Kolmogorov Equation for a Stochastic Reaction–Diffusion Equation with Multiplicative Noise

Abstract

Reaction–diffusion equations can model complex systems where randomness plays a role, capturing the interaction between diffusion processes and random fluctuations. The Kolmogorov equations associated with these systems play an important role in understanding the long-term behavior, stability, and control of such complex systems. In this paper, we investigate the existence of a classical solution for the Kolmogorov equation associated with a stochastic reaction–diffusion equation driven by nonlinear multiplicative trace-class noise. We also establish the existence of an invariant measure $\nu$ for the corresponding transition semigroup $P_t$, where the infinitesimal generator in $L^2(H,\nu )$ is identified as the closure of the Kolmogorov operator $K_0$.

1. Introduction

In recent years, stochastic reaction–diffusion equations have been widely applied in fields such as physics, chemistry, biology, economics, and various branches of engineering. As a result, the corresponding Kolmogorov equations and related problems have also attracted significant attention from researchers. For instance, Cerrai [1] investigated stochastic partial differential equations with additive noise, focusing on the regularity of solutions, the smoothness of the transition semigroup in bounded and uniformly continuous function spaces, and various aspects of the Kolmogorov equation, including the long-term behavior of its solutions and the stochastic optimal control problem related to the Hamilton–Jacobi–Bellman equation. Building on earlier work on stochastic reaction–diffusion equations and stochastic Navier–Stokes equations with additive noise, Da Prato [2] explored stochastic differential equations with multiplicative noise. He demonstrated the existence of classical solutions for the associated Kolmogorov equation, as well as the existence and uniqueness of invariant measures, and he examined the properties of the Kolmogorov operator. Nevertheless, challenges related to stochastic reaction–diffusion equations driven by multiplicative noise continue to be a key area for further research.

In this paper, we consider the following stochastic reaction–diffusion equation in the domain D with Dirichlet bounded conditions,

$$\left\{\begin{array}{c}dX(t,\xi )=[{\Delta }_{\xi }X(t,\xi )+f(\xi ,X(t,\xi ))]dt+\sigma (\xi ,X(t,\xi ))dW(t,\xi ),\hfill \\ X(t,\xi )=0,t\ge 0,\xi \in \partial D,\hfill \\ X(0,\xi )=x\left(\xi \right),x\in H,\xi \in D,\hfill \end{array}\right.$$

(1)

where $D={(0,1)}^d\subseteq {\mathbb{R}}^d$, $d\le 3$, $H=L^2\left(D\right)$, $f,\sigma \in D\times \mathbb{R}\to \mathbb{R}$ are two measurable functions with proper conditions and $W:[0,T]\times \mathrm{\Omega }\to H$ is a standard Q-Wiener process with respect to ${\left({\mathcal{F}}_t\right)}_{t\in [0,T]}$ (see below for a precise definition) with covariance operator $Q:H\to H$. The existence and uniqueness of a mild solution of $\left(1\right)$ are well known [3,4].

We reformulate problem (1) as an abstract form

$$\left\{\begin{array}{c}dX\left(t\right)=\left[AX\right(t)+F(X\left(t\right)\left)\right]dt+\Sigma \left(X\right(t\left)\right)dW\left(t\right),\hfill \\ X\left(0\right)=x,\hfill \end{array}\right.$$

(2)

where the operator A is expressed as

$$Ax={\Delta }_{\xi }x,x\in D\left(A\right)=H^2\left(D\right)\cap H_0^1\left(D\right),$$

(3)

and F, $\mathrm{\Sigma }$ are the Nemytskii operators, defined by

$$\left(F\right(x\left)\right)\left(\xi \right)=f(\xi ,x(\xi \left)\right),\xi \in D,x\in H,$$

(4)

$$\left(\mathrm{\Sigma }\right(x\left)y\right)\left(\xi \right)=\sigma (\xi ,x(\xi \left)\right)y\left(\xi \right),\xi \in D,x,y\in H.$$

(5)

The mild solution of Equation (2) is denoted by $X(t,x)$, and we then define the two deterministic functions

$$u(t,x)=:P_t\phi \left(x\right)=\mathbb{E}\left[\phi \left(X(t,x)\right)\right],t\ge 0,x\in H,$$

(6)

and

$$\psi \left(x\right)={\int }_0^{\infty }{e}^{-\lambda t}\mathbb{E}\left[g\left(X(t,x)\right)\right]dt,x\in H.$$

(7)

Here, $\phi$ is a real bounded Borel function and g is a continuous bounded function. If u is sufficiently regular, we show that it is a classical solution of the parabolic Kolmogorov equation

$$\left\{\begin{array}{c}{u}_t(t,x)=K_{0}u(t,x),t\ge 0,x\in D\left(A\right),\hfill \\ u(0,x)=\phi \left(x\right),x\in H,\hfill \end{array}\right.$$

(8)

with the Kolmogorov operator

$$K_0\phi \left(x\right)={\displaystyle \frac{1}{2}}\mathrm{Tr}\left[D^2\phi \left(x\right){\mathrm{\Sigma }}^2\left(x\right)Q\right]+\langle Ax+F\left(x\right),D\phi \left(x\right)\rangle ,x\in D\left(A\right),$$

(9)

where $D\phi$ and $D^2\phi$ denote the first and second Fréchet derivatives of function $\phi$.

Moreover, by combining with (8), we can show that $\psi$ is a classical solution of the elliptic Kolmogorov equation

$$\lambda \psi \left(x\right)-K_0\psi \left(x\right)=g\left(x\right),\lambda >0,x\in D\left(A\right).$$

(10)

So far, significant progress has been made on Kolmogorov equations associated with stochastic partial differential equations [5,6,7,8,9,10,11,12,13,14,15]. For Kolmogorov equations associated with stochastic partial differential equations driven by additive noise, S. Cerrai [1] studied the equation expressed as

$$\left\{\begin{array}{c}dX(t,\xi )=[{\Delta }_{\xi }X(t,\xi )+f\left(X(t,\xi )\right)]dt+QdW(t,\xi ),\hfill \\ X(t,\xi )=0,t\ge 0,\xi \in \partial D,\hfill \\ X(0,\xi )=x\left(\xi \right),x\in H,\xi \in D,\hfill \end{array}\right.$$

(11)

where the noise $W(t,\xi )$ is a cylindrical wiener process, studying the existence, uniqueness, and optimal regularity of the solutions of $\left(8\right)$ and $\left(10\right)$ in Hölder spaces with the Kolmogorov operator

$$K_0\phi \left(x\right)={\displaystyle \frac{1}{2}}\mathrm{Tr}\left[D^2\phi \left(x\right)Q{Q}^{*}\right]+\langle Ax+F\left(x\right),D\phi \left(x\right)\rangle ,x\in D\left(A\right).$$

(12)

In the context of Kolmogorov equations with multiplicative noise, G. Da Prato [2] addressed the scenario where $f=0$, $d=1$, and W is a cylindrical Wiener process, such as

$$\left\{\begin{array}{c}dX(t,\xi )={\Delta }_{\xi }X(t,\xi )dt+\sigma (\xi ,X(t,\xi ))dW(t,\xi ),\hfill \\ X(t,0)=X(t,1)=0,t\ge 0,\hfill \\ X(0,\xi )=x\left(\xi \right),x\in H,\xi \in D,\hfill \end{array}\right.$$

(13)

proving the existence of a smooth solution of $\left(8\right)$ with the Kolmogorov operator

$$K_0\phi \left(x\right)={\displaystyle \frac{1}{2}}\mathrm{Tr}\left[D^2\phi \left(x\right){\mathrm{\Sigma }}^2\left(x\right)\right]+\langle Ax,D\phi \left(x\right)\rangle ,x\in D\left(A\right).$$

(14)

Additionally, Zhou [15] investigated the existence and uniqueness of mild solutions for Kolmogorov equations associated with stochastic evolution equations and applied these results to optimal control problems.

Moreover, it is worth noting that several studies have addressed the Kolmogorov equation (8) as a parabolic equation. G. Da Prato and J. Zabczyk [16,17] analyze Equation (8) under the assumption that the coefficients F and $\mathrm{\Sigma }$ are k-order Fréchet differentiable on H. However, since F and $\mathrm{\Sigma }$ are defined in (4) and (5) as Nemytskii operators associated with the functions f and $\sigma$, it is evident that F may be not Fréchet differentiable in H, even if f and $\sigma$ are smooth. As a result, the methods used in [16] are not effective in this context.

Early research on invariant measures for stochastic partial differential equations focused primarily on additive noise. Seminal works by G. Da Prato and J. Zabczyk [8], M. Röckner [12], and M. Hairer [18] established frameworks using tools like the Krylov–Bogoliubov theorem (relying on tightness and Feller properties), dissipative systems theory, and asymptotic strong Feller property. For reaction–diffusion equations with multiplicative noise, S. Cerrai [19] studied the existence of an invariant measure constructed via the asymptotic behavior of solutions in spaces of Hölder continuous functions and showing the family of probability measures is tight in Banach space. The same method has been applied to some problems (cf. Refs. [20,21] and the references therein).

Inspired by the existing research, the aim of this paper is to investigate the existence of classical solutions to the Kolmogorov equation and invariant measure for the corresponding transition semigroup associated with stochastic reaction–diffusion equations. This work distinguishes itself from previous studies by replacing additive noise with trace-class multiplicative noise (cf. Ref. [1] id devoted to Hölder continuous perturbations of the infinite dimensional Heat semigroup) and introducing a nonlinear reaction term (as opposed to the linear formulation in Ref. [2]), and it is more general (details are shown in

Table 1. Multidimensional theoretical comparative analysis table.

Criterion	Cerrai [1]	Da Prato [2]	Our Work
Solution Type	Existence, uniqueness and optimal regularity of the solutions	Existence of a smooth solution	Existence of a classical solution for the Kolmogorov equation
Noise type	Additive cylindrical Wiener process	Multiplicative cylindrical Wiener process	Multiplicative trace-class noise
Spatial dimension	$d=n$	$d=1$	$d=1,2,3$
Reaction term	Nonlinear	zero	Nonlinear

). Therefore, the method in [1,2] does not work and some new technique needs to be proposed. Using the singular form of the Gronwall inequality and generalized Mittag-Leffler function, the regularity of the mild solution can be obtained. Our main contributions include establishing the existence conditions for the classical solutions of Equations $\left(8\right)$ and $\left(10\right)$ and proving that the transition semigroup $P_t$ admits an invariant measure $\nu$ by the Krylov–Bogoliubov theorem, which can be extended to a strongly continuous contraction semigroup on $L^2(H,\nu )$.

The remainder of this paper is organized as follows. Section 2 introduces the notation and some preliminary results. In Section 3, we study the stochastic differential equation and provide estimates for the derivative of the solution. Section 4 presents the existence of classical solutions for the Kolmogorov equations. Finally, we demonstrate the existence of an invariant measure for the transition semigroup $P_t$, and we show that the infinitesimal generator of $P_t$ in $L^2(H,\nu )$ is characterized as the closure of the Kolmogorov operator $K_0$ in Section 5.

2. Preliminaries

We introduce some notations used throughout this paper (details are shown in

Table 2. Nomenclature table.

Symbol	Meaning
D	Domain of variable x
$F,\mathrm{\Sigma }$	Nemytskii operators
H	Separable real Hilbert space
$\left(e_k\right)$	Complete orthonormal system of H
$\phi$	Bounded Borel function
g	A continuous bounded function
$L\left(H\right)$	The Banach algebra of all bounded linear operators
$L_1\left(H\right)$	The space of all nuclear operators
$D_{\phi }$	The first Fréchet derivatives of function $\phi$
$D_{\phi }^2$	The second Fréchet derivatives of function $\phi$
$W(t,\xi )$	Standard Q-Wiener process
${\beta }_k\left(t\right)$	One-dimensional standard Brownian motion
$\psi$	A classical solution of the elliptic Kolmogorov equation
$C_b(H;H)$	The space of all continuous and bounded mappings
$C_W([0,T];H)$	The space of all mean-square continuous adapted stochastic processes
u	A classical solution of the parabolic Kolmogorov equation
Q	Covariance operator
T	Time horizon
p	Order of Lebesgue space
d	Spatial dimension

). Let H denote the separable real Hilbert space $L^2\left(D\right)$ with norm $|·|$ and inner product $\langle ·,·\rangle$. The space $L\left(H\right)$, with norm $\left|\right|·\left|\right|$, represents the Banach algebra of all bounded linear operators from H to itself, i.e.,

$$\parallel T\parallel =sup\left\{\right|Tx|;x\in H,|x|=1\},T\in L\left(H\right),$$

(15)

and $L_1\left(H\right)$ represents the space of all nuclear operators. We denote by $C_b(H;H)$ the space of all continuous and bounded mappings $\phi :H\to H$, equipped with the norm

$${\parallel \phi \parallel }_0=\underset{x\in H}{sup}\left|\phi \left(x\right)\right|,$$

(16)

which forms a Banach space. Moreover, $C_b^1(H;H)$ denotes the subspace of $C_b(H;H)$ consisting of all functions $\phi :H\to H$ that are Fréchet differentiable on H, with continuous and bounded derivatives $D\phi$. The norm is defined by

$${\parallel \phi \parallel }_1={\parallel \phi \parallel }_0+\underset{x\in H}{sup}\left|D\phi \left(x\right)\right|,$$

(17)

and the space $C_b^k(H;H)$ for any $k\ge 2$ is defined analogously. For brevity, we write $C_b^k(H;\mathbb{R})$ as $C_b^k\left(H\right)$, where $k\in \mathbb{N}$.

Let $T\in (0,\infty )$ be a real number, and let $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$ be a complete probability space with a normal filtration ${\left({\mathcal{F}}_t\right)}_{t\ge 0}$. Let W: $[0,T]\times \mathrm{\Omega }\to H$ be a standard Q-Wiener process with respect to ${\left({\mathcal{F}}_t\right)}_{t\in [0,T]}$, with covariance operator Q: $H\to H.$ Define $C_W([0,T];H)$ as the space of all mean-square continuous adapted stochastic processes $X(·)$ defined on $[0,T]$, taking values in H, such that $X\left(t\right)$ is ${\mathcal{F}}_t$ measurable for every $t\in [0,T]$. It is evident that $C_W([0,T];H)$, equipped with the norm

$${\parallel X\parallel }_{C_W([0,T];H)}={\left(\underset{t\in [0,T]}{sup}{\mathbb{E}\left(\right|X\left(t\right)|}^2)\right)}^{1/2},$$

(18)

is a Banach space.

For the sake of simplicity in computation, we assume that $d=1$ and let $\left(e_k\right)$ denote a complete orthonormal system in H, given by

$$e_k\left(\xi \right)=\sqrt{2}sin\left(k\pi \xi \right),$$

(19)

for all $\xi \in [0,1]$ and $k\in \mathbb{N}$, and we have

$$A{e}_k=-{\pi }^2{k}^2{e}_k,$$

(20)

that is, $\left(e_k\right)$ are the eigenfunctions of A and

$$\parallel e^{tA}\parallel \le e^{-{\pi }^{2}t}.$$

(21)

Regarding the covariance operator of the Q-Wiener process, there is a classical result [22] stating that the covariance operator has a standard orthonormal basis in H. Without loss of generality, we can assume that Q and A share a common set of eigenfunctions, with $Q{e}_k={\mu }_k{e}_k.$ Thus, we can express it as follows

$$W(t,\xi )=\sum _{k=1}^{\infty }\sqrt{{\mu }_k}{e}_k\left(\xi \right){\beta }_k\left(t\right),$$

(22)

which is uniformly convergent on $[0,T]$$\mathbb{P}$-a.s, where $\left({\beta }_k\left(t\right)\right)$ is a sequence of independent one-dimensional standard Brownian motions.

Hypothesis 1.

(1) 

Let $f:D\times \mathbb{R}\to \mathbb{R}$ be a measurable function with ${\int }_D{\left|f(\xi ,0)\right|}^{2}d\xi <\infty$ and

$$\underset{\xi \in D}{sup}\underset{z_1,z_2\in \mathbb{R},z_1\ne z_2}{sup}{\displaystyle \frac{|f(\xi ,z_1)-f(\xi ,z_2)|}{|z_1-z_2|}}<\infty .$$

(23)

(2) 

Let $\sigma :D\times \mathbb{R}\to \mathbb{R}$ be a function that there exists a real number q such that

$$|\sigma ({\xi }_1,z_1)-\sigma ({\xi }_2,z_2)|\le q(|{\xi }_1-{\xi }_2|+|z_1-z_2|),$$

(24)

for all ${\xi }_1,{\xi }_2\in D$ and all $z_1,z_2\in \mathbb{R}$; in addition, ${\int }_D{\left|\sigma (\xi ,0)\right|}^{2}d\xi <q^2$.

(3) 

Let the covariance operator $Q:H\to H$ of the Q-Wiener process be a nonnegative bounded linear operator with the complete orthonormal system $\left(e_k\right)$, and the corresponding eigenvalues $\left({\mu }_k\right)$ satisfy ${\sum }_{k=1}^{\infty }{\mu }_k<\infty$.

Definition 1.

A mild solution $X(t,x)\in C_W([0,T];H)$ of Equation (2) is an ${\mathcal{F}}_t$-adapted process, such that

$$X(t,x)=e^{tA}x+{\int }_0^t{e}^{(t-s)A}F\left(X(s,x)\right)ds+{\int }_0^t{e}^{(t-s)A}\mathrm{\Sigma }\left(X(s,x)\right)dW\left(s\right),$$

(25)

for $\mathbb{P}$-a.e. $\omega \in \mathrm{\Omega }$ and all $t\ge 0$, $x\in H$.

Notably, under the given assumptions, the stochastic convolution

$$W_A\left(t\right)={\int }_0^t{e}^{(t-s)A}\mathrm{\Sigma }\left(X(s,x)\right)dW\left(s\right)$$

(26)

is well defined in H, and

$$\mathbb{E}{\left(\underset{t\in [0,T]}{sup}|{\int }_0^t{e}^{(t-s)A}\mathrm{\Sigma }\left(X(s,x)\right)dW\left(s\right){|}^p\right)}^{{\displaystyle \frac{1}{p}}}<\infty$$

(27)

for all $p\ge 2$.

The following result addresses the existence and uniqueness of a mild solution to Equation (2), with the proof primarily based on Theorem 1 in [4].

Proposition 1.

Assume Hypothesis 1(1)–(3). For any initial condition $x\in H$, Equation $\left(2\right)$ has a unique mild solution $X(t,x)\in C_W([0,T];H)$.

In this work, we consider Galerkin approximations for problem (25). For any $n\in \mathbb{N}$, let ${\mathrm{\Pi }}_n$ denote the projector

$${\mathrm{\Pi }}_{n}x=\sum _{k=1}^n\langle x,e_k\rangle e_k,x\in H,$$

(28)

and define $A_n=A{\mathrm{\Pi }}_n$, $F_n\left(x\right)={\mathrm{\Pi }}_{n}F\left({\mathrm{\Pi }}_{n}x\right)$, ${\mathrm{\Sigma }}_n\left(x\right)={\mathrm{\Pi }}_n\mathrm{\Sigma }\left({\mathrm{\Pi }}_{n}x\right)$. We then consider the equation

$$X^n(t,x)=e^{t{A}_n}x+{\int }_0^t{e}^{(t-s)A_n}{F}_n\left(X^n(s,x)\right)ds+{\int }_0^t{e}^{(t-s)A_n}{\mathrm{\Sigma }}_n\left(X^n(s,x)\right)dW\left(s\right),$$

(29)

and obtain the following standard result.

Proposition 2.

Assume Hypothesis 1(1)–(3). For any $T>0$, $x\in H$, and $n\in \mathbb{N}$, there exists a unique solution $X^n(t,x)$ to Equation $\left(29\right)$. Furthermore,

$$\underset{n\to \infty }{lim}{X}^n(t,x)=X(t,x),in{C}_W([0,T];H),$$

(30)

where $X(t,x)$ is the solution to Equation (2).

3. Estimates for the Solution

In this section, we consider the stochastic partial differential Equation (2). Based on Proposition 2, which guarantees the existence and uniqueness of solutions for the Galerkin approximations and their convergence to the solution of (2), we denote its unique solution by $X(t,x)$. This provides the fundamental basis for our subsequent derivative analysis. Next, we will estimate the derivatives of $X(t,x)$, which will play a crucial role in the remainder of this section. These derivative estimates are essential building blocks for Proposition 3 and Proposition 4. Proposition 3 will utilize these estimates to establish the first-order Gâteaux differentiability of $X(t,x)$, and Proposition 4 will further extend the results to the second-order Gâteaux differentiability under stronger assumptions.

We will use a singular form of the Gronwall inequality [23], which plays a key role in proving the following results. Lemma 1, based on this singular Gronwall inequality, will be a vital tool in deriving the estimates for the derivatives in both Proposition 3 and Proposition 4. It helps us bound the growth of the derivative-related terms over time. For completeness, we provide a brief proof here.

Lemma 1.

Assume that $a\left(t\right)$ is a non-negative, locally integrable function on ${\mathbb{R}}^{+}$ and $g\left(t\right)$ is a non-negative, non-decreasing, continuous function on ${\mathbb{R}}^{+}$. If the function $f\left(t\right)$ is non-negative and locally integrable on ${\mathbb{R}}^{+}$, and it satisfies the following inequality for a constant for a constant $\alpha >0$

$$f\left(t\right)\le a\left(t\right)+g\left(t\right){\int }_0^t{(t-s)}^{\alpha -1}f\left(s\right)ds.$$

(31)

Then,

$$f\left(t\right)\le a\left(t\right)+{\int }_0^t\left(\sum _{n=1}^{\infty }{\displaystyle \frac{{\left(g\left(t\right)\mathrm{\Gamma }\left(\alpha \right)\right)}^n}{\mathrm{\Gamma }\left(n\alpha \right)}}{(t-s)}^{n\alpha -1}\right)a\left(s\right)ds$$

(32)

holds for any $t\in [0,T]$ and $T\ge 0$.

Proof. 

Let

$$Bf\left(t\right)=g\left(t\right){\int }_0^t{(t-s)}^{\alpha -1}f\left(s\right)ds,$$

(33)

for any locally integrable non-negative function $f\left(t\right)$. Through iteration, we thus have

$$f\left(t\right)\le \sum _{i=0}^{n-1}{B}^{i}a\left(t\right)+B^{n}f\left(t\right).$$

(34)

Next, we proceed to prove the result in the following two steps:

$\left(i\right)$

${\displaystyle B^{n}f\left(t\right)\le {\int }_0^t\frac{{\left(g\left(t\right)\mathrm{\Gamma }\left(\alpha \right)\right)}^n}{\mathrm{\Gamma }\left(n\alpha \right)}{(t-s)}^{n\alpha -1}f\left(s\right)ds}$, which can be obtained by the method of induction.

$\left(ii\right)$

For any $t\in [0,T]$, $\underset{n\to \infty }{lim}{B}^{n}f\left(t\right)=0$, and ${\displaystyle \sum _{n=1}^{\infty }{B}^{n}a\left(t\right)<\infty }$.

□

Proposition 3.

In addition, to assume Hypothesis 1(1)–(3), assume that for all $\xi \in D$, the functions $f(\xi ,·)$ and $\sigma (\xi ,·)$ belong to $C^1\left(\mathbb{R}\right)$ and satisfy the following conditions

$$\underset{\xi \in D}{sup}\underset{z\in \mathbb{R}}{sup}\left|{\partial }_{z}f(\xi ,z)\right|<\infty ,$$

(35)

$$\underset{\xi \in D}{sup}\underset{z\in \mathbb{R}}{sup}\left|{\partial }_z\sigma (\xi ,z)\right|<\infty .$$

(36)

Then, $X(t,x)$ is Gâteaux differentiable at any point $x\in H$ and for any $h\in H$ the directional derivative $D_{x}X(t,x)h=:{\eta }^h(t,x)$ is the unique mild solution of equation

$$\left\{\begin{array}{c}d{\eta }^h(t,x)=[A{\eta }^h(t,x)+DF\left(X(t,x)\right){\eta }^h(t,x)]dt\hfill \\ +D\mathrm{\Sigma }\left(X(t,x)\right){\eta }^h(t,x)dW\left(t\right),\hfill \\ {\eta }^h(0,x)=h.\hfill \end{array}\right.$$

(37)

Moreover, for any $m\ge 1$, $h\in H$,

$$\mathbb{E}\left(\right|{\eta }^h{(t,x)|}^{2m})\le a_1{e}^{{\lambda }_{1}t}{\left|h\right|}^{2m},t\ge 0,$$

(38)

for some constants ${\lambda }_1$ and $a_1$.

Remark 1.

Since the function f (or σ) satisfies the conditions in Proposition 3, the corresponding Nemytskii operators F (or Σ) are Gâteaux differentiable at any point $x\in H$ along any direction $y\in H$. The Gâteaux derivative is given by

$$\left(DF\left(x\right)y\right)\left(\xi \right)={\partial }_{z}f(\xi ,x\left(\xi \right))y\left(\xi \right),\xi \in D.$$

(39)

Proof of Proposition 3.

We consider the approximating problem (29). With the help of Proposition 2 and the results from the finite-dimensional case [24], the unique solution $X^n(t,x)$ is differentiable at any point $x\in H$ along any direction $h\in H$ in $C_W([0,T];H)$. The derivative, denoted by $D_x{X}^n(t,x)h=:{\eta }_n^h(t,x)$, solves the following problem

$$\left\{\begin{array}{c}d{\eta }_n^h(t,x)=[A_n{\eta }_n^h(t,x)+D{F}_n\left(X^n(t,x)\right){\eta }_n^h(t,x)]dt\hfill \\ +D{\mathrm{\Sigma }}_n\left(X^n(t,x)\right){\eta }_n^h(t,x)dW\left(t\right),\hfill \\ {\eta }_n^h(0,x)=h.\hfill \end{array}\right.$$

(40)

Indeed, the existence and uniqueness of the mild solution to Equation $\left(40\right)$ are ensured [3], and we obtain

$$\underset{n\to \infty }{lim}{\eta }_n^h(t,x)={\eta }^h(t,x),\mathrm{in}{C}_W([0,T];H),$$

(41)

by proving that ${\left\{{\eta }_n^h\right\}}_{n\in \mathbb{N}}$ is a Cauchy sequence in $C_W([0,T];H)$, which is similar to the proof of Theorem 3.5 in [7].

For any $n\in \mathbb{N}$, $ϵ>0$, and $h\in H$, we have that for $\mathbb{P}$-a.s., the following holds

$$X^n(t,x+ϵh)-X^n(t,x)={\int }_0^1\langle D_x{X}^n(t,\rho (x+ϵh)+(1-\rho )x),ϵh\rangle d\rho .$$

(42)

Then, by applying $\left(30\right)$ and $\left(41\right)$ and taking the limit as $n\to \infty$, we obtain

$$X(t,x+ϵh)-X(t,x)=ϵ{\int }_0^1{\eta }^h(t,x+ϵ\rho h)d\rho .$$

(43)

Note that if the estimate (38) holds, we can conclude that $X(t,x)$ is Gâteaux differentiable at any point $x\in H$, and its derivative in any direction $h\in H$ is the unique mild solution of (37).

To prove $\left(38\right)$, we rewrite ${\eta }^h(t,x)$ in integral form

$$\begin{array}{cc}\hfill {\eta }^h(t,x)& =e^{tA}h+{\int }_0^t{e}^{(t-s)A}DF\left(X(s,x)\right){\eta }^h(s,x)ds\hfill \\ & +{\int }_0^t{e}^{(t-s)A}D\mathrm{\Sigma }\left(X(s,x)\right){\eta }^h(s,x)dW\left(s\right).\hfill \end{array}$$

(44)

In light of (22), we have

$$\begin{array}{cc}\hfill {\eta }^h(t,x)& =e^{tA}h+{\int }_0^t{e}^{(t-s)A}DF\left(X(s,x)\right){\eta }^h(s,x)ds\hfill \\ & +\sum _{k=1}^{\infty }{\int }_0^t{e}^{(t-s)A}D\mathrm{\Sigma }\left(X(s,x)\right){\eta }^h(s,x)\sqrt{{\mu }_k}{e}_{k}d{\beta }_k\left(s\right).\hfill \end{array}$$

(45)

Then,

$$\begin{array}{cc}& \mathbb{E}\left(\right|{\eta }^h(t,x){|}^{2m})\hfill \\ & \le 3^{2m-1}\mathbb{E}[{\left|e^{tA}h\right|}^{2m}+|{\int }_0^t{e}^{(t-s)A}DF\left(X(s,x)\right){\eta }^h(s,x)ds{|}^{2m}\hfill \\ & +\left|\sum _{k=1}^{\infty }{\int }_0^t{e}^{(t-s)A}D\mathrm{\Sigma }\left(X(s,x)\right){\eta }^h(s,x)\sqrt{{\mu }_k}{e}_{k}d{\beta }_k\left(s\right){|}^{2m}\right]\hfill \\ & \le 3^{2m-1}\{e^{-2m{\pi }^{2}t}{\left|h\right|}^{2m}+\mathbb{E}\left[{\left({\int }_0^t\left|e^{(t-s)A}DF\left(X(s,x)\right){\eta }^h(s,x)\right|ds\right)}^{2m}\right]\hfill \\ & +c\mathbb{E}\left[{\left({\int }_0^t\sum _{k=1}^{\infty }\right|e^{(t-s)A}D\mathrm{\Sigma }\left(X(s,x)\right){\eta }^h(s,x)\sqrt{{\mu }_k}{e}_k{|}^{2}ds)}^m\right]\}\hfill \\ & \le 3^{2m-1}\{e^{-2m{\pi }^{2}t}{\left|h\right|}^{2m}+c_1\mathbb{E}\left[{\left({\int }_0^t{e}^{-{\pi }^2(t-s)}\left|{\eta }^h(s,x)\right|ds\right)}^{2m}\right]\hfill \\ & +c{c}_2\mathbb{E}{\left[({\int }_0^t\sum _{k=1}^{\infty }{e}^{-2{\pi }^2(t-s)k^2}|{\eta }^h(s,x){|}^{2}ds\sum _{k=1}^{\infty }{\mu }_k\right)}^m\left]\right\}\hfill \\ & \le 3^{2m-1}\{e^{-2m{\pi }^{2}t}{\left|h\right|}^{2m}+c_1\mathbb{E}\left[{\left({\int }_0^{t}J\left({\pi }^2(t-s)\right)\left|{\eta }^h(s,x)\right|ds\right)}^{2m}\right]\hfill \\ & +c{c}_2{\left(\sum _{k=1}^{\infty }{\mu }_k\right)}^m\mathbb{E}{\left[({\int }_0^{t}J\left(2{\pi }^2(t-s)\right)|{\eta }^h(s,x){|}^{2}ds\right)}^m\left]\right\}.\hfill \end{array}$$

(46)

Here, we have applied Minkowski’s inequality and Burkholder’s inequality, where c is a given positive constant, $c_1=({sup}_{\xi \in D,z\in \mathbb{R}}|{\partial }_{z}f(\xi ,z){|)}^{2m}$, $c_2=({sup}_{\xi \in D,z\in \mathbb{R}}|{\partial }_z\sigma (\xi ,z){|)}^{2m}$, and $J\left(t\right)={\sum }_{n=1}^{\infty }{e}^{-t{n}^2},t>0$.

Taking notice of

$$\begin{array}{cc}\hfill J\left(t\right)=& e^{-t}+\sum _{n=1}^{\infty }{e}^{-t{(n+1)}^2}\le e^{-t}+{\int }_1^{\infty }{e}^{-t{x}^2}dx\hfill \\ \hfill \le & e^{-t}\left(1+{\int }_1^{\infty }{e}^{-t(x^2-1)}dx\right)\le e^{-t}\left(1+{\int }_0^{\infty }{e}^{-t{y}^2}{\displaystyle \frac{y}{\sqrt{y^2+1}}}dy\right)\hfill \\ \hfill \le & e^{-t}\left(1+{\int }_0^{\infty }{e}^{-t{y}^2}\right)=e^{-t}(1+\sqrt{\pi }{t}^{-{\displaystyle \frac{1}{2}}})\hfill \\ \hfill \le & 4t^{-1/2}{e}^{-t/2},\hfill \end{array}$$

(47)

and the fact

$$\begin{array}{c}\hfill {\int }_0^t{s}^{-{\displaystyle \frac{1}{2}}}{e}^{-\theta s}ds\le {\int }_0^{\infty }{s}^{-{\displaystyle \frac{1}{2}}}{e}^{-\theta s}ds={\displaystyle \frac{1}{\sqrt{\theta }}}\mathrm{\Gamma }\left({\displaystyle \frac{1}{2}}\right)=\sqrt{{\displaystyle \frac{\pi }{\theta }}},t>0,\theta >0,\end{array}$$

(48)

applying the Hölder inequality with weight, we obtain

$$\begin{array}{cc}& \mathbb{E}\left[\right|{\eta }^h{(t,x)|}^{2m}]\le 3^{2m-1}\{{\left|h\right|}^{2m}+c_1{\left({\displaystyle \frac{4}{\pi }}\right)}^{2m}\mathbb{E}{\left[{\int }_0^t{(t-s)}^{-{\displaystyle \frac{1}{2}}}{e}^{-{\displaystyle \frac{{\pi }^2(t-s)}{2}}}\left|{\eta }^h(s,x)\right|ds\right]}^{2m}\hfill \\ & +c{c}_2{\left({\displaystyle \frac{4}{\sqrt{2}\pi }}\right)}^m{\left(\sum _{k=1}^{\infty }{\mu }_k\right)}^m\mathbb{E}{\left[{\int }_0^t{(t-s)}^{-{\displaystyle \frac{1}{2}}}{e}^{-{\pi }^2(t-s)}{\left|{\eta }^h(s,x)\right|}^{2}ds\right]}^m\}\hfill \\ & \le 3^{2m-1}\{{\left|h\right|}^{2m}+c_1{\left({\displaystyle \frac{4}{\pi }}\right)}^{2m}\mathbb{E}[\left({\int }_0^t{(t-s)}^{-{\displaystyle \frac{1}{2}}}{e}^{-{\displaystyle \frac{{\pi }^2(t-s)}{2}}}{\left|{\eta }^h(s,x)\right|}^{2m}ds\right)\hfill \\ & \times {\left({\int }_0^t{(t-s)}^{-{\displaystyle \frac{1}{2}}}{e}^{-{\displaystyle \frac{{\pi }^2(t-s)}{2}}}ds\right)}^{2m-1}]+c{c}_2{\left({\displaystyle \frac{4}{\sqrt{2}\pi }}\right)}^m{\left(\sum _{k=1}^{\infty }{\mu }_k\right)}^m\hfill \\ & \times \mathbb{E}\left[\left({\int }_0^t{(t-s)}^{-{\displaystyle \frac{1}{2}}}{e}^{-{\pi }^2(t-s)}{\left|{\eta }^h(s,x)\right|}^{2m}ds\right){\left({\int }_0^t{(t-s)}^{-{\displaystyle \frac{1}{2}}}{e}^{-{\pi }^2(t-s)}ds\right)}^{m-1}\right]\}\hfill \\ & \le 3^{2m-1}{\left\{\right|h|}^{2m}+c_1{\left(\sqrt{{\displaystyle \frac{2}{\pi }}}\right)}^{2m-1}{\left({\displaystyle \frac{4}{\pi }}\right)}^{2m}{\int }_0^t{(t-s)}^{-{\displaystyle \frac{1}{2}}}\mathbb{E}\left[\right|{\eta }^h(s,x){|}^{2m}]ds\hfill \\ & +c{c}_2{\left(\sqrt{{\displaystyle \frac{1}{\pi }}}\right)}^{m-1}{\left({\displaystyle \frac{4}{\sqrt{2}\pi }}\right)}^m{\left(\sum _{k=1}^{\infty }{\mu }_k\right)}^m{\int }_0^t{(t-s)}^{-{\displaystyle \frac{1}{2}}}\mathbb{E}\left[\right|{\eta }^h(s,x){|}^{2m}]ds\}\hfill \\ & \le 3^{2m-1}\left\{{\left|h\right|}^{2m}+c_3{\int }_0^t{(t-s)}^{-{\displaystyle \frac{1}{2}}}\mathbb{E}\left[\right|{\eta }^h(s,x){|}^{2m}]ds\right\},\hfill \end{array}$$

(49)

where $c_3=c_1{\left(\sqrt{{\displaystyle \frac{2}{\pi }}}\right)}^{2m-1}{\left({\displaystyle \frac{4}{\pi }}\right)}^{2m}+c{c}_2{\left(\sqrt{{\displaystyle \frac{1}{\pi }}}\right)}^{m-1}{\left({\displaystyle \frac{4}{\sqrt{2}\pi }}\right)}^m{\left({\displaystyle \sum _{k=1}^{\infty }}{\mu }_k\right)}^m$.

Due to Lemma 1, we obtain

$$\begin{array}{cc}\hfill \mathbb{E}\left(\right|{\eta }^h(t,x){|}^{2m})\le & 3^{2m-1}{\left|h\right|}^{2m}\left(1+{\int }_0^t\sum _{n=1}^{\infty }{\displaystyle \frac{{\left(3^{2m-1}{c}_3\mathrm{\Gamma }(1/2)\right)}^n}{\mathrm{\Gamma }(n/2)}}{(t-s)}^{n/2-1}ds\right)\hfill \\ \hfill =& 3^{2m-1}{\left|h\right|}^{2m}\sum _{n=0}^{\infty }{\displaystyle \frac{{\left(3^{2m-1}{c}_3\mathrm{\Gamma }(1/2)\right)}^n}{\mathrm{\Gamma }(n/2)}}·{\displaystyle \frac{2}{n}}{t}^{n/2}\hfill \\ \hfill =& 3^{2m-1}{\left|h\right|}^{2m}\sum _{n=0}^{\infty }{\displaystyle \frac{{\left(3^{2m-1}{c}_3\mathrm{\Gamma }(1/2)t^{1/2}\right)}^n}{\mathrm{\Gamma }(n/2+1)}}.\hfill \end{array}$$

(50)

Employing the notion presented in [25]

$$E_{\alpha ,\beta }\left(z\right)=\sum _{n=0}^{\infty }{\displaystyle \frac{z^n}{\mathrm{\Gamma }(\alpha n+\beta )}},\alpha >0,\beta >0$$

(51)

and for a constant $\rho >0$, the estimate

$$|E_{{\displaystyle \frac{1}{2}},1}\left(z\right)|\le \rho e^{{\left|z\right|}^2}$$

(52)

holds; see the Theorem 1.5 in [25].

Thus,

$$\begin{array}{cc}\hfill \mathbb{E}\left(\right|{\eta }^h(t,x){|}^{2m})\le & 3^{2m-1}{\left|h\right|}^{2m}{E}_{{\displaystyle \frac{1}{2}},1}\left(3^{2m-1}{c}_3\mathrm{\Gamma }(1/2)t^{1/2}\right)\hfill \\ \hfill \le & a_1{\left|h\right|}^{2m}{e}^{{\lambda }_{1}t},\hfill \end{array}$$

(53)

where $a_1=3^{2m-1}\rho$ and ${\lambda }_1={\left(3^{2m-1}{c}_3\mathrm{\Gamma }(1/2)\right)}^2$. This completes the proof. □

Proposition 4.

In addition, to assume Hypothesis 1(1)–(3), assume functions $f(\xi ,·)$, $\sigma (\xi ,·)\in C^2\left(\mathbb{R}\right)$, for all $\xi \in D$, and

$$\underset{\xi \in D}{sup}\underset{z\in \mathbb{R}}{sup}\left|{\partial }_z^{j}f(\xi ,z)\right|<\infty ,j=1,2,$$

(54)

$$\underset{\xi \in D}{sup}\underset{z\in \mathbb{R}}{sup}\left|{\partial }_z^j\sigma (\xi ,z)\right|<\infty ,j=1,2.$$

(55)

Then, $X(t,x)$ is twice Gâteaux differentiable in x, and for any $h,k\in H$, the second-order directional derivative

$${\zeta }^{h,k}(t,x):=D_x^{2}X(t,x)(h,k)$$

(56)

is the unique mild solution of equation

$$\left\{\begin{array}{c}d{\zeta }^{h,k}(t,x)=[A{\zeta }^{h,k}(t,x)+DF\left(X(t,x)\right){\zeta }^{h,k}(t,x)]dt\hfill \\ +D\mathrm{\Sigma }\left(X(t,x)\right){\zeta }^{h,k}(t,x)dW\left(t\right)\hfill \\ +D^{2}F\left(X(t,x)\right)({\eta }^h(t,x),{\eta }^k(t,x))dt\hfill \\ +D^2\mathrm{\Sigma }\left(X(t,x)\right)({\eta }^h(t,x),{\eta }^k(t,x))dW\left(t\right),\hfill \\ {\zeta }^{h,k}(0,x)=0.\hfill \end{array}\right.$$

(57)

In addition, for any $h,k\in H$

$$\mathbb{E}\left(\right|{\zeta }^{h,k}(t,x){|}^2)\le a_2{e}^{{\lambda }_{2}t}{\left|h\right|}^2{\left|k\right|}^2.t\ge 0,$$

(58)

for two constants ${\lambda }_2$ and $a_2$.

Proof. 

Under the assumptions of Proposition 3 and leveraging the first-order derivative estimates for ${\eta }^h(t,x)$ (Proposition 3), we now establish the existence and regularity of the second-order derivatives ${\zeta }^{h,k}(t,x)$. Similar to the proof of Proposition 3, we can demonstrate that $X(t,x)$ is twice Gâteaux differentiable with respect to x, and the second-order directional derivative is the unique mild solution of Equation $\left(57\right)$. Here, we omit the detailed description and focus on the proof of (58).

From the definition of the derivative of the Nemytskii operators F (and $\mathrm{\Sigma }$) (see Remark 2 below), we need to ensure that there exists $p_1>2$ such that ${\eta }^h(t,x)\in L^{p_1}(0,1)$ and ${\eta }^k(t,x)\in L^{p_2}(0,1)$, where $p_2=2p_1/(p_1-2)$. Fortunately, with the help of the Sobolev embedding theorem $D\left({(-A)}^{1/8}\right)\subset L^4(0,1)$, and since ${\eta }^h(t,x)\in D\left({(-A)}^{1/8}\right)$ for any $h\in H$, we can choose $p_1=p_2=4$.

Now, based on (22) and (57), we derive

$$\begin{array}{cc}\hfill {\zeta }^{h,k}(t,x)& ={\int }_0^t{e}^{(t-s)A}DF\left(X(s,x)\right){\zeta }^{h,k}(s,x)ds\hfill \\ & +{\int }_0^t{e}^{(t-s)A}{D}^{2}F\left(X(s,x)\right)({\eta }^h(s,x),{\eta }^k(s,x))dt\hfill \\ & +\sum _{n=1}^{\infty }{\int }_0^t\sqrt{{\mu }_n}{e}^{(t-s)A}D\mathrm{\Sigma }\left(X(s,x)\right){\zeta }^{h,k}(s,x)e_{n}d{\beta }_n\left(s\right)\hfill \\ & +\sum _{n=1}^{\infty }{\int }_0^t\sqrt{{\mu }_n}{e}^{(t-s)A}{D}^2\mathrm{\Sigma }\left(X(s,x)\right)({\eta }^h(s,x),{\eta }^k(s,x))e_{n}d{\beta }_n\left(s\right).\hfill \end{array}$$

(59)

Hence,

$$\begin{array}{cc}& \mathbb{E}\left[\right|{\zeta }^{h,k}(t,x){|}^2]\le 4\mathbb{E}\left[\right|{\int }_0^t{e}^{(t-s)A}DF\left(X(s,x)\right){\zeta }^{h,k}(s,x)ds{|}^2\hfill \\ & +|{\int }_0^t{e}^{(t-s)A}{D}^{2}F\left(X(s,x)\right)({\eta }^h(s,x),{\eta }^k(s,x))ds{|}^2\hfill \\ & +|\sum _{n=1}^{\infty }{\int }_0^t\sqrt{{\mu }_n}{e}^{(t-s)A}D\mathrm{\Sigma }\left(X(s,x)\right){\zeta }^{h,k}(s,x)e_{n}d{\beta }_n\left(s\right){|}^2\hfill \\ & +\left|\sum _{n=1}^{\infty }{\int }_0^t\sqrt{{\mu }_n}{e}^{(t-s)A}{D}^2\mathrm{\Sigma }\left(X(s,x)\right)({\eta }^h(s,x),{\eta }^k(s,x))e_{n}d{\beta }_n\left(s\right){|}^2\right]\hfill \\ & =:4(I_1+I_2+I_3+I_4).\hfill \end{array}$$

(60)

Observe that $\parallel e^{tA}\parallel \le e^{-{\pi }^{2}t}\le J\left({\pi }^{2}t\right)$, and in combination with $\left(54\right)$ and $\left(55\right)$, we have

$$\begin{array}{cc}& I_1=\mathbb{E}\left[|{\int }_0^t{e}^{(t-s)A}DF\left(X(s,x)\right){\zeta }^{h,k}(s,x)ds{|}^2\right]\hfill \\ & \le c_1^2\mathbb{E}\left[{\left({\int }_0^{t}J\left({\pi }^2(t-s)\right)\left|{\zeta }^{h,k}(s,x)\right|ds\right)}^2\right]\hfill \\ & \le {\displaystyle \frac{16c_1^2}{{\pi }^2}}\mathbb{E}\left[{\left({\int }_0^t{(t-s)}^{-1/2}{e}^{-{\pi }^2(t-s)/2}\left|{\zeta }^{h,k}(s,x)\right|ds\right)}^2\right]\hfill \\ & \le {\displaystyle \frac{16c_1^2}{{\pi }^2}}\mathbb{E}\left[{\int }_0^t{(t-s)}^{-1/2}{e}^{-{\pi }^2(t-s)/2}{\left|{\zeta }^{h,k}(s,x)\right|}^{2}ds\right]\left[{\int }_0^t{(t-s)}^{-1/2}{e}^{-{\pi }^2(t-s)/2}ds\right]\hfill \\ & \le {\displaystyle \frac{16\sqrt{2}{c}_1^2}{{\pi }^2\sqrt{\pi }}}{\int }_0^t{(t-s)}^{-1/2}\mathbb{E}\left[\right|{\zeta }^{h,k}(s,x){|}^2]ds,\hfill \end{array}$$

(61)

$$\begin{array}{cc}& I_2=\mathbb{E}\left[|{\int }_0^t{e}^{(t-s)A}{D}^{2}F\left(X(s,x)\right)({\eta }^h(s,x),{\eta }^k(s,x))ds{|}^2\right]\hfill \\ & \le c_2^2\mathbb{E}\left[{\left({\int }_0^t\parallel e^{(t-s)A}\parallel |{\eta }^h{(s,x)|}_{L^4(0,1)}{\left|{\eta }^k(s,x)\right|}_{L^4(0,1)}ds\right)}^2\right]\hfill \\ & \le c_2^2\mathbb{E}\left[{\left({\int }_0^{t}J\left({\pi }^2(t-s)\right)|{\eta }^h{(s,x)|}_{L^4(0,1)}{\left|{\eta }^k(s,x)\right|}_{L^4(0,1)}ds\right)}^2\right]\hfill \\ & \le {\displaystyle \frac{16c_2^2}{{\pi }^2}}\mathbb{E}\left[{\left({\int }_0^t{(t-s)}^{-{\displaystyle \frac{1}{2}}}{e}^{-{\displaystyle \frac{{\pi }^2(t-s)}{2}}}|{\eta }^h{(s,x)|}_{L^4(0,1)}{\left|{\eta }^k(s,x)\right|}_{L^4(0,1)}ds\right)}^2\right]\hfill \\ & \le {\displaystyle \frac{16c_2^2}{{\pi }^2}}\mathbb{E}\left[{\int }_0^t{(t-s)}^{-{\displaystyle \frac{1}{2}}}{e}^{-{\displaystyle \frac{{\pi }^2(t-s)}{2}}}|{\eta }^h{(s,x)|}_{L^4(0,1)}^2{\left|{\eta }^k(s,x)\right|}_{L^4(0,1)}^{2}ds\right]\hfill \\ & \times \left({\int }_0^t{(t-s)}^{-{\displaystyle \frac{1}{2}}}{e}^{-{\displaystyle \frac{{\pi }^2(t-s)}{2}}}ds\right)\hfill \\ & \le {\displaystyle \frac{16\sqrt{2}{c}_2^2}{{\pi }^2\sqrt{\pi }}}{\int }_0^t{(t-s)}^{-{\displaystyle \frac{1}{2}}}{\left(\mathbb{E}\left[\right|{\eta }^h(s,x){|}_{L^4(0,1)}^4]\right)}^{{\displaystyle \frac{1}{2}}}{\left(\mathbb{E}\left[\right|{\eta }^k(s,x){|}_{L^4(0,1)}^4]\right)}^{{\displaystyle \frac{1}{2}}}ds,\hfill \end{array}$$

(62)

$$\begin{array}{cc}& I_3=\mathbb{E}\left[|\sum _{n=1}^{\infty }{\int }_0^t\sqrt{{\mu }_n}{e}^{(t-s)A}D\mathrm{\Sigma }\left(X(s,x)\right){\zeta }^{h,k}(s,x)e_{n}d{\beta }_n\left(s\right){|}^2\right]\hfill \\ & \le \mathbb{E}\left[\sum _{n=1}^{\infty }{\int }_0^t|\sqrt{{\mu }_n}{e}^{(t-s)A}D\mathrm{\Sigma }\left(X(s,x)\right){\zeta }^{h,k}(s,x)e_n{|}^{2}ds\right]\hfill \\ & \le c_3^2\sum _{n=1}^{\infty }{\mu }_n{\int }_0^{t}J\left(2{\pi }^2(t-s)\right)\mathbb{E}\left[\right|{\zeta }^{h,k}(s,x){|}^2]ds\hfill \\ & \le {\displaystyle \frac{4c_3^2}{\sqrt{2}\pi }}\sum _{n=1}^{\infty }{\mu }_n{\int }_0^t{(t-s)}^{-{\displaystyle \frac{1}{2}}}\mathbb{E}\left[\right|{\zeta }^{h,k}(s,x){|}^2]ds,\hfill \end{array}$$

(63)

$$\begin{array}{cc}& I_4=\mathbb{E}\left[|\sum _{n=1}^{\infty }{\int }_0^t\sqrt{{\mu }_n}{e}^{(t-s)A}{D}^2\mathrm{\Sigma }\left(X(s,x)\right)({\eta }^h(s,x),{\eta }^k(s,x))e_{n}d{\beta }_n\left(s\right){|}^2\right]\hfill \\ & \le \mathbb{E}\left[\sum _{n=1}^{\infty }{\int }_0^t|\sqrt{{\mu }_n}{e}^{(t-s)A}{D}^2\mathrm{\Sigma }\left(X(s,x)\right)({\eta }^h(s,x),{\eta }^k(s,x))e_n{|}^{2}ds\right]\hfill \\ & \le c_4\sum _{n=1}^{\infty }{\mu }_n{\int }_0^{t}J\left(2{\pi }^2(t-s)\right)\mathbb{E}\left[\right|{\eta }^h{(s,x)|}_{L^4(0,1)}^2|{\eta }^k(s,x){|}_{L^4(0,1)}^2]ds\hfill \\ & \le c_5\sum _{n=1}^{\infty }{\mu }_n{\int }_0^t{(t-s)}^{-{\displaystyle \frac{1}{2}}}{\left(\mathbb{E}\left[\right|{\eta }^h(s,x){|}_{L^4(0,1)}^4]\right)}^{{\displaystyle \frac{1}{2}}}{\left(\mathbb{E}\left[\right|{\eta }^k(s,x){|}_{L^4(0,1)}^4]\right)}^{{\displaystyle \frac{1}{2}}}ds,\hfill \end{array}$$

(64)

where $c_1=\underset{\xi \in D,z\in \mathbb{R}}{sup}\left|{\partial }_{z}f(\xi ,z)\right|$, $c_2=\underset{\xi \in D,z\in \mathbb{R}}{sup}\left|{\partial }_z^{2}f(\xi ,z)\right|$, $c_3=c·\underset{\xi \in D,z\in \mathbb{R}}{sup}\left|{\partial }_z\sigma (\xi ,z)\right|$, $c_4=c·{\left(\underset{\xi \in D}{sup}\underset{z\in \mathbb{R}}{sup}\left|{\partial }_z^2\sigma (\xi ,z)\right|\right)}^2$, $c_5={\displaystyle \frac{4c_4}{\sqrt{2}\pi }}$.

Analogously to the argument of Proposition 3, the following assertion can be established.

• Assertion. For any $h\in H$, there exist two constants $a_0>0$ and ${\lambda }_0>0$ such that

$$\begin{array}{c}\hfill {\mathbb{E}\left[\right|(-A)}^{1/8}{\eta }^h{(t,x)|}^4]\le a_0{e}^{{\lambda }_{0}t}{\left|{(-A)}^{1/8}{e}^{tA}h\right|}^4,\end{array}$$

(65)

and

$$\begin{array}{c}\hfill \mathbb{E}\left[\right|{\eta }^h{(t,x)|}_{L^4(0,1)}^4]\le a_0{e}^{{\lambda }_{0}t}{\left|h\right|}^4.\end{array}$$

(66)

In combination with (60)–(66), we obtain

$$\begin{array}{c}\hfill \mathbb{E}\left[\right|{\zeta }^{h,k}{(t,x)|}^2]\le 2c_6{e}^{{\lambda }_{0}t}{\left|h\right|}^2{\left|k\right|}^2+2c_7{\int }_0^t{(t-s)}^{-{\displaystyle \frac{1}{2}}}\mathbb{E}\left[\right|{\zeta }^{h,k}(s,x){|}^2]ds,\end{array}$$

(67)

where

$$c_6=4max\left\{{\displaystyle \frac{16\sqrt{2}{c}_2}{{\pi }^2}}{a}_0{\lambda }_0^{-{\displaystyle \frac{1}{2}}},{\displaystyle \frac{4c_4}{\sqrt{2\pi }}}\sum _{n=1}^{\infty }{\mu }_n{a}_0{\lambda }_0^{-{\displaystyle \frac{1}{2}}}\right\},$$

(68)

$$c_7=4max\left\{{\displaystyle \frac{16\sqrt{2}{c}_1}{{\pi }^2\sqrt{\pi }}},{\displaystyle \frac{4c_3}{\sqrt{2}\pi }}\sum _{n=1}^{\infty }{\mu }_n\right\}.$$

(69)

Thus, the estimate $\left(58\right)$ follows from $\left(67\right)$ by Lemma 1. □

Remark 2.

Since the function f satisfies the conditions in Proposition 3, if we fix $h\in L^{p_1}\left(D\right)$ with $p_1>2$, then the mapping $DF(·)h:H\to H$ is differentiable at any point $x\in H$ along any direction $k\in L^{p_2}\left(D\right)$, where $p_2=2p_1/(p_1-2)$, and the following holds:

$$\begin{array}{c}\hfill (D^{2}F\left(x\right)(h,k)\left(\xi \right)={\partial }_z^{2}f(\xi ,x\left(\xi \right))\left(hk\right)\left(\xi \right),\xi \in D,\end{array}$$

(70)

and it is suitable for $D^2\mathrm{\Sigma }$.

4. The Kolmogorov Equation

The purpose of this section is to demonstrate the existence of classical solutions to the Kolmogorov equation associated with stochastic differential equations driven by multiplicative noise.

Let $X(t,x)$ denote the mild solution of the stochastic partial differential Equation (2). Based on the definitions (6), (7), and (9), we will demonstrate that if u and $\psi$ are sufficiently regular, then $u(t,x)$ is a classical solution to the parabolic Kolmogorov equation (8), and $\psi \left(x\right)$ is a solution to the elliptic Kolmogorov equation (10).

Definition 2.

A function $u:[0,\infty )\times H\to \mathbb{R}$ is called a classical solution of the problem $\left(8\right)$ if

$\left(i\right)$

for any $t>0$, the function $u(t,·)\in C_b^1\left(H\right)$ and its second-order derivatives exist and are bounded in all directions in H;

$\left(ii\right)$

for any $x\in D\left(A\right)$, the function $u(·,x)$ is differentiable on $(0,\infty )$ and satisfies Equation $\left(8\right)$.

Theorem 1.

Suppose that $\phi \in C_b^2\left(H\right)$ and $D^2\phi \left(x\right)\in L_1\left(H\right)$ for all $x\in H$ and $Tr\left[D^2\phi \right]\in C_b\left(H\right)$. Under the assumptions of Proposition 3, the function $u(t,x)$ defined by $\left(6\right)$ is a classical solution to $\left(8\right)$. Furthermore, for all $t>0$ and $x\in H$, we have

$$\begin{array}{c}\hfill |D_{x}u(t,x)|\le a_3{e}^{{\lambda }_{3}t}{\parallel \phi \parallel }_1,\end{array}$$

(71)

$$\begin{array}{c}\hfill \parallel D_x^{2}u(t,x)\parallel \le a_3{e}^{{\lambda }_{3}t}{\parallel \phi \parallel }_2,\end{array}$$

(72)

and

$$\begin{array}{c}\hfill |Tr\left(D_x^{2}u(t,x){\mathrm{\Sigma }}^2\left(x\right)Q\right)|\le a_3{e}^{{\lambda }_{3}t}{\parallel \phi \parallel }_2.\end{array}$$

(73)

for some constants $a_3$ and ${\lambda }_3$.

Proof. 

For all $x,h\in H$ and $t\ge 0$, by Proposition 3, we have that

$$\langle D_{x}u(t,x),h\rangle =\mathbb{E}\left[\langle D\phi \left(X(t,x)\right),{\eta }^h(t,x)\rangle \right].$$

(74)

Thus,

$$\begin{array}{c}\hfill \langle D_{x}u(t,x),h\rangle \le {\parallel \phi \parallel }_1\mathbb{E}\left[\right|{\eta }^h(t,x)\left|\right]\le a_1^{1/2}{e}^{{\displaystyle \frac{{\lambda }_1}{2}}t}\left|h\right|·{\parallel \phi \parallel }_1,\end{array}$$

(75)

which implies $\left(71\right)$.

In the same way, for all $x,h,k\in H$ and $t\ge 0$, one deduces

$$\langle D_x^{2}u(t,x)h,k\rangle =\mathbb{E}\left[\langle D\phi \left(X(t,x)\right),{\zeta }^{h,k}(t,x)\rangle \right]+\mathbb{E}\left[\langle D^2\phi \left(X(t,x)\right){\eta }^h(t,x),{\eta }^k(t,x)\rangle \right].$$

(76)

By Remark 2, we find that

$$\begin{array}{c}|\langle D_x^{2}u(t,x)h,k\rangle {|\le \parallel \phi \parallel }_1\mathbb{E}\left[\right|{\zeta }^{h,k}(t,x)\left|\right]+\hfill \\ {\parallel \phi \parallel }_2\left(\mathbb{E}\right[|{\eta }^h{(t,x)|}_{L_4(0,1)}^2{\left]\right)}^{1/2}\left(\mathbb{E}\right[|{\eta }^k{(t,x)|}_{L_4(0,1)}^2{\left]\right)}^{1/2},\hfill \end{array}$$

(77)

and then, from $\left(38\right)$ and $\left(58\right)$, we derive the estimate $\left(72\right)$.

Indeed,

$$\begin{array}{cc}\hfill \mathrm{Tr}\left(D_x^{2}u(t,x){\mathrm{\Sigma }}^2\left(x\right)Q\right)=& \sum _{n=1}^{\infty }\langle D_x^{2}u(t,x)\mathrm{\Sigma }\left(x\right)Q{e}_n,\mathrm{\Sigma }\left(x\right)e_n\rangle \hfill \\ \hfill =& \sum _{n=1}^{\infty }{\mu }_n\langle D_x^{2}u(t,x)\mathrm{\Sigma }\left(x\right)e_n,\mathrm{\Sigma }\left(x\right)e_n\rangle ,\hfill \end{array}$$

(78)

which leads to

$$\begin{array}{cc}\hfill \left|\mathrm{Tr}\right(D_x^{2}u(t,x){\mathrm{\Sigma }}^2\left(x\right)Q\left)\right|\le & \sum _{n=1}^{\infty }{\mu }_n\left|\langle D_x^{2}u(t,x)\mathrm{\Sigma }\left(x\right)e_n,\mathrm{\Sigma }\left(x\right)e_n\rangle \right|\hfill \\ \hfill \le & \sum _{n=1}^{\infty }{\mu }_n\parallel D_x^{2}u(t,x)\parallel ·|\mathrm{\Sigma }\left(x\right)e_n{|}^2\hfill \\ \hfill \le & \sum _{n=1}^{\infty }{\mu }_n{\parallel \mathrm{\Sigma }\parallel }_0^2·\parallel D_x^{2}u(t,x)\parallel .\hfill \end{array}$$

(79)

Hence, $\left(73\right)$ follows from $\left(55\right)$ and $\left(72\right)$. This implies that the condition $\left(i\right)$ of Definition 2 holds by Remark 3 below.

Due to the differentiability of $X(·,x)$ in $(0,\infty )$, it is clear that $u(·,x)$ is also differentiable in $(0,\infty )$ for all $x\in D\left(A\right)$. To prove that $u(t,x)=\mathbb{E}\left[\phi \right(X(t,x)\left)\right]$ satisfies Equation $\left(8\right)$, we consider the Galerkin approximations of Equation $\left(8\right)$. For any $n\in \mathbb{N}$, setting $A_n=A{\mathrm{\Pi }}_n$, $F_n\left(x\right)={\mathrm{\Pi }}_{n}F\left({\mathrm{\Pi }}_{n}x\right)$, and ${\mathrm{\Sigma }}_n\left(x\right)={\mathrm{\Pi }}_n\mathrm{\Sigma }\left({\mathrm{\Pi }}_{n}x\right)$, it follows that

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{u}_t^n(t,x)={\displaystyle \frac{1}{2}}\mathrm{Tr}\left(D_x^2{u}^n(t,x){\mathrm{\Sigma }}_n^2\left(x\right)Q\right)+\langle A_{n}x+F_n\left(x\right),D_x{u}^n(t,x)\rangle ,\hfill \\ u^n(0,x)=\phi \left({\mathrm{\Pi }}_{n}x\right).\hfill \end{array}\right.\end{array}$$

(80)

By a standard result in finite dimensions [24], Equation $\left(80\right)$ has a unique classical solution, denoted by $u^n(t,x)$. Additionally, we can show that the function $\mathbb{E}\left[\phi \left(X^n(t,x)\right)\right]$ is a solution to Equation $\left(80\right)$ by applying the Itô formula to the Galerkin approximations to Equation $\left(25\right)$. Therefore, $u^n(t,x)=\mathbb{E}\left[\phi \left(X^n(t,x)\right)\right]$ is the unique classical solution of Equation $\left(80\right)$.

Given that for any $n\in \mathbb{N}$ the analogous estimates $\left(71\right)$, $\left(72\right)$, and $\left(73\right)$ hold for $u^n(t,x)$, with the corresponding constants independent of n, and for all $x,h,k\in H$, we are also able to obtain

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\underset{n\to \infty }{lim}{u}^n(t,x)=u(t,x),\hfill \\ \underset{n\to \infty }{lim}{D}_x{u}^n(t,x)h=D_{x}u(t,x)h,\hfill \\ \underset{n\to \infty }{lim}{D}_x^2{u}^n(t,x)(h,k)=D_x^{2}u(t,x)(h,k),\hfill \end{array}\right.\end{array}$$

(81)

and

$$\underset{n\to \infty }{lim}\mathrm{Tr}\left(D_x^2{u}^n(t,x){\mathrm{\Sigma }}_n^2\left(x\right)Q\right)=\mathrm{Tr}\left(D_x^{2}u(t,x){\mathrm{\Sigma }}^2\left(x\right)Q\right),$$

(82)

uniformly for $t\in [0,T]$. So

$$\underset{n\to \infty }{lim}{u}_t^n(t,x)=u_t(t,x),t>0,x\in D\left(A\right),$$

(83)

and by taking the limit $n\to \infty$ to $\left(80\right)$, we obtain that $u(t,x)$ fulfills $\left(8\right)$, that is, the condition $\left(ii\right)$ of Definition 2 also holds. □

Remark 3.

Let $\phi :H\to \mathbb{R}$, $x_0\in H$, and $R>0$, φ has a bounded and linear Gâteaux derivative in $B_R\left(x_0\right)$; here, $B_R\left(x_0\right)$ is the ball of center $x_0$ and radius R in H, and the Gâteaux derivative operator is continuous at $x_0$. Then, φ is Fréchet differentiable at the point $x_0$.

Theorem 2.

Suppose that $g\in C_b^2\left(H\right)$, $D^{2}g\left(x\right)\in L_1\left(H\right)$ for all $x\in H$, and $Tr\left[D^{2}g\right]\in C_b\left(H\right)$. Then, under the hypothesis for Proposition 3, for any $\lambda >{\lambda }_3$, the function $\psi \left(x\right)={\int }_0^{\infty }{e}^{-\lambda t}\mathbb{E}\left[g\left(X(t,x)\right)\right]dt$ is a classical solution of $\left(10\right)$.

Proof. 

Applying Theorem 1 we derive that $u(t,x)=\mathbb{E}\left[g\right(X(t,x)\left)\right]$ is the solution of the equation

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{u}_t(t,x)=K_{0}u(t,x),t\ge 0,x\in D\left(A\right),\hfill \\ u(0,x)=g\left(x\right),x\in H,\hfill \end{array}\right.\end{array}$$

(84)

and the estimates $\left(71\right)$ and $\left(72\right)$ ensure that

$$\psi \left(x\right)={\int }_0^{\infty }{e}^{-\lambda t}u(t,x)dt\in C_b^1\left(H\right),\mathrm{for}\lambda >{\lambda }_3,$$

(85)

and it has bounded second-order derivatives along any directions of H.

Actually, the formula of integration by parts can be used as follows,

$$\begin{array}{cc}\hfill \psi \left(x\right)=& {\int }_0^{\infty }{e}^{-\lambda t}u(t,x)dt\hfill \\ \hfill =& -{\displaystyle \frac{1}{\lambda }}[-u(0,x)-{\int }_0^{\infty }{e}^{-\lambda t}{u}_t(t,x)dt]\hfill \\ \hfill =& {\displaystyle \frac{1}{\lambda }}g\left(x\right)+{\displaystyle \frac{1}{\lambda }}{\int }_0^{\infty }{e}^{-\lambda t}{K}_{0}u(t,x)dt\hfill \\ \hfill =& {\displaystyle \frac{1}{\lambda }}(g\left(x\right)+K_0\psi \left(x\right)),\hfill \end{array}$$

(86)

that is, $\psi \left(x\right)={\int }_0^{\infty }{e}^{-\lambda t}\mathbb{E}\left[g\left(X(t,x)\right)\right]dt$ fulfills the Equation $\left(10\right)$. □

5. Invariant Measure

The existence of an invariant measure. This section is dedicated to the study of the invariant measure for the semigroup $P_t$, $t\ge 0$ associated with Equation $\left(25\right)$, given by

$$P_t\phi \left(x\right)=\mathbb{E}\left[\phi \left(X(t,x)\right)\right],t\ge 0,x\in H,\phi \in C_b\left(H\right).$$

(87)

Obviously $P_t$ satisfies the semigroup law and is a Feller semigroup, meaning that $P_t\phi \in C_b\left(H\right)$ for all $\phi \in C_b\left(H\right)$ and $t\ge 0$; see [7].

We say that a Borel probability measure $\nu$ in H is invariant for $P_t$ if

$${\int }_H{P}_t\phi \left(x\right)\nu \left(dx\right)={\int }_H\phi \left(x\right)\nu \left(dx\right),\mathrm{for} \mathrm{all}t\ge 0,\phi \in C_b\left(H\right).$$

(88)

Theorem 3.

Assume Hypothesis 1(1)–(3). There exists an invariant measure ν for $P_t$. Moreover, for any $m\in \mathbb{N}$, we have that

$${\int }_H{\left|x\right|}^{2m}\nu \left(dx\right)<\infty .$$

(89)

Proof. 

Let $X(t,x)$ denote the mild solution of $\left(25\right)$, defined as

$$\begin{array}{c}\hfill X(t,x)=e^{tA}x+{\int }_0^t{e}^{(t-s)A}F\left(X(s,x)\right)ds+\sum _{n=1}^{\infty }{\int }_0^t{e}^{(t-s)A}\mathrm{\Sigma }\left(X(s,x)\right)\sqrt{{\mu }_n}{e}_{n}d{\beta }_n\left(s\right).\end{array}$$

(90)

Hence,

$$\begin{array}{cc}\hfill \mathbb{E}\left[\right|X(t,x){|}^2]\le & 3|e^{tA}{x|}^2+3\mathbb{E}\left[|{\int }_0^t{e}^{(t-s)A}F\left(X(s,x)\right)ds{|}^2\right]\hfill \\ & +3\mathbb{E}\left[|\sum _{n=1}^{\infty }{\int }_0^t{e}^{(t-s)A}\mathrm{\Sigma }\left(X(s,x)\right)\sqrt{{\mu }_n}{e}_{n}d{\beta }_n\left(s\right){|}^2\right]\hfill \\ \hfill \le & 3e^{-2{\pi }^{2}t}{\left|x\right|}^2+3{\parallel F\parallel }_0{\left({\int }_0^t{e}^{-{\pi }^2(t-s)}ds\right)}^2\hfill \\ & +{3\parallel \mathrm{\Sigma }\parallel }_0{\int }_0^t|\sum _{n=1}^{\infty }{e}^{(t-s)A}\sqrt{{\mu }_n}{e}_n{|}^{2}ds\hfill \\ \hfill \le & 3e^{-2{\pi }^{2}t}{\left|x\right|}^2+{\displaystyle \frac{3}{{\pi }^2}}{\parallel F\parallel }_0+3\sum _{n=1}^{\infty }{\mu }_n{\parallel \mathrm{\Sigma }\parallel }_0{\int }_0^t\sum _{n=1}^{\infty }{e}^{-2{\pi }^2{n}^2(t-s)}ds.\hfill \end{array}$$

(91)

By combining the above inequality with $J\left(2{\pi }^2(t-s)\right)={\sum }_{n=1}^{\infty }{e}^{-2{\pi }^2{n}^2(t-s)}$, as defined in the proof of Proposition 3, there exists a constant $c>0$ such that

$$\begin{array}{c}\hfill {\mathbb{E}\left[\right|X(t,x)|}^2]\le c(1+e^{-2{\pi }^{2}t}{\left|x\right|}^2).\end{array}$$

(92)

Let ${\pi }_t(x,dy)$ denote the law of $X(t,x)$, for any fixed $x\in H$, and any $t\ge 0$, $R>0$,

$$\begin{array}{cc}\hfill {\pi }_t(x,B_R^c)\le & {\displaystyle \frac{1}{R^2}}{\int }_H{\left|y\right|}^2{\pi }_t(x,dy)\hfill \\ \hfill =& {\displaystyle \frac{1}{R^2}}{\mathbb{E}\left[\right|X(t,x)|}^2]\le {\displaystyle \frac{c}{R^2}}{(1+|x|}^2),\hfill \end{array}$$

(93)

Here, $B_R^c$ denotes the complement of the ball $B_R$, centered at 0 with radius R in H. Since $P_t$ is Feller, the existence of an invariant measure for $P_t$ follows from the Krylov–Bogoliubov Theorem.

Similarly, for any $m\in \mathbb{N}$, there exists a constant $c_m>0$ such that

$$\begin{array}{c}\hfill {\mathbb{E}\left[\right|X(t,x)|}^{2m}]\le c_m(1+e^{-2m{\pi }^{2}t}{\left|x\right|}^{2m}),x\in H,t\ge 0.\end{array}$$

(94)

For any $\delta >0$, set

$$\begin{array}{c}\hfill {\phi }_{\delta }\left(x\right)={\displaystyle \frac{{\left|x\right|}^{2m}}{1+{\delta \left|x\right|}^{2m}}},x\in H,\end{array}$$

(95)

then, ${\phi }_{\delta }\in C_b\left(H\right)$, and for any $x\in H$, $t\ge 0$, we have

$$\begin{array}{cc}& P_t\left({\phi }_{\delta }\right)\left(x\right)=\mathbb{E}\left[{\phi }_{\delta }\left(X(t,x)\right)\right]={\int }_H{\displaystyle \frac{{\left|y\right|}^{2m}}{1+{\delta \left|y\right|}^{2m}}}{\pi }_t(x,dy)\hfill \\ & \le {\int }_H{\left|y\right|}^{2m}{\pi }_t(x,dy)={\mathbb{E}\left[\right|X(t,x)|}^{2m}]\hfill \\ & \le c_m(1+e^{-2m{\pi }^{2}t}{\left|x\right|}^{2m}).\hfill \end{array}$$

(96)

Since $\nu$ is invariant for $P_t$, we derive that

$${\int }_H{\phi }_{\delta }\left(x\right)\nu \left(dx\right)={\int }_H{P}_t\left({\phi }_{\delta }\right)\left(x\right)\nu \left(dx\right)\le c_m\left(1+e^{-2m{\pi }^{2}t}{\int }_H{\left|x\right|}^{2m}\nu \left(dx\right)\right).$$

(97)

Then, taking the limit $t\to \infty$, it holds that

$${\int }_H{\displaystyle \frac{{\left|x\right|}^{2m}}{1+{\delta \left|x\right|}^{2m}}}\nu \left(dx\right)\le c_m,$$

(98)

and letting $\delta$ tend to 0 yields the conclusion $\left(89\right)$. □

The transition semigroup in $L^2(H,\nu )$. By Theorem 3, the invariant measure for $P_t$ exists and is denoted by $\nu$. Although it is well known that the semigroup $P_t$ is not generally strongly continuous in $C_b\left(H\right)$, it can be uniquely extended to a strongly continuous semigroup of contractions on $L^p(H,\nu )$, $p\ge 1$ (see [5] and the references therein), with its infinitesimal generator in $L^p(H,\nu )$ denoted by $K_p$. The following result is primarily studied in $L^2(H,\nu )$.

Theorem 4.

Assume that the conditions of Proposition 3 hold, then $K_2$ is the closure of the Kolmogorov operator $K_0$ in $L^2(H,\nu )$.

Proof. 

First, we show that $K_2\phi =K_0\phi$ for any $\phi \in D\left(K_0\right)$. Indeed, by Theorem 2 and the estimate $\left(89\right)$, it follows that for any $\phi \in D\left(K_0\right)$ and $x\in H$

$$K_2\phi \left(x\right)=\underset{t\to 0}{lim}{\displaystyle \frac{1}{t}}(P_t\phi \left(x\right)-\phi \left(x\right))=\underset{t\to 0}{lim}{\displaystyle \frac{u(t,x)-u(0,x)}{t}}=K_0\phi \left(x\right),$$

(99)

in $L^2(H,\nu )$; thus, $K_2$ extends $K_0$. In fact, $K_0$ is dissipative due to the dissipativeness of $K_2$, making $K_0$ closable. Its closure, denoted by $(\overline{K_0},D\left(\overline{K_0}\right))$, with respect to the $\pi$-convergence defined by Priola in [26], remains to be shown as equal to $K_2=\overline{K_0}$.

Let $\lambda >{\lambda }_3$, $g\in C_b^2\left(H\right)$, from Theorem 3, we know that

$$\psi \left(x\right)={\int }_0^{\infty }{e}^{-\lambda t}{P}_{t}g\left(x\right)dt,x\in H$$

(100)

belongs to $D\left(\overline{K_0}\right)$ and satisfies the following elliptic equation

$$\lambda \psi \left(x\right)-\overline{K_0}\psi \left(x\right)=g\left(x\right),\forall x\in H.$$

(101)

Therefore, the closure of the range of $(\lambda -K_0)$ includes $C_b^2\left(H\right)$, which is dense in $L^2(H,\nu )$. By the Lumer–Phillips theorem [27], we conclude that $\overline{K_0}$ is m-dissipative. On the other hand, since $K_2$ is the infinitesimal generator of a strongly continuous semigroup of contractions, it is also m-dissipative. Given that $K_2$ extends $K_0$, we deduce $K_2=\overline{K_0}$. This completes the proof. □