# Numerical Ergodicity and Uniform Estimate of Monotone SPDEs Driven by
Multiplicative Noise

Zhihui Liu  Department of Mathematics & National Center for Applied
Mathematics Shenzhen, Southern University of Science and Technology, Shenzhen
518055, China<EMAIL_ADDRESS>
###### Abstract.

We analyze the long-time behavior of numerical schemes for a class of monotone
SPDEs driven by multiplicative noise. By deriving several time-independent a
priori estimates for the numerical solutions, combined with the ergodic theory
of Markov processes, we establish the exponential ergodicity of these schemes
with a unique invariant measure, respectively. Applying these results to the
stochastic Allen–Cahn equation indicates that these schemes always have at
least one invariant measure, respectively, and converge strongly to the exact
solution with sharp time-independent rates. We also show that these numerical
invariant measures are exponentially ergodic and thus give an affirmative
answer to a question proposed in (J. Cui, J. Hong, and L. Sun, Stochastic
Process. Appl. (2021): 55–93), provided that the interface thickness is not
too small.

###### Key words and phrases:

monotone stochastic partial differential equation, stochastic Allen–Cahn
equation, numerical invariant measure, numerical ergodicity, time-independent
strong error estimate

###### 2010 Mathematics Subject Classification:

Primary 60H35; 60H15, 65L60

The author is supported by National Natural Science Foundation of China, No.
12101296, and Shenzhen Basic Research Special Project (Natural Science
Foundation) Basic Research (General Project), No. JCYJ20220530112814033.

##  1\. Introduction

Recently, a lot of researchers investigated numerical analysis of the
following stochastic partial differential equations (SPDE) under homogeneous
Dirichlet boundary condition in a finite time horizon:

(1.1) $\displaystyle\begin{split}&{\rm d}X(t,\xi)=(\Delta
X(t,\xi)+f(X(t,\xi))){\rm d}t+g(X(t,\xi)){\rm d}W(t,\xi),\\\
&X(t,\xi)=0,\quad(t,\xi)\in\mathbb{R}_{+}\times\partial\mathscr{O};\\\
&X(0,\xi)=X_{0}(\xi),\quad\xi\in\mathscr{O},\end{split}$

where the physical domain $\mathscr{O}\subset\mathbb{R}^{d}$ ($d=1,2,3$) is a
bounded open set with smooth boundary. Here $f$ is assumed to be of monotone
type with polynomial growth, $g$ satisfies the usual Lipschitz condition, in
an infinite-dimensional setting, and $W$ is an infinite-dimensional Wiener
process, see Section 2.1. Eq. (1.1) includes the following semiclassical
stochastic Allen–Cahn equation, arising from phase transition in materials
science by stochastic perturbation, as a special case:

(1.2) $\displaystyle{\rm d}X=\Delta X{\rm d}t+\alpha^{-2}(X-X^{3}){\rm
d}t+{\rm d}W,\quad X(0)=X_{0},$

where $\alpha>0$ is the interface thickness; see, e.g., [BGJK23], [BJ19],
[BCH19], [CH19], [GM09], [LQ19], [LQ20], [LQ21], [MP18] and references
therein.

The numerical behavior in the infinite time horizon, especially, the numerical
ergodicity of SPDEs, including Eq. (1.1) (and Eq. (1.2)), is a natural and
intriguing question. As a significant asymptotic behavior, the ergodicity
characterizes the case of temporal average coinciding with spatial average,
which has vital applications in quantum mechanics, fluid dynamics, financial
mathematics, and many other fields [DPZ96], [HW19]. The spatial average, i.e.,
the mean of a given function for the stationary law, called the invariant
measure of the diffusion is also known as the ergodic limit that is desirable
to compute in many practical applications. Then one has to investigate a
stochastic system over long-time intervals, which is one of the main
difficulties from the computational perspective. Generally, the explicit
expression of the invariant measure for a nonlinear infinite-dimensional
stochastic system is rarely available. Therefore, it is usually impossible to
exactly compute the ergodic limit for a nonlinear SPDE driven by
multiplicative noise; exceptional examples are gradient Langevin systems
(driven by additive noise), see, e.g., [BRVE10], [LV22]. For this reason, it
motivates and fascinates a lot of investigations in the recent decade for
constructing numerical algorithms that can inherit the ergodicity of the
original system and approximate the ergodic limit efficiently.

In finite-dimensional case, much progress has been made in the design and
analysis of numerical approximations of the desired ergodic limits, see, e.g.,
[MSH02], [WL19], [LMW23] and references therein for numerical ergodicity of
SODEs. Quite the contrary, the construction and analysis of numerical ergodic
limits for SPDEs are still in their early stages. The authors in [Bré14],
[BK17], [BV16] firstly studied Galerkin-based linear implicit Euler scheme and
high order integrator to approximate the invariant measures of a parabolic
SPDE driven by additive white noise; see also [CHW17] for the ergodic limit of
a spectral Galerkin modified implicit Euler scheme of the damping stochastic
Schrödinger equation driven by additive trace-class noise. Recently,
exponential Euler-type schemes of full discretization and tamed semi-
discretization were used in [CGW20] and [Bré22], respectively, to approximate
the invariant measure of a stochastic Allen–Cahn type equation driven by
additive colored noise. The authors in [CHS21] investigated the spectral
Galerkin drift implicit Euler (DIE) scheme to approximate the invariant
measure of the same type of equation and proposed a question of whether the
invariant measure of the temporal DIE scheme is unique. We note that almost
all of the above literature focus on the numerical ergodicity of SPDEs driven
by additive noise, the numerical ergodicity in the multiplicative noise case
is more subtle and challenging.

Apart from the approximation of the invariant measure, the strong
approximation, i.e., in the moments’ sense, is also an important issue. There
exists a general theory of strong error analysis of numerical approximations
for Lipschitz SPDEs in a finite time horizon; see, e.g., [ACLW16], [CHL17],
[JR15] and references therein. Recently, we [LQ21] give a theory of strong
error analysis of numerical approximations for monotone SPDEs in a finite time
horizon. Whether there is a theory of strong error analysis in the infinite
time horizon is a natural question.

The above two questions on long-time behaviors of numerical approximations for
Eq. (1.1) motivate the present study. On one hand, we aim to establish the
exponential ergodicity of the numerical schemes studied in [GM09], [LQ21],
[MP18] for Eq. (1.1). It should be pointed out that there is no further
restriction on the time step size for the DIE scheme (3.1) in addition to the
usual requirement of implicit scheme (see Theorem 3.1); see [LMW23] for an
upper bound requirement on the time step size for numerical ergodicity of
monotone SODEs. Meanwhile, applying one of our main results, Theorem 3.1, we
conclude that the DIE scheme (3.1) is unique ergodic (and exponentially
mixing) and thus give an affirmative answer to the question proposed in
[CHS21] (see Remark 4.1), provided that the monotone coefficient of the drift
function is not too large. On the other hand, we develop a theory of time-
independent strong error analysis for these fully discrete schemes, see
Theorem 3.3, which generalizes the main results of [LQ21] to the infinite time
horizon.

The paper is organized as follows. In Section 2, we give several necessary
uniform estimates and exponential ergodicity of Eq. (1.1). These uniform
estimates are utilized in Section 3 to establish the exponential ergodicity of
the proposed semi-discrete and fully discrete schemes. In Section 3, we also
derive time-independent strong error estimates for the aforementioned fully
discrete schemes. Finally, the main results in Section 3 are applied to the
stochastic Allen–Cahn equation (1.2) in Section 4.

##  2\. Uniform A Priori Estimates and Exponential Ergodicity of Monotone
SPDEs

In this section, we first present the required preliminaries and main
assumptions that will be used throughout the paper. Then we derive several
necessary uniform a priori estimates of Eq. (1.1).

###  2.1. Main Assumptions

Denote $a\vee b:=\max(a,b)$ and $a\wedge b:=\min(a,b)$ for $a,b\in\mathbb{R}$.
For $r\in[2,\infty]$ and $\theta=-1$ or $1$, we use the notations
$(L^{r}=L^{r}(\mathscr{O}),\|\cdot\|_{L^{r}})$ and
$(\dot{H}^{\theta}=\dot{H}^{\theta}(\mathscr{O}),\|\cdot\|_{\theta})$ to
denote the usual Lebesgue and Sobolev interpolation spaces, respectively. In
particular, we denote $H:=L^{2}$, $V:=\dot{H}^{1}$, and $V^{*}:=\dot{H}^{-1}$.
The space $H$ is equipped with Borel $\sigma$-algebra $\mathcal{B}(H)$ and
with the inner product and norm given by $\|\cdot\|$ and
$\langle\cdot,\cdot\rangle$, respectively. The norms in $\dot{H}^{1}$ and
$\dot{H}^{-1}=(\dot{H}^{1})^{*}$ and the duality product between them are
denoted by $\|\cdot\|_{1}$, $\|\cdot\|_{-1}$, and
${}_{1}\langle\cdot,\cdot\rangle_{-1}$, respectively. We use
$\mathcal{C}_{b}(H)$ and ${\rm Lip}_{b}(H)$ to denote the class of bounded,
continuous and Lipschitz continuous, respectively, functions on $H$. For
$\phi\in{\rm Lip}_{b}(H)$, define $|\phi|_{\rm Lip}:=\sup_{x\neq
y}|\phi(x)-\phi(y)|/\|x-y\|$.

Let ${\bf Q}$ be a self-adjoint and positive definite linear operator on $H$.
Denote $(H_{0}:=Q^{1/2}H,\|\cdot\|_{H_{0}}:=\|Q^{-1/2}\cdot\|)$ and denote by
$(\mathcal{L}_{2}^{0}:=HS(H_{0};H),\|\cdot\|_{\mathcal{L}_{2}^{0}})$ and
$(\mathcal{L}_{2}^{1}:=HS(H_{0};\dot{H}^{1}),\|\cdot\|_{\mathcal{L}_{2}^{1}})$
the space of Hilbert–Schmidt operators from $H_{0}$ to $H$ and $\dot{H}^{1}$,
respectively. Let $W$ be an $H$-valued Q-Wiener process on a filtered
probability space $(\Omega,\mathscr{F},\mathbb{F}:=(\mathscr{F}_{t})_{t\geq
0},\mathbb{P})$, which has the Karhunen–Loève expansion

$W(t)=\sum_{k\in\mathbb{N}}\sqrt{q_{k}}g_{k}\beta_{k}(t),\quad t\geq 0.$

Here $\\{g_{k}\\}_{k=1}^{\infty}$ forming an orthonormal basis of $H$ are the
eigenvectors of Q subject to the eigenvalues $\\{q_{k}\\}_{k=1}^{\infty}$, and
$\\{\beta_{k}\\}_{k=1}^{\infty}$ are mutually independent one-dimensional
Brownian motions in $(\Omega,\mathscr{F},\mathbb{F},\mathbb{P})$. We mainly
focus on trace-class noise, i.e., Q is a trace-class operator, or
equivalently, ${\rm Trace}({\bf Q}):=\sum_{k\in\mathbb{N}}q_{k}<\infty$.

Our main conditions on the coefficients of Eq. (1.1) are the following two
assumptions.

######  Assumption 2.1.

$f:\mathbb{R}\rightarrow\mathbb{R}$ is differentiable and there exist scalars
$L_{i}\in\mathbb{R}$, $i=1,2,3,4,5$, and $q\geq 1$ such that

(2.1) $\displaystyle(f(\xi)-f(\eta))(\xi-\eta)\leq
L_{1}(\xi-\eta)^{2},\quad\xi,\eta\in\mathbb{R},$ (2.2) $\displaystyle
f(\xi)\xi\leq L_{2}|\xi|^{2}+L_{3},\quad\xi\in\mathbb{R},$ (2.3)
$\displaystyle|f^{\prime}(\xi)|\leq
L_{4}|\xi|^{q-1}+L_{5},\quad\xi\in\mathbb{R}.$

######  Example 2.1.

A concrete example such that all the conditions in Assumption 2.1 holds is an
odd polynomial (with order $q$ and) with a negative leading coefficient.

By the condition (2.1) and the differentiability of $f$, it is clear that

(2.4) $\displaystyle f^{\prime}(\xi)\leq L_{1},\quad\xi\in\mathbb{R}.$

We also note that one can always assume that $L_{1}\leq L_{2}$, as one can
take $L_{2}=L_{1}+\varepsilon$ in (2.2) from (2.1) and Young inequality; here
and in the rest of the paper, $\varepsilon$ denotes an arbitrary small
positive constant which would be different in each appearance.

Define by $F:L^{q+1}\rightarrow L^{(q+1)^{\prime}}$ the Nemytskii operator
associated with $f$:

(2.5) $\displaystyle F(x)(\xi):=f(x(\xi)),\quad x\in\dot{H}^{1},\
\xi\in\mathscr{O},$

where $(q+1)^{\prime}$ denotes the conjugation of $q+1$, i.e.,
$1/(q+1)^{\prime}+1/(q+1)=1$. Then it follows from the monotonicity condition
(2.1) and the coercivity condition (2.2) that the operator $F$ defined in
(2.5) has a continuous extension from $L^{q+1}$ to $L^{(q+1)^{\prime}}$ and
satisfies

(2.6) ${}_{L^{q+1}}\langle x-y,F(x)-F(y)\rangle_{L^{(q+1)^{\prime}}}\leq
L_{1}\|x-y\|^{2},$ $\displaystyle\quad x,y\in L^{q+1},$ (2.7)
${}_{L^{q+1}}\langle x,F(x)\rangle_{L^{(q+1)^{\prime}}}\leq
L_{2}\|x\|^{2}+L_{3},$ $\displaystyle\quad x\in L^{q+1},$

where ${}_{L^{(q+1)^{\prime}}}\langle\cdot,\cdot\rangle_{L^{q+1}}$ denotes the
duality product between $L^{(q+1)^{\prime}}$ and $L^{q+1}$.

Throughout, we assume that $q\geq 1$ when $d=1,2$ and $q\in[1,3]$ when $d=3$,
so that the following frequently used Sobolev embeddings holds:

(2.8) $\displaystyle\dot{H}^{1}\subset L^{2q}\subset L^{q+1}\subset H\subset
L^{(q+1)^{\prime}}\subset L^{(2q)^{\prime}}\subset\dot{H}^{-1}.$

In this case, (2.6) and (2.7) can be expressed as

(2.9) ${}_{1}\langle x-y,F(x)-F(y)\rangle_{-1}\leq L_{1}\|x-y\|^{2},$
$\displaystyle\quad x,y\in\dot{H}^{1},$ (2.10) ${}_{1}\langle
x,F(x)\rangle_{-1}\leq L_{2}\|x\|^{2}+L_{3},$ $\displaystyle\quad
x\in\dot{H}^{1}.$

The inequality (2.10), in combination with the Poincaré inequality that

(2.11) $\displaystyle\|\nabla x\|^{2}\geq\lambda_{1}\|x\|^{2},\quad
x\in\dot{H}^{1},$

where $\lambda_{1}$ denotes the first eigenvalue of the Dirichlet Laplacian
operator, denoted by $A$ with domain ${\rm Dom}(A)=H^{2}\cap\dot{H}^{1}$,
implies that

${}_{1}\langle x-y,A(x-y)+F(x)-F(y)\rangle_{-1}$ (2.12)
$\displaystyle\leq-(\lambda_{1}-L_{1})\|x-y\|^{2},\quad x,y\in\dot{H}^{1},$

and that

(2.13) ${}_{1}\langle x,Ax+F(x)\rangle_{-1}$
$\displaystyle\leq-(\lambda_{1}-L_{2})\|x\|^{2}+L_{3},\quad x\in\dot{H}^{1}.$

Similarly, noting that $L_{1}\leq L_{2}$, we have

(2.14) $\displaystyle\langle x,Ax+F(x)\rangle_{1}$
$\displaystyle\leq-(\lambda_{1}-L_{2})\|x\|_{1}^{2}+L_{3},\quad
x\in\dot{H}^{2}.$

Denote by $G:H\rightarrow\mathcal{L}_{2}^{0}$ the Nemytskii operator
associated with $g:\mathbb{R}\rightarrow\mathbb{R}$:

(2.15) $\displaystyle G(x)g_{k}(\xi):=g(x(\xi))g_{k}(\xi),\quad x\in
H,~{}k\in\mathbb{N},~{}\xi\in\mathscr{O}.$

Our main condition on the diffusion operator $G$ defined in (2.15) is the
following usual Lipschitz continuity and linear growth conditions.

######  Assumption 2.2.

There exist positive constants $L_{6}$, $L_{7}$, and $L_{8}$ such that

(2.16) $\displaystyle\|G(x)-G(y)\|_{\mathcal{L}_{2}^{0}}^{2}\leq
L_{6}\|x-y\|^{2},$ $\displaystyle\quad x,y\in H,$ (2.17)
$\displaystyle\|G(x)\|^{2}_{\mathcal{L}_{2}^{0}}\leq L_{7}\|x\|^{2}+L_{8},$
$\displaystyle\quad x\in H.$

Moreover, $G(\dot{H}^{1})\subset\mathcal{L}_{2}^{1}$ and there exist positive
constants $L_{9}$ and $L_{10}$ such that

(2.18) $\displaystyle\|G(z)\|^{2}_{\mathcal{L}_{2}^{1}}\leq
L_{9}\|z\|^{2}_{1}+L_{10},\quad z\in\dot{H}^{1}.$

######  Remark 2.1.

In the additive noise case $G\equiv{\rm Id}$, the identity operator in $H$,
the conditions (2.16)–(2.18) reduce to the assumption $\|(-A)^{1/2}{\bf
Q}^{1/2}\|_{HS(H;H)}<\infty$ which was imposed in [QW19] and [CHS21]; see
[LQ21, Remark 2.2] for an example of multiplicative noise case.

With these preliminaries, (1.1) is equivalent to the following infinite-
dimensional stochastic evolution equation:

(2.19) $\displaystyle\begin{split}{\rm d}X_{t}&=(AX_{t}+F(X_{t})){\rm
d}t+G(X_{t}){\rm d}W,\quad t\geq 0;\\\ X(0)&=X_{0},\end{split}$

where the random initial datum $X_{0}$ is assumed to vanish on the boundary
$\partial\mathscr{O}$ of the physical domain throughout the present paper.

###  2.2. Uniform Moments’ Estimates of The Exact Solution

The existence of a unique variational solution of Eq. (2.19) in $H$ and
$\dot{H}^{1}$ was shown in [Cer01] or [KR07] and [Liu13], respectively, under
similar conditions as in Assumptions 2.1–2.2. This solution is also proved to
be the mild solution of (2.19) in [LQ21], where the author also derived that
the solution is indeed continuous a.s. and satisfies certain moments’
estimates in a finite time horizon.

Taking advantage of the dissipativity conditions (2.13) and (2.14), we can
show the following uniform stability estimate both in $H$ and $\dot{H}^{1}$ of
Eq. (2.19). Throughout, we assume without loss of generality that the initial
datum $X_{0}$ of Eq. (2.19) is a deterministic function; $X_{t}^{x}$ denotes
the solution of Eq. (2.19) starting from $X_{0}=x$. Note that we will
frequently use $C$ to denote generic time-independent constants independent of
various discrete parameters that would be different in each appearance.

######  Proposition 2.1.

Let $p\geq 2$, $X_{0}\in H$, Assumption 2.1 and conditions (2.16)–(2.17) hold.
Assume that $L_{2}+\frac{p-1}{2}L_{7}<\lambda_{1}$. Then there exist positive
constants $\gamma_{1}$ and $C_{\gamma_{1}}$ such that

(2.20) $\displaystyle\mathbb{E}\|X_{t}\|^{p}$ $\displaystyle\leq
e^{-\gamma_{1}t}\|X_{0}\|^{p}+C_{\gamma_{1}},\quad t\geq 0.$

Moreover, if $X_{0}\in\dot{H}^{1}$, $L_{2}+\frac{p-1}{2}L_{9}<\lambda_{1}$,
and condition (2.18) hold, then there exist positive constants $\gamma_{2}$
and $C_{\gamma_{2}}$ such that

(2.21) $\displaystyle\mathbb{E}\|X_{t}\|^{p}_{1}$ $\displaystyle\leq
e^{-\gamma_{2}t}\|X_{0}\|^{p}_{1}+C_{\gamma_{2}},\quad t\geq 0.$

###### Proof.

Applying Itô formula (see, e.g., [LR15, Theorem 4.2.5]) to the functional
$\|X\|^{p}$ leads to

$\displaystyle{\rm d}\|X\|^{p}$
$\displaystyle=\Big{[}\frac{p(p-2)}{2}\|X\|^{p-4}\sum_{k\in\mathbb{N}_{+}}~{}\langle
X,G(X){\bf Q}^{\frac{1}{2}}g_{k}\rangle^{2}$
$\displaystyle\quad+\frac{p}{2}\|X\|^{p-2}\Big{(}2~{}_{1}\langle
X,AX+F(X)\rangle_{-1}+\|G(X)\|^{2}_{\mathcal{L}_{2}^{0}}\Big{)}\Big{]}{\rm
d}t$ (2.22) $\displaystyle\quad+p\|X\|^{p-2}\langle X,G(X){\rm d}W\rangle.$

To lighten the notations, here and after we omit the temporal variable when an
integral appears. For $\gamma_{1}\in\mathbb{R}$, it follows from the product
rule that

$\displaystyle e^{\gamma_{1}t}\|X_{t}\|^{p}-\|X_{0}\|^{p}$
$\displaystyle=\int_{0}^{t}e^{\gamma_{1}r}\Big{[}\frac{p(p-2)}{2}\|X\|^{p-4}\sum_{k\in\mathbb{N}_{+}}\langle
X,G(X){\bf Q}^{\frac{1}{2}}g_{k}\rangle^{2}$
$\displaystyle\quad+\frac{p}{2}\|X\|^{p-2}(2~{}_{1}\langle
X,AX+F(X)\rangle_{-1}+\|G(X)\|^{2}_{\mathcal{L}_{2}^{0}})+\gamma_{1}\|X\|^{p}\Big{]}{\rm
d}r$ (2.23) $\displaystyle\quad+p\int_{0}^{t}e^{\gamma_{1}r}\|X\|^{p-2}\langle
X,G(X){\rm d}W\rangle.$

By the condition (2.17) and Young inequality, we have

$\displaystyle\|X\|^{p-4}\sum_{k\in\mathbb{N}_{+}}\langle X,G(X){\bf
Q}^{\frac{1}{2}}g_{k}\rangle^{2}\leq(L_{7}+\varepsilon)\|X\|^{p}+C_{\varepsilon},$
$\displaystyle\|X\|^{p-2}\|G(X)\|^{2}_{\mathcal{L}_{2}^{0}}\leq(L_{7}+\varepsilon)\|X\|^{p}+C_{\varepsilon},$

for $X\in H$. Combining the above two estimates and (2.13), taking expectation
on Eq. (2.2), and noting that $\int_{0}^{t}e^{\gamma_{1}r}\|X\|^{p-2}\langle
X,G(X){\rm d}W\rangle$ is a (real-valued) martingale, we obtain

(2.24) $\displaystyle e^{\gamma_{1}t}\mathbb{E}\|X_{t}\|^{p}$
$\displaystyle\leq\|X_{0}\|^{p}-\gamma_{1}^{*}\int_{0}^{t}e^{\gamma_{1}r}\mathbb{E}\|X\|^{p}{\rm
d}r+C_{\varepsilon}\int_{0}^{t}e^{\gamma_{1}r}{\rm d}r,$

with
$\gamma_{1}^{*}:=p(\lambda_{1}-L_{2}-\frac{p-1}{2}(L_{7}+\varepsilon))-\gamma_{1}$.
As $L_{2}+\frac{p-1}{2}L_{7}<\lambda_{1}$, one can take $\varepsilon>0$ and
$\gamma_{1}>0$ to be sufficiently small such that $\gamma_{1}^{*}>0$. Then it
follows that

$\displaystyle\mathbb{E}\|X_{t}\|^{p}$ $\displaystyle\leq
e^{-\gamma_{1}t}\|X_{0}\|^{p}+C_{\varepsilon}\int_{0}^{t}e^{-\gamma_{1}r}{\rm
d}r,$

from which we obtain (2.20).

Now let $X_{0}\in\dot{H}^{1}$. Applying Itô formula to the functional
$e^{\gamma_{1}t}\|X_{t}\|^{p}_{1}$ leads to

$\displaystyle e^{\gamma_{2}t}\|X_{t}\|^{p}_{1}-\|X_{0}\|^{p}_{1}$
$\displaystyle=\int_{0}^{t}e^{\gamma_{2}r}\Big{[}\frac{p(p-2)}{2}\|X\|_{1}^{p-4}\sum_{k\in\mathbb{N}_{+}}\langle
X,G(X){\bf Q}^{\frac{1}{2}}g_{k}\rangle_{1}^{2}$
$\displaystyle\quad+\frac{p}{2}\|X\|_{1}^{p-2}(2\langle
X,AX+F(X)\rangle_{1}+\|G(X)\|^{2}_{\mathcal{L}_{2}^{1}})+\gamma_{2}\|X\|^{p}_{1}\Big{]}{\rm
d}r$ $\displaystyle\quad+p\int_{0}^{t}e^{\gamma_{2}r}\|X\|_{1}^{p-2}\langle
X,G(X){\rm d}W\rangle_{1}.$

By the conditions (2.16) and (2.18), we have

$\displaystyle\|X\|_{1}^{p-4}\sum_{k\in\mathbb{N}_{+}}\langle X,G(X){\bf
Q}^{\frac{1}{2}}g_{k}\rangle_{1}^{2}\leq(L_{9}+\varepsilon)\|X\|_{1}^{p}+C_{\varepsilon},\quad
u\in\dot{H}^{1},$
$\displaystyle\|X\|_{1}^{p-2}\|G(X)\|^{2}_{\mathcal{L}_{2}^{1}}\leq(L_{9}+\varepsilon)\|X\|_{1}^{p}+C_{\varepsilon},\quad
u\in\dot{H}^{1}.$

Using techniques similar to those used to establish (2.24), together with the
above two estimates, (2.14), and the fact that
$\int_{0}^{t}e^{\gamma_{2}r}\|X\|_{1}^{p-2}\langle X,G(X){\rm d}W\rangle_{1}$
is a martingale, implies that

$\displaystyle e^{\gamma_{2}t}\mathbb{E}\|X_{t}\|^{p}_{1}$
$\displaystyle\leq\|X_{0}\|^{p}_{1}-\gamma_{2}^{*}\int_{0}^{t}e^{\gamma_{2}r}\mathbb{E}\|X\|_{1}^{p}{\rm
d}r+C_{\varepsilon}\int_{0}^{t}e^{\gamma_{2}r}{\rm d}r,$

with
$\gamma_{2}^{*}:=p(\lambda_{1}-L_{2}-\frac{p-1}{2}(L_{9}+\varepsilon))-\gamma_{2}>0$
for some sufficiently small and positive $\varepsilon$ and $\gamma_{2}$, as
$L_{2}+\frac{p-1}{2}L_{9}<\lambda_{1}$. Then it follows that

$\displaystyle\mathbb{E}\|X_{t}\|_{1}^{p}$ $\displaystyle\leq
e^{-\gamma_{2}t}\|X_{0}\|^{p}+C_{\varepsilon}\int_{0}^{t}e^{-\gamma_{2}(t-r)}{\rm
d}r,$

from which we conclude (2.21). ∎∎

As a by product of the uniform stability estimate (2.21), we have the
following uniform temporal Hölder continuity estimate; see [BIK16, Proposition
5.4] for a similar result in finite-dimensional case. It will be used in
Section 4 to derive the time-independent strong error estimates of the
proposed implicit numerical schemes.

######  Corollary 2.1.

Let $p\geq 2$, $X_{0}\in\dot{H}^{1}$, and Assumptions 2.1–2.2 hold. Assume
that $L_{1}+\frac{p-1}{2}L_{9}<\lambda_{1}$. Then there exists a constant $C$
such that

(2.25) $\displaystyle\sup_{t>s\geq
0}\frac{\mathbb{E}\|X_{t}-X_{s}\|^{p}}{(t-s)^{p/2}}\leq
C(1+\|X_{0}\|_{1}^{pq}).$

###### Proof.

For any $t>s\geq 0$, it was shown in [LQ21, Theorem 3.2] that

$\displaystyle\mathbb{E}\|X_{t}-X_{s}\|^{p}$ $\displaystyle\leq
C(t-s)^{p/2}(1+\|X_{0}\|_{1}^{p}+\sup_{t\geq
0}\mathbb{E}\|F(X_{t})\|_{-1}^{p}+\sup_{t\geq
0}\mathbb{E}\|G(X_{t})\|_{\mathcal{L}_{2}^{0}}^{p}).$

This estimate, together with the conditions (2.3), (2.17), the embeddings
(2.8), and the stability estimate (2.21), implies the desired estimate (2.25).
∎∎

###  2.3. Exponential Ergodicity of The Exact Solution

It is clear that the solution of Eq. (2.19) is a Markov process, see, e.g.,
[LR15, Proposition 4.3.5], so we define the family of Markov semigroup
$\\{P_{t}\\}_{t\geq 0}$ associated with Eq. (2.19) as

$\displaystyle(P_{t}\phi)(x):=\mathbb{E}\phi(X_{t}^{x}),\quad x\in H,~{}t\geq
0.$

A probability measure $\mu$ on $H$ is called invariant for
$\\{P_{t}\\}_{t\in\mathbb{I}}$ with either $\mathbb{I}=\mathbb{R}_{+}$ or
$\mathbb{I}=\mathbb{N}$, if

$\displaystyle\int_{H}P_{t}\phi(x)\mu({\rm
d}x)=\mu(\phi):=\int_{H}\phi(x)\mu({\rm d}x),\quad\phi\in\mathcal{C}_{b}(H),\
t\in\mathbb{I}.$

An invariant (probability) measure $\mu$ is called ergodic for
$\\{P_{t}\\}_{t\geq 0}$ (or Eq. (2.19)) or $\\{P_{m}\\}_{m\in\mathbb{N}}$ if
for all $\phi\in L^{2}(H;\mu)$, it holds, respectively, that

$\displaystyle\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}P_{t}\phi{\rm
d}t\Big{(}\text{or}~{}\lim_{m\to\infty}\frac{1}{m}\sum_{k=0}^{m}P_{k}\phi\Big{)}=\mu(\phi)\quad\text{in}~{}L^{2}(H;\mu).$

The following unique ergodicity of Eq. (2.19), with an exponential convergence
to the equilibrium, was proved using a remote initial condition method in
[LR15, Theorem 4.3.9].

######  Theorem 2.1.

Let Assumption 2.1 and conditions (2.16)-(2.17) hold. Assume that
$(L_{1}+L_{6}/2)\vee(L_{2}+L_{7}/2)<\lambda_{1}$. Then there exists a unique
invariant measure $\pi$ in $H$ for Eq. (2.19). Moreover, for any $t\geq 0$,
$x\in H$, and $\phi\in{\rm Lip}_{b}(H)$, there exist a positive constant
$\gamma_{3}$ such that

(2.26)
$\displaystyle|P_{t}\phi(x)-\pi(\phi)|\leq\exp(-\gamma_{3}t/2)|\phi|_{\rm
Lip}\int_{H}\|x-y\|\pi({\rm d}y).$

######  Remark 2.2.

A Markov semigroup $\\{P_{t}\\}_{t\in\mathbb{I}}$ with
$\mathbb{I}=\mathbb{R}_{+}$ or $\mathbb{I}=\mathbb{N}$ satisfying an
exponential convergence to the equilibrium, like (2.26), is called
exponentially mixing. It is clear that once $\\{P_{t}\\}$ is exponentially
mixing for the invariant measure $\mu$, then the following strong law of large
numbers (implying the ergodicity) holds:

$\displaystyle\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}P_{t}\phi(x){\rm
d}t~{}\Big{(}\text{or}~{}\lim_{m\to\infty}\frac{1}{m}\sum_{k=0}^{m}P_{k}\phi(x)\Big{)}=\mu(\phi),\quad\forall~{}x\in
H,~{}\mu\text{-a.s.}$

##  3\. Exponential Ergodicity and Uniform Estimates of Galerkin-based Euler
Scheme

In this section, we will first establish the exponential ergodicity of both
the drift-implicit Euler (DIE) semi-discrete scheme and Galerkin-based fully
discrete scheme. Then we will derive uniform strong error estimates of these
fully discrete scheme. At the end of this section, some generalization to the
Milstein-type scheme and its Galerkin discretization is given.

###  3.1. Exponential Ergodicity of DIE Scheme

Let $\tau\in(0,1)$. For an $m\in\mathbb{N}$, define $t_{m}:=m\tau$. We
discrete Eq. (2.19) in time with a DIE scheme and the resulting temporal semi-
discrete problem is to find an $\dot{H}^{1}$-valued Markov chain $\\{X_{m}:\
m\in\mathbb{N}_{+}\\}$ such that

(3.1) $\displaystyle X_{m+1}=X_{m}+\tau AX_{m+1}+\tau
F(X_{m+1})+G(X_{m})\delta_{m}W,$

starting from $X_{0}\in H$, where $\delta_{m}W=W(t_{m+1})-W(t_{m})$,
$m\in\mathbb{N}$. This DIE scheme had been widely studied; see, e.g., [CHS21],
[FLZ17], [LQ21], [QW19], and references therein. Throughout we assume that
$\tau\in(0,1)$ with $(L_{1}-\lambda_{1})\tau<1$ so that (3.1) is well-
posedness.

We begin with the following uniform moments’ stability of the above DIE scheme
(3.1). It should be pointed out that there is no other restriction on the time
step size $\tau\in(0,1)$; see [LMW23, Theorems 4.1 and 4.2] for an upper bound
requirement on $\tau$ for the numerical ergodicity of monotone SODEs.

######  Lemma 3.1.

Let $X_{0}\in H$, Assumption 2.1 and conditions (2.16)-(2.17) hold, and
$\\{X_{m}:m\in\mathbb{N}\\}$ be the solution of the DIE scheme (3.1) starting
from $X_{0}$. Assume that $L_{2}+L_{7}/2<\lambda_{1}$. Then there exist
positive constants $\gamma_{5}$, $C_{\gamma_{5}}$, and $C$ such that for any
$\tau\in(0,1)$,

(3.2) $\displaystyle\mathbb{E}\|X_{m}\|^{2}\leq
e^{-\gamma_{5}t_{m}}\|X_{0}\|^{2}+C_{\gamma_{5}},\quad m\in\mathbb{N},$ (3.3)
$\displaystyle\sup_{m\in\mathbb{N}_{+}}\frac{1}{m}\sum_{k=0}^{m}\mathbb{E}\|X_{k}\|_{1}^{2}\leq
C(1+\|X_{0}\|^{2}).$

###### Proof.

For simplicity, set $F_{k}=F(X_{k})$ and $G_{k}=G(X_{k})$ for
$k\in\mathbb{N}$. Testing (3.1) using $\langle\cdot,\cdot\rangle$-inner
product with $X_{k+1}$ and using the elementary equality

(3.4) $\displaystyle 2\langle
x-y,x\rangle=\|x\|^{2}-\|y\|^{2}+\|x-y\|^{2},\quad x,y\in H,$

we have

$\displaystyle\|X_{k+1}\|^{2}-\|X_{k}\|^{2}+\|X_{k+1}-X_{k}\|^{2}$
$\displaystyle=2~{}_{1}\langle X_{k+1},\Delta X_{k+1}+F_{k+1}\rangle_{-1}\tau$
(3.5) $\displaystyle\quad+2\langle
X_{k+1}-X_{k},G_{k}\delta_{k}W\rangle+2\langle X_{k},G_{k}\delta_{k}W\rangle.$

By the estimate (2.13) and Cauchy–Schwarz inequality, we have

$\displaystyle\|X_{k+1}\|^{2}-\|X_{k}\|^{2}\leq-2(\lambda_{1}-L_{2})\|X_{k+1}\|^{2}\tau+\|G_{k}\delta_{k}W\|^{2}+2\langle
X_{k},G_{k}\delta_{k}W\rangle+2L_{3}\tau.$

It follows that

$\displaystyle(1+2(\lambda_{1}-L_{2})\tau)\|X_{k+1}\|^{2}\leq\|X_{k}\|^{2}+\|G_{k}\delta_{k}W\|^{2}+2\langle
X_{k},G_{k}\delta_{k}W\rangle+2L_{3}\tau.$

Taking expectations on both sides, noting the fact that both $G_{k}$ and
$X_{k}$ are independent of $\delta_{k}W$, using Itô isometry and (2.17), we
get

$\displaystyle(1+2(\lambda_{1}-L_{2})\tau)\mathbb{E}\|X_{k+1}\|^{2}\leq(1+L_{7}\tau)\mathbb{E}\|X_{k}\|^{2}+(2L_{3}+L_{8})\tau,$

from which we obtain

$\displaystyle\mathbb{E}\|X_{k}\|^{2}$
$\displaystyle\leq\Big{(}\frac{1+L_{7}\tau}{1+2(\lambda_{1}-L_{2})\tau}\Big{)}^{k}\|X_{0}\|^{2}$
$\displaystyle\quad+\frac{(2L_{3}+L_{8})\tau}{1+2(\lambda_{1}-L_{2})\tau}\sum_{j=0}^{k-1}\Big{(}\frac{1+L_{7}\tau}{1+2(\lambda_{1}-L_{2})\tau}\Big{)}^{j},\quad
k\in\mathbb{N}_{+}.$

Note that

(3.6) $\displaystyle a^{k}<e^{-(1-a)k},\quad\forall~{}a\in(0,1),$

and that $\frac{1+L_{7}\tau}{1+2(\lambda_{1}-L_{2})\tau}<1$ for any
$\tau\in(0,1)$, which is ensured by the condition $L_{2}+L_{7}/2<\lambda_{1}$,
we conclude (3.2) with
$\gamma_{5}:=\frac{2(\lambda_{1}-L_{2})-L_{7}}{1+2(\lambda_{1}-L_{2})}$ and
$C_{\gamma_{5}}:=\frac{2L_{3}+L_{8}}{2(\lambda_{1}-L_{2})-L_{7}}$.

Substracting $2\varepsilon~{}_{1}\langle X_{k+1},\Delta X_{k+1}\rangle_{-1}$,
with small $\varepsilon>0$, from the first term on the right hand side in Eq.
(3.1), and using the arguments in the above, we obtain

$\displaystyle(1+2(\lambda_{1}-L_{2}-\varepsilon)\tau)\mathbb{E}\|X_{k+1}\|^{2}+2\varepsilon\mathbb{E}\|\nabla
X_{k+1}\|^{2}\tau$
$\displaystyle\leq(1+L_{7}\tau)\mathbb{E}\|X_{k}\|^{2}+(2L_{3}+L_{8})\tau,$

from which we derive

$\displaystyle\mathbb{E}\|X_{k}\|^{2}+\frac{2\varepsilon}{1+2(\lambda_{1}-L_{2}-\varepsilon)\tau}\sum_{j=0}^{k}\mathbb{E}\|\nabla
X_{j}\|^{2}\tau$
$\displaystyle\leq\Big{(}\frac{1+L_{7}\tau}{1+2(\lambda_{1}-L_{2}-\varepsilon)\tau}\Big{)}^{k}\|X_{0}\|^{2}$
$\displaystyle\quad+\frac{(2L_{3}+L_{8})\tau}{1+2(\lambda_{1}-L_{2})\tau}\sum_{j=0}^{k-1}\Big{(}\frac{1+L_{7}\tau}{1+2(\lambda_{1}-L_{2})\tau}\Big{)}^{j},\quad
k\in\mathbb{N}_{+}.$

Then we conclude (3.3) by the above estimate, the elementary estimate (3.6),
and (3.2). ∎∎

In the discrete case, there is no Itô formula anymore. Whether $\\{e^{\gamma
t_{m}}\|X_{m}^{x}-X_{m}^{y}\|^{2}:m\in\mathbb{N}\\}$, for some $\gamma$, is
discrete supermartingale is unknown, where $\\{X_{m}^{x}:m\in\mathbb{N}\\}$
and $\\{X_{m}^{y}:m\in\mathbb{N}\\}$ are denoted by the solution of the DIE
scheme (3.1) starting from $x$ and $y$, respectively. Fortunately, using the
monotonicity conditions in Assumptions 2.1-2.2, one can show the following
discrete uniform continuity dependence estimate similar to the finite-
dimensional case in [LMY19, Theorem 7.6] or [LMW23, Lemma 4.2].

######  Lemma 3.2.

Let $x,y\in H$, Assumption 2.1 and conditions (2.16)-(2.17) hold. Assume that
$L_{1}+L_{6}/2<\lambda_{1}$. Then there exist a positive constant $\gamma_{6}$
such that for any $\tau\in(0,1)$,

(3.7) $\displaystyle\mathbb{E}\|X_{m}^{x}-X_{m}^{y}\|^{2}\leq
e^{-\gamma_{6}t_{m}}\|x-y\|^{2},\quad m\in\mathbb{N}.$

Consequently, $\\{P_{m}\\}_{m\in\mathbb{N}_{+}}$ is Feller, i.e., if
$\phi\in\mathcal{C}_{b}(H)$, then $P_{m}\phi\in\mathcal{C}_{b}(H)$ for all
$m\in\mathbb{N}_{+}$ (or equivalently, $P_{1}\phi\in\mathcal{C}_{b}(H)$).

###### Proof.

For $k\in\mathbb{N}$, set $F_{k}^{x}=F(X_{k}^{x})$, $F_{k}^{y}=F(X_{k}^{y})$,
$G_{k}^{x}=G(X_{k}^{x})$, $G_{k}^{y}=G(X_{k}^{y})$, and
$E_{k}=X_{k}^{x}-X_{k}^{y}$. Using the same technique in the proof of Lemma
3.1 to derive (3.1), we have

$\displaystyle\|E_{k+1}\|^{2}-\|E_{k}\|^{2}+\|E_{k+1}-E_{k}\|^{2}$
$\displaystyle=2~{}_{1}\langle E_{k+1},\Delta
E_{k+1}+F^{x}_{k+1}-F^{y}_{k+1}\rangle_{-1}\tau$ $\displaystyle\quad+2\langle
E_{k+1}-E_{k},(G^{x}_{k}-G^{y}_{k})\delta_{k}W\rangle+2\langle
E_{k},(G^{x}_{k}-G^{y}_{k})\delta_{k}W\rangle.$

By the estimate (2.12) and Cauchy–Schwarz inequality, there exists a constant
$C$ such that

$\displaystyle\|E_{k+1}\|^{2}-\|E_{k}\|^{2}\tau$
$\displaystyle\leq-2(\lambda_{1}-L_{1})\|E_{k+1}\|^{2}\tau+\|(G^{x}_{k}-G^{y}_{k})\delta_{k}W\|^{2}$
$\displaystyle\quad+2\langle E_{k},(G^{x}_{k}-G^{y}_{k})\delta_{k}W\rangle.$

Taking expectations on both sides, noting the fact that both
$G^{x}_{k}-G^{y}_{k}$ and $X_{k}$ are independent of $\delta_{k}W$, using Itô
isometry and (2.16), we obtain

$\displaystyle(1+2(\lambda_{1}-L_{1})\tau)\mathbb{E}\|E_{k+1}\|^{2}\leq(1+L_{6}\tau)\mathbb{E}\|E_{k}\|^{2},$

from which we obtain

$\displaystyle\mathbb{E}\|E_{k}\|^{2}\leq\Big{(}\frac{1+L_{6}\tau}{1+2(\lambda_{1}-L_{1})\tau}\Big{)}^{k}\|X-Y\|^{2},\quad
k\in\mathbb{N}.$

Then we conclude (3.7) with
$\gamma_{6}:=\frac{2(\lambda_{1}-L_{1})-L_{6}}{2(\lambda_{1}-L_{1})+1}$ by the
above estimate and (3.6). ∎∎

Now we can show that the DIE scheme (3.1) is ergodic with a unique invariant
measure. Moreover, it converges exponentially to the equilibrium.

######  Theorem 3.1.

Let Assumption 2.1 and conditions (2.16)-(2.17) hold. Assume that
$(L_{1}+L_{6}/2)\vee(L_{2}+L_{7}/2)<\lambda_{1}$. Then for any $\tau\in(0,1)$,
the DIE scheme (3.1) possesses a unique invariant measure $\pi_{\tau}$ in $H$
which is exponentially mixing:

(3.8)
$\displaystyle|P_{m}\phi(x)-\pi_{\tau}(\phi)|\leq\exp(-\gamma_{6}t_{m}/2)|\phi|_{\rm
Lip}\int_{H}\|x-y\|\pi_{\tau}({\rm d}y),$

for any $m\in\mathbb{N}$, $x\in H$, and $\phi\in{\rm Lip}_{b}(H)$, where
$\gamma_{6}$ appeared at (3.7).

###### Proof.

Define a sequence $\\{\mu^{\tau}_{m}\\}_{m\in\mathbb{N}_{+}}$ of probability
measures by

$\displaystyle\mu^{\tau}_{m}(B):=\frac{1}{m}\sum_{k=0}^{m}P_{k}\chi_{B}(0)dt,\quad
B\in\mathcal{B}(H),~{}m\in\mathbb{N}_{+}.$

By Chebyshev inequality and the temporal average estimate (3.3) (with
$X_{0}=0$), for any $\varepsilon>0$, there exists a positive scalar
$I_{\varepsilon}$ and a compact set $B=\\{x\in H:~{}\|x\|_{1}\leq
I_{\varepsilon}\\}\subset\mathcal{B}(H)$ such that

$\displaystyle\mu^{\tau}_{m}(B^{c})$
$\displaystyle=\frac{1}{m}\sum_{k=0}^{m}\mathbb{P}(\|X_{k}^{0}\|_{1}>I_{\varepsilon})dt\leq\frac{I^{-2}_{\varepsilon}}{m}\sum_{k=0}^{m}\mathbb{E}\|X_{k}^{0}\|_{1}^{2}\leq
CI^{-2}_{\varepsilon}<\varepsilon,$

from which we have that $\\{\mu^{\tau}_{m}\\}_{m\in\mathbb{N}_{+}}$ is tight.
Then we conclude the existence of an invariant measure $\pi_{\tau}$ of
$\\{P_{m}\\}_{m\in\mathbb{N}}$ with $\pi_{\tau}(\|\cdot\|^{2})<\infty$, via
Krylov–Bogoliubov theorem (see, e.g., [HW19, Theorem 1.2]), by the tightness
of $\\{\mu^{\tau}_{m}\\}_{m\in\mathbb{N}_{+}}$ and the Feller property of
$\\{P_{m}\\}_{m\in\mathbb{N}}$ in Lemma 3.2.

Now let $\mu$ be any invariant measure of $\\{P_{m}\\}_{m\geq 0}$. By the
property of invariant measure and continuous dependence estimate (3.7),

$\displaystyle|P_{m}\phi(x)-\mu(\phi)|$
$\displaystyle=\Big{|}\int_{H}(P_{m}\phi(x)-P_{m}\phi(y))\mu({\rm d}y)\Big{|}$
$\displaystyle\leq|\phi|_{\rm
Lip}\int_{H}\mathbb{E}\|X_{t}^{x}-X_{t}^{y}\|\mu({\rm d}y)$
$\displaystyle\leq\exp(-\gamma_{6}t_{m}/2)|\phi|_{\rm
Lip}\int_{H}\|x-y\|\mu({\rm d}y),$

for any $x\in H$ and $m\geq 0$. Taking $m\to\infty$, we have that all such
invariant measures coincide, i.e., $\mu=\pi_{\tau}$, and thus (3.8) holds. ∎∎

###  3.2. Exponential Ergodicity of DIEG Scheme

Next, let us generalize the arguments in the previous two parts to spatial
discretizations via Galerkin approximations of the temporal DIE scheme (3.1).
Then we derive the time-independent strong convergence rate for the fully
discrete scheme towards the exact solution of Eq. (2.19).

Let $h\in(0,1)$, $\mathcal{T}_{h}$ be a regular family of partitions of
$\mathscr{O}$ with maximal length $h$, and $V_{h}\subset\dot{H}^{1}$ be the
space of continuous functions on $\bar{\mathscr{O}}$ which are piecewise
linear over $\mathcal{T}_{h}$ and vanish on the boundary
$\partial\mathscr{O}$. Let $A_{h}:V_{h}\rightarrow V_{h}$ and
$\mathcal{P}_{h}:\dot{H}^{-1}\rightarrow V_{h}$ be the discrete Laplacian and
generalized orthogonal projection operators, respectively, defined by

$\displaystyle\langle v^{h},A_{h}x^{h}\rangle$ $\displaystyle=-\langle\nabla
x^{h},\nabla v^{h}\rangle,\quad x^{h},v^{h}\in V_{h},$ $\displaystyle\langle
v^{h},\mathcal{P}_{h}z\rangle$ $\displaystyle=_{1}\langle
v^{h},z\rangle_{-1},\quad z\in\dot{H}^{-1},\ v^{h}\in V_{h}.$

We discrete DIE (3.1) in space with Galerkin method, called the DIEG scheme,
and the resulting fully discrete problem is to find a $V^{h}$-valued discrete
process $\\{X^{h}_{m}:\ m\in\mathbb{N}\\}$ such that

(DIEG) $\displaystyle\begin{split}&X^{h}_{m+1}=X^{h}_{m}+\tau
A_{h}X^{h}_{m+1}+\tau\mathcal{P}_{h}F(X^{h}_{m+1})+\mathcal{P}_{h}G(X^{h}_{m})\delta_{m}W,\\\
&X^{h}_{0}=\mathcal{P}_{h}X_{0},\end{split}$

$m\in\mathbb{N}$, which had been widely studied; see, e.g., [FLZ17], [QW19],
[LQ21]. It is clear that the DIEG scheme (DIEG) is equivalent to the scheme

$\displaystyle X^{h}_{m+1}=S_{h,\tau}X^{h}_{m}+\tau
S_{h,\tau}\mathcal{P}_{h}F(X^{h}_{m+1})+S_{h,\tau}\mathcal{P}_{h}G(X^{h}_{m})\delta_{m}W,\quad
m\in\mathbb{N},$

with initial datum $X^{h}_{0}=\mathcal{P}_{h}X_{0}$, where $S_{h,\tau}:=({\rm
Id}-\tau A_{h})^{-1}$ is a space-time approximation of the continuous
semigroup $S$ in one step. Iterating the above equation for $m$-times, we
obtain

$\displaystyle X^{h}_{m+1}$
$\displaystyle=S_{h,\tau}^{m+1}X^{h}_{0}+\tau\sum_{i=0}^{m}S_{h,\tau}^{m+1-i}\mathcal{P}_{h}F(X^{h}_{i+1})$
$\displaystyle\quad+\sum_{i=0}^{m}S_{h,\tau}^{m+1-i}\mathcal{P}_{h}G(X^{h}_{i})\delta_{i}W,\quad
m\in\mathbb{N}.$

As in Lemmas 3.1 and 3.2, we can show the uniform stability and continuous
dependence on initial data independent of $h$, where the resulting constants
appeared at Lemmas 3.1 and 3.2. The proof is analogous to that of Lemmas 3.1
and 3.2 just noting that $\mathcal{P}_{h}$ is an orthogonal projection
operator which is also contracted, so we omit the details.

######  Lemma 3.3.

Let $X_{0},x,y\in H$, Assumption 2.1 and conditions (2.16)-(2.17) hold. Denote
by $\\{X^{h}_{m}:m\in\mathbb{N}\\}$, $\\{X_{m}^{h,x}:m\in\mathbb{N}\\}$, and
$\\{X_{m}^{h,y}:m\in\mathbb{N}\\}$ the solutions of the DIEG scheme (DIEG)
starting from $X_{0}$, $x$, and $y$, respectively.

  * i)

Assume that $L_{2}+L_{7}/2<\lambda_{1}$. Then for any $\tau\in(0,1)$,

(3.9) $\displaystyle\mathbb{E}\|X_{k}\|^{2}\leq
e^{-\gamma_{5}t_{k}}\|X_{0}\|^{2}+C_{\gamma_{5}},\quad k\in\mathbb{N}.$

  * ii)

Assume that $L_{1}+L_{6}/2<\lambda_{1}$. Then for any $\tau\in(0,1)$,

(3.10) $\displaystyle\mathbb{E}\|X_{m}^{h,x}-X_{m}^{h,y}\|^{2}\leq
e^{-\gamma_{6}t_{m}}\|X-Y\|^{2}.\quad m\in\mathbb{N}.$

Consequently, the corresponding fully discrete Markov semigroup, denoted by
$\\{P_{m}^{h}\\}_{m\in\mathbb{N}}$, of the DIEG scheme (DIEG) is Feller.

Now we can show that the DIEG scheme (DIEG) is exponentially mixing. On the
contrary to the case of DIE scheme (3.1), the dimension of state space $V_{h}$
of the DIEG scheme (DIEG) is finite.

######  Theorem 3.2.

Let Assumption 2.1 and conditions (2.16)-(2.17) hold. Assume that
$(L_{1}+L_{6}/2)\vee(L_{2}+L_{7}/2)<\lambda_{1}$. Then for any $\tau\in(0,1)$,
the DIEG scheme (DIEG) possesses a unique invariant measure $\pi_{\tau}^{h}$
in $V_{h}$ which is exponentially mixing:

(3.11)
$\displaystyle|P_{m}\phi(x)-\pi_{\tau}^{h}(\phi)|\leq\exp(-\gamma_{6}t_{m}/2)|\phi|_{\rm
Lip}\int_{H}\|x-y\|\pi_{\tau}^{h}({\rm d}y),$

for any $m\in\mathbb{N}$, $x\in H$, and $\phi\in{\rm Lip}_{b}(H)$, where
$\gamma_{6}$ appeared at (3.7).

###### Proof.

Consider the sequence $\\{\mu^{\tau}_{m}\\}_{m\in\mathbb{N}_{+}}$ of
probability measures

$\displaystyle\mu^{\tau,h}_{m}(B):=\frac{1}{m}\sum_{k=0}^{m}P_{k}^{h}\chi_{B}(0)dt,\quad
B\in\mathcal{B}(V_{h}),~{}m\in\mathbb{N}_{+}.$

By Chebyshev inequality and the uniform estimate (3.9), for any
$\varepsilon>0$, there exists a positive scalar $J_{\varepsilon}$ and a
compact set $B=\\{x\in V_{h}:~{}\|x\|\leq
J_{\varepsilon}\\}\subset\mathcal{B}(H)$ such that

$\displaystyle\mu^{\tau,h}_{m}(B^{c})$
$\displaystyle=\frac{1}{m}\sum_{k=0}^{m}\mathbb{P}(\|X_{k}^{h,0}\|>J_{\varepsilon})dt$
$\displaystyle\leq
J^{-2}_{\varepsilon}\sup_{k\in\mathbb{N}}\mathbb{E}\|X_{k}^{h,0}\|^{2}\leq
CJ^{-2}_{\varepsilon}<\varepsilon,$

from which we have that $\\{\mu_{t}\\}_{T\geq 1}$ is tight. The exponential
mixing estimate (3.11) and the existence of a unique invariant measure follow
from the continuous dependence estimate (3.10) and similar arguments in the
proof of Theorem 3.1. ∎∎

###  3.3. Uniform Strong Error estimate of DIEG Scheme

In this part, we aim to derive time-independent strong error estimates of the
proposed DIEG scheme (DIEG) towards to the exact solution of Eq. (2.19).

We will combine semigroup theory and variational theory as in [LQ21]. To this
end, for $m\in\mathbb{N}$, let us denote by
$E_{h,\tau}(t)=S(t)-S_{h,\tau}^{m+1}\mathcal{P}_{h}$ for
$t\in(t_{m},t_{m+1}]$, and define the auxiliary process (see [LQ21] and
[QW19])

$\displaystyle\widetilde{X}^{h}_{m+1}$
$\displaystyle=S_{h,\tau}^{m+1}X^{h}_{0}+\tau\sum_{i=0}^{m}S_{h,\tau}^{m+1-i}\mathcal{P}_{h}F(X(t_{i+1}))$
(3.12)
$\displaystyle\quad+\sum_{i=0}^{m}S_{h,\tau}^{m+1-i}\mathcal{P}_{h}G(X(t_{i}))\delta_{i}W.$

In view of the well-known uniform boundedness of $S_{h,\tau}$, the conditions
(2.3) and (2.18), and the uniform stability (2.21), it is not difficult to
show that for any $p\geq 2$,

(3.13)
$\displaystyle\sup_{k\in\mathbb{N}}\mathbb{E}\|\widetilde{X}^{h}_{k}\|_{1}^{p}\leq
C(1+\sup_{t\geq 0}\mathbb{E}\|X_{t}\|_{1}^{pq})\leq C(1+\|X_{0}\|_{1}^{pq}).$

We begin with the following uniform strong error estimate between the exact
solution $u$ of Eq. (2.19) and the auxiliary process
$\\{\widetilde{X}^{h}_{m}\\}_{m\in\mathbb{N}}$ defined by (3.3). It is proved
using the idea in [LQ21, Lemma 4.2] and the uniform moments’ bound (2.21).

######  Lemma 3.4.

Let $X_{0}\in\dot{H}^{1}$ and Assumptions 2.1-2.2 hold. Assume that $p\geq 2$
and $L_{1}+\frac{p(2q-1)-1}{2}L_{9}<\lambda_{1}$. Then there exist a positive
constant $C$ such that for any $\tau\in(0,1)$,

(3.14)
$\displaystyle\sup_{m\in\mathbb{N}}\mathbb{E}\|X(t_{m})-\widetilde{X}^{h}_{m}\|^{p}\leq
C(1+\|X_{0}\|_{1}^{p(2q-1)})(h^{p}+\tau^{p/2}).$

###### Proof.

Let $m\in\mathbb{N}$. In [LQ21, Lemma 4.2], the error

$\displaystyle
J^{m+1}:=(\mathbb{E}\|X(t_{m+1})-\widetilde{X}^{h}_{m+1}\|^{p})^{1/p}$

is divided into three parts $J^{m+1}_{i}$, $i=1,2,3$, respectively, which
satisfies

$\displaystyle J^{m+1}_{1}$
$\displaystyle=(\mathbb{E}\|E_{h,\tau}(t_{m+1})X_{0}\|^{p})^{1/p}\leq
C(h+\tau^{1/2})\|X_{0}\|_{1},$ $\displaystyle J^{m+1}_{2}$
$\displaystyle=\Big{(}\mathbb{E}\Big{\|}\int_{0}^{t_{m+1}}S_{t_{m+1}-r}F(X_{r}){\rm
d}r-\tau\sum_{i=0}^{m}S_{h,\tau}^{m+1-i}\mathcal{P}_{h}F(X(t_{i+1}))\Big{\|}^{p}\Big{)}^{1/p}$
$\displaystyle\leq C\Big{(}1+\sup_{t\geq
0}(\mathbb{E}\|X_{t}\|_{1}^{2p(q-1)})^{\frac{1}{2p}}\Big{)}\Big{(}\sup_{t\neq
s}\frac{\mathbb{E}\|X_{t}-X_{s}\|^{2p}}{|t-s|^{p}}\Big{)}^{\frac{1}{2p}}\tau^{\frac{1}{2}}$
$\displaystyle\quad+C(h+\tau^{1/2})\sup_{t\geq
0}(\mathbb{E}\|F(X_{t})\|^{p})^{1/p},$ $\displaystyle J^{m+1}_{3}$
$\displaystyle=\Big{(}\mathbb{E}\Big{\|}\int_{0}^{t_{m+1}}S_{t_{m+1}-r}G(X_{r}){\rm
d}W_{r}-\sum_{i=0}^{m}S_{h,\tau}^{m+1-i}\mathcal{P}_{h}G(X(t_{i}))\delta_{i}W\Big{\|}^{p}\Big{)}^{1/p}$
$\displaystyle\leq C\Big{(}\sup_{t\neq
s}\frac{\mathbb{E}\|X_{t}-X_{s}\|^{2p}}{|t-s|^{p}}\Big{)}^{\frac{1}{2p}}\tau^{\frac{1}{2}}+C(h+\tau^{\frac{1}{2}})(1+\sup_{t\geq
0}(\mathbb{E}\|X_{t}\|_{1}^{p})^{\frac{1}{p}}).$

Putting the above three estimates together results in

$\displaystyle J^{m+1}$ $\displaystyle\leq
C\Big{(}1+\|X_{0}\|_{1}+\Big{(}1+\sup_{t\geq
0}(\mathbb{E}\|X_{t}\|_{1}^{2p(q-1)})^{\frac{1}{2p}}\Big{)}$
$\displaystyle\qquad\times\Big{(}\sup_{t\neq
s}\frac{\mathbb{E}\|X_{t}-X_{s}\|^{2p}}{|t-s|^{p}}\Big{)}^{\frac{1}{2p}}$
$\displaystyle\quad+\sup_{t\geq
0}(\mathbb{E}\|F(X_{t})\|^{p})^{\frac{1}{p}}+\sup_{t\geq
0}(\mathbb{E}\|X_{t}\|_{1}^{p})^{\frac{1}{p}}\Big{)}(h+\tau^{\frac{1}{2}}).$

This inequality in combination with the estimate on $J^{0}$ that

$\displaystyle J^{0}=\|X_{0}-\mathcal{P}_{h}X_{0}\|_{L_{\omega}^{p}L^{2}}\leq
Ch\|X_{0}\|_{1},$

the condition (2.3), the embeddings (2.8), the uniform stability estimate
(2.21), and the uniform Hölder estimate (2.25), completes the proof of (3.14).
∎∎

Now we are at the position deriving the uniform strong error estimate between
the exact solution $u$ of Eq. (2.19) and the DIEG scheme (DIEG).

######  Theorem 3.3.

Let $X_{0}\in\dot{H}^{1}$, Assumptions 2.1-2.2 hold. Assume that
$L_{1}+\frac{2(q^{2}+q-1)-1}{2}L_{9}<\lambda_{1}$. Then for any $\tau\in(0,1)$
there exist a positive constant $C$ such that

(3.15)
$\displaystyle\sup_{k\in\mathbb{N}}\mathbb{E}\|X_{t_{k}}-X^{h}_{k}\|^{2}\leq
C(1+\|X_{0}\|_{1}^{2(q^{2}+q-1)})(h^{2}+\tau).$

###### Proof.

For $k\in\mathbb{N}$, denote $e^{h}_{k}:=\widetilde{X}^{h}_{k}-X^{h}_{k}$.
From (3.3), it is clear that

$\displaystyle\widetilde{X}^{h}_{m+1}=\widetilde{X}_{m}^{h}+\tau
A_{h}\widetilde{X}^{h}_{m+1}+\tau\mathcal{P}_{h}F(X_{t_{m+1}})+\mathcal{P}_{h}G(X_{t_{m}})\delta_{m}W.$

Then by (DIEG) and the above equality, we have

$\displaystyle e^{h}_{k+1}-e^{h}_{k}$ $\displaystyle=(\Delta
e^{h}_{k+1}+(F(X_{t_{k+1}})-F(X^{h}_{k+1}))\tau$
$\displaystyle\quad+(G(X_{t_{k}})-G(X^{h}_{k}))\delta_{k}W.$

Testing with $e^{h}_{k+1}$ and using the inequality (3.4), the condition
(2.12), Cauchy–Schwarz inequality, and the elementary inequality
$\|X+Y\|^{2}\leq(1+\varepsilon)\|X\|^{2}+(1+\varepsilon^{-1})\|Y\|^{2}$ for
$X,Y\in H$, we obtain

$\displaystyle\|e^{h}_{k+1}\|^{2}-\|e^{h}_{k}\|^{2}+\|e^{h}_{k+1}-e^{h}_{k}\|^{2}$
$\displaystyle=2~{}_{1}\langle
e^{h}_{k+1},F(X_{t_{k+1}})-F(\widetilde{X}^{h}_{k+1})\rangle_{-1}\tau$
$\displaystyle\quad+2~{}_{1}\langle e^{h}_{k+1},\Delta
e^{h}_{k+1}+F(\widetilde{X}^{h}_{k+1})-F(X^{h}_{k+1})\rangle_{-1}\tau$
$\displaystyle\quad+2\langle
e^{h}_{k+1}-e^{h}_{k},(G(X_{t_{k}})-G(X^{h}_{k}))\delta_{k}W\rangle$
$\displaystyle\quad+2\langle
e^{h}_{k},(G(X_{t_{k}})-G(X^{h}_{k}))\delta_{k}W\rangle$
$\displaystyle\leq-2(\lambda_{1}-L_{1}-\varepsilon)\tau\|e^{h}_{k+1}\|^{2}+\|e^{h}_{k+1}-e^{h}_{k}\|^{2}$
$\displaystyle\quad+2\langle
e^{h}_{k},(G(X_{t_{k}})-G(X^{h}_{k}))\delta_{k}W\rangle+C_{\varepsilon}\tau\|F(X_{t_{k+1}})-F(\widetilde{X}^{h}_{k+1})\|^{2}_{-1}$
$\displaystyle\quad+(1+\varepsilon^{-1})\|(G(X_{t_{k}})-G(\widetilde{X}^{h}_{k}))\delta_{k}W\|^{2}$
$\displaystyle\quad+(1+\varepsilon)\|(G(\widetilde{X}^{h}_{k})-G(X^{h}_{k}))\delta_{k}W\|^{2}.$

It follows that

$\displaystyle(1-2(\lambda_{1}-L_{1}-\varepsilon)\tau)\|e^{h}_{k+1}\|^{2}$
$\displaystyle\leq\|e^{h}_{k}\|^{2}+2\langle
e^{h}_{k},(G(X_{t_{k}})-G(X^{h}_{k}))\delta_{k}W\rangle$
$\displaystyle\quad+(1+\varepsilon^{-1})\|(G(X_{t_{k}})-G(\widetilde{X}^{h}_{k}))\delta_{k}W\|^{2}$
$\displaystyle\quad+C_{\varepsilon}\tau\|F(X_{t_{k+1}})-F(\widetilde{X}^{h}_{k+1})\|^{2}_{-1}$
$\displaystyle\quad+(1+\varepsilon)\|(G(\widetilde{X}^{h}_{k})-G(X^{h}_{k}))\delta_{k}W\|^{2}.$

Now taking expectation on both sides of the above inequality and using the
fact $\mathbb{E}\langle
e^{h}_{k},(G(X_{t_{k}})-G(X^{h}_{k}))\delta_{k}W\rangle=0$ and the estimates

$\displaystyle\sup_{k\in\mathbb{N}}\mathbb{E}\|F(X_{t_{k+1}})-F(\widetilde{X}^{h}_{k+1})\|^{2}_{-1}$
$\displaystyle\leq C(1+\|X_{0}\|_{1}^{2(q^{2}+q-1)})(h^{2}+\tau),$
$\displaystyle\sup_{k\in\mathbb{N}}\mathbb{E}\|G(X_{t_{k}})-G(\widetilde{X}^{h}_{k})\|_{\mathcal{L}_{2}^{0}}^{2}$
$\displaystyle\leq C(1+\|X_{0}\|_{1}^{2(2q-1)})(h^{2}+\tau),$

which follow from (3.14) and the conditions (2.3) and (2.16) (see [Liu22,
Theorem 4.1] for details), the embeddings (2.8), and the uniform stability
estimates (2.20) and (3.13), we get

$\displaystyle(1+2(\lambda_{1}-L_{1}-\varepsilon)\tau)\mathbb{E}\|e^{h}_{k+1}\|^{2}$
$\displaystyle\leq(1+L_{6}(1+\varepsilon)\tau)\mathbb{E}\|e^{h}_{k}\|^{2}$
$\displaystyle\quad+C_{\varepsilon}(1+\|X_{0}\|_{1}^{2(2q-1)})(h^{2}+\tau)\tau,$

from which we obtain

$\displaystyle\mathbb{E}\|e^{h}_{k}\|^{2}$
$\displaystyle\leq\Big{(}\frac{1+L_{6}(1+\varepsilon)\tau}{1+2(\lambda_{1}-L_{1}-\varepsilon)\tau}\Big{)}^{k}\|e^{h}_{0}\|^{2}$
$\displaystyle\quad+\frac{C_{\varepsilon}(1+\|X_{0}\|_{1}^{2(2q-1)})(h^{2}+\tau)\tau}{1+2(\lambda_{1}-L_{1}-\varepsilon)\tau}\sum_{j=0}^{k-1}\Big{(}\frac{1+L_{6}(1+\varepsilon)\tau}{1+2(\lambda_{1}-L_{1}-\varepsilon)\tau}\Big{)}^{j},$

for $k\in\mathbb{N}$. Now we can conclude (3.15) by the above estimate, the
inequality (3.6), the estimate (3.14), and the estimate
$\mathbb{E}\|X_{t_{k}}-X^{h}_{k}\|^{2}\leq
2\mathbb{E}\|X_{t_{k}}-\widetilde{X}^{h}_{k}\|^{2}+2\mathbb{E}\|e^{h}_{k}\|^{2}$.
∎∎

###  3.4. Application to Milstein-type Schemes

Finally, let us apply the previous arguments to the following DIE Milstein
(DIEM) scheme and DIEM Galerkin (DIEMG) scheme, which is frequently used to
construct temporal high order scheme (see [LQ21]):

(DIEM) $\displaystyle Y_{m+1}$ $\displaystyle=Y_{m}+\tau AY_{m+1}+\tau
F(X_{m+1})+G(X_{m})\delta_{m}W$
$\displaystyle\quad+DG(Y_{m})G(Y_{m})\Big{[}\int_{t_{m}}^{t_{m+1}}(W(r)-W(t_{m})){\rm
d}W_{r}\Big{]},\quad Y_{0}:=X_{0};$ $\displaystyle Y^{h}_{m+1}$
$\displaystyle=Y^{h}_{m}+\tau
A_{h}Y^{h}_{m+1}+\tau\mathcal{P}_{h}F(Y^{h}_{m+1})+\mathcal{P}_{h}G(Y^{h}_{m})\delta_{m}W$
$\displaystyle\quad+\mathcal{P}_{h}DG(Y^{h}_{m})G(Y^{h}_{m})\Big{[}\int_{t_{m}}^{t_{m+1}}(W(r)-W(t_{m})){\rm
d}W_{r}\Big{]},\quad Y_{0}^{h}:=\mathcal{P}_{h}X_{0}.$

To this end, we need the following infinite-dimensional analog of the
commutativity condition in the SODE setting. It was first introduced in [JR15]
where the authors observed that a certain class of semilinear SPDEs with
multiplicative trace-class noise naturally fulfills such commutativity
condition. It holds true automatically in the case of the additive noise case;
see [LQ21, Example 5.1] for an example of $g$ such that its Nemytskii operator
$G$ satisfies Assumption 3.1 (and Assumption 2.2).

######  Assumption 3.1.

$G:H\rightarrow\mathcal{L}_{2}^{0}$ is continuously Fréchet differentiable and
there exist two positive constants $L_{11}$ and $L_{12}$ such that

(3.16) $\displaystyle\|DG(x)G(x)-DG(y)G(y)\|_{HS(U_{0},\mathcal{L}_{2}^{0})}$
$\displaystyle\leq L_{11}\|x-y\|,\quad x,y\in H,$ (3.17)
$\displaystyle\|DG(z)\|_{\mathcal{L}(H,\mathcal{L}_{2}^{0})}$
$\displaystyle\leq L_{12},\quad z\in\dot{H}^{1}.$

Similar to the DIE scheme (3.1), we have the following uniform moments’
stability and continuity dependence estimates for the above DIEM scheme
(DIEM). In this case, there is an upper bound restriction on the time step
size $\tau$. We note that such restriction appeared also in [WL19] where the
authors derived numerical ergodicity of the Milstein scheme for Lipschitz
SODEs, as one needs to control the last term on the right hand side of the
DIEM scheme (DIEM).

######  Lemma 3.5.

Let $X_{0}\in H$, Assumptions 2.1 and 3.1 and conditions (2.16)-(2.17) hold.
Assume that $L_{2}+L_{7}/2<\lambda_{1}$. Then there exist positive constants
$\gamma_{7}$, $C_{\gamma_{7}}$, $C$, and $\tau_{1}$ such that for any
$\tau\in(0,\tau_{1})$,

(3.18) $\displaystyle\mathbb{E}\|Y_{m}\|^{2}\leq
e^{-\gamma_{7}t_{m}}\|X_{0}\|^{2}+C_{\gamma_{7}},\quad m\in\mathbb{N},$ (3.19)
$\displaystyle\sup_{m\in\mathbb{N}_{+}}\frac{1}{m}\sum_{k=0}^{m}\mathbb{E}\|Y_{k}\|_{1}^{2}\leq
C(1+\|X_{0}\|^{2}).$

###### Proof.

Using Itô isometry twice and the condition (3.17), we obtain

$\displaystyle\mathbb{E}\Big{\|}DG(Y_{k})G(Y_{k})\Big{[}\int_{t_{k}}^{t_{k+1}}(W(r)-W(t_{k})){\rm
d}W_{r}\Big{]}\Big{\|}^{2}$
$\displaystyle=\int_{t_{k}}^{t_{m+1}}\mathbb{E}\|DG(Y_{k})G(Y_{k})(W(r)-W(t_{k}))\|_{\mathcal{L}_{2}^{0}}^{2}{\rm
d}r$ $\displaystyle\leq
L_{12}\mathbb{E}\|G(Y_{k})\|_{\mathcal{L}_{2}^{0}}^{2}\int_{t_{k}}^{t_{k+1}}(r-t_{k}){\rm
d}r$
$\displaystyle\leq\frac{1}{2}L_{12}(L_{7}\mathbb{E}\|Y_{k}\|^{2}+L_{8})\tau^{2}.$

Based on the proof of Lemma 3.1 and the fact that $Y_{k}$ is
$\mathcal{F}_{t_{k}}$-measurable which is independent of the multiple Itô
integral in $[t_{k},t_{k+1}]$, we have

$\displaystyle(1+2(\lambda_{1}-L_{2})\tau)\mathbb{E}\|Y_{k+1}\|^{2}$
$\displaystyle\leq\mathbb{E}\|Y_{k}\|^{2}+2L_{3}\tau+\mathbb{E}\|G_{k}\delta_{k}W\|^{2}+\mathbb{E}\Big{\|}DG(Y_{k})G(Y_{k})\Big{[}\int_{t_{k}}^{t_{k+1}}(W(r)-W(t_{k})){\rm
d}W_{r}\Big{]}\Big{\|}^{2}$
$\displaystyle\quad+2\mathbb{E}\Big{\langle}G_{k}\delta_{k}W,DG(Y_{k})G(Y_{k})\Big{[}\int_{t_{k}}^{t_{k+1}}(W(r)-W(t_{k})){\rm
d}W_{r}\Big{]}\Big{\rangle}$
$\displaystyle\leq(1+L_{7}\tau+\frac{1}{2}L_{12}L_{7}\tau^{2})\mathbb{E}\|Y_{k}\|^{2}+(2L_{3}+L_{8}+\frac{1}{2}L_{12}L_{8}\tau)\tau,$

and thus

$\displaystyle\mathbb{E}\|X_{k}\|^{2}$
$\displaystyle\leq\Big{(}\frac{1+L_{7}\tau+\frac{1}{2}L_{12}L_{7}\tau^{2}}{1+2(\lambda_{1}-L_{2})\tau}\Big{)}^{k}\|X_{0}\|^{2}$
$\displaystyle\quad+\frac{(2L_{3}+L_{8}+\frac{1}{2}L_{12}L_{8}\tau)\tau}{1+2(\lambda_{1}-L_{2})\tau}\sum_{j=0}^{k-1}\Big{(}\frac{1+L_{7}\tau+\frac{1}{2}L_{12}L_{7}\tau^{2}}{1+2(\lambda_{1}-L_{2})\tau}\Big{)}^{j},$

for $k\in\mathbb{N}_{+}$. Noting that
$\frac{1+L_{7}\tau+\frac{1}{2}L_{12}L_{7}\tau^{2}}{1+2(\lambda_{1}-L_{2})\tau}<1$
for any $\tau<\tau_{1}:=\frac{4(\lambda_{1}-L_{2}-L_{7}/2)}{L_{7}L_{12}}$,
which is ensured by the condition $L_{2}+L_{7}/2<\lambda_{1}$. Now we can
conclude (3.18) and (3.19) by the above estimate, (3.6), and the proof of
Lemma 3.1. ∎∎

######  Lemma 3.6.

Let $x,y\in H$, Assumptions 2.1 and 3.1 and conditions (2.16)-(2.17) hold.
Assume that $L_{1}+L_{6}/2<\lambda_{1}$. Then there exist positive constants
$\gamma_{8}$ and $\tau_{2}$ such that for any $\tau\in(0,\tau_{2})$,

(3.20) $\displaystyle\mathbb{E}\|X_{m}^{x}-X_{m}^{y}\|^{2}\leq
e^{-\gamma_{8}t_{m}}\|x-y\|^{2},\quad m\in\mathbb{N}.$

Consequently, the corresponding discrete Markov semigroup, denoted by
$\\{\hat{P}_{m}\\}_{m\in\mathbb{N}}$, of the DIEM scheme (DIEM) is Feller.

###### Proof.

As in the proof of Lemma 3.2 and 3.5, it suffices to handle the last term in
the DIEM scheme (DIEM). Using Itô isometry twice and the condition (3.16), we
obtain

$\displaystyle\mathbb{E}\Big{\|}[DG(Y_{k}^{x})G(Y_{k}^{x})-DG(Y_{k}^{y})G(Y_{k}^{y})]\Big{[}\int_{t_{k}}^{t_{k+1}}(W(r)-W(t_{k})){\rm
d}W_{r}\Big{]}\Big{\|}^{2}$
$\displaystyle=\mathbb{E}\|DG(Y_{k}^{x})G(Y_{k}^{x})-DG(Y_{k}^{y})G(Y_{k}^{y})\|_{HS(U_{0};\mathcal{L}_{2}^{0})}^{2}\int_{t_{k}}^{t_{m+1}}(r-t_{k}){\rm
d}r$
$\displaystyle\leq\frac{1}{2}L_{11}\mathbb{E}\|Y_{k}^{x}-Y_{k}^{y}\|^{2}\tau^{2}.$

Then

$\displaystyle(1+2(\lambda_{1}-L_{1})\tau)\mathbb{E}\|Y_{k+1}^{x}-Y_{k+1}^{y}\|^{2}\leq(1+L_{6}\tau+\frac{1}{2}L_{11}\tau^{2})\mathbb{E}\|Y_{k}^{x}-Y_{k}^{y}\|^{2},$

from which we obtain

$\displaystyle\mathbb{E}\|E_{k}\|^{2}\leq\Big{(}\frac{1+L_{6}\tau+\frac{1}{2}L_{11}\tau^{2}}{1+2(\lambda_{1}-L_{1})\tau}\Big{)}^{k}\|X-Y\|^{2},\quad
k\in\mathbb{N},$

Then we conclude (3.7), for any
$\tau<\tau_{2}:=\frac{4(\lambda_{1}-L_{1}-L_{6}/2)}{L_{6}L_{11}}$, by the
above estimate and (3.6). ∎∎

Now we can show that the following exponential mixing property of the DIEM
scheme (DIEM) and the DIEMG scheme (3.4), where the resulting constants
appeared at Lemmas 3.5 and 3.6. The proof is analogous to (3.1), based on the
uniform stability and continuous dependence estimates in Lemmas 3.5 and 3.6,
respectively, and their versions for (3.4), so we omit the details.

######  Theorem 3.4.

In addition to the conditions in Theorem 3.2, assume that Assumption 3.1
holds. Then for any $\tau\in(0,\tau_{1}\wedge\tau_{2})$, the DIEM scheme
(DIEM) and the DIEMG scheme (3.4) possess unique invariant measures
$\hat{\pi}_{\tau}$ in $H$ and $\hat{\pi}_{\tau}^{h}$ in $V_{h}$, respectively,
which are exponentially mixing:

$\displaystyle|\hat{P}_{m}\phi(x)-\hat{\pi}_{\tau}(\phi)|\leq\exp(-\gamma_{8}t_{m}/2)|\phi|_{\rm
Lip}\int_{H}\|x-y\|\hat{\pi}_{\tau}({\rm d}y),$
$\displaystyle|\hat{P}_{m}^{h}\phi(x)-\hat{\pi}_{\tau}^{h}(\phi)|\leq\exp(-\gamma_{8}t_{m}/2)|\phi|_{\rm
Lip}\int_{H}\|x-y\|\pi_{\tau}^{h}({\rm d}y),$

for any $m\in\mathbb{N}$, $x\in H$, and $\phi\in{\rm Lip}_{b}(H)$. ∎

We also have the following parallel uniform strong error estimate for the
DIEMG scheme (3.4), following the proofs of Theorem 3.3 and Theorem 1.2 in
[LQ21].

######  Theorem 3.5.

Let $X_{0}\in\dot{H}^{1}$, Assumptions 2.1, 2.2, and 3.1 hold. Assume that
$L_{1}+\frac{2(q^{2}+q-1)-1}{2}L_{9}<\lambda_{1}$. Then there exist a positive
constant $C$ such that for any $\tau\in(0,1)$,

(3.21)
$\displaystyle\sup_{k\in\mathbb{N}}\mathbb{E}\|X_{t_{k}}-Y^{h}_{k}\|^{2}\leq
C(1+\|X_{0}\|_{1}^{2(q^{2}+q-1)})(h^{2}+\tau).$

∎

######  Remark 3.1.

One can also consider spectral Galerkin approximations applied to the DIE
scheme (3.1) and DIEM scheme (DIEM) and get the spectral Galerkin DIE and DIEM
schemes, respectively, as Eq. (DIEG) and Eq. (3.4), where one just need to
transform the finite element index to spectral Galerkin index. All the
exponential ergodicity and uniform strong error estimate results, e.g.,
Theorems 3.2, 3.3, 3.4, and 3.5, also holds true for spectral Galerkin DIE or
DIEM schemes.

######  Remark 3.2.

Using the method in [Liu22, Theorem 4.1] and the arguments in the proof of
Theorem 3.3, one can show similar uniform strong error estimate as (3.15) and
(3.21) in $p$-th moment, for any $p\geq 2$, for the DIEG scheme (DIEG) and
DIEMG scheme (3.4). In particular, if $X_{0}\in\dot{H}^{1+\gamma}$ with some
$\gamma\in[0,1]$, then one could show that the DIEMG scheme (3.4) possesses
higher strong order of convergence in both spatial and temporal directions:

$\displaystyle\sup_{k\in\mathbb{N}}\mathbb{E}\|X_{t_{k}}-Y^{h}_{k}\|^{p}\leq
C(1+\|X_{0}\|_{1+\gamma}^{p(q^{2}+q-1)})(h^{1+\gamma}+\tau^{\frac{1+\gamma}{2}})^{p},$

provided more regular and growth conditions (see [LQ21, Assumptions 5.1 and
5.2]) on the coefficients of Eq. (2.19) hold true.

##  4\. Applications to Stochastic Allen–Cahn Equation

In the last section, let us apply the exponential ergodicity and uniform
strong error analysis results, developed in Section 3, to the semiclassical
stochastic Allen–Cahn equation (1.2).

We note that as the driven noise in Eq. (1.2) is additive, the DIE scheme
(3.1) and DIEG scheme (DIEG) coincide with the DIEM scheme (DIEM) and DIEMG
scheme (3.4), respectively. Therefore, it suffices to consider the DIE scheme
(3.1) and DIEG scheme (DIEG) in this part.

Let us first check the conditions in the main Assumptions 2.1 and 2.2 for Eq.
(1.2) with $f(x)=\alpha^{-2}(x-x^{3})$ and $G\equiv{\rm Id}$. At first, the
growth order of $f^{\prime}$ in (2.4) is $q-1=2$, so that $q=3$. Secondly, it
is not difficult to show that $L_{1}=\alpha^{-2}$ in (2.1) and
$L_{4}=2\alpha^{-2}$ and $L_{5}=\alpha^{-2}$ in (2.3). For (2.2), note that by
Young inequality,

$\displaystyle\alpha^{-2}(x-x^{3})x=-\alpha^{-2}x^{4}+\alpha^{-2}x^{2}\leq-
Cx^{2}+C_{\alpha},$

for any positive $C$ and certain positive constant $C_{\alpha}$. So one take
$L_{2}$ to be any negative scalar. We emphasize that the choice of $L_{2}$ to
be any negative scalar is only needed in this section to ensure the existence
of the invariant measures of Eq. (1.2) and its DIEG scheme (DIEG).

For the conditions on $G$ of Assumption 2.2, $L_{6}=L_{7}=L_{9}=0$,
$L_{8}={\rm Trace}({\bf Q})$, and $L_{10}=\|(-A)^{1/2}{\bf
Q}^{1/2}\|^{2}_{HS(H;H)}$. Similarly, as $G\equiv{\rm Id}$, the bounded
constants in Assumption 3.1 are both zero: $L_{11}=L_{12}=0$. Therefore,
$L_{2}+L_{7}/2=L_{2}<\lambda_{1}$ is always valid as one can take $L_{2}$ to
be any negative scalar, so that the main conditions in Proposition 2.1 and
Lemma 3.1 hold and thus Eq. (1.2) and its DIEG scheme (DIEG) always possess at
least one invariant measure, respectively. Meanwhile,
$L_{1}+L_{6}/2=\alpha^{-2}<\lambda_{1}$ for $\alpha>\lambda_{1}^{-1/2}$, so
that the main conditions in Theorems 2.1 and 3.1 hold and thus Eq. (1.2) and
its DIEG scheme (DIEG) are both exponentially ergodic, respectively, provided
that the interface thickness is not too small.

Finally, when $\alpha>\lambda_{1}^{-1/2}$, we have
$L_{1}+21L_{9}/2=\alpha^{-2}<\lambda_{1}$, so that the main condition in
Theorem 3.3 hold. Therefore, we have the following exponential ergodicity and
uniform strong error estimate for the stochastic Allen–Cahn equation (1.2).

######  Theorem 4.1.

Eq. (1.2) as well as its DIE scheme (3.1) and DIEG scheme (DIEG) always
possess at least one invariant measure, respectively. They are all
exponentially ergodic, provided that $\alpha>\lambda_{1}^{-1/2}$. Moreover,
there exist a positive constant $C$ such that for any $\tau\in(0,1)$,

$\displaystyle\sup_{k\in\mathbb{N}}\mathbb{E}\|X_{t_{k}}-X^{h}_{k}\|^{2}\leq
C(1+\|X_{0}\|_{1}^{22})(h^{2}+\tau).$

######  Remark 4.1.

The authors in [CHS21, Page 88] proposed a question whether the invariant
measure of the DIE scheme (3.1), applied to Eq. (1.1) with polynomial drift of
order 3 with negative leading coefficients (as in Remark 2.1), driven by
additive noise, is unique. In this case, $L_{1}$ is just the bound
$\sup_{\xi\in\mathbb{R}}f^{\prime}(x)$ and $L_{6}=0$. Applying Theorem 3.1, we
conclude that, in this case, the DIE scheme (3.1) is exponentially ergodic and
thus give an affirmative answer to the question proposed in [CHS21], provided
that $\sup_{\xi\in\mathbb{R}}f^{\prime}(x)<\lambda_{1}$.

######  Remark 4.2.

Combining the uniform ergodicity of the DIEG scheme (DIEG) with the uniform
weak error estimate in [CHS21], one can show the error estimate between the
fully discrete invariant measure $\pi^{h}_{\tau}$ of (DIEG) and the exact
invariant measure $\pi$ of Eq. (1.2).

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