text:
# Stability and moment estimates for the stochastic singular $\Phi$-Laplace
equation

Florian Seib  ,  Wilhelm Stannat  Technische Universität Berlin (MA 7-5)
Institut für Mathematik
Straße des 17. Juni 136
10623 Berlin
Germany<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Jonas M. Tölle
Aalto University
Department of Mathematics and Systems Analysis
PO Box 11100 (Otakaari 1, Espoo)
00076 Aalto
Finland<EMAIL_ADDRESS>
(Date: August 27, 2024)

###### Abstract.

We study stability, long-time behavior and moment estimates for stochastic
evolution equations with additive Wiener noise and with singular drift given
by a divergence type quasilinear diffusion operator which may not necessarily
exhibit a homogeneous diffusivity. Our results cover the singular stochastic
$p$-Laplace equations and, more generally, singular stochastic $\Phi$-Laplace
equations with zero Dirichlet boundary conditions. We obtain improved moment
estimates and quantitative convergence rates of the ergodic semigroup to the
unique invariant measure, classified in a systematic way according to the
degree of local degeneracy of the potential at the origin. We obtain new
concentration results for the invariant measure and establish maximal
dissipativity of the associated Kolmogorov operator. In particular, we recover
the results for the curve shortening flow in the plane by Es-Sarhir, von
Renesse and Stannat, NoDEA 16(9), 2012, and improve the results by Liu and
Tölle, ECP 16, 2011.

###### Key words and phrases:

Singular drift stochastic partial differential equations, stochastic
$p$-Laplace equation, stochastic non-homogeneous $\Phi$-Laplace equation,
long-time behavior of solutions, ergodic semigroup, maximal dissipative
Kolmogorov operator.

###### 2010 Mathematics Subject Classification:

Primary: 35K67, 60H15; Secondary: 37A25, 37L40, 47D07

##  1\. Introduction

The study of long-time behavior for Markovian dynamics associated to
stochastic evolution equations has a long history. Its significance to
mathematical physics is emphasized in the equations of Fokker and Planck, and
Kolmogorov [34], which are discussed in detail in the monographs [12, 48].
Extended to stochastic partial differential equations (SPDEs), the analysis of
invariant probability distributions and Kolmogorov operators in infinite
dimensions [17, 18] provides improved insight into the averaged dynamics of
PDEs perturbed by random noise with infinitely many frequency modes [19].

We are interested in stability and moment estimates for nonlinear SPDEs of the
following type

(1.1) $\partial_{t}X_{t}=\operatorname{div}[\phi(\nabla
X_{t})]+B\,\partial_{t}W_{t},\quad X_{t}|_{\partial{\mathcal{O}}}=0,\quad
t>0,\quad X_{0}=u_{0},$

where $u_{0}\in L^{2}({\mathcal{O}})$ for a bounded, open and convex domain
${\mathcal{O}}\subset\mathbb{R}^{d}$ with piecewise $C^{2}$-boundary, $d\geq
1$. We require that the solution $X_{t}$, $t>0$ vanishes on the boundary
$\partial{\mathcal{O}}$, that is, we impose zero Dirichlet boundary
conditions. Here, $B$ is a linear operator that is regularizing the spatial
component and $\\{W_{t}\\}_{t\geq 0}$ a cylindrical Wiener process on some
filtered probability space $(\Omega,\mathcal{F},\\{\mathcal{F}_{t}\\}_{t\geq
0},\mathbb{P})$ with values in some separable Hilbert space $U$. The
nonlinearity $\phi:\mathbb{R}^{d}\to\mathbb{R}^{d}$ is monotone in the sense
that

(1.2) $\langle\phi(x)-\phi(y),x-y\rangle\geq 0,\quad x,y\in\mathbb{R}^{d}.$

Existence and uniqueness of solutions to SPDEs with nonlinear monotone drift
were first systematically studied in [36, 47].

In this work, we shall study the singular drift case, including examples where
the Jacobian of $\phi$ has a singularity in zero. We assume that $\phi$ has
sublinear growth, i.e. for some $C>0$,

(1.3) $|\phi(x)|\leq C(1+|x|).$

We refer to [29, 40, 51] for a well-posedness theory for such equations.
Examples of such $\phi$ that satisfy our main Assumption 1.1, include, among
others, the following ones.

#### Stochastic singular $p$-Laplace

For $p\in(1,2)$, let $\phi(x)=|x|^{p-2}x$ and yield the stochastic singular
$p$-Laplace equation. It has been first addressed in [38]. The results have
been extended in [40, 29, 31, 28]. The multi-valued stochastic $1$-Laplace
equation, which is also known as the _total variation flow_ is not within the
scope of this work, instead, we refer to [7, 28, 29, 30, 31]. The
$L^{1}$-theory of the stochastic $p$-Laplace equation has been discussed in
[52].

#### Stochastic non-Newtonian fluids

For $p\in(1,2)$, let $\phi(x)=\left(1+|x|^{2}\right)^{\frac{p-2}{2}}x$, which
yields a simplified model for stochastic singular power law fluids without
convection term. It exhibits a similar nonlinearity as in an equation modeling
non-Newtonian fluids with stochastic perturbation in the shear-thinning case,
see [13, 53, 8].

#### Stochastic curve shortening flow

In one spatial dimension $d=1$, we may formally differentiate
$\phi(x)=\arctan(x)$ in order to obtain the so-called (additive noise)
_stochastic curve shortening flow_

$\partial_{t}X_{t}=\frac{\partial_{xx}X_{t}}{1+(\partial_{x}X_{t})^{2}}+B\,\partial_{t}W_{t},\quad
t>0,\quad X_{0}=u_{0}.$

Which is a formulation of the stochastic mean curvature flow in the plane, see
[25, 28]. It has been generalized toward two spatial dimensions in [32].

#### Further examples

Furthermore, we have the minimal surface flow
$\phi(x)=\frac{x}{\sqrt{1+|x|^{2}}}$, which is related to the mean curvature
flow, see [27, 10], and the logarithmic diffusion
$\phi(x)=\log(1+|x|)\frac{x}{|x|}$, as addressed e.g. in [29].

The main problem with these kinds of nonlinear diffusivities remains the lack
of coercivity and dissipativity in the general situation. This may be achieved
by Sobolev embeddings in certain cases [39], however, energy methods as e.g.
stochastic variational inequalities [31, 28, 54, 7] seem to deal with well-
posedness issues better without loosing for instance the Markov property of
the semigroup [30].

Other variational approaches to divergence type drift SPDEs can be found in
[44, 43, 39, 7, 40]. The case of gradient type Stratonovich noise is addressed
in [16, 57, 3, 4].

### Long-time behavior

Even in the case of analytically weak variational solutions, we may define a
_Feller semigroup_ $\\{P_{t}\\}_{t\geq 0}$ associated to the dynamics of (1.1)
via extension of

$P_{t}f(u):=\mathbb{E}[f(X_{t}^{u})],\quad
f\in\operatorname{Lip}_{b}(L^{2}({\mathcal{O}})),\quad t\geq 0,$

where $\\{X_{t}^{u}\\}_{t\geq 0}$ denotes a solution to (1.1) with $X_{0}=u$,
see [18]. We call a Borel probability measure $\mu$ on $L^{2}({\mathcal{O}})$
_invariant_ with respect to $\\{P_{t}\\}_{t\geq 0}$ if

$\int_{L^{2}({\mathcal{O}})}P_{t}f\,d\mu=\int_{L^{2}({\mathcal{O}})}f\,d\mu$

for every $t\geq 0$ and every $f\in C_{b}(L^{2}({\mathcal{O}}))$. We refer to
[56] for an overview article on this topic. Uniqueness of invariant measures
for singular stochastic equations in low spatial dimensions was first proved
with methods from [35] for the stochastic curve shortening flow in [25]. These
results were then extended to the stochastic singular $p$-Laplace equation in
[41] and to more general equations in [29, 46]. In particular, for additive
noise, the decay behavior of deterministic solutions (that is, for $B\equiv
0$) plays an important role, see [49, 50, 20] for results of this type. We
would like to point out that our result is novel even for the deterministic
case. Second order estimates for the deterministic parabolic $p$-Laplace
system have been obtained in [14, 15]. Ergodicity of the semigroup and
uniqueness of invariant measures for local and nonlocal singular $p$-Laplace
equations in all spatial dimensions was proved in [30].

We are also interested in improved moment estimates for singular monotone
drift SPDEs which yield more explicit convergence rates in the spirit of [24,
26, 41]. To this end, we may assume that the singularity is of a certain
polynomial type, more precisely, we shall assume that $\phi$ has a radially
symmetric potential, that is, $\phi:\mathbb{R}^{d}\to\mathbb{R}^{d}$ is of the
form

(1.4) $\phi(z)=\begin{cases}\psi(|z|)\frac{z}{|z|},&\text{if}\quad z\not=0,\\\
0,&\text{if}\quad z=0,\end{cases}$

where $\psi:\mathbb{R}\to\mathbb{R}$ satisfies the following

######  Assumption 1.1.

Suppose that $\psi\in W_{\operatorname{loc}}^{1,1}(\mathbb{R})$ such that
there exists $C>0$ with

  1. (C1)

$\psi^{\prime}(r)>0$ for every $r>0$,

  2. (C2)

$\psi(r)=-\psi(-r)$ for every $r\in\mathbb{R}$,

  3. (C3)

$\psi(r)\leq C(1+|r|)$ for every $r\geq 0$,

  4. (C4)

there exist $a\geq 0$, $b>0$, $s\in(0,2]$ with

$(a+b|r|^{s})|\psi^{\prime}(r)|\geq 1$

for every $r\geq 0$,

  5. (C5)

there exist $c>0$ and $K\geq 0$ such that

$\psi(r)\cdot r\geq c|r|-K,\quad r\geq 0.$

  6. (C6)

$0<s\leq\frac{4}{d}\wedge 2,$

We are ready to formulate our main result, see Theorems 4.3 and 5.1, which
generalizes the results of [24, 26, 41, 29], which assume the more restrictive
assumption $s<\frac{4}{d+2}\wedge 1$ compared to (C6).

######  Theorem 1.2.

Assume (C1)–(C6) and that111We denote the _space of Hilbert-Schmidt operators_
from the separable Hilbert space $U$ to a separable Hilbert space $S$ by
$L_{2}(U,S)$. $B\in L_{2}(U,H_{0}^{1}({\mathcal{O}}))$. Set

$s^{\ast}=(2-s)\vee\frac{4-s}{2+s}.$

  1. (i)

Then there exists a unique invariant Borel probability measure $\mu$ of
$\\{P_{t}\\}_{t\geq 0}$ that is concentrated on the subset
$W^{2,\alpha}_{0}({\mathcal{O}})\cap H^{1}_{0}({\mathcal{O}})\subset
L^{2}({\mathcal{O}})$ for any $1<\alpha\leq d\frac{2-s}{d-s}$ if $d\geq 2$ and
$\alpha=1$ if $d=1$ with

$\int_{L^{2}({\mathcal{O}})}\|u\|_{W_{0}^{2,\alpha}({\mathcal{O}})}^{s^{\ast}}\,\mu(du)+\int_{L^{2}({\mathcal{O}})}\lVert
u\rVert_{H_{0}^{1}({\mathcal{O}})}^{s^{\ast}}\,\mu(du)<+\infty.$

  2. (ii)

Let $d_{0}:=1\vee\frac{d}{2}$ and

$0<\beta\leq\beta^{\star}:=1\wedge\frac{8-2s}{s(2+s)d_{0}}\in\left[\frac{1}{2d_{0}},1\right].$

Then there exist non-negative constants $C_{1}$, $C_{2}$ and $C_{3}$ such that

$\limsup_{t\to+\infty}\left[t^{\frac{s^{\ast}}{s}}\frac{\lvert
P_{t}f(u)-P_{t}f(v)\rvert}{\lVert
u-v\rVert_{L^{2}({\mathcal{O}})}^{\beta}}\right]\leq
C_{1}|f|_{\beta}\left(C_{2}+C_{3}\int_{L^{2}({\mathcal{O}})}\lVert
u\rVert_{H_{0}^{1}({\mathcal{O}})}^{s^{\ast}}\,\mu(du)\right)$

for every $u,v\in H_{0}^{1}({\mathcal{O}})$ and for every
$f:L^{2}({\mathcal{O}})\to\mathbb{R}$ that is bounded and $\beta$-Hölder-
continuous222$f:L^{2}({\mathcal{O}})\to\mathbb{R}$ is called _$\beta$ -Hölder-
continuous_ if $\sup_{u\not=v}\frac{\lvert f(u)-f(v)\rvert}{\lVert
u-v\rVert_{L^{2}({\mathcal{O}})}^{\beta}}=:\lvert f\rvert_{\beta}<+\infty$..

Finally, we shall address the maximal dissipativity of the associated
Kolmogorov operator with infinitely many variables, see [55, 6, 17, 23, 26,
56, 11, 45] for related works.

We note that the degenerate $p$-Laplace, i.e. the case $p>2$, may be treated
more easily with the Krylov-Bogoliubov method [18], as is done in the
variational setup in [5]. Previous approaches include dimension-free Harnack
inequalities [37, 33, 42] and the more direct approach via irreducibility
[58]. In the case of multiplicative noise, existence of invariant measures for
monotone drift equations has been proved in [21].

### Structure of the paper

We start with collecting some preliminary results in Section 2 including a
known well-posedness result for (1.1) and results on invariant measures of
Markovian Feller semigroups. We conclude Section 2 with a discussion of our
assumptions and a collection of our main examples for $\phi$. In Section 3, as
one the main results, we shall prove a stability result for the semigroup, see
Proposition 3.3. In Section 4, we shall first record a result for the case of
one spatial dimension, see Lemma 4.1. Next, we prove the improved moment
estimates and a concentration result for the unique invariant measure, see
Theorem 4.3, which implies our main result Theorem 1.2 (i). In Section 5, the
main result Theorem 1.2 (ii) is proved and applied to the main examples, see
Theorem 5.1 and the subsequent examples. In Section 6, under an additional
assumption, we prove the maximal dissipativity of the Kolmogorov operator
associated to the semigroup, see Theorem 6.4.

##  2\. Preliminary results

###  2.1. Existence and uniqueness of solutions

Let us recall the following conditions from [29], simplified with regard to
the time-dependence of the drift coefficients, which is not needed here. The
results are adapted to the Gelfand triple333Let $H$ be a separable Hilbert
space and let $V$ be another Hilbert space (with topological dual $V^{\ast}$)
which is embedded densely and continuously into $H$. Then, $H$ is identified
with its dual via the Riesz isometry. The triple $V\subset H\subset V^{\ast}$
is called _Gelfand triple_.

(2.1) $V:=H_{0}^{1}({\mathcal{O}})\subset H:=L^{2}({\mathcal{O}})\subset
V^{\ast}=H^{-1}({\mathcal{O}})$

that also encodes the zero Dirichlet boundary condition. Here, we use the
shorthand notation $H_{0}^{1}({\mathcal{O}})=W^{1,2}_{0}({\mathcal{O}})$ and
the notation $H^{-1}({\mathcal{O}})$ for the topological dual space of
$H_{0}^{1}({\mathcal{O}})$.

Suppose that $A:V\to V^{\ast}$ satisfies the following conditions: There
exists a constant $C>0$ such that

  1. (A1)

The map $u\mapsto A(u)$ is maximal monotone444That is, the graph of $A$ is
maximal in the class of monotone graphs, ordered by set-inclusion, see [2] for
details..

  2. (A2)

For all $u\in V$:

$\|A(u)\|_{V^{\ast}}\leq C\|u\|_{V}.$

  3. (A3)

For all $u\in V$, and for all $n\in\mathbb{N}$:

$2\sideset{{}_{V^{\ast}}}{{}_{V}}{\mathop{\langle{A(u)},{T_{n}(u)}\rangle}}\geq-C\|u\|_{V}^{2},$

such that $C$ is independent of $n$, where
$T_{n}:=n\left(\operatorname{Id}-(\operatorname{Id}-\frac{\Delta}{n})^{-1}\right)$
denotes the Yosida-approximation of the negative Dirichlet Laplace operator
$-\Delta=-\sum_{i=1}^{d}\partial_{x_{i}}\partial_{x_{i}}$.

Let $U$ be a separable Hilbert space. For a separable Hilbert space $S$,
denote the _space of Hilbert-Schmidt operators_ from $U$ to $S$ by
$L_{2}(U,S)$. Denote by $\\{W_{t}\\}_{t\geq 0}$ a cylindrical Wiener
process555See [19] for the notion of a cylindrical Wiener process in Hilbert
space. in $U$ for a stochastic basis
$(\Omega,\mathcal{F},\\{\mathcal{F}_{t}\\}_{t\geq 0},\mathbb{P})$ that
satisfies the standard assumptions.

######  Assumption 2.1.

Assume that

  1. (B1)

$B\in L_{2}(U,V)$.

######  Definition 2.2.

We say that a continuous $\\{\mathcal{F}_{t}\\}_{t\geq 0}$-adapted stochastic
process $X:[0,T]\times\Omega\to H$ is a _solution_ to

(2.2) $dX_{t}+A(X_{t})\,dt=B\,dW_{t},\quad X_{0}=u_{0},$

if $X\in L^{2}(\Omega;C([0,T];H))\cap L^{2}([0,T]\times\Omega;V)$ and solves
the following integral equation in $V^{\ast}$

$X_{t}=u_{0}-\int_{0}^{t}A(X_{\tau})\,d\tau+B\,W_{t},$

$\mathbb{P}$-a.s. for all $t\in[0,T]$.

######  Theorem 2.3.

Suppose that conditions (A1)–(A3), (B1) hold. Let $u_{0}\in
L^{2}(\Omega,\mathcal{F}_{0},\mathbb{P};V)$. Then there exists a unique
solution in the sense of the previous definition to the equation

(2.3) $dX_{t}+A(X_{t})\,dt=B\,dW_{t},\quad X_{0}=u_{0},$

that satisfies

$\mathbb{E}\left[\sup_{t\in[0,T]}\|X_{t}\|_{V}^{2}\right]<+\infty.$

###### Proof.

See [29, Theorem 4.4]. ∎

######  Definition 2.4.

An $\\{\mathcal{F}_{t}\\}_{t\geq 0}$-adapted stochastic process $X\in
L^{2}(\Omega;C([0,T];H))$ is called a _generalized solution_ 666Also referred
to as _limit solution_.__ to (2.2) with starting point $u_{0}\in H$ if for all
approximations $u_{0}^{m}\in V$, $m\in\mathbb{N}$ with
$\|u_{0}^{m}-u_{0}\|_{H}\to 0$ as $m\to+\infty$ and all $B_{m}\in L_{2}(U,V)$
such that $B_{m}\to B\in L_{2}(U,H)$ strongly in $L_{2}(U,H)$ as
$m\to+\infty$, we have that

$X^{m}\to X\quad\text{strongly
in$\quad$$L^{2}(\Omega;C([0,T];H))$$\quad$as$\quad$$m\to+\infty$.}$

######  Theorem 2.5.

Suppose that conditions (A1)–(A3) hold and that $B\in L_{2}(U,H)$. Let
$u_{0}\in L^{2}(\Omega,\mathcal{F}_{0},\mathbb{P};H)$. Then there exists a
unique generalized solution in the sense of the previous definition to the
equation

(2.4) $dX_{t}+A(X_{t})\,dt=B\,dW_{t},\quad X_{0}=u_{0}.$

###### Proof.

See [29, Theorem 4.6]. ∎

On a bounded, open and convex domain ${\mathcal{O}}\subset\mathbb{R}^{d}$ with
piecewise $C^{2}$-boundary, we shall consider the $\Phi$-Laplace SPDE

(2.5) $dX_{t}=\operatorname{div}[\phi(\nabla X_{t})]\,dt+B\,dW_{t},\quad
X_{0}=u_{0},$

with zero Dirichlet boundary conditions for $t\in[0,T]$ and $u_{0}\in
H:=L^{2}({\mathcal{O}})$, where $\phi$ is of the form (1.4). We sometimes use
the notation

$A(u):=-\operatorname{div}[\phi(\nabla u)],$

which is made rigorous in the compact Gelfand triple (2.1). Indeed,

(2.6)
$\sideset{{}_{V^{\ast}}}{{}_{V}}{\mathop{\langle{A(u)},{v}\rangle}}=\int_{\mathcal{O}}\langle\phi(\nabla
u),\nabla v\rangle\,dx,\quad u,v\in V.$

######  Lemma 2.6.

Suppose that $\phi$ satisfies (C1)–(C3). Then $A$ satisfies conditions
(A1)–(A3).

###### Proof.

Set

$\Psi(x):=\int_{0}^{|x|}\psi(r)\,dr,$

which is a radially symmetric, convex, continuous and convex function with
$\Psi(0)=0$ and at most subquadratic growth. We have the representation

$A(u)=\nabla^{\ast}\circ\partial\left(\int_{{\mathcal{O}}}\Psi(\cdot)\,dx\right)\circ\nabla
u,\quad u\in V,$

where $\partial$ denotes the subgradient and
$\nabla^{\ast}=(-\Delta)^{-1}\circ\operatorname{div}$ denotes the adjoint
operator of $\nabla:V=H_{0}^{1}({\mathcal{O}})\to
L^{2}({\mathcal{O}};\mathbb{R}^{d})$. We may apply the results from [29,
Proposition 7.1 and Section 7.2.1] to yield the claim. See, in particular,
[29, Example 7.9]. We note that we use the convexity assumption on the
boundary here, cf. [29, Proposition D.2] and [7, Proposition 8.2]. ∎

Applying Theorem 2.5, we obtain the following existence and uniqueness result.

######  Corollary 2.7.

Suppose that $\phi$ satisfies (C1)–(C3), (C5) and that $B$ satisfies (B1).
Then there exists a unique generalized solution to (2.5) in the sense of
Definition 2.4.

###  2.2. Feller semigroups and invariant measures

Following [18], we set

(2.7) $P_{t}f(u):=\mathbb{E}[f(X_{t}^{u})],\quad
f\in\operatorname{Lip}_{b}(H),\quad t\geq 0,$

where $\\{X_{t}^{u}\\}_{t\geq 0}$ denotes a (generalized) solution to (2.5)
with initial datum $X_{0}=u$. By the results in [29, Section 6.4], the
extension of $\\{P_{t}\\}_{t\geq 0}$ to $C_{b}(H)$ defines a Markovian
transition semigroup. It can easily be seen that $\\{P_{t}\\}_{t\geq 0}$ is
stochastically continuous, that is,

$\lim_{t\to 0+}P_{t}f(u)=f(u),\quad\text{for
all}\;\;f\in\operatorname{Lip}_{b}(H),\;\;u\in H,$

cf. [18, Proposition 2.1.1], and satisfies the Feller property, that is, for
any $f\in C_{b}(H)$ and $t\geq 0$ one has $P_{t}f\in C_{b}(H)$. Denote its
dual semigroup restricted to finite Borel measures by
$\\{P^{\ast}_{t}\\}_{t\geq 0}$. A probability measure $\mu$ is called
_invariant_ for $\\{P_{t}\\}_{t\geq 0}$ if $P_{t}^{\ast}\mu=\mu$ for any
$t\geq 0$.

We recall the following concepts defined e.g. in [35].

######  Definition 2.8.

We say that $\\{P_{t}\\}_{t\geq 0}$ is _weak- $\ast$-mean ergodic_ if there
exists a Borel probability measure $\mu_{\ast}$ on ${\mathcal{B}}(H)$, that is
the Borel sets of $H$, such that

(2.8)
$\text{w-}\lim_{T\to+\infty}\frac{1}{T}\int_{0}^{T}P_{t}^{\ast}\nu\,dt=\mu_{\ast}\quad\text{for
every Borel probability measure $\nu$ on ${\mathcal{B}}(H)$,}$

where the limit is in the sense of weak convergence of probability measures.

We say that the _weak law of large numbers_ holds for $\\{P_{t}\\}_{t\geq 0}$,
for a function $f\in\operatorname{Lip}_{b}(H)$ and for a probability measure
$\nu$ on ${\mathcal{B}}(H)$ if

$\mathbb{P}_{\nu}\text{-}\lim_{T\to+\infty}\frac{1}{T}\int_{0}^{T}f(X(t))\,dt=\int_{H}f\,d\mu_{\ast},$

where $\mu_{\ast}$ denotes the invariant measure of $\\{P_{t}\\}_{t\geq 0}$
and $\\{X(t)\\}_{t\geq 0}$ denotes the Markov process related to
$\\{P_{t}\\}_{t\geq 0}$ whose initial distribution is $\nu$ and whose path
measure is $\mathbb{P}_{\nu}$, and where the convergence takes place in
$\mathbb{P}_{\nu}$-probability.

As noted in [35, Remark 3], (2.8) implies uniqueness of the invariant measure.

######  Theorem 2.9.

Suppose that $\phi$ satisfies (C1)–(C5) and that

(2.9) $s<\frac{4}{d+2}\wedge 1.$

Then there exists a unique invariant measure $\mu$ for the semigroup
$\\{P_{t}\\}_{t\geq 0}$ such that $\\{P_{t}\\}_{t\geq 0}$ is weak-$\ast$-mean
ergodic and the weak law of large numbers holds for any
$f\in\operatorname{Lip}_{b}(H)$ and any probability measure $\nu$.

###### Proof.

We would like to use [29, Remark 5.5 and Proposition 7.1] to see that the
hypotheses of [29, Theorem 5.6] are satisfied. For that we only have to verify
that our equation admits a Lyapunov function with compact sublevel sets and
that the solution to the deterministic counterpart of our equation (that is,
$B\equiv 0$) vanishes for $t\to+\infty$.

By Lemma 2.10, we get for $u\in V$,

(2.10)
$2\sideset{{}_{V^{\ast}}}{{}_{V}}{\mathop{\langle{A(u)},{u}\rangle}}=2\int_{\mathcal{O}}\langle\phi(\nabla
u),\nabla u\rangle\,dx\geq 2\int_{{\mathcal{O}}}|\nabla
u|^{2-s}\,dx-2K|{\mathcal{O}}|=:\Theta(u),$

where we denote $|{\mathcal{O}}|:=\int_{{\mathcal{O}}}\,dx$. By (2.9) and the
Rellich-Kodrachov theorem, $W_{0}^{1,2-s}({\mathcal{O}})\subset
L^{2}({\mathcal{O}})$ compactly, thus $\Theta$ is a Lyapunov function with
compact sublevels. Let $(v_{t})_{t\geq 0}$ be a solution to the deterministic
equation,

$dv_{t}+A(v_{t})\,dt=0,\quad t>0,\quad v_{0}=u.$

It remains to prove that

$\lim_{t\to+\infty}\|v_{t}\|_{H}=0$

for every initial datum $u\in H$. By the chain rule,

$\frac{d}{dt}\|v_{t}\|_{H}^{2}\leq-2C\left(\|v_{t}\|_{H}^{2}\right)^{\frac{2-s}{2}}+2K|{\mathcal{O}}|.$

In analogy to [29, Proof of Remark 5.5], we see that
$f(t):=\|v_{t}\|_{H}^{2}e^{-2K|{\mathcal{O}}|t}$ is a subsolution to the
ordinary differential equation

$f^{{}^{\prime}}(t)=-2Cf(t)^{\frac{2-s}{2}}.$

Hence

$\|v_{t}\|_{H}^{2}\leq
e^{2K|{\mathcal{O}}|t}\left(\left(\|u\|_{H}^{s}-Cst\right)\vee
0\right)^{\frac{2}{s}}\longrightarrow 0,$

as $t\to+\infty$. Now, we may apply [29, Remark 5.5 and Proposition 7.1] and
see that the hypotheses for [29, Theorem 5.6] are satisfied. Thus the claimed
result follows. ∎

The above result is improved in Theorems 4.3 and 3.3 below, as we just need to
assume (C6) instead of (2.9). We refer to [9] for a new unified two-step
approach to proving the existence of invariant measures.

###  2.3. Discussion of the assumptions

If $s\in(0,1]$, (C5) follows readily from the other assumptions.

######  Lemma 2.10.

Assumptions (C1)–(C4), together with the assumption that (C4) holds with
$s\in(0,1]$, imply that there exist constants $c=c(a,b)>0$ and
$K=K(a,b,s,C)\geq 0$ such that

$\psi(r)\cdot r\geq c|r|^{2-s}-K,\quad r\geq 0.$

In particular, then (C5) follows.

###### Proof.

Let $s\in(0,1]$. Integrating $(a+b|r|^{s})|\psi^{\prime}(r)|\geq 1$ by parts
over $[0,x]$, $x\geq 0$ yields

$(a+b|x|^{s})\psi(x)\geq x+bs\int_{0}^{x}\psi(r)|r|^{s-1}\,dr\geq x.$

Hence,

$a|x|\psi(x)+a\psi(x)+b|x|\psi(x)\geq
a|x|^{1-s}\psi(x)+b|x|\psi(x)\geq|x|^{2-s}.$

Using (C3) and rearranging terms yields

$K(a,s,C)+\frac{1}{2}|x|^{2-s}+(a+b)|x|\psi(x)\geq|x|^{2-s},$

which proves the claim for $s\in(0,1]$. ∎

In Section 5, we shall discuss the following examples.

######  Example 2.11.

The following examples satisfy Assumptions (C1)–(C5).

Singular $p$-Laplace:

Let $p\in(1,2)$. Then $\psi(r)=|r|^{p-2}r$ satisfies (C1)–(C5) with $C=c=K=1$,
$a=0$, $b=(p-1)^{-1}$, $s=2-p$.

Non-Newtonian fluids:

Let $p\in(1,2)$. Then $\psi(r)=(1+|r|^{2})^{\frac{p-2}{2}}r$ satisfies
(C1)–(C5) with $C=K=c=1$, $a=b=(p-1)^{-1}$, $s=2-p$.

$\log$-diffusion:

Let $\psi(r)=\log(1+|r|)\operatorname{sgn}(r)$. Then $\psi$ satisfies
(C1)–(C5) with $C=a=b=s=c=1$, $K=2$.

Minimal surface flow:

Let $\psi(r)=\frac{r}{\sqrt{1+|r|^{2}}}$. Then $\psi$ satisfies (C1)–(C5) with
$C=a=b=s=c=K=1$.

Curve shortening flow:

Let $\psi(r)=\arctan(r)$. Then $\psi$ satisfies (C1)–(C5) with $C=a=b=c=K=1$,
$s=2$.

######  Remark 2.12.

Note that for the first two examples, (2.9) translates to $p>\frac{2d}{d+2}$,
and $d<\frac{2p}{2-p}$ respectively, which was assumed e.g. in [41]. However,
for the first two examples, we just need to assume condition (C6) which is
$p\geq 2-\frac{4}{d}$, and $d\leq\frac{4}{2-p}$ respectively.

##  3\. Stability

In this section, stability, that is, rates of convergence for large times for
the solutions to (2.5) starting at two distinct initial data will be
established. Unless otherwise stated, we assume that conditions (C1)–(C6)
hold. On a bounded, open and convex domain
${\mathcal{O}}\subset\mathbb{R}^{d}$ with piecewise $C^{2}$-boundary, we shall
consider the $\Phi$-Laplace SPDE (2.5) such that $B$ satisfies (B1).

Let us first record a lemma for the situation that $d=1$. Let
${\mathcal{O}}=(0,L)\subset\mathbb{R}$, for some $L>0$. Set
$I:=\overline{{\mathcal{O}}}=[0,L]$. Let $\Delta=\partial_{xx}$ be the
Dirichlet Laplace on ${\mathcal{O}}$.

######  Lemma 3.1.

Let $u,v\in V=H_{0}^{1}(I)$. For the line segment $\gamma:[0,1]\to
L^{2}(0,L),\ \lambda\mapsto\gamma(\lambda,u,v)$, where
$\gamma(\lambda,u,v)(x):=\partial_{x}u(x)+\lambda(\partial_{x}v(x)-\partial_{x}u(x))$,
we have that

(3.1) ${}_{H_{\phantom{0}}^{-1}}\langle
A(u)-A(v),u-v\rangle_{H_{0}^{1}}\geq\frac{\lVert
u-v\rVert_{H}^{2}}{L}\int_{0}^{1}\left(\int_{0}^{L}\left(\psi^{\prime}(\gamma(\lambda,u,v)(x))\right)^{-1}\,dx\right)^{-1}\,d\lambda.$

###### Proof.

First note that (C4) guarantees that the expression on the RHS of (3.1) is
almost surely less or equal zero as for $d\lambda$-a.e. $\lambda\in[0,1]$,

$\int_{0}^{L}\left(\psi^{\prime}(\gamma(\lambda,u,v)(x))\right)^{-1}\,dx\leq
2^{(s-1)\vee
0}\int_{0}^{L}\left(a+b|\partial_{x}u(x)|^{s}+b|\partial_{x}v(x)|^{s}\right)\,dx<+\infty.$

We find the identity

${}_{H_{\phantom{0}}^{-1}}\langle
A(u)-A(v),u-v\rangle_{H_{0}^{1}}=\int_{0}^{L}\left(\phi(\partial_{x}u(x))-\phi(\partial_{x}v(x))\right)(\partial_{x}u(x)-\partial_{x}v(x))\,dx$
$\displaystyle=$
$\displaystyle-\int_{0}^{L}\int_{0}^{1}\frac{d}{d\lambda}\phi(\gamma(\lambda,u,v)(x))(\partial_{x}u(x)-\partial_{x}v(x))\,d\lambda\,dx$
$\displaystyle=$
$\displaystyle\int_{0}^{L}\int_{0}^{1}\psi^{\prime}(\gamma(\lambda,u,v)(x))(\partial_{x}u(x)-\partial_{x}v(x))^{2}\,d\lambda\,dx$

Let $\tilde{u},\tilde{v}$ denote continuous representatives of $u,v\in
H_{0}^{1}(0,L)$. Note that $\tilde{u}(0)=\tilde{v}(0)=0$. Using the
fundamental theorem of calculus and Hölder inequality yields

$\displaystyle\left|\tilde{u}(x)-\tilde{v}(x)\right|^{2}=\left(\int_{0}^{x}(\partial_{y}u(y)-\partial_{y}v(y))\,dy\right)^{2}$
$\displaystyle\leq$
$\displaystyle\left(\int_{0}^{L}\psi^{\prime}(\gamma(\lambda,u,v)(x))(\partial_{x}u(x)-\partial_{x}v(x))^{2}\,dx\right)\times\left(\int_{0}^{L}\left(\psi^{\prime}(\gamma(\lambda,u,v)(x))\right)^{-1}\,dx\right).$

Hence,

$\displaystyle\frac{1}{L}\int_{0}^{L}\left|u(x)-v(x)\right|\,dx$
$\displaystyle\leq$
$\displaystyle\left(\int_{0}^{L}\psi^{\prime}(\gamma(\lambda,u,v)(x))(\partial_{x}u(x)-\partial_{x}v(x))^{2}\,dx\right)\times\left(\int_{0}^{L}\left(\psi^{\prime}(\gamma(\lambda,u,v)(x))\right)^{-1}\,dx\right).$

By dividing and integrating with respect to $\lambda$, we achieve

$\begin{gathered}\frac{\lVert
u-v\rVert_{H}^{2}}{L}\int_{0}^{1}\left(\int_{0}^{L}\left(\psi^{\prime}(\gamma(\lambda,u,v)(x))\right)^{-1}\,dx\right)^{-1}\,d\lambda\\\
\leq\int_{0}^{1}\int_{0}^{L}\psi^{\prime}(\gamma(\lambda,u,v)(x))(\partial_{x}u(x)-\partial_{x}v(x))^{2}\,dx\,d\lambda.\end{gathered}$

∎

Now, consider the general case that $d\geq 1$. Let
${\mathcal{O}}\subset\mathbb{R}^{d}$ be a bounded, open and convex domain with
piecewise $C^{2}$-boundary. We denote
$|{\mathcal{O}}|:=\int_{{\mathcal{O}}}\,dx$. We define
$\Psi:\mathbb{R}^{d}\to\mathbb{R}$ by

(3.2) $\Psi(x):=\int_{0}^{\left\lvert x\right\rvert}\psi(r)\,dr.$

We see that $\Psi$ is a radially symmetric convex function of at most
quadratic growth, and its first and second derivatives are given by

$D\Psi(x)=\phi(x)=\begin{cases}0,&\text{if }x=0,\\\ \psi(\left\lvert
x\right\rvert)\frac{x}{\left\lvert x\right\rvert},&\text{if }x\neq
0,\end{cases}$

and by

$(D^{2}\Psi(x))_{(i,j)}=\frac{\psi(\left\lvert x\right\rvert)}{\left\lvert
x\right\rvert}\delta_{ij}+\left(\psi^{\prime}(\left\lvert
x\right\rvert)-\frac{\psi(\left\lvert x\right\rvert)}{\left\lvert
x\right\rvert}\right)\frac{x_{i}x_{j}}{\left\lvert x\right\rvert^{2}},$

with special case $(D^{2}\Psi(x))_{(1,1)}=\psi^{\prime}(|x|)$ for $d=1$.

Using the Gram matrix, one obtains that the eigenvalues of $D^{2}\Psi(x)$ are
$\psi^{\prime}(\left\lvert x\right\rvert)$ and $\frac{\psi(\left\lvert
x\right\rvert)}{\left\lvert x\right\rvert}$ with respective multiplicities $1$
and $d-1$. In particular,

(3.3) $\langle D^{2}\Psi(x)h,h\rangle\geq\psi^{\prime}(\left\lvert
x\right\rvert)\wedge\left(\frac{\psi(\left\lvert x\right\rvert)}{\left\lvert
x\right\rvert}\right)|h|^{2}\quad\text{for every}\quad h\in\mathbb{R}^{d}.$

As above, for $u,v\in V,$ the operator $A(u)$ can be defined via

$\sideset{{}_{V^{\ast}}}{{}_{V}}{\mathop{\langle{A(u)},{v}\rangle}}=\int_{\mathcal{O}}\langle\phi(\nabla
u),\nabla v\rangle\,dx.$

For $u\in C^{2}(\mathcal{O})$ we have the representation

(3.4) $A(u)=-\frac{\psi(\left\lvert\nabla u\right\rvert)}{\left\lvert\nabla
u\right\rvert}\Delta u-\left(\psi^{\prime}(\left\lvert\nabla
u\right\rvert)-\frac{\psi(\left\lvert\nabla u\right\rvert)}{\left\lvert\nabla
u\right\rvert}\right)\sum_{i,j=1}^{d}\frac{\partial_{x_{i}}\partial_{x_{j}}u\partial_{x_{i}}u\partial_{x_{j}}u}{|\nabla
u|^{2}}.$

Also, on a formal level,

$\displaystyle\langle A(u),v\rangle_{V}=$
$\displaystyle-\sum_{i=1}^{d}\int_{\mathcal{{\mathcal{O}}}}\partial_{x_{i}}(\operatorname{div}(\phi(\nabla
u))\,\partial_{x_{i}}v\,dx$ $\displaystyle=$
$\displaystyle\int_{\mathcal{{\mathcal{O}}}}\operatorname{div}(\phi(\nabla
u))\,\Delta v\,dx$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{d}\int_{\mathcal{O}}\left\langle D^{2}\Psi(\nabla
u)\nabla\partial_{x_{i}}u,\nabla\partial_{x_{i}}v\right\rangle\,dx.$

######  Lemma 3.2.

Let $d\geq 2$. For $u,v\in H_{0}^{1}({\mathcal{O}})$, define $\gamma:[0,1]\to
L^{2}({\mathcal{O}};\mathbb{R}^{d})$, $\lambda\mapsto\gamma(\lambda,u,v)$ via
$\gamma(\lambda,u,v)(x)=\nabla u(x)+\lambda(\nabla v(x)-\nabla u(x))$. Set
$\Lambda_{\min}(x):=\psi^{\prime}(\left\lvert
x\right\rvert)\wedge\frac{\psi(\left\lvert x\right\rvert)}{\left\lvert
x\right\rvert}$. Then there exists $C=C({\mathcal{O}},d)>0$,

${}_{H_{\phantom{0}}^{-1}}\langle A(u)-A(v),u-v\rangle_{H_{0}^{1}}\geq
C\left\lVert
u-v\right\rVert_{L^{2}}^{2}\int_{0}^{1}\left(\int_{\mathcal{O}}\left(\Lambda_{\min}(\gamma(\lambda,u,v)(x))\right)^{-\frac{d}{2}}\,dx\right)^{-\frac{2}{d}}\,d\lambda.$

###### Proof.

As in the assertion, $\Lambda_{\min}(x):=\psi^{\prime}(\left\lvert
x\right\rvert)\wedge\frac{\psi(\left\lvert x\right\rvert)}{\left\lvert
x\right\rvert}$. First note that (C4) and (C6) guarantee that the expression
on the RHS is well-defined as for $d\lambda$-a.e. $\lambda\in[0,1]$,

$\int_{\mathcal{O}}\left(\Lambda_{\min}(\gamma(\lambda,u,v)(x))\right)^{-\frac{d}{2}}\,dx\leq\int_{\mathcal{O}}\left(a+b|\nabla
u(x)|^{\frac{ds}{2}}+b|\nabla v(x)|^{\frac{ds}{2}}\right)\,dx<+\infty.$

We find the identity

${}_{H_{\phantom{0}}^{-1}}\langle
A(u)-A(v),u-v\rangle_{H_{0}^{1}}=\int_{\mathcal{O}}\left\langle\psi(\nabla
u(x))-\psi(\nabla v(x)),\nabla u(x)-\nabla v(x)\right\rangle\,dx$
$\displaystyle=$
$\displaystyle-\int_{\mathcal{O}}\int_{0}^{1}\frac{d}{d\lambda}\psi(\gamma(\lambda,u,v)(x))(\nabla
u(x)-\nabla v(x))\,d\lambda\,dx$ $\displaystyle=$
$\displaystyle\int_{\mathcal{O}}\int_{0}^{1}\left\langle
D^{2}\Psi(\gamma(\lambda,u,v)(x))(\nabla u(x)-\nabla v(x)),\nabla u(x)-\nabla
v(x)\right\rangle\,d\lambda\,dx,$

where $\Psi$ is as in (3.2). From the embedding of
$W_{0}^{1,\frac{2d}{d+2}}({\mathcal{O}})$ into $L^{2}({\mathcal{O}})$ and
Hölder’s inequality, we derive

$\displaystyle\left\lVert u-v\right\rVert_{L^{2}}\leq C\left\lVert
u-v\right\rVert_{W_{0}^{1,\frac{2d}{d+2}}}$ $\displaystyle\leq$ $\displaystyle
C\left(\int_{\mathcal{O}}\left\langle D^{2}\Psi(\gamma(\lambda,u,v)(x))(\nabla
u(x)-\nabla v(x)),\nabla u(x)-\nabla
v(x)\right\rangle^{\frac{d}{d+2}}\Lambda_{\min}(\gamma(\lambda,u,v)(x))^{-\frac{d}{d+2}}\,dx\right)^{\frac{d+2}{2d}}$
$\displaystyle\leq$ $\displaystyle C\left(\int_{\mathcal{O}}\left\langle
D^{2}\Psi(\gamma(\lambda,u,v)(x))(\nabla u(x)-\nabla v(x)),\nabla u(x)-\nabla
v(x)\right\rangle\,dx\right)^{\frac{1}{2}}$
$\displaystyle\times\left(\int_{\mathcal{O}}\Lambda_{\min}(\gamma(\lambda,u,v)(x))^{-\frac{d}{2}}\,dx\right)^{\frac{1}{d}},$

where $C>0$ may change from line to line. Integration with respect to
$\lambda$ yields

$\displaystyle\left\lVert
u-v\right\rVert_{L^{2}}^{2}\int_{0}^{1}\left(\int_{\mathcal{O}}\Lambda_{\min}(\gamma(\lambda,u,v)(x))^{-\frac{d}{2}}\,dx\right)^{-\frac{2}{d}}\,d\lambda$
$\displaystyle\leq$ $\displaystyle C\int_{0}^{1}\int_{\mathcal{O}}\left\langle
D^{2}\Psi(\gamma(\lambda))(\nabla u(x)-\nabla v(x)),\nabla u(x)-\nabla
v(x)\right\rangle\,dx\,d\lambda$

Rearranging terms yields the result. ∎

######  Proposition 3.3.

Suppose that $d\geq 1$ with ${\mathcal{O}}\subset\mathbb{R}^{d}$ bounded, open
and convex. Let $u$ and $v$ be solutions to (1.1) with $\phi$ as in (1.4) and
let $u_{0},v_{0}\in V=H_{0}^{1}({\mathcal{O}})$. Let
$d_{0}:=1\vee\frac{d}{2}$. For $0\leq\alpha\leq 1$, $l\geq 1$. and $s\in(0,2]$
as in (C4), we have that

$\displaystyle\lVert u_{t}-v_{t}\rVert_{H}^{2\alpha}\leq t^{-\alpha
l}C(l,\alpha,{\mathcal{O}},d)\left(\frac{1}{t}\int_{0}^{t}(a|{\mathcal{O}}|)^{\alpha
d_{0}}+b^{\alpha d_{0}}\lVert u_{\tau}\rVert_{V}^{\alpha d_{0}s}+b^{\alpha
d_{0}}\lVert v_{\tau}\rVert_{V}^{\alpha
d_{0}s}\,d\tau\right)^{\frac{l}{d_{0}}}\lVert
u_{0}-v_{0}\rVert_{H}^{2\alpha}.$

###### Proof.

Starting from Lemmas 3.1 and 3.2, where $\gamma(\lambda,u,v)(x)=\nabla
u(x)+\lambda(\nabla v(x)-\nabla u(x))$,

$\frac{1}{2}\frac{d}{dt}\lVert u_{t}-v_{t}\rVert_{H}^{2}\leq-C\lVert
u_{t}-v_{t}\rVert_{H}^{2}\int_{0}^{1}\left(\int_{{\mathcal{O}}}\left(\psi^{\prime}(\gamma(\lambda,u_{t},v_{t})(x))\right)^{-d_{0}}\,dx\right)^{-\frac{1}{d_{0}}}\,d\lambda,$

where in the $d=1$ case, $C=L^{-1}$. Gronwall’s Lemma yields

$\lVert u_{t}-v_{t}\rVert_{H}^{2}\leq\lVert
u_{0}-v_{0}\rVert_{H}^{2}\exp\left(-2C\int_{0}^{t}\int_{0}^{1}\left(\int_{{\mathcal{O}}}\left(\psi^{\prime}(\gamma(\lambda,u_{\tau},v_{\tau})(x))\right)^{-d_{0}}\,dx\right)^{-\frac{1}{d_{0}}}\,d\lambda\,d\tau\right).$

For $l\geq 1$, $x\geq 0$, the elementary estimate
$\log(\frac{x}{l})\leq\frac{x}{l}-1$ implies

$e^{x}\geq c_{l}x^{l}$

with $c_{l}:=\left(\frac{e}{l}\right)^{l}$. Applying Jensen’s inequality
thrice, we get by (C4) that

$\displaystyle\lVert u_{0}-v_{0}\rVert_{H}^{2\alpha}\geq
c_{l}^{\alpha}\left(\int_{0}^{t}2C\int_{0}^{1}\left(\int_{{\mathcal{O}}}\left(\psi^{\prime}(\gamma(\lambda,u_{\tau},v_{\tau})(x))\right)^{-d_{0}}\,dx\right)^{-\frac{1}{d_{0}}}\,d\lambda\,d\tau\right)^{\alpha
l}\lVert u_{t}-v_{t}\rVert_{H}^{2\alpha}$ $\displaystyle\geq$ $\displaystyle\
c_{l}^{\alpha}t^{\alpha
l}\left(\frac{1}{t}\int_{0}^{t}\left(2C\int_{0}^{1}\left(\int_{{\mathcal{O}}}\left(\psi^{\prime}(\gamma(\lambda,u_{\tau},v_{\tau})(x))\right)^{-d_{0}}\,dx\right)^{-\frac{1}{d_{0}}}\,d\lambda\right)^{\alpha}\,d\tau\right)^{l}\lVert
u_{t}-v_{t}\rVert_{H}^{2\alpha}$ $\displaystyle=$ $\displaystyle\
c_{l}^{\alpha}(2C)^{\alpha l}t^{\alpha
l}\left(\frac{1}{t}\int_{0}^{t}\left(\int_{0}^{1}\left(\int_{{\mathcal{O}}}\left(\psi^{\prime}(\gamma(\lambda,u_{\tau},v_{\tau})(x))\right)^{-d_{0}}\,dx\right)^{-\frac{1}{d_{0}}}\,d\lambda\right)^{\alpha}\,d\tau\right)^{l}\lVert
u_{t}-v_{t}\rVert_{H}^{2\alpha}.$ $\displaystyle\geq$ $\displaystyle
c_{l}^{\alpha}(2C)^{\alpha l}t^{\alpha
l}\left(\frac{1}{t}\int_{0}^{t}\left(\int_{0}^{1}\int_{{\mathcal{O}}}\left(\psi^{\prime}(\gamma(\lambda,u_{\tau},v_{\tau})(x))\right)^{-d_{0}}dx\,d\lambda\right)^{\alpha}\,d\tau\right)^{-\frac{l}{d_{0}}}\lVert
u_{t}-v_{t}\rVert_{H}^{2\alpha}$ $\displaystyle\geq$ $\displaystyle
c_{l}^{\alpha}(2C)^{\alpha l}t^{\alpha l}\frac{\lVert
u_{t}-v_{t}\rVert_{H}^{2\alpha}}{\left(\frac{1}{t}\int_{0}^{t}\left((a|{\mathcal{O}}|)^{\alpha
d_{0}}+b^{\alpha d_{0}}\lVert u_{\tau}\rVert_{V}^{\alpha d_{0}s}+b^{\alpha
d_{0}}\lVert v_{\tau}\rVert_{V}^{\alpha
d_{0}s}\right)\,d\tau\right)^{\frac{l}{d_{0}}}}.$

∎

##  4\. Moment estimates

Consider equation (2.5) with the same assumptions as stated in the beginning
of Section 3, in other words, we assume (C1)–(C6) and (B1).

Let us start with recording a second order functional inequality for the one-
dimensional case that we will utilize later. Let $d=1$. Let
${\mathcal{O}}=(0,L)\subset\mathbb{R}$, for some $L>0$. Set
$I:=\overline{{\mathcal{O}}}=[0,L]$. Let $\Delta=\partial_{xx}$ be the
Dirichlet Laplace on $I$. Let us equip $W_{0}^{1,1}(I)$ with the norm
$\|u\|_{W_{0}^{1,1}(I)}=\int_{I}|\partial_{x}u|\,dx.$

######  Lemma 4.1 (compare with Lemma 2.2. in [26]).

For $u\in C_{0}^{\infty}(I)$ and $a\geq 0$, $b>0$ and $s\in(0,2]$ as in
Assumption (C4), there exists constants $C(L,a,s)>0$ and $C(L,b,s)>0$ such
that

$\left(\int_{I}|\partial_{xx}u|\,dx\right)^{s^{\ast}}\leq
s^{\ast}\int_{I}\psi^{\prime}(\partial_{x}u)|\partial_{xx}u|^{2}\,dx+C(L,b,s)\|u\|_{W_{0}^{1,1}(I)}^{1\vee(2-s)}+C(L,a,s),$

where

(4.1) $s^{\ast}:=(2-s)\vee\frac{4-s}{2+s}.$

###### Proof.

Let $u\in C_{0}^{\infty}(I)$. Let $\alpha\in[\frac{1}{2},2-\frac{s}{2})$. By
Hölder inequality, Young inequality and (C4),

$\displaystyle\left(\int_{I}|\partial_{xx}u|\,dx\right)^{\alpha}$
$\displaystyle\leq$
$\displaystyle\left(\int_{I}\psi^{\prime}(\partial_{x}u)|\partial_{xx}u|^{2}\,dx\right)^{\frac{\alpha}{2}}\left(\int_{I}\frac{1}{\psi^{\prime}(\partial_{x}u)}\,dx\right)^{\frac{\alpha}{2}}$
$\displaystyle\leq$
$\displaystyle\frac{\alpha}{2}\int_{I}\psi^{\prime}(\partial_{x}u)|\partial_{xx}u|^{2}\,dx+\frac{2-\alpha}{2}\left(\int_{I}\left(a+b|\partial_{x}u|^{s}\right)\,dx\right)^{\frac{\alpha}{2-\alpha}}$
$\displaystyle\leq$
$\displaystyle\frac{\alpha}{2}\int_{I}\psi^{\prime}(\partial_{x}u)|\partial_{xx}u|^{2}\,dx+2^{\frac{2\alpha-2}{2-\alpha}\vee
0}\left[\frac{2-\alpha}{2}(aL)^{\frac{\alpha}{2-\alpha}}+b^{\frac{\alpha}{2-\alpha}}\frac{2-\alpha}{2}\left(\int_{I}|\partial_{x}u|^{s}\,dx\right)^{\frac{\alpha}{2-\alpha}}\right].$

Furthermore, by Jensen’s inequality and the embedding
$W_{0}^{1,1}(I)\hookrightarrow C(I)$ (note that $u(0)=0$),

$\begin{aligned}
&L^{\frac{2-s}{2}}\left(\int_{I}|\partial_{x}u|^{s}\,dx\right)^{\frac{2}{s}}\\\
\leq&\int_{I}|\partial_{x}u|^{2}\,dx=-\int_{I}u\partial_{xx}u\,dx\\\
\leq&\|\partial_{x}u\|_{L^{1}(I)}\int_{I}|\partial_{xx}u|\,dx.\end{aligned}\leq\|u\|_{\infty}\int_{I}|\partial_{xx}u|\,dx$

Since $\alpha<2-\frac{s}{2}$, we can apply Young’s inequality again and get
altogether,

$\displaystyle 2\left(\int_{I}|\partial_{xx}u|\,dx\right)^{\alpha}$
$\displaystyle\leq$
$\displaystyle\alpha\int_{I}\psi^{\prime}(\partial_{x}u)|\partial_{xx}u|^{2}\,dx+C(L,b,\alpha,s)\left(\int_{I}|\partial_{x}u|^{s}\,dx\right)^{\frac{\alpha}{2-\alpha}}+C(L,a,\alpha)$
$\displaystyle\leq$
$\displaystyle\alpha\int_{I}\psi^{\prime}(\partial_{x}u)|\partial_{xx}u|^{2}\,dx+C(L,b,\alpha,s)\left(\int_{I}|\partial_{xx}u|\,dx\right)^{\frac{\alpha
s}{4-2\alpha}}\|\partial_{x}u\|_{L^{1}(I)}^{\frac{\alpha
s}{4-2\alpha}}+C(L,a,\alpha)$ $\displaystyle\leq$
$\displaystyle\alpha\int_{I}\psi^{\prime}(\partial_{x}u)|\partial_{xx}u|^{2}\,dx+\left(\int_{I}|\partial_{xx}u|\,dx\right)^{\alpha}+C(L,b,\alpha,s)\|\partial_{x}u\|_{L^{1}(I)}^{\frac{\alpha
s}{4-2\alpha-s}}+C(L,a,\alpha).$

Choosing $\alpha=s^{\ast}=(2-s)\vee\frac{4-s}{2+s}$ yields the proof. ∎

Now, consider the general multivariate case $d\geq 1$. Let
${\mathcal{O}}\subset\mathbb{R}^{d}$ be a bounded, open and convex domain with
piecewise $C^{2}$-boundary. To achieve the main result, we first show a
pathwise regularity result for (2.5). This result combined with moment
estimates for the finite dimensional Galerkin approximations of (2.5) yields
the existence of a unique invariant measure for (2.5).

######  Lemma 4.2.

For every $0<s\leq 2$, and every $1<\alpha\leq d\frac{2-s}{d-s}$, there exists
$C=C(d,s,{\mathcal{O}})>0$ such that for every $u\in
C_{0}^{\infty}(\mathcal{O})$,

$\left(\int_{\mathcal{O}}\sum_{i=1}^{d}\sum_{i=1}^{d}\lvert\partial_{x_{i}}\partial_{x_{j}}u\rvert^{\alpha}\,dx\right)^{\frac{s^{\ast}}{\alpha}}\leq
C\left(1+\sum_{i=1}^{d}\int_{\mathcal{O}}\left\langle D^{2}\Psi(\nabla
u)\nabla\partial_{x_{i}}u,\nabla\partial_{x_{i}}u\right\rangle\,dx\right),$

where $s^{\ast}$ is as in (4.1) and $\Psi$ is defined as in (3.2).

###### Proof.

Set $\Lambda_{\min}(x):=\psi^{\prime}(\left\lvert
x\right\rvert)\wedge\frac{\psi(\left\lvert x\right\rvert)}{\left\lvert
x\right\rvert}$. Let $1<\alpha\leq d\frac{2-s}{d-s}$. First, by using Hölder’s
and Young’s inequalities, we get for $\varepsilon>0$,

$\displaystyle\left(\int_{\mathcal{O}}\sum_{i=1}^{d}\sum_{j=1}^{d}\lvert\partial_{x_{i}}\partial_{x_{j}}u\rvert^{\alpha}dx\right)^{\frac{2-s}{\alpha}}\
$ $\displaystyle\leq$ $\displaystyle
C\left(\int_{\mathcal{O}}\sum_{i=1}^{d}\left\langle D^{2}\Psi(\nabla
u)\nabla\partial_{x_{i}}u,\nabla\partial_{x_{i}}u\right\rangle^{\frac{\alpha}{2}}\cdot\Lambda_{\min}^{-\frac{\alpha}{2}}(\nabla
u)\,dx\right)^{\frac{2-s}{\alpha}}$ $\displaystyle\leq$ $\displaystyle
C\left(\int_{\mathcal{O}}\sum_{i=1}^{d}\left\langle D^{2}\Psi(\nabla
u)\nabla\partial_{x_{i}}u,\nabla\partial_{x_{i}}u\right\rangle\,dx\right)^{\frac{2-s}{2}}\times\left(\int_{\mathcal{O}}\Lambda_{\min}^{-\frac{\alpha}{2-\alpha}}(\nabla
u)\,dx\right)^{\frac{(2-s)(2-\alpha)}{2\alpha}}$ $\displaystyle\leq$
$\displaystyle
C\left[\frac{2-s}{2}\varepsilon^{-\frac{2}{2-s}}\int_{\mathcal{O}}\sum_{i=1}^{d}\left\langle
D^{2}\Psi(\nabla
u)\nabla\partial_{x_{i}}u,\nabla\partial_{x_{i}}u\right\rangle\,dx+\frac{s}{2}\varepsilon^{\frac{2}{s}}\left(\int_{\mathcal{O}}\Lambda_{\min}^{-\frac{\alpha}{2-\alpha}}(\nabla
u)\,dx\right)^{\frac{(2-s)(2-\alpha)}{s\alpha}}\right].$

We note that $W_{0}^{2,\alpha}({\mathcal{O}})$ is continuously embedded into
$W_{0}^{1,\frac{s\alpha}{2-\alpha}}({\mathcal{O}})$ for $1<\alpha\leq
d\frac{2-s}{d-s}$. This embedding, together with (C4), yields

$\displaystyle\left(\int_{\mathcal{O}}\Lambda_{\min}^{-\frac{\alpha}{2-\alpha}}(\nabla
u)\,dx\right)^{\frac{(2-s)(2-\alpha)}{s\alpha}}\leq
C\left(1+\int_{\mathcal{O}}|\nabla
u|^{\frac{s\alpha}{2-\alpha}}\,dx\right)^{\frac{(2-s)(2-\alpha)}{s\alpha}}\leq
C\left(1+\left\lVert
u\right\rVert_{W_{0}^{2,d\frac{2-s}{d-s}}({\mathcal{O}})}^{s^{\ast}}\right).$

Choosing $\varepsilon$ small enough completes the proof. ∎

We shall prove the existence of the unique invariant measure for the ergodic
semigroup and the main concentration result now.

######  Theorem 4.3.

There exists a unique invariant Borel probability measure $\mu$ for the
semigroup $\\{P_{t}\\}$ of equation (2.5) that is concentrated on the subset
$W^{2,\alpha}_{0}({\mathcal{O}})\cap W^{1,2}_{0}({\mathcal{O}})\subset
L^{2}({\mathcal{O}})$ for any $1<\alpha\leq d\frac{2-s}{d-s}$ if $d\geq 2$ and
$\alpha=1$ if $d=1$ with

$\int_{H}\|u\|_{W_{0}^{2,\alpha}({\mathcal{O}})}^{s^{\ast}}\,\mu(du)+\int_{H}\lVert
u\rVert_{V}^{s^{\ast}}\,\mu(du)+\sum_{i=1}^{d}\int_{H}\langle
D^{2}\Psi\left(\nabla u\right)\partial_{x_{i}}\nabla u,\partial_{x_{i}}\nabla
u\rangle_{H}\,\mu(du)<+\infty,$

where $s^{\ast}$ is as in (4.1) and $s\in(0,2]$ is as in assumption (C4) and
$\Psi$ is defined as in (3.2).

###### Proof.

For a complete orthonormal system $\\{e_{k}\\}_{k\in\mathbb{N}}$ of
$H=L^{2}({\mathcal{O}})$, and for
$H_{m}:=\operatorname{span}\\{e_{k}\;\colon\;1\leq k\leq m\\}$, we consider
the (finite dimensional) approximating stochastic differential equation in $H$

(4.2) $\begin{split}du^{m}(t)+Au^{m}(t)\,dt&=P_{m}B\,dW_{t}\\\
u_{0}^{m}(x)&=P_{m}u_{0}(x),\end{split}$

where $A$ is as in (2.6). Note that here, we cannot use the standard Galerkin
approximation as in [25], for there is no guarantee that the nonlinear
operator $A$ maps $H_{m}$ into $V=H_{0}^{1}({\mathcal{O}})$. One option would
be to add an extra smoothing step for the operator $A$ as e.g. in [31, Proof
of Theorem 3.1]. Instead, we follow the strategy from [29, Section 6.3.1] to
find solutions $u_{m}$ converging to $u$ in $L^{2}(\Omega,C([0,T];H))$ with
the aim to consider the corresponding invariant measures $\nu_{m}$. For
$n\in\mathbb{N}$, let
$J_{n}:=\left(\operatorname{Id}-\frac{\Delta}{n}\right)^{-1}$ and
$T_{n}:=-\Delta J_{n}=n(\operatorname{Id}-J_{n})$, respectively, be the
resolvent and the smooth Yosida approximation of the nonnegative definite
differential operator
$-\Delta=-\sum_{i=1}^{d}\partial_{x_{i}}\partial_{x_{i}}$, respectively. Let
$\langle u,v\rangle_{n}:=\langle u,T_{n}v\rangle_{H}$. The induced norms
$\lVert\cdot\rVert_{n}$ are equivalent to the $H$-norm and converge
monotonically as follows

$\lVert u\rVert_{n}\underset{n\to+\infty}{\uparrow}\begin{cases}\lVert
u\rVert_{V},&u\in V,\\\ +\infty,&\text{otherwise.}\end{cases}$

Applying Itô’s formula to $\lVert u^{m}(t)\rVert_{n}^{2}$, we get

(4.3) $\lVert u_{t}^{m}\rVert_{n}^{2}=\lVert
u_{0}^{m}\rVert_{n}^{2}-2\int_{0}^{t}\langle
A(u_{\tau}^{m}),T_{n}(u_{\tau}^{m})\rangle\,d\tau+\int_{0}^{t}\lVert
P_{m}B\rVert_{L_{2}(U,H_{m})}^{2}\,d\tau+M_{t}^{n}$

where $t\mapsto M_{t}^{n}=2\int_{0}^{t}\langle
u_{\tau}^{m},P_{m}B\,dW_{\tau}\rangle_{n}$ is a local martingale. From [7,
Proposition 8.2] and (C3), applied to [29, Proposition 7.1], it follows that

$\langle A(u_{\tau}^{m}),T_{n}(u_{\tau}^{m})\rangle_{H}\geq 0\quad
d\tau\text{-a.e}.$

We note that the convexity assumption of for the domain ${\mathcal{O}}$ enters
here, as it is needed to apply [7, Proposition 8.2]. By taking expectation,
together with Fatou’s lemma, employing the bounds $\lVert
u_{0}^{m}\rVert_{n}\leq\lVert u_{0}\rVert_{V}$ and $\lVert
P_{m}B\rVert_{L_{2}(U,H^{m})}^{2}\leq\lVert B\rVert_{L_{2}(U,V)}^{2}$, it
follows from (4.3) for every $t\geq 1$ that

(4.4)
$\frac{1}{t}\mathbb{E}\left(\int_{0}^{t}\sum_{i=1}^{d}\int_{{\mathcal{O}}}\langle
D^{2}\Psi(\nabla
u_{\tau}^{m})\nabla\partial_{x_{i}}u_{\tau}^{m},\nabla\partial_{x_{i}}u_{\tau}^{m}\rangle\,dx\,d\tau\right)\leq\frac{1}{2}\left(\mathbb{E}\lVert
u_{0}\rVert_{V}^{2}+\lVert B\rVert_{L_{2}(U,V)}^{2}\right)=:K_{1},$

independently of $m$. On the other hand, the application of Itô’s formula to
$t\mapsto\lVert u_{t}^{m}\rVert_{H}^{2}$ yields

$\lVert u_{t}^{m}\rVert_{H}^{2}=\lVert
u_{0}^{m}\rVert_{H}^{2}-2\int_{0}^{t}\langle
A(u_{\tau}^{m}),u_{\tau}^{m}\rangle_{H}\,d\tau+\int_{0}^{t}\lVert
P_{m}B\rVert_{L_{2}(U,H_{m})}^{2}\,d\tau+N_{t}^{m}$

where $t\mapsto N_{t}^{m}=2\int_{0}^{t}\langle
u_{\tau}^{m},P_{m}B\,dW_{\tau}\rangle_{H}$ and $\langle
A(u^{m}),u^{m}\rangle=\int_{{\mathcal{O}}}\langle\phi(\nabla u^{m}),\nabla
u^{m}\rangle\,dx$. Note that by (C5) and Lemma 2.10, we get

$\psi(r)\cdot r\geq c\lvert r\rvert^{1\vee(2-s)}-K,$

for suitable constants $c>0$ and $K\geq 0$. This combined with the same
arguments as before entails that for every $t\geq 1$

(4.5) $\frac{1}{t}\mathbb{E}\left(\int_{0}^{t}\int_{{\mathcal{O}}}\lvert\nabla
u_{\tau}^{m}\rvert^{1\vee(2-s)}\,dx\,d\tau\right)\leq\frac{1}{2c}\left(\lVert
u_{0}\rVert_{H}^{2}+\lVert B\rVert_{L_{2}(U,H)}^{2}+K\right)=:K_{2}$

for a positive constant $K_{2}>0$.

Let us treat the case $d=1$ separately first, so let us assume that
${\mathcal{O}}=(0,L)$. By the above estimates, it is easy to prove that the
family of probability measures
$\left\\{\frac{1}{t}\int_{0}^{t}P_{\tau}^{\ast}\delta_{u_{0}}\,d\tau\right\\}_{t\geq
1}$ (where $\delta_{u_{0}}$ denotes the Dirac measure in $u_{0}$) is tight in
the finite dimensional space $H_{m}$, and we thus find that by the Krylov-
Bogoliubov existence theorem [18, Theorem 3.1.1] that there exists an
invariant measure $\nu_{m}$ for the Markov semigroup of (4.2) such that for
arbitrary $N>0$

(4.6) $\int_{H}\int_{I}\left(\lvert\partial_{x}u\rvert^{1\vee(2-s)}dx\wedge
N\right)\nu_{m}(du)+\int_{H}\int_{I}\left(\psi^{\prime}(\partial_{x}u)|\partial_{xx}u|^{2}dx\wedge
N\right)\nu_{m}(du)\leq K_{3},$

where $K_{3}=K_{1}+K_{2}$. An application of the monotone convergence lemma
yields

(4.7) $\sup_{m\geq
1}\int_{H}\int_{I}\lvert\partial_{x}u\rvert^{1\vee(2-s)}\,dx\,\nu_{m}(du)+\sup_{m\geq
1}\int_{H}\int_{I}\psi^{\prime}(\partial_{x}u)|\partial_{xx}u|^{2}\,dx\,\nu_{m}(du)\leq
K_{3}.$

Combining (4.6) with Lemma 4.1, we obtain

(4.8) $\sup_{m\geq
1}\int_{H}\left(\int_{0}^{L}\lvert\partial_{xx}u\rvert\,dx\right)^{s^{\ast}}\,\nu_{m}(du)<+\infty.$

Due to the compactness of the embedding $W_{0}^{2,1}(0,L)\subset
W_{0}^{1,1}(0,L)$, we get that the sequence of measures $\\{\nu_{m}\\}_{m\geq
1}$ is tight w.r.t. the $W^{1,1}$-topology. Let $\nu$ denote the limit of a
weakly convergent subsequence in the sense of weak convergence of probability
measures. For a function $u\in L_{\operatorname{loc}}^{1}(B)$, we define its
total variation on an open subset $B\subset\mathbb{R}^{d}$ by

$[Du](B):=\sup\left\\{\int_{B}u\operatorname{div}v\,dx\;:\;v\in
C^{1}(\overline{B},\mathbb{R}^{d}),\;\lVert v\rVert_{\infty}\leq 1\right\\}.$

Recall that $u\in L^{1}(B)$ belongs to the space $BV(B)$ of functions of
bounded variation if $[Du](B)<+\infty$. Note that

$u\mapsto\begin{cases}[D\partial_{x}u](\mathbb{R}),&\text{ if
}\partial_{x}u\in BV(\mathbb{R})\cap L^{1}(0,L),\;u\in W^{1,1}_{0}(0,L),\\\
+\infty,&\text{ if }\partial_{x}u\in L^{1}(0,L)\setminus BV(\mathbb{R}),\;u\in
W^{1,1}_{0}(0,L),\end{cases}$

is the lower semi-continuous envelope of

$u\mapsto\int_{0}^{L}|\partial_{xx}u|\,dx,\quad u\in W^{2,1}_{0}(0,L),$

in $W^{1,1}_{0}(0,L)$. The lower semi-continuity of
$u\mapsto[D\partial_{x}u](\mathbb{R})$ w.r.t the strong
$W^{1,1}_{0}(0,L)$-topology, together with (4.8) yields by [1, Proposition
1.62 (a)],

$\int_{H}\left([D\partial_{x}u](\mathbb{R})\right)^{s^{\ast}}\,\nu(du)<+\infty.$

However, by the tightness, it can be seen by standard arguments that the
support of $\nu$ is contained in $W_{0}^{2,1}(0,L)$ and thus

$\int_{H}\left(\int_{0}^{L}\partial_{xx}u\,dx\right)^{s^{\ast}}\,\nu(du)<+\infty.$

From the boundedness of the embedding of $W_{0}^{2,1}(0,L)$ into
$W_{0}^{1,2}(0,L)$ it follows that

$\int_{H}\lVert u\|_{V}^{s^{\ast}}\,\nu(du)<+\infty.$

Hence the support of $\nu$ is included in $W_{0}^{2,1}(0,L)\cap
H_{0}^{1}(0,L)$.

Let us suppose $d\geq 1$ now. By the above estimates, it is easy to prove that
the family of probability measures
$\left\\{\frac{1}{t}\int_{0}^{t}P_{\tau}^{\ast}\delta_{u_{0}}\,d\tau\right\\}_{t\geq
1}$ is tight in the finite dimensional space $H_{m}$ and thus by the Krylov-
Bogoliubov existence theorem [18, Theorem 3.1.1], as the bound (4.5) is
independent of $t$, there exists an invariant measure $\nu_{m}$ for (4.2). To
show uniqueness, note that for any Lipschitz function $F$ and $\nu_{m}$-almost
every $u_{0},v_{0}$ in $H_{m}$, Proposition 3.3 yields (for
$\alpha=\frac{1}{2}$ and $l=d_{0}=1\vee\frac{d}{2})$ that

(4.9)
$\begin{split}&\left\lvert\mathbb{E}\left(\frac{1}{t}\int_{0}^{t}F(u_{\tau})\,d\tau\right)-\mathbb{E}\left(\frac{1}{t}\int_{0}^{t}F(v_{\tau})\,d\tau\right)\right\rvert\\\
\leq&\operatorname{Lip}(F)\,\mathbb{E}\left(\frac{1}{t}\int_{0}^{t}\lVert
u_{\tau}-v_{\tau}\rVert_{H}\,d\tau\right)\\\
\leq&t^{-\frac{d_{0}}{2}}C\left(1+\frac{1}{t}\mathbb{E}\left(\int_{0}^{t}\lVert
u_{\tau}\rVert_{V}^{\frac{sd_{0}}{2}}+\lVert
v_{\tau}\rVert_{V}^{\frac{sd_{0}}{2}}\,d\tau\right)\right)\lVert
u_{0}-v_{0}\rVert_{H}\\\ \leq&t^{-\frac{d_{0}}{2}}C\left(1+\lVert
u_{0}\rVert_{V}^{\frac{sd_{0}}{2}}+\lVert
v_{0}\rVert_{V}^{\frac{sd_{0}}{2}}\right)\lVert
u_{0}-v_{0}\rVert_{H}\longrightarrow 0,\end{split}$

as $t\to+\infty,$ where $C>0$ may change from line to line. The latter implies
uniqueness of the invariant measure $\nu_{m}$ of the ergodic semigroup of
(4.2) by standard arguments, see e.g. [41, Section 2.1, proof of Theorem 1.2
(iii)].

Now, it follows from ergodicity that for arbitrary $L>0$

(4.10) $\int_{H}\sum_{i=1}^{d}\int_{\mathcal{O}}\left(\left\langle
D^{2}\Psi(\nabla u)\nabla u_{x_{i}},\nabla u_{x_{i}}\right\rangle dx\wedge
L\right)\,\nu_{m}(du)\leq C.$

The monotone convergence lemma yields

(4.11) $\sup_{m\geq
1}\int_{H}\sum_{i=1}^{d}\int_{\mathcal{O}}\left(\left\langle D^{2}\Psi(\nabla
u)\nabla u_{x_{i}},\nabla u_{x_{i}}\right\rangle dx\right)\,\nu_{m}(du)\leq
C.$

Combining (4.11) with Lemma 4.2, we obtain

(4.12) $\sup_{m\geq 1}\int_{H}\left\lVert
u\right\rVert_{W_{0}^{2,(2-s)\frac{d}{d-s}}}^{s^{\ast}}\,\nu_{m}(du)<+\infty.$

Keeping assumption (C6) in mind, the compactness of the embedding
$W_{0}^{2,d\frac{2-s}{d-s}}(\mathcal{O})$ into $V=W_{0}^{1,2}({\mathcal{O}})$
for $s<\frac{4}{d}$ implies that the sequence of measures
$\\{\nu_{m}\\}_{m\geq 1}$ is tight in $V$. Thus, there exists an invariant
measure $\mu$ such that

$\int_{H}\lVert u\|_{V}^{s^{\ast}}\,\mu(du)<+\infty.$

The moment estimate for the $W_{0}^{2,\alpha}({\mathcal{O}})$-norm, where
$\alpha$ is as in the statement of the theorem, follows by lower semi-
continuity from (4.12) and [1, Proposition 1.62 (a)].

The uniqueness of $\mu$ can be proved by the same arguments as for $\nu_{m}$,
see the computation (4.9) combined with [41, Section 2.1, proof of Theorem 1.2
(iii)]. ∎

As a consequence, we obtain the proof of Theorem 1.2 (i).

######  Corollary 4.4 (Stochastic $p$-Laplace).

Suppose that $\psi(r)=|r|^{p-2}r$, $p\in(1,2)$. Then, under assumptions (C6)
and (B1) we get that the unique invariant measure $\mu$ is concentrated on
$W_{0}^{2,\frac{dp}{d+p-2}}({\mathcal{O}})\cap H^{1}_{0}({\mathcal{O}})\subset
L^{2}({\mathcal{O}})$ and satisfies

$\int_{L^{2}({\mathcal{O}})}\|u\|_{W_{0}^{2,\frac{dp}{d+p-2}}({\mathcal{O}})}^{p}\,\mu(du)+\int_{L^{2}({\mathcal{O}})}\lVert
u\rVert_{H^{1}_{0}({\mathcal{O}})}^{p}\,\mu(du)+\int_{L^{2}({\mathcal{O}})}\int_{\mathcal{O}}|\nabla
u|^{p-2}|D^{2}u|^{2}\,dx\,\mu(du)<+\infty.$

###### Proof.

Note that here, $s^{\ast}=p$. The result follows from the fact that for
$\psi(r)=|r|^{p-2}r$, $\Psi(x)=\frac{1}{p}|x|^{p}$,

$(D^{2}\Psi(x))_{(i,j)}=|x|^{p-2}\delta_{ij}-(2-p)|x|^{p-2}\frac{x_{i}x_{j}}{|x|^{2}}.$

∎

##  5\. Decay estimate and examples

Proposition 3.3 is applied to prove the following main decay estimate for the
semigroup $\\{P_{t}\\}$.

######  Theorem 5.1.

Let $\\{P_{t}\\}$ be the semigroup from (2.7) and let $\mu$ denote its unique
invariant measure. Let $d_{0}:=1\vee\frac{d}{2}$. Let
$s^{\ast}=(2-s)\vee\frac{4-s}{2+s}$. Let
$0<\beta\leq\beta^{\star}:=1\wedge\frac{8-2s}{s(2+s)d_{0}}\in\left[\frac{1}{2d_{0}},1\right]$.
Then, under the assumptions of Proposition 3.3, there exist non-negative
constants $C_{1}$, $C_{2}$ and $C_{3}$ such that

$\limsup_{t\to+\infty}\left[t^{\frac{s^{\ast}}{s}}\frac{\lvert
P_{t}f(u)-P_{t}f(v)\rvert}{\lVert u-v\rVert_{H}^{\beta}}\right]\leq
C_{1}|f|_{\beta}\left(C_{2}+C_{3}\int_{H}\lVert
u\rVert_{V}^{s^{\ast}}\,\mu(du)\right),$

for every $u,v\in V$ and every $f:H\to\mathbb{R}$ that is bounded and
$\beta$-Hölder-continuous, i.e.,

$\sup_{\begin{subarray}{c}u,v\in H\\\ u\not=v\end{subarray}}\frac{\lvert
f(u)-f(v)\rvert}{\lVert u-v\rVert_{H}^{\beta}}=:\lvert
f\rvert_{\beta}<+\infty.$

###### Proof.

We use Proposition 3.3 and get for
$\frac{\beta}{2}=:\alpha\leq\frac{1}{2}\wedge\frac{s^{\ast}}{sd_{0}}=\frac{\beta^{\star}}{2}$,
$l:=\frac{s^{\ast}}{\alpha d_{0}s}\geq 1$,

$\displaystyle\lvert
P_{t}f(u)-P_{t}f(v)\rvert=\lvert\mathbb{E}[f(u_{t})-f(v_{t})]\rvert\leq\lvert
f\rvert_{2\alpha}\mathbb{E}\left[\lVert
u_{t}-v_{t}\rVert_{H}^{2\alpha}\right]$
$\displaystyle\leq|f|_{2\alpha}t^{-\frac{s^{\ast}}{s}}C(l,\alpha,{\mathcal{O}},d)\mathbb{E}\left[\left(\frac{1}{t}\int_{0}^{t}\left((a|{\mathcal{O}}|)^{\alpha
d_{0}}+b^{\alpha d_{0}}\lVert u_{\tau}\rVert_{V}^{\alpha d_{0}s}+b^{\alpha
d_{0}}\lVert v_{\tau}\rVert_{V}^{\alpha
d_{0}s}\right)\,d\tau\right)^{\frac{l}{d_{0}}}\right]\lVert
u_{0}-v_{0}\rVert_{H}^{2\alpha}$
$\displaystyle\leq|f|_{\beta}t^{-\frac{s^{\ast}}{s}}C(l,\alpha,{\mathcal{O}},d)3^{(\frac{l}{d_{0}}-1)\vee
0}\mathbb{E}\left(\frac{1}{t}\int_{0}^{t}\left((a|{\mathcal{O}}|)^{\frac{s^{\ast}}{s}}+b^{\frac{s^{\ast}}{s}}\lVert
u_{\tau}\rVert_{V}^{s^{\ast}}+b^{\frac{s^{\ast}}{s}}\lVert
v_{\tau}\rVert_{V}^{s^{\ast}}\right)\,d\tau\right)\lVert
u_{0}-v_{0}\rVert_{H}^{\beta}.$

Note that $\mathbb{E}\left(\frac{1}{t}\int_{0}^{t}\lVert
u_{\tau}\rVert_{V}^{s^{\ast}}\,d\tau\right)$ converges to $\int_{H}\lVert
u\rVert_{V}^{s^{\ast}}\,\mu(du)$ due to the ergodicity of $\\{P_{t}\\}$. ∎

As a consequence, we obtain the proof of Theorem 1.2 (ii). As an application,
we find the following convergence rates for the main examples.

###  5.1. Stochastic mean curvature flow

For the stochastic mean curvature flow equation in $1+1$ dimensions, i.e.

$A(u)=-\frac{\partial_{xx}u}{1+(\partial_{x}u)^{2}}=-\frac{\partial}{\partial
x}\arctan(\partial_{x}u),\quad\psi(r)=\arctan(r),$

we have $s=2$, $\beta^{\star}=\frac{1}{2}$, so our result recovers the result
in [26].

######  Corollary 5.2.

For $f$ Hölder continuous with $\beta=\frac{1}{2}$, we have that

$\limsup_{t\to+\infty}\left[t^{\frac{1}{4}}\frac{\lvert
P_{t}f(u)-P_{t}f(v)\rvert}{\lVert u-v\rVert_{L^{2}(0,L)}^{\beta}}\right]\leq
C_{1}\lvert f\rvert_{\frac{1}{2}}\left(C_{2}+C_{3}\int_{H}\lVert
u\rVert_{V}^{\frac{1}{2}}\,\mu(du)\right),$

where the constants do not depend on $f$.

###  5.2. Singular $p$-Laplace

Let $d_{0}:=1\vee\frac{d}{2}$. Consider, for $p\in(1,2)$, the operator

$A(u)=-\operatorname{div}\left(|\nabla u|^{p-2}\nabla
u\right),\quad\psi(r)=|r|^{p-2}r.$

With $s=2-p$, we get $\beta^{\star}=\frac{8-2(2-p)}{(2-p)(4-p)d_{0}}\wedge 1$,
$s^{\ast}=p$ and therefore get the following result which improves the result
in [41], see also Remark 2.12.

######  Corollary 5.3.

For $f$ Hölder continuous with $0<\beta\leq\beta^{\star}$, we have that

$\limsup_{t\to+\infty}\left[t^{\frac{p}{2-p}}\frac{\lvert
P_{t}f(u)-P_{t}f(v)\rvert}{\lVert
u-v\rVert_{L^{2}({\mathcal{O}})}^{\beta}}\right]\leq C_{1}\lvert
f\rvert_{\beta}\left(C_{2}+C_{3}\int_{H}\lVert
u\rVert_{V}^{p}\,\mu(du)\right),$

where the constants do not depend on $f$.

###  5.3. Non-Newtonian Fluids

Let $d_{0}:=1\vee\frac{d}{2}$. Consider the following operator for
$p\in(1,2)$,

$A(u)=-\operatorname{div}\left(\left(1+|\nabla
u|^{2}\right)^{\frac{p-2}{2}}\nabla
u\right),\quad\psi(r)=\left(1+r^{2}\right)^{\frac{p-2}{2}}r.$

As before, we have $s=2-p$, and get
$\beta^{\star}=\frac{8-2(2-p)}{(2-p)(4-p)d_{0}}\wedge 1$, $s^{\ast}=p$.
Therefore, we obtain the following.

######  Corollary 5.4.

For $f$ Hölder continuous with $0<\beta\leq\beta^{\star}$, we have that

$\limsup_{t\to+\infty}\left[t^{\frac{p}{2-p}}\frac{\lvert
P_{t}f(u)-P_{t}f(v)\rvert}{\lVert
u-v\rVert_{L^{2}({\mathcal{O}})}^{\beta}}\right]\leq C_{1}\lvert
f\rvert_{\beta}\left(C_{2}+C_{3}\int_{H}\lVert
u\rVert_{V}^{p}\,\mu(du)\right),$

where the constants do not depend on $f$.

##  6\. Maximal dissipativity of the associated Kolmogorov operator

Set

$\kappa(r)=\psi^{\prime}(\left\lvert
r\right\rvert)\wedge\frac{\psi(\left\lvert r\right\rvert)}{\left\lvert
r\right\rvert},$

i.e. $\Lambda_{\min}(x)=\kappa(|x|)$, where $\Lambda_{\min}$ is defined as in
Lemma 3.2, and set

$D_{0}:=\left\\{u\in W_{0}^{1,1}(\mathcal{O})\;:\;\sqrt{\kappa\left(|\nabla
u|\right)}\,\partial_{x_{i}}\partial_{x_{j}}u\in L^{2}(\mathcal{O}),1\leq
i,j\leq d\right\\}.$

Recall that Theorem 4.3 yields that the support of the invariant measure $\mu$
is included in $D_{0}$, so that

$\|\sqrt{\kappa\left(|\nabla
u|\right)}\,\partial_{x_{i}}\partial_{x_{j}}u\|^{2}_{H}$

is well-defined (and finite) $\mu$-a.e., and

(6.1) $\sum_{i,j=1}^{d}\int_{H}\|\sqrt{\kappa\left(|\nabla
u|\right)}\,\partial_{x_{i}}\partial_{x_{j}}u\|^{2}_{H}\,\mu(du)<+\infty.$

The Kolmogorov operator associated with (1.1) on sufficiently smooth functions
$F$ is given by

$J_{0}F(u):=\frac{1}{2}\operatorname{Tr}_{H}\left(BB^{\ast}D^{2}F(u)\right)-\langle
A(u),DF(u)\rangle.$

In order to realize $J_{0}$ as an operator in $L^{1}(H,\mu)$ with domain
$D(J_{0}):=C_{b}^{2}(H)$, we need to impose conditions on $A$ ensuring that
$\|A(u)\|\in H$. To this end we now pose the following additional assumption.

######  Assumption 6.1.

Suppose that

  1. (C7)

there exists a constant $M>0$ such that for every $r\in\mathbb{R},$

$\left|\psi^{\prime}(\left\lvert r\right\rvert)-\frac{\psi(\left\lvert
r\right\rvert)}{\left\lvert r\right\rvert}\right|+\frac{\psi(\left\lvert
r\right\rvert)}{\left\lvert r\right\rvert}\leq M\sqrt{\kappa(r)}.$

######  Example 6.2.

We shall collect the following examples for (C7) and leave their verification
to the reader.

  1. (i)

For $p\in(1,2)$, there exists as constant $C(p)>0$ such that
$\psi(r)=(1+|r|^{2})^{\frac{p-2}{p}}r$ satisfies (C7) with $M=C(p)$.

  2. (ii)

$\psi(r)=\log(1+|r|)\operatorname{sgn}(r)$ satisfies (C7) with $M=2$.

  3. (iii)

$\psi(r)=\arctan(r)$ satisfies (C7) with $M=4$.

######  Remark 6.3.

We would like to point out that for the singular $p$-Laplace
$\psi(r)=|r|^{p-2}r$, (C7) reduces to

$\sqrt{p-1}|r|^{p-2}\leq M|r|^{\frac{p-2}{2}}\quad\text{for every
$r\in\mathbb{R}$.}$

Clearly, the existence of such $M>0$ cannot be proved for $p\not=2$. Neither
does the minimal surface flow satisfy (C7).

An immediate implication of (C7) is that for $u\in D_{0}$ due to (3.4)

$\displaystyle|A(u)|$ $\displaystyle\leq\left|\psi^{\prime}(\left\lvert\nabla
u\right\rvert)-\frac{\psi(\left\lvert\nabla u\right\rvert)}{\left\lvert\nabla
u\right\rvert}\right|\left(\sum_{i,j=1}^{d}\left(\partial_{x_{i}}\partial_{x_{j}}u\right)^{2}\right)^{\frac{1}{2}}+\frac{\psi(\left\lvert\nabla
u\right\rvert)}{\left\lvert\nabla u\right\rvert}\left\lvert\Delta
u\right\rvert$ $\displaystyle\leq M\sqrt{\kappa\left(|\nabla
u|\right)}\left(\sum_{i,j=1}^{d}\left(\partial_{x_{i}}\partial_{x_{j}}u\right)^{2}\right)^{\frac{1}{2}}$

for some finite constant. Moreover, (6.1) now implies that $\|A(u)\|\in
L^{2}(H,\mu)$, so that for $F\in C_{b}^{2}(H)$ it follows that $\langle
A(u),DF(u)\rangle\in L^{2}(H,\mu)$, hence $J_{0}F(u)\in L^{1}(H,\mu)$, too.
From Itô’s formula for applied to $F(u(t))$ with regular initial condition, it
follows that $\mu$ is infinitesimally invariant for $J_{0}$. Therefore, since

$J_{0}F^{2}=2FJ_{0}F+\langle BB^{\ast}DF,DF\rangle,\quad F\in D(J_{0}),$

also,

$\int_{H}FJ_{0}F\,d\mu=-\frac{1}{2}\int_{H}\lVert B^{\ast}DF\rVert^{2}\,d\mu.$

Thus $J_{0}$ is dissipative in the Hilbert space $L^{2}(H,\mu)$. By similar
arguments as in [22] one can prove that $J_{0}$ is also dissipative in
$L^{1}(H,\mu)$. As a consequence, it is closable and its closure
$J:=\bar{J_{0}}$ with the domain $D(J)$ is dissipative.

######  Theorem 6.4 (compare with Theorem 4.1 in [26]).

The operator $(J,D(J))$ generates a $C_{0}$-semigroup of contractions on
$L^{1}(H,\mu)$.

###### Proof.

We shall prove that for $\lambda>0$, $\operatorname{range}(\lambda-J)$ is
dense in $L^{1}(H,\mu)$. To this aim, for $\alpha>0$, consider the Yosida
approximation of $A$ defined by

$A_{\alpha}(u)=A(J_{\alpha}(u)),\quad\text{where$\quad
J_{\alpha}(u)=(\operatorname{Id}+\alpha A)^{-1}(u)$.}$

For the sequence $A_{\alpha}$ we have the following:

  1. (i)

For any $\alpha>0$, $A_{\alpha}$ is dissipative and Lipschitz continuous,

  2. (ii)

$\lVert A_{\alpha}(u)\rVert_{H}\leq\lVert A(u)\rVert_{H}$ for any $u\in D(A)$.

Note that the function $A_{\alpha}$ is not differentiable in general.
Therefore we consider a $C^{1}$-approximation. For $\alpha,\beta>0$ we set

$A_{\alpha,\beta}(u):=\int_{H}e^{\beta\Delta}A_{\alpha}(e^{\beta\Delta}u+v)\,\mathcal{N}_{0,\sigma_{\beta}}(dv)$

where $\mathcal{N}_{0,\sigma_{\beta}}$ is the Gaussian measure on $H$ with
mean $0$ and covariance operator
$\sigma_{\beta}:=\int_{0}^{\beta}e^{2\tau\Delta}\,d\tau$, and
$e^{\tau\Delta}$, $\tau>0$, denotes the semigroup generated by the Dirichlet
Laplace operator on $L^{2}(\mathcal{O})$. Then, $A_{\alpha,\beta}$ is
dissipative since

$\langle
A_{\alpha,\beta}(u)-A_{\alpha,\beta}(v),u-v\rangle=\int_{H}\left\langle
A_{\alpha}(e^{\beta\Delta}u+w)-A_{\alpha}(e^{\beta\Delta}v+w),e^{\beta\Delta}(u-v)\right\rangle\,\mathcal{N}_{0,\sigma_{\beta}}(dw)\geq
0.$

We would like to use the Cameron-Martin theorem to show that
$A_{\alpha,\beta}$ is differentiable. To this end we first need to check that
$e^{\beta\Delta}u\in\operatorname{range}(\sigma_{\beta}^{\frac{1}{2}})$.
According to Proposition B.1 in [19] it suffices to prove that there exists a
constant $M$ such that $\|e^{\beta\Delta}v\|^{2}_{H}\leq
M\|\sigma_{\beta}^{\frac{1}{2}}v\|_{H}^{2}$ for all $v\in H$. But this follows
from the fact that

$\displaystyle\|e^{\beta\Delta}v\|_{H}^{2}$
$\displaystyle=\frac{1}{\beta}\int_{0}^{\beta}\|e^{(\beta-\tau)\Delta}e^{\tau\Delta}v\|_{H}^{2}\,d\tau\leq\frac{1}{\beta}\int_{0}^{\beta}\|e^{\tau\Delta}v\|_{H}^{2}\,d\tau$
$\displaystyle=\frac{1}{\beta}\int_{0}^{\beta}\langle
e^{2\tau\Delta}v,v\rangle\,d\tau=\frac{1}{\beta}\langle\sigma_{\beta}v,v\rangle=\frac{1}{\beta}\|\sigma_{\beta}^{\frac{1}{2}}v\|_{H}^{2}.$

An application of the Cameron-Martin theorem yields

$\langle
DA_{\alpha,\beta}(u),h\rangle=\int_{H}e^{\beta\Delta}A_{\alpha}(e^{\beta\Delta}u+v)\langle\sigma_{\beta}^{-1}e^{\beta\Delta}h,v\rangle\,\mathcal{N}_{0,\sigma_{\beta}}(dv).$

and by the Cameron-Martin formula it is $C^{\infty}$ differentiable. Moreover,
as $\alpha,\beta\to 0,A_{\alpha,\beta}\to A$ pointwise. Let us introduce the
following approximating equation

(6.2) $\displaystyle
du_{\alpha,\beta}(t)=-A_{\alpha,\beta}(u_{\alpha,\beta}(t))\,dt+B\,dW_{t},\quad
t>0,\quad u_{\alpha,\beta}(0)=u.$

Since $A_{\alpha,\beta}$ is globally Lipschitz, equation (6.2) has a unique
strong solution $(u_{\alpha,\beta}(t))_{t\geq 0}$. Moreover, by the regularity
of $A_{\alpha,\beta}$ the process $(u_{\alpha,\beta}(t))_{t\geq 0}$ is
differentiable on $H$. For any $h\in H$ we set
$\eta_{h}(t,u):=Du_{\alpha,\beta}(t,u)\cdot h$. It holds that

(6.3)
$\displaystyle\frac{d}{dt}\eta_{h}(t,u)=-DA_{\alpha,\beta}(u_{\alpha,\beta}(t,u))\cdot\eta_{h}(t,u),\quad\eta_{h}(0,u)=h.$

From the dissipativity of $A_{\alpha,\beta}$, we have that

$\langle DA_{\alpha,\beta}(z)h,h\rangle\geq 0,\quad h\in H,z\in D(A).$

Hence by multiplying both sides of 6.3 with $\eta_{h}(t,u)$ and integrating
with respect to $t$, we get

(6.4) $\lVert\eta_{h}(t,u)\rVert^{2}\leq\lVert h\rVert^{2}.$

Now, for $\lambda>0$ and $F\in C_{b}^{2}(H)$ consider the following elliptic
equation

(6.5) $(\lambda-J_{A_{\alpha,\beta}})\varphi_{\alpha,\beta}=F,\ \lambda>0,$

where $J_{A_{\alpha,\beta}}$ is the Kolmogorov operator corresponding to the
stochastic differential equation (6.2). It is well-known that this equation
has a solution $\varphi_{\alpha,\beta}\in C_{b}^{2}(H)$ and can be written in
the form $\varphi_{\alpha,\beta}=R(\lambda,J_{A_{\alpha,\beta}})F$, where

$R(\lambda,J_{A_{\alpha,\beta}})F(u)=\int_{0}^{+\infty}e^{-\lambda
t}\mathbb{E}(F(u_{\alpha,\beta})(t,u))\,dt$

is the pseudo resolvent associated with $J_{A_{\alpha,\beta}}$. Thus we have

(6.6) $\lVert\lambda\varphi_{\alpha,\beta}\rVert_{\infty}\leq\lVert
f\rVert_{\infty}.$

We have, moreover, for all $h\in H$,

$D{\varphi_{\alpha,\beta}}(u)h=\int_{0}^{+\infty}e^{-\lambda
t}\mathbb{E}\left(DF(u_{\alpha,\beta})(t,u)((Du_{\alpha,\beta})(t,u)h)\right)dt.$

Consequently, using (6.4), it follows that

$\lVert D\varphi_{\alpha,\beta}\rVert\leq\frac{1}{\lambda}\lVert
DF\rVert_{\infty}.$

From (6.5), we have

$\displaystyle\lambda\varphi_{\alpha,\beta}(u)$
$\displaystyle-\frac{1}{2}\operatorname{Tr}_{H}\left(BB^{\ast}D^{2}\varphi(u)\right)+\langle
A(u),D\varphi_{\alpha,\beta}(u)\rangle$ $\displaystyle=F(u)+\langle
A(u)-A_{\alpha,\beta}(u),D\varphi_{\alpha,\beta}(u)\rangle,\ \lambda>0,\ u\in
D(A).$

Using gradient bound (6.6), we deduce that for $\alpha,\beta\to 0$,

$\int_{H}\lvert\langle
A(u)-A_{\alpha,\beta}(u),D\varphi_{\alpha,\beta}(u)\rangle\rvert\,\mu(du)\leq\frac{1}{\lambda}\lVert
DF\rVert_{\infty}\lVert A_{\alpha,\beta}-A\rVert_{L^{2}(H,\mu)}.$

By Lebesgue’s theorem $\lVert A_{\alpha,\beta}-A\rVert_{L^{2}(H,\mu)}$
converges to $0$ as $\alpha,\beta\to 0$. Therefore we deduce that for
$\alpha,\beta\to 0$

$\lambda\varphi_{\alpha,\beta}(u)-\frac{1}{2}\operatorname{Tr}_{H}\left(BB^{\ast}D^{2}\varphi_{\alpha,\beta}(u)\right)+\langle
A(u),D\varphi_{\alpha,\beta}(u)\rangle\to F$

strongly in $L^{1}(H,\mu)$. This implies that

$C_{b}^{2}(H)\subset\overline{(\lambda-J_{0})(D(J_{0}))}.$

Since $C_{b}^{2}(H)$ is dense in $L^{1}(H,\mu)$, the proof is complete.∎

## Statements and Declarations

The authors have no relevant financial or non-financial interests to disclose.
The authors have no competing interests to declare that are relevant to the
content of this article.

The work of the first author was partially supported by the German Science
Foundation (DFG) via Research Unit FOR 2402.

The second author acknowledges support by the Academy of Finland and the
European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme (grant agreements no. 741487 and no.
818437).

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license: Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/

authors: Florian Seib, Wilhelm Stannat and Jonas M. T\"olle

submitter: Jonas M. T\"olle Dr. math.