N}}_{0}}\frac{2^{n+1}}{(2^{n}+\kappa)^{2}}=:\phi(\kappa).$

In particular, for every $\varepsilon>0$ there exists a $\kappa>0$ such that
the above right-hand side is smaller than $\varepsilon$.

###### Proof.

For $n\in{\mathbb{N}}_{0}$, let $T_{n}:=\inf\\{t\geq 0:\langle
M\rangle_{t}>2^{n}\\}$. Then we have

$\displaystyle\sup_{t\in[T_{n},T_{n+1})}(M_{t}-\langle
M\rangle_{t})\leq\sup_{t\in[0,T_{n+1})}M_{t}-2^{n}$

and

$\sup_{t\in[0,T_{0})}(M_{t}-\langle
M\rangle_{t})\leq\sup_{t\in[0,T_{0})}M_{t}.$

We use Doob’s $L^{2}$-inequality to deduce that

${\mathbb{P}}\Bigl{(}\sup_{t\in[0,T_{n+1})}M_{t}\geq
2^{n}+\kappa\Bigr{)}\leq(2^{n}+\kappa)^{-2}{\mathbb{E}}[M_{T_{n+1}}^{2}]\leq\frac{2^{n+1}}{(2^{n}+\kappa)^{2}}$

and

${\mathbb{P}}\Bigl{(}\sup_{t\in[0,T_{0})}M_{t}\geq\kappa\Bigr{)}\leq\frac{1}{\kappa^{2}}.$

Therefore,

${\mathbb{P}}\Bigl{(}\sup_{t\geq 0}(M_{t}-\langle
M\rangle_{t})\geq\kappa\Bigr{)}\leq\frac{1}{\kappa^{2}}+\sum_{n\in{\mathbb{N}}_{0}}\frac{2^{n+1}}{(2^{n}+\kappa)^{2}}.$

∎

######  Lemma 3.5.

Let $t_{0}\geq 0$, $\rho,\kappa:[t_{0},\infty)\to(0,\infty)$ be functions and
$t_{0}\leq T^{\prime}\leq T$ be two stopping times such that, for every $t\geq
t_{0}$, on $\\{T^{\prime}\leq t\leq T\\}$,

$|\beta_{t}^{\dagger}f(X_{t})|^{2}\leq\rho_{T^{\prime}}\langle
f(X_{t}),-\alpha_{t}\rangle\text{ \ and \
}\frac{1}{2}\int_{T^{\prime}}^{t}\mathrm{tr}(\beta_{s}^{\dagger}H(X_{s})\beta_{s})\,ds\leq\kappa_{T^{\prime}}.$

Then,

$\displaystyle{\mathbb{P}}\Bigl{(}\sup_{t\in[T^{\prime},T]}$ $\displaystyle
F(X_{t})-F(X_{T^{\prime}})\geq
2\kappa_{T^{\prime}}\Bigr{)}\leq\phi\Bigl{(}\inf\limits_{t\geq
t_{0}}\frac{\kappa_{t}}{\rho_{t}}\Bigr{)}.$

###### Proof.

We use again representation (5):

$\displaystyle dF(X_{t})$ $\displaystyle=\langle
f(X_{t}),\alpha_{t}\rangle\,dt+\langle
f(X_{t}),\beta_{t}\,dW_{t}\rangle+\frac{1}{2}\mathrm{tr}(\beta_{t}^{\dagger}H(X_{t})\beta_{t})\,dt.$

Consider the local martingale

$(M_{t})_{t\geq t_{0}}:=\Bigl{(}\int_{0}^{t}1\mathrm{l}_{\\{T^{\prime}\leq
s\leq T\\}}\langle f(X_{s}),\beta_{s}\,dW_{s}\rangle\Bigr{)}_{t\geq t_{0}}$

and note that

$\langle M\rangle_{t}=\int_{0}^{t}1\mathrm{l}_{\\{T^{\prime}\leq s\leq
T\\}}|\beta_{s}^{\dagger}f(X_{s})|^{2}\,ds\leq\rho_{T^{\prime}}\int_{0}^{t}1\mathrm{l}_{\\{T^{\prime}\leq
s\leq T\\}}\langle f(X_{s}),-\alpha_{s}\rangle\,ds.$

Hence,

$\displaystyle\int_{0}^{t}1\mathrm{l}_{\\{T^{\prime}\leq s\leq T\\}}\langle
f(X_{s}),\alpha_{s}\rangle\,ds$
$\displaystyle+\int_{0}^{t}1\mathrm{l}_{\\{T^{\prime}\leq s\leq T\\}}\langle
f(X_{s}),\beta_{s}\,dW_{s}\rangle$ $\displaystyle\leq
M_{t}-\frac{1}{\rho_{T^{\prime}}}\langle M\rangle_{t}=:\Xi_{t}.$

Note that
$\Xi_{t}=\rho_{T^{\prime}}(\frac{1}{\rho_{T^{\prime}}}M_{t}-\langle\frac{1}{\rho_{T^{\prime}}}M\rangle_{t})$
with $(\frac{1}{\rho_{T^{\prime}}}M_{t})_{t\geq t_{0}}$ being a continuous
local martingale started in zero so that with Lemma 3.4

$\displaystyle{\mathbb{P}}\Bigl{(}\sup_{t\geq
t_{0}}\Xi_{t}\geq\kappa_{T^{\prime}}\Bigr{)}$
$\displaystyle={\mathbb{P}}\Bigl{(}\sup_{t\geq
t_{0}}\frac{1}{\rho_{T^{\prime}}}M_{t}-\langle\frac{1}{\rho_{T^{\prime}}}M\rangle_{t}\geq\frac{\kappa_{T^{\prime}}}{\rho_{T^{\prime}}}\Bigr{)}$
$\displaystyle\leq{\mathbb{P}}\Bigl{(}\sup_{t\geq
t_{0}}\frac{1}{\rho_{T^{\prime}}}M_{t}-\langle\frac{1}{\rho_{T^{\prime}}}M\rangle_{t}\geq\inf\limits_{t\geq
t_{0}}\frac{\kappa_{t}}{\rho_{t}}\Bigr{)}\leq\phi\Bigl{(}\inf\limits_{t\geq
t_{0}}\frac{\kappa_{t}}{\rho_{t}}\Bigr{)}.$

Altogether, using

$\displaystyle\sup_{T^{\prime}\leq t\leq
T}F(X_{t})-F(X_{T^{\prime}})\leq\sup_{T^{\prime}\leq t\leq
T}\Xi_{t}+\underbrace{\sup_{T^{\prime}\leq t\leq
T}\frac{1}{2}\int_{T^{\prime}}^{t}\mathrm{tr}(\beta_{s}^{\dagger}H(X_{s})\beta_{s})\,ds}_{\leq\kappa_{T^{\prime}}}$

we get that

$\displaystyle{\mathbb{P}}\Bigl{(}\sup_{T^{\prime}\leq t\leq
T}F(X_{t})-F(X_{T^{\prime}})\geq 2\kappa_{T^{\prime}}\Bigr{)}$
$\displaystyle\leq{\mathbb{P}}\Bigl{(}\sup_{t\geq
t_{0}}\Xi_{t}\geq\kappa_{T^{\prime}}\Bigr{)}\leq\phi\Bigl{(}\inf\limits_{t\geq
t_{0}}\frac{\kappa_{t}}{\rho_{t}}\Bigr{)}.$

∎

###  3.4. Proof of Theorem 1.6

We cite Lemma 3.7 from [DK21].

######  Lemma 3.6.

Let $F:{\mathbb{R}}^{d}\to{\mathbb{R}}$ be a Łojasiewicz-function with
differential $f$ and $K\subset{\mathbb{R}}^{d}$ be an arbitrary compact set.

  1. (1)

The set of _critical levels_

$\mathcal{L}_{K}:=\\{F(x):x\in\mathcal{C}_{F}\cap K\\}$

is finite so that $F$ has at most a countable number of critical levels.

  2. (2)

For every critical level $\ell\in\mathcal{L}_{K}$ there exists an open
neighbourhood $U\supset F^{-1}(\\{\ell\\})\cap K$, $\L>0$,
$\theta\in[\frac{1}{2},1)$ such that for every $y\in U$

$|f(y)|\geq\L|F(y)-\ell|^{\theta}.$

  3. (3)

For a neighbourhood as in (2), there exists $\varepsilon>0$ such that

$F^{-1}((\ell-\varepsilon,\ell+\varepsilon))\cap K\subset U.$

Now we are able to prove the main result of this article.

###### Proof of Theorem 1.6.

For $\rho,C,t_{0}>0$ and a set $K\subset{\mathbb{R}}^{d}$ let

$\displaystyle T_{\rho,C,t_{0},K}:=\inf\Bigl{\\{}t\geq t_{0}:\ $
$\displaystyle{\langle f(X_{t}),-\alpha_{t}\rangle}<\rho|f(X_{t})|^{2},\
|\alpha_{t}|>C|f(X_{t})|,\ |f(X_{t})|>C,$
$\displaystyle|\beta_{t}|^{2}>C\sigma_{t}^{2},\
\frac{1}{2}\mathrm{tr}(\beta_{t}^{\dagger}H(X_{t})\beta_{t})>C\sigma_{t}^{2}\
\text{ or }\ X_{t}\notin K\Bigr{\\}}.$

Set $\theta_{0}:=\frac{\sigma}{2\sigma-1}\in(\frac{1}{2},1)$.

1) First let $U$ be an open and bounded set such that for a
$\theta\in(\theta_{0},1)$, an $\ell\in{\mathbb{R}}$ and $\L>0$, for all $y\in
U$

$|f(y)|\geq\L\,|F(y)-\ell|^{\theta}.$

1.a) We show that on $\\{T_{\rho,C,t_{0},U}=\infty\\}$, almost surely, the
random set

$S:=\\{s\geq 0:F(X_{s})<\ell-(s+1)^{-\frac{1}{2\theta-1}}\\}$

is bounded.

By Proposition 2.1 we have on $\\{T_{\rho,C,t_{0},U}=\infty\\}$ that, almost
surely, $(F(X_{t}))_{t\geq 0}$ converges and $f(X_{t})\to 0$, which together
with the Łojasiewicz-inequality implies that $F(X_{t})\to\ell$.

By choice of $\theta$ one has $2\sigma>\frac{2\theta}{2\theta-1}$. Thus, for
$t_{1}\geq t_{0}$ large enough we get for all $t\geq t_{1}$

$C\sigma_{t}^{2}\leq\frac{1}{4\theta-2}(t+1)^{-\frac{2\theta}{2\theta-1}}.$

Let $T^{\prime}_{t_{1}}:=\inf\\{t\geq
t_{1}:F(X_{t})<\ell-(t+1)^{-\frac{1}{2\theta-1}}\\}$, then one has

$\sup_{T^{\prime}_{t_{1}}\leq t\leq
T_{\rho,C,t_{0},U}}\int_{T^{\prime}_{t_{1}}}^{t}\frac{1}{2}\mathrm{tr}(\beta_{s}^{\dagger}H(X_{s})\beta_{s})\,ds\leq\frac{1}{4\theta-2}\int_{T^{\prime}_{t_{1}}}^{\infty}(s+1)^{-\frac{2\theta}{2\theta-1}}\,ds=\frac{1}{2}(T^{\prime}_{t_{1}}+1)^{-\frac{1}{2\theta-1}}.$

Furthermore, we have for $t$ with $T^{\prime}_{t_{1}}\leq
t<T_{\rho,C,t_{0},U}$

$|\beta_{t}^{\dagger}f(X_{t})|^{2}\leq|\beta_{t}^{\dagger}|^{2}|f(X_{t})|^{2}\leq\frac{C}{\rho}(T^{\prime}_{t_{1}}+1)^{-2\sigma}\langle
f(X_{t}),-\alpha_{t}\rangle$

so that, using Lemma 3.5, we get

$\displaystyle{\mathbb{P}}(S\cap[t_{1},\infty)\not=\emptyset,T_{\rho,C,t_{0},U}=\infty)$
$\displaystyle={\mathbb{P}}(T^{\prime}_{t_{1}}<\infty,T_{\rho,C,t_{0},U}=\infty,F(X_{t})\to\ell)$
$\displaystyle\leq{\mathbb{P}}\Bigl{(}\sup_{t\in[T^{\prime}_{t_{1}},T_{\rho,C,t_{0},U})}F(X_{t})-F(X_{T^{\prime}_{t_{1}}})\geq(T^{\prime}_{t_{1}}+1)^{-\frac{1}{2\theta-1}}\Bigr{)}$
$\displaystyle\leq\phi\Bigl{(}\frac{\rho}{2C}(t_{1}+1)^{2\sigma-\frac{1}{2\theta-1}}\Bigr{)}\overset{t_{1}\to\infty}{\longrightarrow}0$

and for all $t_{1}\geq t_{0}$

${\mathbb{P}}(S\text{ is
unbounded},T_{\rho,C,t_{0},U}=\infty)\leq{\mathbb{P}}(S\cap[t_{1},\infty)\not=\emptyset,T_{\rho,C,t_{0},U}=\infty)\to
0.$

1.b) Let

$\bar{T}_{\rho,C,t_{0},U}:=T_{\rho,C,t_{0},U}\wedge\inf\\{t\geq
t_{0}:X_{t}<\ell-(t+1)^{-\frac{1}{2\theta-1}}\\}.$

We show that on $\\{\bar{T}_{\rho,C,t_{0},U}=\infty\\}$ the process
$(X_{t})_{t\geq 0}$ converges, almost surely.

We set $\sigma^{\prime}_{t}:=\sqrt{C}\sigma_{t}$ and
$v_{t}:=w_{t}:=(t+1)^{-\frac{1}{2\theta-1}}$ (so that $C_{w}=1$) and
$T:=\bar{T}_{\rho,C,t_{0},U}$ and show that the functions satisfy the
Assumptions of Proposition 3.1, Proposition 3.3 and the right-hand side of the
estimate in Proposition 3.3 is finite.

First note that one can apply Proposition 3.1 for the process $(X_{t})_{t\geq
t_{0}}$ started at time $t_{0}$ in $X_{t_{0}}$. In particular, we have

$-\dot{v}_{t}=\frac{1}{2\theta-1}(t+1)^{-\frac{2\theta}{2\theta-1}}=\frac{1}{2\theta-1}v_{t}^{2\theta}$

and with $2\sigma>\frac{2\theta}{2\theta-1}$

$v_{t}^{2\theta}=(t+1)^{-\frac{2\theta}{2\theta-1}}\geq\frac{1}{C}(\sigma^{\prime}_{t})^{2}$

so that (6) is satisfied. Further, with the definition of $T_{\rho,C,t_{0},K}$

${\mathbb{E}}\Bigl{[}\int_{t_{0}}^{T}|\beta_{s}^{\dagger}f(X_{s})|^{2}\,ds\Bigr{]}\leq\int_{t_{0}}^{\infty}C^{3}\sigma_{t}^{2}\,dt<\infty,$

so that (7) is satisfied. Now, we can choose $\eta>0$ sufficiently small and
$\alpha>0$ sufficiently large so that (11) is satisfied and with the
boundedness of $U$ and Proposition 3.1 we get for sufficiently large
$R,C^{\prime}>0$ that for all $t\geq t_{0}$

${\mathbb{E}}\bigl{[}1\mathrm{l}_{\\{T>t\\}}(F(X_{t})-\ell+w_{t})\bigr{]}\leq\Phi^{(R)}_{t}+\alpha
v_{t}\leq C^{\prime}(t-t_{0}+1)^{-\frac{1}{2\theta-1}}.$

Regarding Proposition 3.3 we recall that, on $\\{T=\infty\\}$, we almost
surely have $F(X_{t})\to\ell$ so that $F(X_{T})$ is well-defined. By the
definition of $T$ we have $-F(X_{T})\leq-\ell+w_{T}$, where $w_{\infty}=0$,
and the monotonicity of $(w_{t})$ gives

${\mathbb{E}}[1\mathrm{l}_{\\{T>t\\}}(F(X_{t})-F(X_{T}))]\leq{\mathbb{E}}[1\mathrm{l}_{\\{T>t\\}}(F(X_{t})-\ell+w_{t})]\leq
C^{\prime}(t-t_{0}+1)^{-\frac{1}{2\theta-1}}.$

Thus, we have

$\displaystyle{\mathbb{E}}[1\mathrm{l}_{\\{T>t\\}}$
$\displaystyle(F(X_{t})-F(X_{T}))]+\frac{1}{2}{\mathbb{E}}\Bigl{[}\int_{t}^{\infty}1\mathrm{l}_{\\{s\leq
T\\}}\mathrm{tr}(\beta_{s}^{\dagger}H(X_{s})\beta_{s})\,ds\Bigr{]}$
$\displaystyle\leq
C^{\prime}(t-t_{0}+1)^{-\frac{1}{2\theta-1}}+C\int_{t}^{\infty}\sigma_{s}^{2}\,ds\leq\Phi_{t},$

where $\Phi_{t}:=C^{\prime\prime}(t-t_{0}+1)^{-\frac{1}{2\theta-1}}$ for
sufficiently large $C^{\prime\prime}$. Moreover, $\dot{\Phi}_{t}$ is an
increasing function so that $t\mapsto-\dot{\Phi}_{t}$ is decreasing and using
that $\theta<1$ we get that

$\Phi_{t}/\sqrt{-\dot{\Phi}_{t}}=\sqrt{2\theta-1}\,(C^{\prime\prime})^{\theta-1/2}\,(\Phi_{t})^{1-\theta}\to
0.$

Moreover, $(M_{t})_{t\geq t_{0}}:=\bigl{(}\int_{t_{0}}^{t\wedge T}\langle
f(X_{s}),\beta_{s}\,dW_{s}\rangle\bigr{)}_{t\geq t_{0}}$ is a regular
martingale as

$\langle
M_{t}\rangle_{\infty}=\int_{t_{0}}^{T}|\beta_{s}^{\dagger}f(X_{s})|^{2}\,ds\leq\int_{t_{0}}^{\infty}C^{3}\sigma_{t}^{2}\,dt<\infty.$

Therefore, Proposition 3.3 is applicable and we get

$\displaystyle{\mathbb{E}}\Bigl{[}\int_{t_{0}}^{\infty}1\mathrm{l}_{\\{s<T\\}}|\alpha_{s}|\,ds\Bigr{]}$
$\displaystyle\leq
C\,{\mathbb{E}}\Bigl{[}\int_{t_{0}}^{\infty}1\mathrm{l}_{\\{s<T\\}}|f(X_{s})|\,ds\Bigr{]}$
$\displaystyle\leq
C\sqrt{\rho}\,{\mathbb{E}}\Bigl{[}\int_{t_{0}}^{\infty}1\mathrm{l}_{\\{s<T\\}}\sqrt{\langle
f(X_{s}),-\alpha_{s}\rangle}\,ds\Bigr{]}$ $\displaystyle\leq
C\sqrt{\rho}\,\int_{t_{0}}^{\infty}\sqrt{-\dot{\Phi}_{s}}\,ds=C\sqrt{\frac{\rho
C^{\prime\prime}}{2\theta-1}}\,\int_{t_{0}}^{\infty}(t-t_{0}+1)^{-\frac{\theta}{2\theta-1}}\,ds$
$\displaystyle=C\sqrt{\rho
C^{\prime\prime}}\frac{\sqrt{2\theta-1}}{1-\theta}<\infty.$

This implies almost sure convergence of
$(\int_{t_{0}}^{t}\alpha_{s}\,ds)_{t\geq t_{0}}$, on
$\\{\bar{T}_{\rho,C,t_{0},U}=\infty\\}$. Note that also
$(\int_{t_{0}}^{\infty}\beta_{t}\,dW_{t})_{t\geq t_{0}}$ converges almost
surely on $\\{\bar{T}_{\rho,C,t_{0},U}=\infty\\}$, since

$\int_{t_{0}}^{\bar{T}_{\rho,C,t_{0},U}}|\beta_{s}|_{F}^{2}\,ds\leq\int_{t_{0}}^{\infty}C\sigma_{s}^{2}\,ds<\infty.$

Hence, we have almost sure convergence of

$X_{t}-X_{t_{0}}=\int_{t_{0}}^{t}\alpha_{t}\,dt+\int_{t_{0}}^{t}\beta_{t}\,dW_{t},$

on $\\{\bar{T}_{\rho,C,t_{0},U}=\infty\\}$.

1.c) Combining 1.a) and 1.b) we conclude that on
$\\{T_{\rho,C,t_{0},U}=\infty\\}$ we have, almost sure, convergence of
$(X_{t})_{t\geq 0}$. Indeed, by 1.a) we have that up to nullsets
$\\{T_{\rho,C,t_{0},U}=\infty\\}$ is contained in

$\bigcup_{n\in{\mathbb{N}}}\\{\bar{T}_{\rho,C,t_{0}+n,U}=\infty\\}$

and by 1.b) on each of the latter sets $(X_{t})_{t\geq 0}$ converges almost
surely.

2) Now let $K\subset{\mathbb{R}}^{d}$ be a compact subset. We show that we
have almost sure convergence of $(X_{t})_{t\geq 0}$ on
$\\{T_{\rho,C,t_{0},K}=\infty\\}$. The statement then follows by observing
that

${\mathbb{C}}\cap{\mathbb{L}}\cap\bigl{\\{}\limsup_{t\to\infty}\frac{|\beta_{t}|}{\sigma_{t}}<\infty\bigr{\\}}=\bigcup_{n\in{\mathbb{N}}}\\{T_{\frac{1}{n},n,n,[-n,n]^{d}}=\infty\\}.$

By Lemma 3.6, $K$ contains finitely many critical levels, say
$\ell_{1},\dots,\ell_{m}\in{\mathbb{R}}$, and there exists $\varepsilon>0$,
$\L>0$, $\theta\in[\frac{\sigma}{2\sigma-1},1)$ and bounded open
neighbourhoods $U_{k}\supset
F^{-1}((\ell_{k}-\varepsilon,\ell_{k}+\varepsilon))\cap K$ such that for each
$k=1,\dots,m$ and $y\in U_{k}$,

$|f(y)|\geq\L|F(y)-\ell_{k}|^{\theta}.$

By Theorem 1.1, we have that on $\\{T_{\rho,C,t_{0},K}=\infty\\}$, almost
surely, the limit $\lim_{t\to\infty}F(X_{t})$ exists and
$\lim_{t\to\infty}f(X_{t})=0$. Since $\inf_{y\in
K\backslash(U_{1}\cup\ldots\cup U_{m})}|f(y)|>0$, we get that on
$\\{T_{\rho,C,t_{0},U}=\infty\\}$, almost surely, $(X_{t})_{t\geq 0}$ attains
from a random time onwards only values in $U_{1}\cup\ldots\cup U_{m}$ and
using the Łojasiewicz-inequality we get that $(F(X_{t}))_{t\geq 0}$ converges
to one of the levels $\ell_{k}$ and, in particular, attains from a random time
onwards only values in a $U_{k}$ (which may be random). Hence, up to nullsets,
$\\{T_{\rho,C,t_{0},K}=\infty\\}$ is contained in

$\bigcup_{n\in{\mathbb{N}}}\bigcup_{k=1}^{m}\\{T_{\rho,C,t_{0}+n,U_{k}}=\infty\\}.$

By 1.c) we have almost sure convergence of $(X_{t})_{t\geq 0}$ on each of the
countably many latter events. ∎

##  4\. Optimality of Theorem 1.6

In this section, we provide an example which satisfies the assumptions of
Theorem 1.6 when choosing $\sigma_{t}=(t+1)^{-1}$, but for which
$(X_{t})_{t\geq 0}$ does not converge.

We consider a rotationally invariant Lyapunov function $F$ on
${\mathbb{R}}^{2}$. Let

$\psi:[0,\infty)\to[0,\infty),r\mapsto\begin{cases}(1-r)^{2},&\text{if
}r\geq\frac{1}{2},\\\
\frac{8}{3}r^{3}-3r^{2}+\frac{2}{3},&\text{else},\end{cases}$

and set $F:{\mathbb{R}}^{2}\to{\mathbb{R}}$, $F(x):=\psi(|x|)$. Note that $F$
is a $C^{2}$-function with critical points

$\mathcal{C}:=f^{-1}(\\{0\\})=\\{x\in{\mathbb{R}}^{2}:|x|\in\\{0,1\\}\\},$

where we again denote by $f:=\nabla F$ the gradient of $F$. For every
$(x_{1},x_{2})\in{\mathbb{R}}^{2}$ we let

$\mathrm{ort}(x_{1},x_{2}):=(-x_{2},x_{1}).$

and denote by $\varphi:{\mathbb{R}}^{2}\to[0,1]$ a rotationally invariant
$C^{\infty}$-function with

$\varphi|_{B(0,2)\backslash B(0,\frac{2}{3})}\equiv 1\text{ \ and \
}\varphi|_{B(0,\frac{1}{2})\cup B(0,3)^{c}}\equiv 0.$

Consider the SDE

$dX_{t}=\Bigl{(}-f(X_{t})+\mathrm{ort}(X_{t})\,||X_{t}|-1|\Bigr{)}\,dt+\frac{1}{t+1}\varphi(X_{t})\,dW_{t}$

started in a point $x_{0}\in{\mathbb{R}}^{2}\backslash\\{0\\}$, where
$(W_{t})_{t\geq 0}$ is a $2$-dimensional Brownian motion with initial value
$W_{0}=0$.

######  Theorem 4.1.

  1. (i)

$F$ is a Łojasiewicz-function.

  2. (ii)

${\mathbb{P}}({\mathbb{C}}\cap{\mathbb{L}})=1$.

  3. (iii)

Almost surely, $(X_{t})_{t\geq 0}$ does not converge.

Figure 1. (a) Graph of $F$

Figure 2. (b) Negative gradient field

Figure 3. (c) Vectorfield of the drift

The proof is based on the following proposition.

######  Proposition 4.2.

For $t\geq 0$ consider the stopping time

$T_{t}:=\inf\bigl{\\{}s\geq t:X_{s}\not\in B(0,2)\backslash
B(0,\mbox{$\frac{2}{3}$})\bigr{\\}}.$

  1. (i)

${\mathbb{P}}({\mathbb{C}}\cap{\mathbb{L}})=1$ and, almost surely,
$\lim_{t\to\infty}|X_{t}|=1$. In particular, almost surely, for all
sufficiently large $t$, $T_{t}=\infty$.

  2. (ii)

For every $t\geq 0$, $(\bar{Z}_{s})_{s\geq t}:=(|X_{s}|-1)_{s\geq t}$ solves
on the random time interval $[t,T_{t}]$ the SDE

$d\bar{Z}_{s}=\Bigl{(}-2\bar{Z}_{s}+\frac{1}{2}\frac{1}{\bar{Z}_{s}+1}\frac{1}{(s+1)^{2}}\Bigr{)}\,ds+\frac{1}{s+1}dB_{s}$

where $B_{s}:=\int_{t}^{s}\frac{1}{|X_{u}|}\langle X_{u},dW_{u}\rangle$ is a
Brownian motion.

  3. (iii)

For $t\geq 0$ let $(Z_{s}^{(t)})_{s\geq t}$ denote the solution to

(14) $\displaystyle Z^{(t)}_{t}=\bar{Z}_{t}\text{ \ and \
}dZ^{(t)}_{s}=-2Z^{(t)}_{s}\,ds+\frac{1}{s+1}dB_{s}$

Then,

$\int_{t}^{T_{t}}|\bar{Z}_{s}-Z^{(t)}_{s}|\,ds\leq\frac{1}{2}\frac{1}{t+1}.$

###### Proof.

(i): For the choice

$\alpha_{t}:=-f(X_{t})+\mathrm{ort}(X_{t})\,||X_{t}|-1|\text{ \ and \
}\beta_{t}:=\frac{1}{t+1}\varphi(X_{t})\,\mathbb{I}_{2}$

$(X_{t})_{t\geq 0}$ solves the SDE (1). We verify that the events $\mathbb{B}$
and ${\mathbb{C}}$ are almost sure events. Note that the diffusivity on
$B(0,\frac{1}{2})\cup B(0,3)^{c}$ is zero and that the vector field pushes
$(X_{t})_{t\geq 0}$ towards values in $\overline{B(0,3)}\backslash
B(0,\frac{1}{2})$. Hence, from a deterministic time, say $t_{0}$, onwards the
process $(X_{t})_{t\geq 0}$ stays on the compact set
$\overline{B(0,3)}\backslash B(0,\frac{1}{2})$. On the latter compact set $F$,
$\nabla F$ and the Hessian $H$ of $F$ are uniformly bounded since $F$ is
$C^{2}$. Therefore, $\mathbb{B}$ is an almost sure set.

For $t\geq t_{0}$, by orthogonality of $f$ and $\mathrm{ort}$ one has

$\langle f(X_{t}),-\alpha_{t}\rangle=|f(X_{t})|^{2}$

and, additionally,

$|\alpha_{t}|=|-f(X_{t})+\mathrm{ort}(X_{t})||X_{t}|-1||\leq\underbrace{|f(X_{t})|}_{=2|1-|X_{t}||}+\underbrace{|X_{t}|}_{\leq
3}\,||X_{t}|-1|\leq 5|1-|X_{t}||=\frac{5}{2}|f(X_{t})|$

Moreover,

$\int_{0}^{\infty}|\beta_{s}|_{F}^{2}\,ds\leq\int_{0}^{\infty}\frac{1}{(s+1)^{2}}|\mathbb{I}_{2}|_{F}^{2}\,ds=2\int_{0}^{\infty}\frac{1}{(s+1)^{2}}\,ds<\infty.$

Consequently, ${\mathbb{C}}$ is an almost sure event. By Theorem 1.1,
$(f(X_{t}))_{t\geq 0}$ converges, almost surely, to zero so that
$\lim_{t\to\infty}|X_{t}|=1$, almost surely.

(ii): Note that $|\cdot|:{\mathbb{R}}^{2}\setminus\\{0\\}\to{\mathbb{R}}$ is
$C^{\infty}$ and, on $[t,T_{t}]$,
$-f(X_{t})=-2\bar{Z}_{t}\frac{X_{t}}{|X_{t}|}$ as well as $\langle
X_{t},\mathrm{ort}(X_{t})\rangle=0$, so that with the Itô-formula we get

$\displaystyle d\bar{Z}_{s}$ $\displaystyle=\frac{1}{|X_{s}|}\langle
X_{s},dX_{s}\rangle+\frac{1}{2}\frac{1}{(s+1)^{2}}\frac{1}{|X_{s}|}ds$
$\displaystyle=\Bigl{(}-2\bar{Z}_{s}+\frac{1}{2}\frac{1}{\bar{Z}_{s}+1}\frac{1}{(s+1)^{2}}\Bigr{)}\,ds+\frac{1}{s+1}dB_{s}.$

(iii): Fix $t\geq 0$ and consider $(\Upsilon^{(t)}_{s})_{s\geq
t}:=(\bar{Z}_{s}-Z^{(t)}_{s})_{s\geq t}$. Then on $[t,T_{t}]$

$d\Upsilon^{(t)}_{s}=-2\Upsilon_{s}^{(t)}\,ds+\frac{1}{2}\frac{1}{\bar{Z}_{s}+1}\frac{1}{(s+1)^{2}}\,ds.$

With $\Upsilon^{(t)}_{t}=0$ we thus get that for $t\leq s\leq T_{t}$

$\Upsilon_{s}^{(t)}=\frac{1}{2}\int_{t}^{s}e^{-2(s-u)}\frac{1}{\bar{Z}_{u}+1}\frac{1}{(u+1)^{2}}\,du\leq\int_{t}^{s}e^{-2(s-u)}\frac{1}{(u+1)^{2}}\,du,$

where we used that for $t\leq u\leq T_{t}$, $\frac{1}{\bar{Z}_{u}+1}\leq 2$.
We thus get with Fubini that

$\displaystyle\int_{t}^{T_{t}}|\bar{Z}_{s}-Z_{s}^{(t)}|\,ds$
$\displaystyle\leq\int_{t}^{\infty}\int_{t}^{s}e^{-2(s-u)}\frac{1}{(u+1)^{2}}\,du\,ds$
$\displaystyle=\int_{t}^{\infty}\underbrace{\int_{u}^{\infty}e^{-2(s-u)}\,ds}_{=\frac{1}{2}}\,\frac{1}{(u+1)^{2}}\,du=\frac{1}{2}\frac{1}{t+1}.$

∎

Now, let $t\geq 0$ and $x_{0}\in{\mathbb{R}}$ and let $(Z_{s})_{s\geq t}$ be
the solution of the SDE

(15) $\displaystyle dZ_{s}=-2Z_{s}\,ds+(s+1)^{-1}\,dB_{s},\text{ with
}Z_{t}=x_{0}.$

Using Itô’s lemma and the Dubins-Schwarz theorem it is straight-forward to see
that one can represent the solution as

$(Z_{s})_{s\geq t}=(e^{-2s}\tilde{B}_{g_{s}})_{s\geq t},$

where $(\tilde{B}_{s})_{s\geq g_{t}}$ is a Brownian motion started at time
$g_{t}$ in $\tilde{B}_{g_{t}}=e^{2t}x_{0}$ and $(g_{s})_{s\geq 0}$ is given by

$g_{s}:=\int_{0}^{s}(u+1)^{-2}e^{4u}\,du\sim\frac{1}{4}(s+1)^{-2}e^{4s}.$

Note that $g_{s}\leq\frac{e^{4s}}{(s+1)^{2}}$.

######  Lemma 4.3.

For every $t\geq 0$ and $x_{0}\in{\mathbb{R}}$ the solution $(Z_{s})_{s\geq
t}$ of (15) satisfies

$\int_{t}^{\infty}|Z_{s}|\,ds=\infty,\qquad\text{almost surely.}$

###### Proof.

We let for $s\geq t$, $\bar{B}_{s}:=\tilde{B}_{g_{s}}-e^{2t}x_{0}$ and note
that

$\Bigl{|}\int_{t}^{u}|Z_{s}|\,ds-\int_{t}^{u}|e^{-2s}\bar{B}_{s}|\,ds\Bigr{|}\leq|x_{0}|\int_{0}^{\infty}e^{-2s}\,ds=\mbox{$\frac{1}{2}$}|x_{0}|.$

Hence, it suffices to show that
$\int_{t}^{\infty}|e^{-2s}\bar{B}_{s}|\,ds=\infty$, almost surely. Set

$\kappa:=\int_{\mathbb{R}}\frac{1}{\sqrt{2\pi}}|x|\,e^{-x^{2}/2}\,dx=\sqrt{\frac{2}{\pi}}.$

One has

$\displaystyle{\mathbb{E}}\Bigl{[}\int_{t}^{u}e^{-2s}|\bar{B}_{s}|\,ds\Bigr{]}$
$\displaystyle=\int_{t}^{u}e^{-2s}{\mathbb{E}}[|\bar{B}_{s}|]\
ds=\kappa\int_{t}^{u}e^{-2s}\sqrt{g_{s}-g_{t}}\,ds.$

Using that $g_{s}\sim\frac{1}{4}(s+1)^{-2}e^{4s}$ we get that

$\kappa\int_{t}^{u}e^{-2s}\sqrt{g_{s}-g_{t}}\,ds\sim\mbox{$\frac{1}{2}$}\kappa\int_{t}^{u}(s+1)^{-1}\,ds\sim\mbox{$\frac{1}{2}$}\kappa\log
u.$

To estimate the variance we first show that for $v\geq u\geq t$ we have

(16) $\displaystyle\mathrm{cov}(|\bar{B}_{u}|,|\bar{B}_{v}|)\leq g_{v}-g_{t}.$

Indeed,

$\displaystyle{\mathbb{E}}[|\bar{B}_{v}\bar{B}_{u}|]$
$\displaystyle\leq{\mathbb{E}}[\bar{B}_{u}^{2}]+{\mathbb{E}}[|\bar{B}_{v}-\bar{B}_{u}|]\,{\mathbb{E}}[|\bar{B}_{u}|]$
$\displaystyle\leq g_{u}-g_{t}+\kappa^{2}\sqrt{g_{v}-g_{u}}\sqrt{g_{u}-g_{t}}$

and

${\mathbb{E}}[|\bar{B}_{v}|]\,{\mathbb{E}}[|\bar{B}_{u}|]=\kappa^{2}\sqrt{g_{v}-g_{t}}\sqrt{g_{u}-g_{t}}.$

Thus (16) follows since $g_{u}\geq g_{t}$. We conclude that for all $\ell\geq
t$

$\displaystyle\mathrm{Var}\Bigl{(}\int_{t}^{\ell}e^{-2s}|\bar{B}_{s}|\,ds\Bigr{)}$
$\displaystyle=\int_{t}^{\ell}\int_{t}^{\ell}e^{-2(u+v)}\mathrm{cov}(|\bar{B}_{u}|,|\bar{B}_{v}|)\
du\,dv$ $\displaystyle\leq 2\int_{t}^{\ell}\int_{u}^{\ell}e^{-2(u+v)}g_{u}\
dv\,du$
$\displaystyle\leq\int_{0}^{\ell}e^{-4u}g_{u}\,du\leq\int_{0}^{\ell}\frac{1}{(u+1)^{2}}\,du\leq
1.$

Hence, as a consequence of the Chebyshev inequality one gets that

$\int_{t}^{\infty}e^{-2s}|\bar{B}_{s}|\,ds=\lim_{u\to\infty}\int_{t}^{u}e^{-2s}|\bar{B}_{s}|\,ds=\infty,\quad\text{almost
surely.}$

∎

###### Proof of Theorem 4.1.

(i): We first verify that $F$ is a Łojasiewicz-function. We need to analyse
the critical points only. One has

$\text{Hess}\,F(0)=\begin{pmatrix}-6&0\\\ 0&-6\end{pmatrix}$

so that the Hessian has full rank at $0$ which implies validity of a
Łojasiewicz-inequality with $\theta=\frac{1}{2}$ on an appropriate
neighbourhood of $0$. Next, let $x\in{\mathbb{R}}^{2}$ with $|x|=1$. Since
$\psi^{\prime\prime}(1)=2$, $\psi$ satisfies a Łojasiewicz-inequality

$|\psi^{\prime}(r)|\geq\L|\psi(r)-\psi(1)|^{\frac{1}{2}}$

for all $r\in[1-\varepsilon,1+\varepsilon)$ and appropriately fixed
$\varepsilon\in(0,\frac{1}{2}]$ and $\L>0$. Consequently, for $y\in
B(0,1+\varepsilon)\backslash B(0,1-\varepsilon)$ one has

$|f(y)|=|\psi^{\prime}(|y|)|\geq\L\,|\psi(|y|)-\psi(1)|^{1/2}=\L\,|F(y)-F(x)|^{1/2}.$

(ii): has been shown in Prop. 4.2.

(iii): We conceive $(X_{t})_{t\geq 0}$ as complex-valued process by letting

$Y_{t}:=X^{(1)}_{t}+iX^{(2)}_{t}$

and note that (since $(Y_{t})_{t\geq 0}$ does not hit $0$) there is a
continuous adapted process $(\Phi_{s})_{s\geq 0}$ satisfying

$Y_{s}=|X_{s}|e^{i\Phi_{s}}.$

Now on $[t,T_{t}]$

$\displaystyle
d\Phi_{s}=||X_{s}|-1|\,ds+\frac{1}{|X_{s}|^{2}}\frac{1}{s+1}\langle\mathrm{ort}(X_{s}),dW_{s}\rangle$

so that

$d\langle\Psi\rangle_{s}=\frac{1}{|X_{s}|^{2}}\frac{1}{(s+1)^{2}}\,ds.$

We thus get that

$\int_{t}^{T_{t}}d\langle\Psi\rangle_{s}=\int_{t}^{T_{t}}\frac{1}{|X_{s}|^{2}}\frac{1}{(s+1)^{2}}\,ds\leq\mbox{$\frac{9}{4}$}\int_{t}^{\infty}\frac{1}{(s+1)^{2}}\,ds<\infty$

so that $\int_{t}^{T_{t}\wedge
s}\frac{1}{|X_{u}|^{2}}\frac{1}{u+1}\langle\mathrm{ort}(X_{u}),dW_{u}\rangle$
converges almost surely as $s\to\infty$, say to $\bar{\Psi}^{(t)}$.
Consequently, on $\\{T_{t}=\infty\\}$ one has as $s\to\infty$

$\Phi_{s}=\int_{t}^{s}||X_{u}|-1|\,du+\bar{\Phi}^{(t)}+o(1),$

where $o(1)$ stands for a term converging to zero as $s\to\infty$. By
Proposition 4.2 (iii) and Lemma 4.3, $\int_{t}^{s}||X_{u}|-1|\,du\to\infty$ as
$s\to\infty$ which shows that $(X_{t})_{t\geq 0}$ does not converge on
$\\{T_{t}=\infty\\}$. The result follows by noticing that
$\bigcup_{t\in{\mathbb{N}}}\\{T_{t}=\infty\\}$ is an almost sure set by
Proposition 4.2 (i). ∎

##  Appendix A Criterion for staying local

Assume that $F$ satisfies $\lim_{|x|\to\infty}F(x)=\infty$ and consider the
set

${\mathbb{L}}_{1}:=\Bigl{\\{}\sup\limits_{t\geq 0}\int_{0}^{t}\bigl{(}\langle
f(X_{s}),\alpha_{s}\rangle+\frac{1}{2}\mathrm{tr}(\beta_{s}^{\dagger}H(X_{s})\beta_{s})\bigr{)}\,ds<\infty\Bigr{\\}}.$

We will show that $\limsup_{t\to\infty}|X_{t}|<\infty$, almost surely, on
${\mathbb{L}}_{1}$. By Itô’s formula

$dF(X_{t})=\langle f(X_{t}),\alpha_{t}\rangle\,dt+\langle
f(X_{t}),\beta_{t}\,dW_{t}\rangle+\frac{1}{2}\mathrm{tr}(\beta_{t}^{\dagger}H(X_{t})\beta_{t})\,dt.$

Now, let $n\in{\mathbb{N}}$ and consider the stopping time

$T_{n}:=\inf\Bigl{\\{}t\geq 0:\int_{0}^{t}\bigl{(}\langle
f(X_{s}),\alpha_{s}\rangle+\frac{1}{2}\mathrm{tr}(\beta_{s}^{\dagger}H(X_{s})\beta_{s})\bigr{)}\,ds\geq
n\Bigr{\\}}.$

and $(Y_{t})_{t\geq 0}$ given by

$Y_{t}:=F(X_{t})-\int_{0}^{t}\bigl{(}\langle
f(X_{s}),\alpha_{s}\rangle+\frac{1}{2}\mathrm{tr}(\beta_{s}^{\dagger}H(X_{s})\beta_{s})\bigr{)}\,ds.$

Then, $(Y_{t}^{T_{n}})_{t\geq 0}$ is a local martingale that is bounded from
below and, thus, a supermartingale. With the martingale convergence theorem we
get almost sure convergence of $(Y_{t}^{T_{n}})_{t\geq 0}$ and, thus, on
$\\{T_{n}=\infty\\}$ we have that $\sup_{t\geq 0}F(X_{t})$ is finite, almost
surely. Finally, note that

${\mathbb{L}}_{1}=\bigcup\limits_{n\in{\mathbb{N}}}\\{T_{n}=\infty\\},$

so that $\limsup_{t\to\infty}|X_{t}|<\infty$, almost surely, on
${\mathbb{L}}_{1}$. Note that for the choice $(\alpha_{t})_{t\geq
0}=(-f(X_{t}))_{t\geq 0}$ we have

$\Bigl{\\{}\sup\limits_{t\geq 0}|H(X_{t})|<\infty\text{ and
}\int_{0}^{\infty}|\beta_{s}|_{F}^{2}\,ds<\infty\Bigr{\\}}\subset{\mathbb{L}}_{1}$

as well as for a compact set $K\subset{\mathbb{R}}^{d}$

$\displaystyle\Bigl{\\{}$ $\displaystyle 1\mathrm{l}_{\\{X_{t}\notin
K\\}}(-|f(X_{t})|^{2}+\frac{1}{2}\mathrm{tr}((\beta_{t})^{\dagger}H(X_{t})\beta_{t}))\leq
0\text{ for all large }t$ $\displaystyle\text{ and
}\int_{0}^{\infty}1\mathrm{l}_{\\{X_{s}\in
K\\}}|\beta_{s}|_{F}^{2}\,ds<\infty\Bigr{\\}}\subset{\mathbb{L}}_{1}.$

Acknowledgement. The authors would like to thank the two anonymous referees
for their valuable comments. Funded by the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) under Germany’s Excellence Strategy EXC
2044–390685587, Mathematics Münster: Dynamics–Geometry–Structure.

## References

  * [AMA05] P.-A. Absil, R. Mahony, and B. Andrews.  Convergence of the iterates of descent methods for analytic cost functions.  SIAM J. Optim., 16(2):531–547, 2005.
  * [Ben99] M. Benaïm.  Dynamics of stochastic approximation algorithms.  In Seminaire de probabilites XXXIII, pages 1–68. Springer, 1999\.
  * [CHS87] T.-S. Chiang, C.-R. Hwang, and S. J. Sheu.  Diffusion for global optimization in $\mathbb{R}^{n}$.  SIAM Journal on Control and Optimization, 25(3):737–753, 1987.
  * [CMI14] T. Colding and W. Minicozzi II.  Lojasiewicz inequalities and applications.  In Surveys in Differential Geometry, XIX, pages 63–82. 2014.
  * [Cur44] H. B. Curry.  The method of steepest descent for non-linear minimization problems.  Quarterly of Applied Mathematics, 2(3):258–261, 1944.
  * [DK21] S. Dereich and S. Kassing.  Convergence of stochastic gradient descent schemes for Łojasiewicz-landscapes.  arXiv:2102.09385, 2021.
  * [FT21] N. Fournier and C. Tardif.  On the simulated annealing in $\mathbb{R}^{d}$.  J. Funct. Anal., 281(5):Paper No. 109086, 30, 2021.
  * [FW12] M. I. Freidlin and A. D. Wentzell.  Random Perturbations of Dynamical Systems.  Grundlehren der mathematischen Wissenschaften. Springer New York, 2012\.
  * [Gar85] C. W. Gardiner.  Handbook of stochastic methods, volume 3.  springer Berlin, 1985.
  * [GH86] S. Geman and C.-R. Hwang.  Diffusions for global optimization.  SIAM Journal on Control and Optimization, 24(5):1031–1043, 1986\.
  * [Gil00] D. T. Gillespie.  The chemical Langevin equation.  The Journal of Chemical Physics, 113(1):297–306, 2000.
  * [Har12] A. Haraux.  Some applications of the Łojasiewicz gradient inequality.  Communications on Pure & Applied Analysis, 11(6):2417, 2012.
  * [HKS89] R. A. Holley, S. Kusuoka, and D. W Stroock.  Asymptotics of the spectral gap with applications to the theory of simulated annealing.  Journal of functional analysis, 83(2):333–347, 1989.
  * [JKO98] R. Jordan, D. Kinderlehrer, and F. Otto.  The variational formulation of the fokker–planck equation.  SIAM journal on mathematical analysis, 29(1):1–17, 1998.
  * [Łoj63] S. Łojasiewicz.  Une propriété topologique des sous-ensembles analytiques réels.  Les équations aux dérivées partielles, 117:87–89, 1963\.
  * [Łoj65] S. Łojasiewicz.  Ensembles semi-analytiques.  Lectures Notes IHES (Bures-sur-Yvette), 1965.
  * [Łoj84] S. Łojasiewicz.  Sur les trajectoires du gradient d’une fonction analytique. (Trajectories of the gradient of an analytic function).  Semin. Geom., Univ. Studi Bologna, 1982/1983:115–117, 1984.
  * [LTE17] Q. Li, C. Tai, and W. E.  Stochastic modified equations and adaptive stochastic gradient algorithms.  In Proceedings of the 34th International Conference on Machine Learning, volume 70, pages 2101–2110. PMLR, 06–11 Aug 2017.
  * [PdM82] J. Palis, Jr. and W. de Melo.  Geometric theory of dynamical systems: an introduction.  Springer Science & Business Media, 1982.
  * [SSG14] D. Schnoerr, G. Sanguinetti, and R. Grima.  The complex chemical Langevin equation.  The Journal of Chemical Physics, 141(2):07B606_1, 2014.
  * [vK92] N. G. van Kampen.  Stochastic processes in physics and chemistry, volume 1.  Elsevier, 1992.