Convergence analysis of a finite difference method for stochastic Cahn–Hilliard equation

Jialin Hong LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, and School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China [email protected] , Diancong Jin School of Mathematics and Statistics, Huazhong University of Science and Technology, and Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan 430074, China [email protected] and Derui Sheng LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, and School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China [email protected]

Abstract.

This paper presents the convergence analysis of the spatial finite difference method (FDM) for the stochastic Cahn–Hilliard equation with Lipschitz nonlinearity and multiplicative noise. Based on fine estimates of the discrete Green function, we prove that both the spatial semi-discrete numerical solution and its Malliavin derivative have strong convergence order $1$ . Further, by showing the negative moment estimates of the exact solution, we obtain that the density of the spatial semi-discrete numerical solution converges in $L^{1}(\mathbb{R})$ to the exact one. Finally, we apply an exponential Euler method to discretize the spatial semi-discrete numerical solution in time and show that the temporal strong convergence order is nearly $\frac{3}{8}$ , where a difficulty we overcome is to derive the optimal Hölder continuity of the spatial semi-discrete numerical solution.

Key words and phrases:

strong convergence rate, finite difference method, exponential Euler method, stochastic Cahn–Hilliard equation

2010 Mathematics Subject Classification:

65C30, 60H35, 60H15, 60H07

This work is supported by the National key R&D Program of China under Grant No. 2020YFA0713701, National Natural Science Foundation of China (Nos. 11971470, 11871068, 12031020, 12022118), and the Fundamental Research Funds for the Central Universities 3004011142.

1. Introduction

Consider the following stochastic Cahn–Hilliard equation

\displaystyle\begin{split}\partial_{t}u+\Delta^{2}u&=\Delta f(u)+\sigma(u)\dot{W},\quad\text{in}~{}[0,T]\times\mathcal{O}\\ \end{split}

(1.1)

with initial condition $u(0,\cdot)=u_{0}$ and homogeneous Dirichlet boundary conditions (DBCs) $u=\Delta u=0$ on $\partial\mathcal{O}$ . Here, $\mathcal{O}:=(0,\pi)$ , $T>0$ , and $\{W(t,x),(t,x)\in[0,T]\times\mathcal{O}\}$ is a Brownian sheet defined on some probability space $(\Omega,\mathscr{F},\mathbb{P})$ . Assume that $u_{0}:\mathcal{O}\rightarrow\mathbb{R}$ is a deterministic continuous function, and $\sigma$ is bounded and globally Lipschitz continuous. Eq. (1.1) is a well-known phenomenological model to describe the complicated phase separation. In the original form, $f$ is the derivative of the homogeneous free energy $F$ which contains a logarithmic term and in some cases can be approximated by an even-degree polynomial with a positive dominant coefficient [4], for example, $F(x)=\frac{1}{4}x^{4}-\frac{1}{2}x^{2}$ . In this case, the truncation technique is usually used to localize (1.1) such that the sequence $\{u_{R}\}_{R\geq 1}$ given by

\displaystyle\begin{split}\partial_{t}u_{R}+\Delta^{2}u_{R}&=\Delta f_{R}(u_{R})+\sigma(u)\dot{W},\quad\text{in}~{}[0,T]\times\mathcal{O}\\ \end{split}

(1.2)

can approximate $u$ in some sense (see e.g., [4, 8]), where $\{f_{R}\}_{R\geq 1}$ satisfies the global Lipschitz condition. Hence, an effective numerical method applied to Eq. (1.2) is expected to approximate Eq. (1.1) well. With this consideration, the present work investigates numerical methods for stochastic Cahn–Hilliard equations with Lipschitz nonlinearity (i.e., $f$ is globally Lipschitz continuous), which includes the linearized Cahn–Hilliard equation ( $f=0$ ).

The existing results on the numerical methods for stochastic Cahn–Hilliard equations mainly focus on the strong convergence analysis. Without being too exhaustive, we mention [5, 21] on the finite element approximation for the case of $f=0$ . For the case of polynomial nonlinearity and additive noise, [19] and [14] respectively obtain the strong convergence of a spatial semi-discretization and a full discretization; [10, 24] establish the strong convergence rates of full discretizations based on the finite element method and spectral Galerkin method in space, respectively. Concerning the case of multiplicative noise, [8] presents the sharp strong convergence rate for a full discretization by using the spectral Galerkin method in space. In addition to the strong convergence analysis, the convergence analysis of densities of numerical solutions is also meaningful, which provides a theoretical foundation to approximate the density of the exact solution of the original system by means of numerical methods. There have been plenty results on density convergence of numerical solutions for various stochastic systems (see e.g., [3, 6, 9, 16, 18, 22]), of which we have not yet found relevant results for stochastic Cahn-Hilliard equations. The present paper aims to approximate the density of the exact solution of Eq. (1.1) via a spatial finite difference method (FDM) and present the strong convergence rate and density convergence of the associated numerical solution.

The spatial FDM has been employed to numerically solve, for instance, stochastic heat equations [15, 11, 1] and stochastic wave equations [7]. First, we give subtle error estimates between the discrete Green function of the spatial FDM and the exact one. Then under the globally Lipschitz condition on $f$ , we obtain the strong convergence order $1$ for the spatial semi-discrete numerical solution $u^{n}(t,x)$ based on the FDM, with $\frac{\pi}{n}$ being the spatial stepsize. Further, it is shown that the Malliavin derivative of $u^{n}(t,x)$ has the strong convergence order $1$ as well. Combining the above results with the negative moment estimates of the exact solution, we deduce that the spatial semi-discrete numerical solution admits a density, which converges in $L^{1}(\mathbb{R})$ to the density of the exact solution.

For more effective computation, we further discretize $u^{n}$ via an exponential Euler method in time and obtain the full discretization $u^{m,n}=\{u^{m,n}(t,x),(t,x)\in[0,T]\times\mathcal{O}\}$ , where $\frac{T}{m}$ denotes the temporal stepsize. As an explicit method, the exponential Euler method is more computationally efficient than the implicit method and does not suffer from the CFL condition. By investigating the temporal Hölder continuity of the spatial semi-discrete numerical solution, we attain the strong convergence rate of the proposed method for Eq. (1.1) with Lipschitz nonlinearity, namely

\|u^{m,n}(t,x)-u(t,x)\|_{L^{p}(\Omega)}\leq C(\epsilon)(n^{-1}+m^{-\frac{3}{8}+\epsilon}),

(1.3)

where $0<\epsilon\ll 1$ . The spatial convergence order $1$ and temporal convergence order nearly $\frac{3}{8}$ in (1.3) are optimal in the sense that they coincide with the mean-square spatial and temporal Hölder continuity exponents of the exact solution, respectively. On the basis of (1.3), a localized argument leads to an $L^{p}(\Omega;\mathbb{R})$ convergence order localized on a set of arbitrarily large probability for Eq. (1.1) with polynomial nonlinearity. With the independent interest, when $f$ is a polynomial of degree $3$ with a positive dominant coefficient, we also establish the Hölder continuity and the uniform moment estimate of the exact solution. These are prepared for the density convergence analysis of the numerical solution of the spatial FDM for Eq. (1.1) with polynomial nonlinearity in our future work.

The rest of this paper is organized as follows. Section 2 gives the Hölder continuity and the uniform moment estimate of the exact solution. Then we introduce the spatial FDM and study its strong convergence order in Section 3. Section 4 presents the density convergence of the spatial semi-discrete numerical solution. In Section 5, we further apply an exponential Euler method to obtain a full discretization and obtain its strong convergence order.

2. Preliminaries

Let $\mathcal{C}^{\alpha}(\mathcal{O})$ be the space of $\alpha$ -Hölder continuous functions on $\mathcal{O}$ if $\alpha\in(0,1)$ , and be the space of $\alpha$ times continuously differentiable functions on $\mathcal{O}$ if $\alpha\in\mathbb{N}$ . For $d\geq 1$ , we denote by $\|\cdot\|$ and $\langle\cdot,\cdot\rangle$ the Euclidean norm and inner product of $\mathbb{R}^{d}$ , respectively. For $1\leq q\leq\infty$ , we denote by $\|\cdot\|_{L^{q}}$ the usual norm of the space $L^{q}(\mathcal{O}):=L^{q}(\mathcal{O};\mathbb{R})$ . For $1\leq p<\infty$ and a Banach space $(H,\|\cdot\|_{H})$ , let $L^{p}(\Omega;H)$ be the space of $H$ -valued random variables with bounded $p$ th moment, endowed with the norm $\|\cdot\|_{L^{p}(\Omega;H)}:=\left(\mathbb{E}[\|\cdot\|_{H}^{p}]\right)^{\frac{1}{p}}.$ Especially, we write $\|\cdot\|_{p}:=\|\cdot\|_{L^{p}(\Omega;\mathbb{R})}$ for short. Hereafter, we use $C$ to denote a generic positive constant that may change from one place to another and depend on several parameters but never on the space and time stepsizes. Without illustrated, the supremum with respect to $t\in[0,T]$ (respectively, $x\in\mathcal{O}$ and $(t,x)\in[0,T]\times\mathcal{O}$ ) is denoted by $\sup_{t}$ (respectively, $\sup_{x}$ and $\sup_{t,x}$ ). In this section, we present the regularity estimate of the exact solution of Eq. (1.1) under Assumption 1 or 2.

Assumption 1.

$f$ satisfies the globally Lipschitz condition, i.e., there is $K>0$ such that

|f(y)-f(z)|\leq K|y-z|,\quad\forall~{}y,z\in\mathbb{R}.

Assumption 2.

$f$ is a polynomial of degree $3$ with a positive dominant coefficient, i.e., $f(x)=a_{0}x^{3}+a_{1}x^{2}+a_{2}x+a_{3}$ with $a_{0}>0$ .

The physical importance of the Dirichlet problem is pointed out to us by M. E. Gurtin: it governs the propagation of a solidification front into an ambient medium which is at rest relative to the front [12]; see [13, 8] and references therein for the study of Cahn–Hilliard equation with DBCs. In this case, the Green function associated to $\partial_{t}+\Delta^{2}$ is given by $G_{t}(x,y)=\sum_{j=1}^{\infty}e^{-\lambda_{j}^{2}t}\phi_{j}(x)\phi_{j}(y),$ $t\in[0,T]$ , $x,y\in\mathcal{O}$ , where $\lambda_{j}=-j^{2}$ , $\phi_{j}(x)=\sqrt{2/\pi}\sin(jx),$ $j\geq 1$ . It is known that $\{\phi_{j},j\geq 1\}$ forms an orthonormal basis of $L^{2}(\mathcal{O})$ . Denote $\mathbb{G}_{t}v(x):=\int_{\mathcal{O}}G_{t}(x,y)v(y)\mathrm{d}y$ , $v\in\mathcal{C}(\mathcal{O}).$ Similar to [4, Lemma 1.2], there exist $C,c>0$ such that

	$\displaystyle\|G_{t}(x,y)\|\leq\frac{C}{t^{1/4}}\exp\Big{(}-c\frac{\|x-y\|^{4/3}}{\|t\|^{1/3}}\Big{)},$		(2.1)
	$\displaystyle\|\Delta G_{t}(x,y)\|\leq\frac{C}{t^{3/4}}\exp\Big{(}-c\frac{\|x-y\|^{4/3}}{\|t\|^{1/3}}\Big{)}.$		(2.2)

The well-posedness of the stochastic Cahn–Hilliard equation under Assumption 2 with NBCs has been established in [4]. Since the Green function with DBCs and NBCs share similar properties, the existence and uniqueness of the solution to Eq. (1.1) under Assumption 2 can be obtained in an almost same way, and we present an outline of the idea here. For $R\geq 1$ , let $K_{R}:\mathbb{R}\rightarrow\mathbb{R}$ be an even smooth cut-off function satisfying

\displaystyle K_{R}(x)=1,\quad\text{if}~{}|x|<R;\qquad K_{R}(x)=0,\quad\text{if}~{}|x|\geq R+1,

(2.3)

and $|K_{R}|\leq 1$ , $|K_{R}^{\prime}|\leq 2$ . Consider a sequence of SPDEs

\displaystyle\partial_{t}\bar{u}_{R}(t,x)+\Delta^{2}\bar{u}_{R}(t,x)=\Delta\big{(}K_{R}(\|\bar{u}_{R}(t,\cdot)\|_{L^{q}})f(\bar{u}_{R}(t,x))\big{)}+\sigma(\bar{u}_{R}(t,x))\dot{W}(t,x)

(2.4)

with DBCs and $\bar{u}_{R}(0,\cdot)=u_{0}\in L^{q}(\mathcal{O})$ for some $q\geq 4$ . Define the stopping times

\tau_{R}:=\inf\{t\geq 0:\|\bar{u}_{R}(t,\cdot)\|_{L^{q}}\geq R\},\quad R\geq 1.

Using the uniqueness of the solution of Eq. (2.4), it is concluded from the local property of stochastic integrals that for $R^{\prime}>R$ , $\bar{u}_{R^{\prime}}(t,\cdot)=\bar{u}_{R}(t,\cdot)$ for $t\leq\tau_{R}$ , so that a process $u$ can be defined by $u(t,\cdot)=\bar{u}_{R}(t,\cdot)$ for $t\leq\tau_{R}$ . Set $\tau_{\infty}=\lim_{R\rightarrow\infty}\tau_{R}$ . Then $u$ is the unique solution of (1.1) on the interval $[0,\tau_{\infty})$ . Further, $\{\bar{u}_{R}\}_{R\geq 1}$ are $\{\mathcal{F}_{t}\}_{t\in[0,T]}$ -adapted stochastic processes such that for $\rho\in[q,\infty)$ ,

\sup_{R\geq 1}\mathbb{E}\Big{[}\sup_{t}\|\bar{u}_{R}(t,\cdot)\|_{L^{q}}^{\rho}\Big{]}\leq C(T,\rho,q)

(2.5)

(see the second inequality in P794 of [4]). Based on (2.5), $\tau_{\infty}=+\infty$ a.s. (see [4, (2.36)]), and thus under Assumption 2, Eq. (1.1) admits a global solution, i.e.,

	$\displaystyle u(t,x)$	$\displaystyle=\mathbb{G}_{t}u_{0}(x)+\int_{0}^{t}\int_{\mathcal{O}}\Delta G_{t-s}(x,y)f(u(s,y))\mathrm{d}y\mathrm{d}s$
		$\displaystyle\quad+\int_{0}^{t}\int_{\mathcal{O}}G_{t-s}(x,y)\sigma(u(s,y))W(\mathrm{d}s,\mathrm{d}y),\quad(t,x)\in[0,T]\times\mathcal{O}.$

It follows from Fatou’s lemma and (2.5) that for $\rho\in[q,\infty)$ ,

\mathbb{E}\Big{[}\sup_{t}\|u(t,\cdot)\|_{L^{q}}^{\rho}\Big{]}\leq\liminf_{R\rightarrow\infty}\mathbb{E}\Big{[}\mathbf{1}_{\{T\leq\tau_{R}\}}\sup_{t}\|\bar{u}_{R}(t,\cdot)\|_{L^{q}}^{\rho}\Big{]}\leq C(T,\rho,q),

(2.6)

which also holds for any $\rho\geq 1$ and $q\geq 1$ in view of the Hölder inequality and the continuity of $u_{0}$ . Besides, under Assumption 1, a standard Picard approximation argument shows that Eq. (1.1) admits a unique solution satisfying (2.6).

Similar to [4, Lemma 1.8], we have the following regularity of $G$ .

Lemma 2.1.

For $\alpha\in(0,1)$ , there exists $C=C_{\alpha}$ such that for $x,y\in\mathcal{O}$ and $t>s$ ,

	$\displaystyle\int_{0}^{t}\int_{\mathcal{O}}\|G_{t-r}(x,z)-G_{t-r}(y,z)\|^{2}\mathrm{d}z\mathrm{d}r\leq C\|x-y\|^{2},$
	$\displaystyle\int_{0}^{s}\int_{\mathcal{O}}\|G_{t-r}(x,z)-G_{s-r}(x,z)\|^{2}\mathrm{d}z\mathrm{d}r+\int_{s}^{t}\int_{\mathcal{O}}\|G_{t-r}(x,z)\|^{2}\mathrm{d}z\mathrm{d}r\leq C\|t-s\|^{\frac{3}{4}\alpha}.$

Based on (2.6) and Lemma 2.1, we present the Hölder continuity of the exact solution. Under either Assumption 1 or Assumption 2, there exists some constant $K_{0}$ such that

\displaystyle|f(x)|\leq K_{0}(1+|x|^{3}).

(2.7)

Lemma 2.2.

Let Assumption 1 or 2 hold, $u_{0}\in\mathcal{C}^{2}(\mathcal{O})$ , and $\alpha\in(0,1)$ . Then for $p\geq 1$ , there exists some constant $C=C(\alpha,p,T,K_{0})$ such that

\displaystyle\|u(t,x)-u(s,y)\|_{p}\leq C(|t-s|^{\frac{3\alpha}{8}}+|x-y|),\quad\forall~{}(t,x),(s,y)\in[0,T]\times\mathcal{O}.

(2.8)

Proof.

We first prove

\displaystyle\sup_{t,x}\|u(t,x)\|_{p}\leq C(p,T).

(2.9)

To this end, we write $u(t,x)=\sum_{i=1}^{3}u_{i}(t,x)$ with

	$\displaystyle u_{1}(t,x):=\mathbb{G}_{t}u_{0}(x),$		(2.10)
	$\displaystyle u_{2}(t,x):=\int_{0}^{t}\int_{\mathcal{O}}\Delta G_{t-r}(x,z)f(u(r,z))\mathrm{d}z\mathrm{d}r,$		(2.11)
	$\displaystyle u_{3}(t,x):=\int_{0}^{t}\int_{\mathcal{O}}G_{t-r}(x,z)\sigma(u(r,z))W(\mathrm{d}r,\mathrm{d}z).$		(2.12)

Let us state a useful property in [4, Lemma 1.6]. For any $\rho\in[1,\infty]$ , $q\in[\rho,+\infty]$ , and $1/\gamma=1/q-1/\rho+1\in[0,1]$ , the linear operator $\mathbb{J}_{t_{0}}$ defined by

\mathbb{J}_{t_{0}}(v)(t,x)=\int_{t_{0}}^{t}\int_{\mathcal{O}}\Delta G_{t-s}(x,y)v(s,y)\mathrm{d}y\mathrm{d}s,\quad 0\leq t_{0}<t\leq T,~{}x\in\mathcal{O},\quad v\in L^{1}(t_{0},T;L^{\rho}(\mathcal{O}))

is a mapping from $L^{1}(t_{0},T;L^{\rho}(\mathcal{O}))$ to $L^{\infty}(t_{0},T;L^{q}(\mathcal{O}))$ with

\displaystyle\|\mathbb{J}_{t_{0}}(v)(t,\cdot)\|_{L^{q}}\leq C\int_{t_{0}}^{t}(t-s)^{-\frac{3}{4}+\frac{1}{4\gamma}}\|v(s,\cdot)\|_{L^{\rho}}\mathrm{d}s.

(2.13)

It follows from (2.6) and (2.7) that

\displaystyle\mathbb{E}[\|f(u(r,\cdot))\|^{p}_{L^{1}}]\leq C\big{(}1+\mathbb{E}[\|u(r,\cdot)\|^{3p}_{L^{3}}]\big{)}\leq C(p,T,K_{0}),\quad\forall~{}r\in[0,T].

(2.14)

Applying (2.13) with $t_{0}=0$ , $q=\gamma=\infty$ and $\rho=1$ leads to

\displaystyle\left\|u_{2}(t,\cdot)\right\|_{L^{\infty}}\leq C\int_{0}^{t}(t-r)^{-\frac{3}{4}}\|f(u(r,\cdot))\|_{L^{1}}\mathrm{d}r,

which combined with the Minkowski inequality and (2.14) implies

\displaystyle\sup_{x}\left\|u_{2}(t,x)\right\|_{p}\leq C\int_{0}^{t}(t-r)^{-\frac{3}{4}}\left(\mathbb{E}[\|f(u(r,\cdot))\|^{p}_{L^{1}}]\right)^{\frac{1}{p}}\mathrm{d}r\leq C(p,T,K_{0})t^{\frac{1}{4}}.

(2.15)

Since $\sigma$ is bounded, the Burkholder inequality [17, Theorem B.1] and (2.1) yield

\displaystyle\left\|u_{3}(t,x)\right\|^{2}_{p}\leq C(p)\int_{0}^{t}\int_{\mathcal{O}}G^{2}_{t-r}(x,z)\mathrm{d}z\mathrm{d}r\leq C(p)t^{\frac{3}{4}},

(2.16)

for $(t,x)\in[0,T]\times\mathcal{O}$ . In addition, (2.1) implies $|u_{1}(t,x)|\leq C\|u_{0}\|_{\mathcal{C}(\mathcal{O})}$ for $x\in\mathcal{O}$ , which together with (2.15) and (2.16) completes the proof of (2.9).

Without loss of generality, assume that $s<t$ . Notice that $u(t,x)-u(t,y)=\mathbb{G}_{t}u_{0}(x)-\mathbb{G}_{t}u_{0}(y)+I_{f}+I_{\sigma}$ with

	$\displaystyle I_{f}:$	$\displaystyle=\int_{0}^{t}\int_{\mathcal{O}}\left[\Delta G_{t-r}(x,z)-\Delta G_{t-r}(y,z)\right]f(u(r,z))\mathrm{d}z\mathrm{d}r,$
	$\displaystyle I_{\sigma}:$	$\displaystyle=\int_{0}^{t}\int_{\mathcal{O}}\left[G_{t-r}(x,z)-G_{t-r}(y,z)\right]\sigma(u(r,z))W(\mathrm{d}r,\mathrm{d}z),$

and $u(t,x)-u(s,x)=\mathbb{G}_{t}u_{0}(x)-\mathbb{G}_{s}u_{0}(x)+J^{-}_{f}+J_{f}+J_{\sigma}+J^{-}_{\sigma}$ with

	$\displaystyle J^{-}_{f}:$	$\displaystyle=\int_{0}^{s}\int_{\mathcal{O}}\left[\Delta G_{t-r}(x,z)-\Delta G_{s-r}(x,z)\right]f(u(r,z))\mathrm{d}z\mathrm{d}r,$
	$\displaystyle J^{-}_{\sigma}:$	$\displaystyle=\int_{0}^{s}\int_{\mathcal{O}}\left[G_{t-r}(x,z)-G_{s-r}(x,z)\right]\sigma(u(r,z))W(\mathrm{d}r,\mathrm{d}z),$
	$\displaystyle J_{f}:$	$\displaystyle=\int_{s}^{t}\int_{\mathcal{O}}\Delta G_{t-r}(x,z)f(u(r,z))\mathrm{d}z\mathrm{d}r,~{}$
	$\displaystyle J_{\sigma}:$	$\displaystyle=\int_{s}^{t}\int_{\mathcal{O}}G_{t-r}(x,z)\sigma(u(r,z))W(\mathrm{d}r,\mathrm{d}z),$

where the explicit dependence of $I_{f}$ , $I_{\sigma}$ , $J^{-}_{f}$ , $J_{f}$ , $J_{\sigma}$ , $J^{-}_{\sigma}$ on $t,s,x,y$ is dropped for simplicity. Using [4, Lemma 2.3] and the assumption $u_{0}\in\mathcal{C}^{2}(\mathcal{O})$ , we get $|\mathbb{G}_{t}u_{0}(x)-\mathbb{G}_{t}u_{0}(y)|+|\mathbb{G}_{t}u_{0}(x)-\mathbb{G}_{s}u_{0}(x)|\leq C(|t-s|^{\frac{1}{2}}+|x-y|).$ By the Burkholder inequality, the boundedness of $\sigma$ , and Lemma 2.1, we obtain $\|I_{\sigma}\|_{p}^{2}+\|J_{\sigma}^{-}\|_{p}^{2}+\|J_{\sigma}\|_{p}^{2}\leq C(|t-s|^{\frac{3}{4}\alpha}+|x-y|^{2}).$ For any $\alpha_{1}>0$ ,

\displaystyle e^{-x}\leq C_{\alpha_{1}}x^{-\alpha_{1}},\quad\forall~{}x>0.

(2.17)

By the orthogonality of $\{\phi_{j}\}_{j=1}^{\infty}$ in $L^{2}(\mathcal{O})$ , (2.17), and $|\phi_{j}(x)-\phi_{j}(y)|\leq j|x-y|$ for $j\geq 1,$

\displaystyle\int_{\mathcal{O}}|\Delta G_{s}(x,z)-\Delta G_{s}(y,z)|^{2}\mathrm{d}z=\sum_{j=1}^{\infty}j^{4}e^{-2j^{4}s}|\phi_{j}(x)-\phi_{j}(y)|^{2}\leq C|x-y|^{2}\sum_{j=1}^{\infty}j^{6-4\rho}s^{-\rho}

with $\rho>0$ . Choosing $\rho\in(\frac{7}{4},2)$ and using the Cauchy–Schwarz inequality,

		$\displaystyle\quad\int_{0}^{t}\int_{\mathcal{O}}\|\Delta G_{s}(x,z)-\Delta G_{s}(y,z)\|\mathrm{d}z\mathrm{d}s\leq\sqrt{\pi}\int_{0}^{t}\left(\int_{\mathcal{O}}\|\Delta G_{s}(x,z)-\Delta G_{s}(y,z)\|^{2}\mathrm{d}z\right)^{\frac{1}{2}}\mathrm{d}s$
		$\displaystyle\leq C\|x-y\|\Big{(}\sum_{j=1}^{\infty}j^{6-4\rho}\Big{)}^{\frac{1}{2}}\int_{0}^{t}s^{-\frac{\rho}{2}}\mathrm{d}s\leq C\|x-y\|.$		(2.18)

Taking advantage of (2) and (2.9), we obtain

\displaystyle\|I_{f}\|_{p}

\displaystyle\leq C|x-y|\sup_{t,x}\|f(u(t,x))\|_{p}\leq C|x-y|(1+\sup_{t,x}\|u(t,x)\|^{3}_{3p})\leq C|x-y|.

Similarly, the orthogonality of $\{\phi_{j}\}_{j=1}^{\infty}$ in $L^{2}(\mathcal{O})$ , the Cauchy–Schwarz inequality, and (2.17) yield that for $\rho>\frac{5}{4}$ ,

		$\displaystyle\quad\int_{s}^{t}\int_{\mathcal{O}}\|\Delta G_{t-r}(x,z)\|\mathrm{d}z\mathrm{d}r\leq\sqrt{\pi}\int_{s}^{t}\left(\int_{\mathcal{O}}\|\Delta G_{t-r}(x,z)\|^{2}\mathrm{d}z\right)^{\frac{1}{2}}\mathrm{d}r$
		$\displaystyle\leq C\int_{s}^{t}\Big{(}\sum_{j=1}^{\infty}j^{4}e^{-2j^{4}(t-r)}\Big{)}^{\frac{1}{2}}\mathrm{d}r\leq C\int_{s}^{t}\Big{(}\sum_{j=1}^{\infty}j^{4-4\rho}(t-r)^{-\rho}\Big{)}^{\frac{1}{2}}\mathrm{d}r\leq C(t-s)^{1-\frac{\rho}{2}},$		(2.19)

which together with (2.9) implies that for $\alpha\in(0,1)$ ,

\displaystyle\|J_{f}\|_{p}\leq\int_{s}^{t}\int_{\mathcal{O}}|\Delta G_{t-r}(x,z)|(1+\|u(r,z)\|^{3}_{3p})\mathrm{d}z\mathrm{d}r\leq C(t-s)^{\frac{3}{8}\alpha}.

Observe that for any $\alpha_{2}\in(0,1]$ ,

\displaystyle 1-e^{-x}\leq C_{\alpha_{2}}x^{\alpha_{2}},\quad x\geq 0.

(2.20)

Then the orthogonality of $\{\phi_{j}\}_{j=1}^{\infty}$ in $L^{2}(\mathcal{O})$ , the Cauchy–Schwarz inequality, (2.20) and (2.17) imply that for $\alpha_{3}\in(0,\frac{3}{8})$ and $\rho=\frac{13}{8}+\alpha_{3}<2$ with $4-4\rho+8\alpha_{3}<-1$ , it holds that

		$\displaystyle\quad\int_{0}^{s}\int_{\mathcal{O}}\|\Delta G_{t-r}(x,z)-\Delta G_{s-r}(x,z)\|\mathrm{d}z\mathrm{d}r$
		$\displaystyle\leq\sqrt{\pi}\int_{0}^{s}\left(\int_{\mathcal{O}}\|\Delta G_{t-r}(x,z)-\Delta G_{s-r}(x,z)\|^{2}\mathrm{d}z\right)^{\frac{1}{2}}\mathrm{d}r$
		$\displaystyle\leq C\int_{0}^{s}\Big{(}\sum_{j=1}^{\infty}j^{4}e^{-2j^{4}(s-r)}\|1-e^{-j^{4}(t-s)}\|^{2}\Big{)}^{\frac{1}{2}}\mathrm{d}r$
		$\displaystyle\leq C\Big{(}\sum_{j=1}^{\infty}j^{4-4\rho+8\alpha_{3}}\Big{)}^{\frac{1}{2}}\int_{0}^{s}(s-r)^{-\frac{\rho}{2}}\mathrm{d}r(t-s)^{\alpha_{3}}\leq C(t-s)^{\alpha_{3}}.$		(2.21)

Hence, it follows from (2.9) that $\|J_{f}^{-}\|_{p}\leq C(t-s)^{\frac{3}{8}\alpha}$ with $\alpha\in(0,1)$ . Finally, combining the above estimates, we obtain (2.8), which completes the proof. ∎

Lemma 2.3.

Under the same conditions of Lemma 2.2, for any $p\geq 1$ ,

\displaystyle\mathbb{E}\Big{[}\sup_{t,x}|u(t,x)|^{p}\Big{]}

\displaystyle\leq C(p,T,K_{0}).

Proof.

Based on (2.8) with $\alpha=\frac{2}{3}$ , we apply [17, Theorem C.6] with $H_{1}=\frac{1}{4},H_{2}=1$ , $H=5$ , $k=p>6$ , $q=1-\frac{6}{p}$ and $\delta=\frac{1}{2}(q+1-H/p)$ to obtain that there exists $C_{1}$ such that

\displaystyle\mathbb{E}\bigg{[}\sup_{(t,x)\neq(s,y)}\frac{|u(t,x)-u(s,y)|^{p}}{(|t-s|^{\frac{1}{4}}+|x-y|)^{p-6}}\bigg{]}\leq C_{1}.

Hence, for $p>6$ ,

\displaystyle\mathbb{E}\Big{[}\sup_{t,x}|u(t,x)|^{p}\Big{]}=\mathbb{E}\Big{[}\sup_{t,x}|u(t,x)-u(0,0)|^{p}\Big{]}

\displaystyle\leq C_{1}\big{(}T^{\frac{1}{4}}+\pi\big{)}^{p-6}.

(2.22)

For $1\leq p\leq 6$ , the desired result follows from (2.22) and the Hölder inequality. The proof is completed. ∎

3. Finite difference method

In this section, we introduce the spatial FDM method for Eq. (1.1) and derive its strong convergence rate. Given a function $w$ on the mesh $\mathcal{O}^{h}:=\{0,h,2h,\ldots,\pi\}$ , we define the difference operator

\delta_{h}w_{i}:=\frac{w_{i-1}-2w_{i}+w_{i+1}}{h^{2}},\quad\delta_{h}^{2}w_{i}=\frac{w_{i-2}-4w_{i-1}+6w_{i}-4w_{i+1}+w_{i+2}}{h^{4}},

for $i\in\mathbb{Z}_{n}:=\{1,2,\ldots,n-1\}$ , where $w_{i}:=w(ih)$ . The compatibility conditions $u_{0}(0)=u_{0}(\pi)=0$ and $u_{0}^{\prime\prime}(0)=u_{0}^{\prime\prime}(\pi)=0$ are direct results of DBCs and the initial condition. One can approximate $u(t,kh)$ via $\{u^{n}(t,kh)\}_{n\geq 2}$ , where $u^{n}(0,kh)=u_{0}(kh)$ and

		$\displaystyle\quad\mathrm{d}u^{n}(t,kh)+\delta_{h}^{2}u^{n}(t,kh)\mathrm{d}t$
		$\displaystyle=\delta_{h}f(u^{n}(t,kh))\mathrm{d}t+n\pi^{-1}\sigma(u^{n}(t,kh))\mathrm{d}(W(t,(k+1)h)-W(t,kh)),$		(3.1)

for $t\in[0,T]$ and $k\in\mathbb{Z}_{n}$ , under the boundary conditions

u^{n}(t,0)=u^{n}(t,\pi)=0,\quad u^{n}(t,-h)+u^{n}(t,h)=u^{n}(t,(n-1)h)+u^{n}(t,(n+1)h)=0,

for $t\in(0,T]$ . For $k\in\mathbb{Z}_{n}\cup\{0\}$ , we use the polygonal interpolation

u^{n}(t,x):=u^{n}(t,kh)+(n\pi^{-1}x-k)(u^{n}(t,(k+1)h)-u^{n}(t,kh)),\quad\forall~{}x\in[kh,(k+1)h].

To solve (3), we introduce

U(t)=(U_{1}(t),\ldots,U_{n-1}(t))^{\top},\qquad\beta_{t}=(\beta^{1}_{t},\ldots,\beta^{n-1}_{t})^{\top}

with $U_{k}(t):=u^{n}(t,kh)$ and $\beta^{k}_{t}:=\sqrt{\frac{n}{\pi}}(W(t,(k+1)h)-W(t,kh))$ for $k\in\mathbb{Z}_{n}$ , where the explicit dependence of $U(t)$ and $\beta_{t}$ on $n$ is omitted. Let

\displaystyle A_{n}:=\frac{n^{2}}{\pi^{2}}\left[\begin{array}[]{cccccc}-2&1&0&\cdots&\cdots&0\\ 1&-2&1&\ddots&\ddots&\vdots\\ 0&1&-2&\ddots&\ddots&\vdots\\ \vdots&0&\ddots&\ddots&\ddots&0\\ \vdots&\ddots&\ddots&1&-2&1\\ 0&\cdots&\cdots&0&1&-2\end{array}\right].

Then (3) can be rewritten into an $(n-1)$ -dimensional SDE

\displaystyle\mathrm{d}U(t)+A_{n}^{2}U(t)\mathrm{d}t=A_{n}F_{n}(U(t))\mathrm{d}t+\sqrt{n/\pi}\Sigma_{n}(U(t))\mathrm{d}\beta_{t}

(3.2)

with the initial condition $U(0)=(u_{0}(h),\ldots,u_{0}((n-1)h))^{\top}$ and the coefficients

F_{n}(U(t))=(f(U_{1}(t)),\ldots,f(U_{n-1}(t)))^{\top},~{}\Sigma_{n}(U(t))=\textrm{diag}(\sigma(U_{1}(t)),\ldots,\sigma(U_{n-1}(t))).

Under Assumption 1, (3.2) admits a unique strong solution which satisfies

	$\displaystyle U(t)$	$\displaystyle=\exp(-A_{n}^{2}t)U(0)+\int_{0}^{t}A_{n}\exp(-A_{n}^{2}(t-s))F_{n}(U(s))\mathrm{d}s$
		$\displaystyle\quad+\sqrt{\frac{n}{\pi}}\int_{0}^{t}\exp(-A_{n}^{2}(t-s))\Sigma_{n}(U(s))\mathrm{d}\beta_{s},\quad t\in[0,T].$		(3.3)

For $j\in\mathbb{Z}_{n}$ , $e_{j}=(e_{j}(1),\ldots,e_{j}(n-1))^{\top}$ given by

e_{j}(k)=\sqrt{\pi/n}\phi_{j}(kh)=\sqrt{2/n}\sin(jkh),\quad k\in\mathbb{Z}_{n},

(3.4)

is an eigenvector of $A_{n}$ associated with the eigenvalue $\lambda_{j,n}=-j^{2}c_{j,n}$ , where

c_{j,n}:=\sin^{2}(\frac{j}{2n}\pi)/(\frac{j}{2n}\pi)^{2}

satisfies $\frac{4}{\pi^{2}}\leq c_{j,n}\leq 1.$ The vectors $\{e_{i}\}_{i=1}^{n-1}$ form an orthonormal basis of $\mathbb{R}^{n-1}$ . In particular,

\displaystyle\langle e_{i},e_{j}\rangle=\frac{\pi}{n}\sum_{k=1}^{n-1}\phi_{i}(kh)\phi_{j}(kh)=\int_{0}^{\pi}\phi_{i}(\kappa_{n}(y))\phi_{j}(\kappa_{n}(y))\mathrm{d}y=\delta_{ij},

(3.5)

where $\kappa_{n}(y)=h\lfloor y/h\rfloor$ with $\lfloor\cdot\rfloor$ being the floor function (see e.g., [15]). It is verified that $1-\frac{\sin a}{a}\leq\frac{1}{6}a^{2}$ for all $a\in[0,\frac{\pi}{2})$ , which indicates that for $j\in\mathbb{Z}_{n}$ ,

\displaystyle 0\leq 1-c_{j,n}=\Big{(}1+\sin(\frac{j}{2n}\pi)/(\frac{j}{2n}\pi)\Big{)}\Big{(}1-\sin(\frac{j}{2n}\pi)/(\frac{j}{2n}\pi)\Big{)}\leq\frac{\pi^{2}j^{2}}{12n^{2}}.

(3.6)

Introduce the discrete kernel

G^{n}_{t}(x,y)=\sum_{j=1}^{n-1}\exp(-\lambda_{j,n}^{2}t)\phi_{j,n}(x)\phi_{j}(\kappa_{n}(y)),

where $\phi_{j,n}(x)=\phi_{j}(kh)+(n\pi^{-1}x-k)(\phi_{j}((k+1)h)-\phi_{j}(kh))$ for $x\in[kh,(k+1)h]$ , $k\in\mathbb{Z}_{n}\cup\{0\}$ . Define the discrete Dirichlet Laplacian $\Delta_{n}$ by $\Delta_{n}w(y)=0$ for $y\in[0,h)$ ,

\displaystyle\Delta_{n}w(y)=\frac{n^{2}}{\pi^{2}}\left(w\big{(}\kappa_{n}(y)+\frac{\pi}{n}\big{)}-2w\big{(}\kappa_{n}(y)\big{)}+w\big{(}\kappa_{n}(y)-\frac{\pi}{n}\big{)}\right),\quad y\in[h,\pi),

(3.7)

where $w:\mathcal{O}\rightarrow\mathbb{R}$ with $w(0)=w(\pi)=0$ . Since $\Delta_{n}\phi_{j}(\kappa_{n}(y))=\lambda_{j,n}\phi_{j}(\kappa_{n}(y))$ ,

\Delta_{n}G^{n}_{t}(x,y)=\sum_{j=1}^{n-1}\lambda_{j,n}\exp(-\lambda_{j,n}^{2}t)\phi_{j,n}(x)\phi_{j}(\kappa_{n}(y)).

Similar to [15, Section 2], based on (3), the diagonalization of the matrix $A_{n}$ , (3.4) and $u^{n}(t,kh)=\sum_{j=1}^{n-1}\langle U(t),e_{j}\rangle e_{j}(k)$ , one has

	$\displaystyle u^{n}(t,x)=$	$\displaystyle\int_{\mathcal{O}}G^{n}_{t}(x,y)u_{0}(\kappa_{n}(y))\mathrm{d}y+\int_{0}^{t}\int_{\mathcal{O}}\Delta_{n}G^{n}_{t-s}(x,y)f(u^{n}(s,\kappa_{n}(y)))\mathrm{d}y\mathrm{d}s$
		$\displaystyle+\int_{0}^{t}\int_{\mathcal{O}}G^{n}_{t-s}(x,y)\sigma(u^{n}(s,\kappa_{n}(y)))W(\mathrm{d}s,\mathrm{d}y),\quad(t,x)\in[0,T]\times\mathcal{O}.$		(3.8)

The follow lemma characterizes the error between $G$ and $G^{n}$ .

Lemma 3.1.

There exists some constant $C=C(T)$ such that for any $x\in\mathcal{O}$ and $t\in(0,T]$ ,

	$\displaystyle\int_{0}^{t}\int_{\mathcal{O}}\|\Delta_{n}G_{s}^{n}(x,y)-\Delta G_{s}(x,y)\|\mathrm{d}y\mathrm{d}s$	$\displaystyle\leq Cn^{-1},$		(3.9)
	$\displaystyle\int_{0}^{t}\int_{\mathcal{O}}\|G^{n}_{s}(x,y)-G_{s}(x,y)\|^{2}\mathrm{d}y\mathrm{d}s$	$\displaystyle\leq Cn^{-2}.$		(3.10)

Proof.

For $s\in[0,T]$ and $x,y\in\mathcal{O}$ , denote $M_{1}^{s,x,y}=\sum_{j=1}^{n-1}\lambda_{j,n}e^{-\lambda_{j,n}^{2}s}\phi_{j,n}(x)\left(\phi_{j}(\kappa_{n}(y))-\phi_{j}(y)\right)$ , $M_{2}^{s,x,y}=\sum_{j=1}^{n-1}(\lambda_{j,n}e^{-\lambda_{j,n}^{2}s}\phi_{j,n}(x)-\lambda_{j}e^{-\lambda_{j}^{2}s}\phi_{j}(x))\phi_{j}(y)$ and $M_{3}^{s,x,y}=\sum_{j=n}^{\infty}-\lambda_{j}e^{-\lambda_{j}^{2}s}\phi_{j}(x)\phi_{j}(y)$ . Since $\{\phi_{j}\}_{j\geq 1}$ is an orthonormal basis of $L^{2}(\mathcal{O})$ , it holds that

		$\displaystyle\quad\int_{\mathcal{O}}\|\Delta_{n}G_{s}^{n}(x,y)-\Delta G_{s}(x,y)\|^{2}\mathrm{d}y=\int_{\mathcal{O}}\|\sum_{i=1}^{3}M_{i}^{s,x,y}\|^{2}\mathrm{d}y$
		$\displaystyle\leq 3\int_{\mathcal{O}}\|M_{1}^{s,x,y}\|^{2}\mathrm{d}y+3\sum_{j=1}^{n-1}\left\|\lambda_{j,n}e^{-\lambda_{j,n}^{2}s}\phi_{j,n}(x)-\lambda_{j}e^{-\lambda_{j}^{2}s}\phi_{j}(x)\right\|^{2}+3\sum_{j=n}^{\infty}j^{4}e^{-2j^{4}s}.$		(3.11)

It follows from the boundedness of $\{\phi_{j}\}_{j\geq 1}$ and (2.17) that for $\alpha_{1}\in(\frac{5}{4},2)$ ,

\displaystyle\int_{0}^{t}\Big{(}\sum_{j=n}^{\infty}j^{4}e^{-2j^{4}s}\Big{)}^{\frac{1}{2}}\mathrm{d}s\leq C\int_{0}^{t}\Big{(}\sum_{j=n}^{\infty}j^{4-4\alpha_{1}}s^{-\alpha_{1}}\Big{)}^{\frac{1}{2}}\mathrm{d}s\leq C(\alpha_{1})n^{\frac{5-4\alpha_{1}}{2}}.

(3.12)

By (3.6), we have $|\lambda_{j}-\lambda_{j,n}|\leq Cj^{4}/n^{2},$ and thus, $\lambda_{j}^{2}-\lambda_{j,n}^{2}=|\lambda_{j}-\lambda_{j,n}||\lambda_{j}+\lambda_{j,n}|\leq Cj^{6}/n^{2}$ , which along with (2.20) yields $|e^{-\lambda_{j,n}^{2}s}-e^{-\lambda_{j}^{2}s}|=e^{-\lambda_{j,n}^{2}s}|1-e^{-(\lambda_{j}^{2}-\lambda_{j,n}^{2})s}|\leq e^{-\lambda_{j,n}^{2}s}\frac{j^{6}}{n^{2}}s$ . Besides, it can be verified that $|\phi_{j,n}(x)-\phi_{j}(x)|\leq Cj/n.$ Therefore, for $\rho,\rho_{1}>0$

	$\displaystyle\quad\sum_{j=1}^{n-1}\left\|\lambda_{j,n}e^{-\lambda_{j,n}^{2}s}\phi_{j,n}(x)-\lambda_{j}e^{-\lambda_{j}^{2}s}\phi_{j}(x)\right\|^{2}$
	$\displaystyle\leq\sum_{j=1}^{n-1}\|\lambda_{j,n}-\lambda_{j}\|^{2}e^{-2\lambda_{j,n}^{2}s}+\sum_{j=1}^{n-1}\lambda_{j}^{2}\|e^{-\lambda_{j,n}^{2}s}-e^{-\lambda_{j}^{2}s}\|^{2}+\sum_{j=1}^{n-1}\lambda_{j}^{2}e^{-2\lambda_{j}^{2}s}\|\phi_{j,n}(x)-\phi_{j}(x)\|^{2}$
	$\displaystyle\leq Cn^{-4}\sum_{j=1}^{n-1}j^{8-4\rho}s^{-\rho}+Cn^{-4}\sum_{j=1}^{n-1}{j^{16-4\rho_{1}}}s^{2-\rho_{1}}+Cn^{-2}\sum_{j=1}^{n-1}j^{6-4\rho}s^{-\rho}.$

Choosing $\rho=\rho_{1}-2\in(0,2)$ , we obtain

\displaystyle\int_{0}^{t}\Big{(}\sum_{j=1}^{n-1}\left|\lambda_{j,n}e^{-\lambda_{j,n}^{2}s}\phi_{j,n}(x)-\lambda_{j}e^{-\lambda_{j}^{2}s}\phi_{j}(x)\right|^{2}\Big{)}^{\frac{1}{2}}\mathrm{d}s\leq C\int_{0}^{t}(n^{5-4\rho}s^{-\rho})^{\frac{1}{2}}\mathrm{d}s\leq Cn^{\frac{5-4\rho}{2}}.

(3.13)

We recall the following inequality in [15, Lemma 3.2]:

\displaystyle\int_{\mathcal{O}}|w(y)-w(\kappa_{n}(y))|^{2}\mathrm{d}y\leq Cn^{-2}\int_{\mathcal{O}}|\frac{\mathrm{d}}{\mathrm{d}y}w(y)|^{2}\mathrm{d}y,\quad\text{for}~{}w\in C^{1}(\mathcal{O}).

(3.14)

Introducing $B_{n}(s,x,y)=\sum_{j=1}^{n-1}\lambda_{j,n}e^{-\lambda_{j,n}^{2}s}\phi_{j,n}(x)\phi_{j}(y)$ and making use of (3.14), we obtain

\displaystyle\quad\int_{\mathcal{O}}|M_{1}^{s,x,y}|^{2}\mathrm{d}y=\int_{\mathcal{O}}|B_{n}(s,x,y)-B_{n}(s,x,\kappa_{n}(y))|^{2}\mathrm{d}y\leq Cn^{-2}\sum_{j=1}^{n-1}j^{6}e^{-2\lambda_{j,n}^{2}s}\phi_{j,n}(x)^{2}.

Hence, it follows from (2.17) that for $\rho\in(\frac{7}{4},2)$ ,

\displaystyle\int_{0}^{t}\Big{(}\int_{\mathcal{O}}|M_{1}^{s,x,y}|^{2}\mathrm{d}y\Big{)}^{\frac{1}{2}}\mathrm{d}s\leq C\int_{0}^{t}\Big{(}\frac{1}{n^{2}}\sum_{j=1}^{n-1}j^{6-4\rho}s^{-\rho}\Big{)}^{\frac{1}{2}}\mathrm{d}s\leq Cn^{-1}.

(3.15)

Combining (3)-(3.13) with (3.15) allows us to deduce

\displaystyle\int_{0}^{t}\int_{\mathcal{O}}|\Delta_{n}G_{s}^{n}(x,y)-\Delta G_{s}(x,y)|\mathrm{d}y\mathrm{d}s\leq\sqrt{\pi}\int_{0}^{t}\left(\int_{\mathcal{O}}|\Delta_{n}G_{s}^{n}(x,y)-\Delta G_{s}(x,y)|^{2}\mathrm{d}y\right)^{\frac{1}{2}}\mathrm{d}s\leq Cn^{-1},

which proves (3.9). It remains to prove (3.10). Set $H_{n}(t,x,y):=\sum_{j=1}^{n-1}\exp(-\lambda_{j}^{2}t)\phi_{j}(x)\phi_{j}(y).$ Then $\int_{\mathcal{O}}|G^{n}_{s}(x,y)-G_{s}(x,y)|^{2}\mathrm{d}y\leq 4\sum_{k=1}^{4}J_{k}(s,x),$ where

	$\displaystyle J_{1}(s,x):=\sum_{j=n}^{\infty}\exp(-2\lambda_{j}^{2}s),\qquad J_{2}(s,x):=\int_{\mathcal{O}}\|H_{n}(s,x,y)-H_{n}(s,x,\kappa_{n}(y))\|^{2}\mathrm{d}y,$
	$\displaystyle J_{3}(s,x):=\sum_{j=1}^{n-1}\|\exp(-\lambda_{j}^{2}s)-\exp(-\lambda_{j,n}^{2}s)\|^{2},\qquad J_{4}(s,x):=\sum_{j=1}^{n-1}e^{-2\lambda_{j,n}^{2}s}\|\phi_{j,n}(x)-\phi_{j}(x)\|^{2}.$

For the first term, we have $\int_{0}^{t}J_{1}(s,x)\mathrm{d}s\leq C\sum_{j=n}^{\infty}j^{-4}\leq Cn^{-3}$ . For $2<\alpha<3$ , $J_{3}(s,x)\leq\sum_{j=1}^{n-1}e^{-2\lambda_{j,n}^{2}s}(1-e^{-(\lambda_{j}^{2}-\lambda_{j,n}^{2})s})^{2}\leq C\sum_{j=1}^{n-1}\frac{j^{12}}{n^{4}}j^{-4\alpha}s^{2-\alpha}\leq Cn^{9-4\alpha}s^{2-\alpha},$ which implies that $\int_{0}^{t}J_{3}(s,x)\mathrm{d}s\leq C_{\epsilon}n^{-3+\epsilon}$ with arbitrarily small $\epsilon>0$ . Since $|\phi_{j,n}(x)-\phi_{j}(x)|\leq Cj/n$ , it holds that $J_{4}(s,x)\leq C\sum_{j=1}^{n-1}j^{-4\alpha}s^{-\alpha}\frac{j^{2}}{n^{2}}\leq Cn^{-2}s^{-\alpha}$ for $\frac{3}{4}<\alpha<1$ , and thus $\int_{0}^{t}J_{4}(s,x)\mathrm{d}s\leq Cn^{-2}.$ Using (3.14), we arrive at

\displaystyle\int_{0}^{t}J_{2}(s,x)\mathrm{d}s\leq Cn^{-2}\int_{0}^{t}\int_{\mathcal{O}}|\frac{\mathrm{d}}{\mathrm{d}y}H_{n}(s,x,y)|^{2}\mathrm{d}y\mathrm{d}s\leq Cn^{-2}\sum_{j=1}^{n-1}\int_{0}^{t}j^{2}\exp(-2j^{4}s)\mathrm{d}s\leq Cn^{-2}.

Combining the above estimates completes the proof of (3.10). ∎

For $n\geq 2$ , denote by $\mathbb{U}(t):=(u(t,h),\ldots,u(t,(n-1)h))^{\top}$ the exact solution of Eq. (1.1) on spatial grid points, where the explicit dependence of $\mathbb{U}(t)$ on $n$ is omitted. We introduce the following auxiliary process $\{\tilde{U}(t),t\in[0,T]\}$ by

\displaystyle\mathrm{d}\tilde{U}(t)+A_{n}^{2}\tilde{U}(t)\mathrm{d}t=A_{n}F_{n}(\mathbb{U}(t))\mathrm{d}t+\sqrt{n/\pi}\Sigma_{n}(\mathbb{U}(t))\mathrm{d}\beta_{t},\quad t\in(0,T]

with initial value $\tilde{U}(0)=U(0)$ . Let $\tilde{u}^{n}=\{\tilde{u}^{n}(t,x),(t,x)\in[0,T]\times\mathcal{O}\}$ satisfy

	$\displaystyle\tilde{u}^{n}(t,x)=$	$\displaystyle\int_{\mathcal{O}}G^{n}_{t}(x,y)u_{0}(\kappa_{n}(y))\mathrm{d}y+\int_{0}^{t}\int_{\mathcal{O}}\Delta_{n}G^{n}_{t-s}(x,y)f(u(s,\kappa_{n}(y)))\mathrm{d}y\mathrm{d}s$
		$\displaystyle+\int_{0}^{t}\int_{\mathcal{O}}G^{n}_{t-s}(x,y)\sigma(u(s,\kappa_{n}(y)))W(\mathrm{d}s,\mathrm{d}y).$		(3.16)

Then $\tilde{U}_{k}(t)=\tilde{u}^{n}(t,kh)$ for $k\in\mathbb{Z}_{n}$ and $t\in[0,T]$ . In order to estimate $\|u^{n}(t,x)-u(t,x)\|_{p}$ , it suffices to estimate $\|\tilde{u}^{n}(t,x)-u(t,x)\|_{p}$ and $\|\tilde{u}^{n}(t,x)-u^{n}(t,x)\|_{p}$ , where the first term is tackled as follows.

Lemma 3.2.

Suppose that Assumption 1 or 2 holds and $u_{0}\in\mathcal{C}^{3}(\mathcal{O})$ . Then for any $p\geq 1$ , there exists some constant $C=C(p,T,K_{0})$ such that for any $(t,x)\in[0,T]\times\mathcal{O}$ ,

\displaystyle\|\tilde{u}^{n}(t,x)-u(t,x)\|_{p}\leq Cn^{-1}.

Proof.

Recall that $u=u_{1}+u_{2}+u_{3}$ , where $u_{i}$ , $i=1,2,3$ , are defined in (2.10)-(2.12), respectively. Similarly, for $(t,x)\in[0,T]\times\mathcal{O}$ , we introduce $\tilde{u}^{n}_{1}(t,x):=\int_{\mathcal{O}}G^{n}_{t}(x,y)u_{0}(\kappa_{n}(y))\mathrm{d}y$ ,

	$\displaystyle\tilde{u}^{n}_{2}(t,x):=\int_{0}^{t}\int_{\mathcal{O}}\Delta_{n}G^{n}_{t-s}(x,y)f(u(s,\kappa_{n}(y)))\mathrm{d}y\mathrm{d}s,$
	$\displaystyle\tilde{u}^{n}_{3}(t,x):=\int_{0}^{t}\int_{\mathcal{O}}G^{n}_{t-s}(x,y)\sigma(u(s,\kappa_{n}(y)))W(\mathrm{d}s,\mathrm{d}y),$

and divide the proof into three parts.

Part 1: Following the proof of [4, Lemma 2.3], we use the PDE satisfied by $G$ to write $u_{1}(t,x)=u_{0}(x)-\int_{0}^{t}\int_{\mathcal{O}}\Delta G_{r}(x,z)u_{0}^{\prime\prime}(z)\mathrm{d}z\mathrm{d}r.$ As a numerical counterpart,

		$\displaystyle\quad\tilde{u}^{n}_{1}(t,x)-\tilde{u}^{n}(0,x)=\int_{\mathcal{O}}\int_{0}^{t}\frac{\partial}{\partial r}G^{n}_{r}(x,z)u_{0}(\kappa_{n}(z))\mathrm{d}z\mathrm{d}r$
		$\displaystyle=-\int_{0}^{t}\int_{\mathcal{O}}\Delta_{n}^{2}G^{n}_{r}(x,z)u_{0}(\kappa_{n}(z))\mathrm{d}z\mathrm{d}r=-\int_{0}^{t}\int_{\mathcal{O}}\Delta_{n}G^{n}_{r}(x,z)\Delta_{n}u_{0}(z)\mathrm{d}z\mathrm{d}r,$		(3.17)

where in the last step we have used the fact that

\int_{\mathcal{O}}\Delta_{n}v(z)w(\kappa_{n}(z))\mathrm{d}z=\int_{\mathcal{O}}v(\kappa_{n}(z))\Delta_{n}w(z)\mathrm{d}z,

for $v,w:\mathcal{O}\rightarrow\mathbb{R}$ with $v=w=0$ on $\partial\mathcal{O}$ . Here, $\tilde{u}^{n}(0,kh)=u_{0}(kh)$ for $k\in\mathbb{Z}_{n}\cup\{0,n\}$ , and $\tilde{u}^{n}(0,x)=u_{0}(kh)+(n\pi^{-1}x-k)(u_{0}((k+1)h)-u_{0}(kh))$ for $x\in[kh,(k+1)h]$ , $k\in\mathbb{Z}_{n}\cup\{0\}$ . In particular, when $u_{0}\in\mathcal{C}^{1}(\mathcal{O})$ , it holds that

\displaystyle|\tilde{u}^{n}(0,x)-\tilde{u}^{n}(0,y)|\leq C|x-y|,\quad x,y\in\mathcal{O}.

(3.18)

By $u_{0}\in\mathcal{C}^{3}(\mathcal{O})$ and (3.7), there exist $\theta_{1},\theta_{2}\in(0,1)$ such that for $z\in[h,\pi)$ ,

|u^{\prime\prime}_{0}(z)-\Delta_{n}u_{0}(z)|=|u^{\prime\prime}_{0}(z)-\frac{1}{2}u_{0}^{\prime\prime}(\kappa_{n}(z)+\theta_{1}\frac{\pi}{n})-\frac{1}{2}u_{0}^{\prime\prime}(\kappa_{n}(z)-\theta_{2}\frac{\pi}{n})|\leq Cn^{-1},

and for $z\in[0,h)$ , $|u^{\prime\prime}_{0}(z)-\Delta_{n}u_{0}(z)|=|u^{\prime\prime}_{0}(z)|=|u^{\prime\prime}_{0}(z)-u^{\prime\prime}_{0}(0)|\leq Cn^{-1}.$ Therefore, using (2.2) and (3.9), a direct calculation gives

	$\displaystyle\|\tilde{u}^{n}_{1}(t,x)-u_{1}(t,x)\|$	$\displaystyle\leq\frac{C}{n}+\int_{0}^{t}\int_{\mathcal{O}}\|\Delta_{n}G^{n}_{r}(x,z)-\Delta G_{r}(x,z)\|\mathrm{d}z\mathrm{d}r$
		$\displaystyle\quad+\int_{0}^{t}\int_{\mathcal{O}}\|\Delta G_{r}(x,z)\|\|u^{\prime\prime}_{0}(z)-\Delta_{n}u_{0}(z)\|\mathrm{d}z\mathrm{d}r\leq Cn^{-1}.$		(3.19)

Part 2: The error $\tilde{u}^{n}_{3}(t,x)-u_{3}(t,x)$ is divided into

	$\displaystyle\tilde{u}^{n}_{3}(t,x)-u_{3}(t,x)=$	$\displaystyle\int_{0}^{t}\int_{\mathcal{O}}[G_{t-s}^{n}(x,y)-G_{t-s}(x,y)]\sigma(u(s,\kappa_{n}(y)))W(\mathrm{d}s,\mathrm{d}y)$
		$\displaystyle+\int_{0}^{t}\int_{\mathcal{O}}G_{t-s}(x,y)[\sigma(u(s,\kappa_{n}(y)))-\sigma(u(s,y))]W(\mathrm{d}s,\mathrm{d}y).$

The Burkholder inequality, the boundedness and Lipschitz continuity of $\sigma$ , (3.10), (2.1), and Lemma 2.2 imply

		$\displaystyle\quad\\|\tilde{u}^{n}_{3}(t,x)-u_{3}(t,x)\\|_{p}^{2}$
		$\displaystyle\leq\int_{0}^{t}\int_{\mathcal{O}}\|G_{t-s}^{n}(x,y)-G_{t-s}(x,y)\|^{2}\mathrm{d}y\mathrm{d}s+C\int_{0}^{t}\int_{\mathcal{O}}G^{2}_{t-s}(x,y)\\|u(s,\kappa_{n}(y))-u(s,y)\\|_{p}^{2}\mathrm{d}y\mathrm{d}s$
		$\displaystyle\leq Cn^{-2}+C\int_{0}^{t}(t-s)^{-\frac{1}{2}}\sup_{y\in\mathcal{O}}\\|u(s,\kappa_{n}(y))-u(s,y)\\|_{p}^{2}\mathrm{d}s\leq Cn^{-2}.$		(3.20)

Part 3: Notice that $\|\tilde{u}^{n}_{2}(t,x)-u_{2}(t,x)\|_{p}\leq I_{1}+I_{2}$ , where

	$\displaystyle I_{1}:=\int_{0}^{t}\int_{\mathcal{O}}\|\Delta_{n}G_{t-s}^{n}(x,y)-\Delta G_{t-s}(x,y)\|\\|f(u(s,\kappa_{n}(y)))\\|_{p}\mathrm{d}y\mathrm{d}s,$
	$\displaystyle I_{2}:=\int_{0}^{t}\int_{\mathcal{O}}\|\Delta G_{t-s}(x,y)\|\left\\|f(u(s,\kappa_{n}(y)))-f(u(s,y))\right\\|_{p}\mathrm{d}y\mathrm{d}s.$

It follows from (3.9), (2.7) and Lemma 2.3 that

\displaystyle I_{1}\leq C\int_{0}^{t}\int_{\mathcal{O}}|\Delta_{n}G_{t-s}^{n}(x,y)-\Delta G_{t-s}(x,y)|\mathrm{d}y\mathrm{d}s\Big{(}1+\sup_{t,x}\|u(t,x)\|_{3p}^{3}\Big{)}\leq Cn^{-1}.

Under Assumption 1 or 2, $|f(a_{1})-f(a_{2})|\leq c_{0}(1+a_{1}^{2}+a_{2}^{2})|a_{2}-a_{1}|$ . Hence, the Hölder inequality together with Lemmas 2.2 and 2.3 yields that for any $(s,x)\in[0,T]\times\mathcal{O}$ ,

	$\displaystyle\quad\\|f(u(s,\kappa_{n}(x)))-f(u(s,x))\\|_{p}$
	$\displaystyle\leq C\\|u(s,\kappa_{n}(x))-u(s,x)\\|_{3p}\big{(}1+\\|u(s,\kappa_{n}(x))\\|^{2}_{3p}+\\|u(s,x)\\|^{2}_{3p}\big{)}\leq Cn^{-1}.$

Therefore, $I_{2}\leq Cn^{-1}\int_{0}^{t}\int_{\mathcal{O}}|\Delta G_{t-s}(x,y)|\mathrm{d}y\mathrm{d}s\leq Cn^{-1}$ , in view of (2.1). In conclusion, $\|\tilde{u}^{n}_{2}(t,x)-u_{2}(t,x)\|_{p}\leq Cn^{-1},$ which along with (3) and (3) finishes the proof. ∎

By Lemma 3.2, we present the strong convergence rate of the spatial FDM for Eq. (1.1). We would like to mention that Theorem 3.3 also holds for stochastic Cahn–Hilliard equations with NBCs.

Theorem 3.3.

Suppose that Assumption 1 holds and $u_{0}\in\mathcal{C}^{3}(\mathcal{O})$ . Then for every $p\geq 1$ , there exists some constant $C=C(p,T,K)$ such that for any $(t,x)\in[0,T]\times\mathcal{O}$ ,

\|u^{n}(t,x)-u(t,x)\|_{p}\leq Cn^{-1}.

Proof.

Denote $e^{n}(t,x):=u^{n}(t,x)-u(t,x)$ . In view of (3) and (3),

	$\displaystyle u^{n}(t,x)-\tilde{u}^{n}(t,x)$	$\displaystyle=\int_{0}^{t}\int_{\mathcal{O}}\Delta_{n}G^{n}_{t-s}(x,y)\big{[}f(u^{n}(s,\kappa_{n}(y)))-f(u(s,\kappa_{n}(y)))\big{]}\mathrm{d}y\mathrm{d}s$
		$\displaystyle\quad+\int_{0}^{t}\int_{\mathcal{O}}G^{n}_{t-s}(x,y)\big{[}\sigma(u^{n}(s,\kappa_{n}(y)))-\sigma(u(s,\kappa_{n}(y)))\big{]}W(\mathrm{d}s,\mathrm{d}y).$

By the expressions of $G_{t}^{n}$ and $\Delta_{n}G_{t}^{n}$ and (2.17), we arrive at that for $0<\epsilon\ll 1$ ,

\displaystyle|G^{n}_{t}(x,y)|\leq C_{\epsilon}t^{-\frac{1}{4}-\epsilon},\qquad|\Delta_{n}G^{n}_{t}(x,y)|\leq C_{\epsilon}t^{-\frac{3}{4}-\epsilon},\quad\forall~{}t\in(0,T],~{}x,y\in\mathcal{O}.

(3.21)

Hence, the Cauchy–Schwarz inequality with respect to the measure $|\Delta_{n}G^{n}_{t-s}(x,y)|\mathrm{d}y\mathrm{d}s$ , Lemma 3.2, Assumption 1, and the Minkowski and Burkholder inequalities yield that for $0<\epsilon\ll 1$ ,

	$\displaystyle\\|e^{n}(t,x)\\|_{p}^{2}$	$\displaystyle\leq Cn^{-2}+C\int_{0}^{t}\int_{\mathcal{O}}\|\Delta_{n}G^{n}_{t-s}(x,y)\|\\|u^{n}(s,\kappa_{n}(y))-u(s,\kappa_{n}(y))\\|_{p}^{2}\mathrm{d}y\mathrm{d}s$
		$\displaystyle\quad+C\int_{0}^{t}\int_{\mathcal{O}}\|G^{n}_{t-s}(x,y)\|^{2}\\|u^{n}(s,\kappa_{n}(y))-u(s,\kappa_{n}(y))\\|_{p}^{2}\mathrm{d}y\mathrm{d}s$
		$\displaystyle\leq Cn^{-2}+C_{\epsilon}\int_{0}^{t}\int_{\mathcal{O}}(t-s)^{-\frac{3}{4}-\epsilon}\\|u^{n}(s,\kappa_{n}(y))-u(s,\kappa_{n}(y))\\|_{p}^{2}\mathrm{d}y\mathrm{d}s,$

in which the second step used (3.21). Taking the supremum over $x$ produces

\displaystyle\sup_{x}\|e^{n}(t,x)\|^{2}_{p}

\displaystyle\leq Cn^{-2}+C_{\epsilon}\int_{0}^{t}(t-s)^{-\frac{3}{4}-\epsilon}\sup_{x}\|e^{n}(s,x)\|_{p}^{2}\mathrm{d}s,

(3.22)

which along with the Gronwall lemma with weak singularities (see e.g., [15, Lemma 3.4]) completes the proof. ∎

4. Convergence of density

For real-valued random variables $X,Y$ , we write $\mathrm{d}_{\mathrm{TV}}(X,Y)$ to indicate the total variation distance between $X$ and $Y$ , i.e.,

\mathrm{d}_{\mathrm{TV}}(X,Y)=2\sup_{A\in\mathscr{B}(\mathbb{R})}\{|\mathbb{P}(X\in A)-\mathbb{P}(Y\in A)|\}=\sup_{\phi\in\Phi}|\mathbb{E}[\phi(X)]-\mathbb{E}[\phi(Y)]|,

where $\Phi$ is the set of continuous functions $\phi:\mathbb{R}\rightarrow\mathbb{R}$ which are bounded by $1$ , and $\mathscr{B}(\mathbb{R})$ is the Borel $\sigma$ -algebra of $\mathbb{R}$ . Furthermore, if $\{X_{n}\}_{n\geq 1}$ and $X_{\infty}$ have the densities $p_{X_{n}}$ and $p_{X_{\infty}}$ respectively, then

\displaystyle\mathrm{d}_{\mathrm{TV}}(X_{n},X_{\infty})=\|p_{X_{n}}-p_{X_{\infty}}\|_{L^{1}(\mathbb{R})}.

(4.1)

In this section, we show that for $k\in\mathbb{Z}_{n}$ , the spatial semi-discrete numerical solution $u^{n}(T,kh)$ admits a density, which converges in $L^{1}(\mathbb{R})$ to the density of the exact solution $u(T,kh)$ .

4.1. Malliavin calculus

We start with introducing some notations in the context of the Malliavin calculus with respect to the space-time white noise (see e.g., [23]). The isonormal Gaussian family $\{W(h),h\in\mathfrak{H}\}$ corresponding to $\mathfrak{H}:=L^{2}([0,T]\times\mathcal{O})$ is given by the Wiener integral $W(h)=\int_{0}^{T}\int_{\mathcal{O}}h(s,y)W(\mathrm{d}s,\mathrm{d}y).$ Denote by $\mathcal{S}$ the class of smooth real-valued random variables of the form

X=\varphi(W(h_{1}),\ldots,W(h_{n})),

(4.2)

where $\varphi\in\mathcal{C}_{p}^{\infty}(\mathbb{R}^{n}),$ $h_{i}\in\mathfrak{H},\,i=1,\ldots,n,\,n\geq 1.$ Here $\mathcal{C}_{p}^{\infty}(\mathbb{R}^{n})$ is the space of all $\mathbb{R}$ -valued smooth functions on $\mathbb{R}^{n}$ whose partial derivatives have at most polynomial growth. The Malliavin derivative of $X\in\mathcal{S}$ of the form (4.2) is an $\mathfrak{H}$ -valued random variable given by $DX=\sum_{i=1}^{n}\partial_{i}\varphi(W(h_{1}),\ldots,W(h_{n}))h_{i},$ which is also a random field $DX=\{D_{\theta,\xi}X,(\theta,\xi)\in[0,T]\times\mathcal{O}\}$ with $D_{\theta,\xi}X\!=\sum_{i=1}^{n}\partial_{i}\varphi(W(h_{1}),\ldots,W(h_{n}))h_{i}(\theta,\xi)$ for almost everywhere $(\theta,\xi,\omega)\in[0,T]\times\mathcal{O}\times\Omega$ . For any $p\geq 1$ , we denote the domain of $D$ in $L^{p}(\Omega;\mathbb{R})$ by $\mathbb{D}^{1,p}$ , meaning that $\mathbb{D}^{1,p}$ is the closure of $\mathcal{S}$ with respect to the norm

\|X\|_{1,p}=\left(\mathbb{E}\left[|X|^{p}+\|DX\|_{\mathfrak{H}}^{p}\right]\right)^{\frac{1}{p}}.

We define the iteration of the operator $D$ in such a way that for $X\in\mathcal{S}$ , the iterated derivative $D^{k}X$ is an $\mathfrak{H}^{\bigotimes k}$ -valued random variable. More precisely, for $k\in\mathbb{N}_{+}$ , $D^{k}X=\{D_{r_{1},\theta_{1}}\cdots D_{r_{k},\theta_{k}}X,(r_{i},\theta_{i})\in[0,T]\times\mathcal{O}\}$ is a measurable function on the product space $([0,T]\times\mathcal{O})^{k}\times\Omega$ . Then for $p\geq 1$ , $k\in\mathbb{N}$ , denote by $\mathbb{D}^{k,p}$ the completion of $\mathcal{S}$ with respect to the norm $\|X\|_{k,p}=\big{(}\mathbb{E}\big{[}|X|^{p}+\sum_{j=1}^{k}\|D^{j}X\|_{\mathfrak{H}^{\bigotimes j}}^{p}\big{]}\big{)}^{\frac{1}{p}}.$ Define $\mathbb{D}^{k,\infty}:=\bigcap_{p\geq 1}\mathbb{D}^{k,p}$ and $\mathbb{D}^{\infty}:=\bigcap_{k\geq 1}\mathbb{D}^{k,\infty}$ to be topological projective limits.

We close this part by the following proposition, which allows us to obtain the convergence of density of a sequence of random variables from the convergence in $\mathbb{D}^{1,2}$ .

Proposition 4.1.

[22, Theorem 4.2] Let $\{X_{n}\}_{n\geq 1}$ be a sequence in $\mathbb{D}^{1,2}$ such that each $X_{n}$ admits a density. Let $X_{\infty}\in\mathbb{D}^{2,4}$ and let $0<\alpha\leq 2$ be such that $\mathbb{E}[\|DX_{\infty}\|_{\mathfrak{H}}^{-\alpha}]<\infty$ . If $X_{n}\rightarrow X_{\infty}$ in $\mathbb{D}^{1,2}$ , then there exists a constant $c>0$ depending only on $X_{\infty}$ such that for any $n\geq 1$ ,

\mathrm{d}_{\mathrm{TV}}(X_{n},X_{\infty})\leq c\|X_{n}-X_{\infty}\|_{1,2}^{\frac{\alpha}{\alpha+2}}.

4.2. Convergence in $\mathbb{D}^{1,2}$

In this part, we extend the strong convergence of the spatial FDM to the convergence in $\mathbb{D}^{1,2}$ . It is shown in [8, Proposition 3.1] or [4, Lemma 3.2] that if $f(x)=(x^{3}-x)K_{R}(x)$ , then for any $(t,x)\in[0,T]\times\mathcal{O}$ , $u(t,x)\in\mathbb{D}^{1,2}$ and satisfies

	$\displaystyle D_{r,z}u(t,x)$	$\displaystyle=G_{t-r}(x,z)\sigma(u(r,z))+\int_{r}^{t}\int_{\mathcal{O}}\Delta G_{t-s}(x,y)f^{\prime}(u(s,y))D_{r,z}u(s,y)\mathrm{d}y\mathrm{d}s$
		$\displaystyle\quad+\int_{r}^{t}\int_{\mathcal{O}}G_{t-s}(x,y)\sigma^{\prime}(u(s,y))D_{r,z}u(s,y)W(\mathrm{d}s,\mathrm{d}y),$		(4.3)

if $r\leq t$ , and $D_{r,z}u(t,x)=0$ , if $r>t$ . Their proofs rely on the global Lipschitz continuity of $f_{R}$ , and thus (4.2) holds naturally whenever $f$ satisfies Assumption 1. Further, we impose Assumption 3 to study the regularity of the exact solution in the Malliavin Sobolev space.

Assumption 3.

For some integer $k\geq 1$ , $f$ and $\sigma$ have bounded derivatives up to order $k$ .

Lemma 4.2.

Under Assumptions 1 and 3, $u(t,x)\in\mathbb{D}^{k,\infty}$ for any $(t,x)\in[0,T]\times\mathcal{O}$ . Moreover, for any $p\geq 1$ , there exists $C=C(k,p,T)$ such that

\sup_{t,x}\|u(t,x)\|_{k,p}\leq C.

Proof.

Define the Picard approximation by $w^{0}(t,x)=u_{0}(x)$ , $(t,x)\in[0,T]\times\mathcal{O}$ , and for $i\in\mathbb{N}$ ,

	$\displaystyle w^{i+1}(t,x)$	$\displaystyle=\mathbb{G}_{t}u_{0}(x)+\int_{0}^{t}\int_{\mathcal{O}}\Delta G_{t-s}(x,y)f(w^{i}(s,y))\mathrm{d}y\mathrm{d}s$
		$\displaystyle\quad+\int_{0}^{t}\int_{\mathcal{O}}G_{t-s}(x,y)\sigma(w^{i}(s,y))W(\mathrm{d}s,\mathrm{d}y),\quad(t,x)\in[0,T]\times\mathcal{O}.$

Fix $(t,x)\in[0,T]\times\mathcal{O}$ . In view of [23, Lemma 1.5.3], the proof of $u(t,x)\in\mathbb{D}^{k,\infty}$ boils down to proving that

(i) $\{w^{i}(t,x)\}_{i\geq 1}$ converges to $u(t,x)$ in $L^{p}(\Omega;\mathbb{R})$ for every $p\geq 1$ .

(ii) for any $p\geq 1$ , $\sup_{i\geq 0}\|w^{i}(t,x)\|_{k,p}<\infty.$

Property (i) and property (ii) with $k=1$ and $p=2$ can be obtained in the same way as in [4, Lemma 3.2] (the sequence $\{w^{i}(t,x)\}_{i\geq 1}$ corresponds to $\{u_{n,k}(t,x)\}_{k\geq 1}$ in [4]). The proof of property (ii) with general $k,p\geq 1$ is omitted since it is standard and similar to those for other kinds of SPDEs with Lipschitz continuous coefficients; see [2, Proposition 4.3] for the case of stochastic heat equations, [25, Theorem 1] for the case of stochastic wave equations. ∎

Similar to properties (i) and (ii), the standard Picard approximation also shows that for any $(t,x)\in[0,T]\times\mathcal{O}$ , $u^{n}(t,x)\in\mathbb{D}^{1,2}$ .

Proposition 4.3.

Suppose that $u_{0}\in\mathcal{C}^{3}(\mathcal{O})$ and Assumptions 1 and 3 hold for $k=2$ . Then there exists some constant $C$ such that for any $(t,x)\in[0,T]\times\mathcal{O}$ ,

\mathbb{E}\left[\|Du^{n}(t,x)-Du(t,x)\|_{\mathfrak{H}}^{2}\right]\leq Cn^{-2}.

Proof.

By the chain rule and (3), we obtain

$\displaystyle D_{r,z}u^{n}(t,x)$	$\displaystyle=G^{n}_{t-r}(x,z)\sigma(u^{n}(r,\kappa_{n}(z)))$
	$\displaystyle\quad+\int_{r}^{t}\int_{\mathcal{O}}\Delta_{n}G^{n}_{t-s}(x,y)f^{\prime}(u^{n}(s,\kappa_{n}(y)))D_{r,z}u^{n}(s,\kappa_{n}(y))\mathrm{d}y\mathrm{d}s$
	$\displaystyle\quad+\int_{r}^{t}\int_{\mathcal{O}}G^{n}_{t-s}(x,y)\sigma^{\prime}(u^{n}(s,\kappa_{n}(y)))D_{r,z}u^{n}(s,\kappa_{n}(y))W(\mathrm{d}s,\mathrm{d}y),$	(4.4)

if $r\leq t$ , and $D_{r,z}u^{n}(t,x)=0$ , if $r>t$ . Combining (4.2) and (4.2), we write

\displaystyle D_{r,z}u^{n}(t,x)-D_{r,z}u(t,x):=I^{n}_{t,x}(r,z)+J^{n}_{t,x}(r,z)+K^{n}_{t,x}(r,z),

(4.5)

where for $r>t$ , $I^{n}_{t,x}(r,z)=J^{n}_{t,x}(r,z)=K^{n}_{t,x}(r,z)=0$ , and for $r\leq t$ ,

	$\displaystyle I^{n}_{t,x}(r,z)$	$\displaystyle=G^{n}_{t-r}(x,z)\sigma(u^{n}(r,\kappa_{n}(z)))-G_{t-r}(x,z)\sigma(u(r,z)),$
	$\displaystyle J^{n}_{t,x}(r,z)$	$\displaystyle=\int_{r}^{t}\int_{\mathcal{O}}\Delta_{n}G^{n}_{t-s}(x,y)f^{\prime}(u^{n}(s,\kappa_{n}(y)))\left[D_{r,z}u^{n}(s,\kappa_{n}(y))-D_{r,z}u(s,\kappa_{n}(y))\right]\mathrm{d}y\mathrm{d}s$
		$\displaystyle\quad+\int_{r}^{t}\int_{\mathcal{O}}\Delta_{n}G^{n}_{t-s}(x,y)f^{\prime}(u^{n}(s,\kappa_{n}(y)))\left[D_{r,z}u(s,\kappa_{n}(y))-D_{r,z}u(s,y)\right]\mathrm{d}y\mathrm{d}s$
		$\displaystyle\quad+\int_{r}^{t}\int_{\mathcal{O}}\Delta_{n}G^{n}_{t-s}(x,y)\left[f^{\prime}(u^{n}(s,\kappa_{n}(y)))-f^{\prime}(u(s,y))\right]D_{r,z}u(s,y)\mathrm{d}y\mathrm{d}s$
		$\displaystyle\quad+\int_{r}^{t}\int_{\mathcal{O}}\left[\Delta_{n}G^{n}_{t-s}(x,y)-\Delta G_{t-s}(x,y)\right]f^{\prime}(u(s,y))D_{r,z}u(s,y)\mathrm{d}y\mathrm{d}s$
		$\displaystyle=:J^{n,1}_{t,x}(r,z)+J^{n,2}_{t,x}(r,z)+J^{n,3}_{t,x}(r,z)+J^{n,4}_{t,x}(r,z),$
	$\displaystyle K^{n}_{t,x}(r,z)$	$\displaystyle=\int_{r}^{t}\int_{\mathcal{O}}G^{n}_{t-s}(x,y)\sigma^{\prime}(u^{n}(s,\kappa_{n}(y)))[D_{r,z}u^{n}(s,\kappa_{n}(y))-D_{r,z}u(s,\kappa_{n}(y))]W(\mathrm{d}s,\mathrm{d}y)$
		$\displaystyle\quad+\int_{r}^{t}\int_{\mathcal{O}}G^{n}_{t-s}(x,y)\sigma^{\prime}(u^{n}(s,\kappa_{n}(y)))\left[D_{r,z}u(s,\kappa_{n}(y))-D_{r,z}u(s,y)\right]W(\mathrm{d}s,\mathrm{d}y)$
		$\displaystyle\quad+\int_{r}^{t}\int_{\mathcal{O}}\left[G^{n}_{t-s}(x,y)\sigma^{\prime}(u^{n}(s,\kappa_{n}(y)))-G_{t-s}(x,y)\sigma^{\prime}(u(s,y))\right]D_{r,z}u(s,y)W(\mathrm{d}s,\mathrm{d}y)$
		$\displaystyle=:K^{n,1}_{t,x}(r,z)+K^{n,2}_{t,x}(r,z)+K^{n,3}_{t,x}(r,z).$

When $r>t$ , we always set $J^{n,i}_{t,x}(r,z)=K^{n,j}_{t,x}(r,z)=0$ for $i=1,2,3,4$ and $j=1,2,3$ . Hereafter, let $\epsilon\ll 1$ be an arbitrarily fixed positive number. A combination of Lemma 2.2 and Theorem 3.3 reveals that for any $p\geq 2$ ,

	$\displaystyle\\|u^{n}(s,\kappa_{n}(y))-u(s,y)\\|_{p}$	$\displaystyle\leq\\|u^{n}(s,\kappa_{n}(y))-u(s,\kappa_{n}(y))\\|_{p}+\\|u(s,\kappa_{n}(y))-u(s,y)\\|_{p}$
		$\displaystyle\leq Cn^{-1},$		(4.6)

for all $(s,y)\in[0,T]\times\mathcal{O}$ . Then the Lipschitz continuity of $f^{\prime}$ , the Minkowski and Cauchy-Schwarz inequalities and (3.21) produce

	$\displaystyle\\|J^{n,3}_{t,x}\\|_{L^{2}(\Omega;\mathfrak{H})}$	$\displaystyle\!\leq\!C_{\epsilon}\!\int_{0}^{t}\int_{\mathcal{O}}(t\!-\!s)^{-\frac{3}{4}-\epsilon}\\|u^{n}(s,\kappa_{n}(y))\!-\!u(s,y)\\|_{4}\\|Du(s,y)\\|_{L^{4}(\Omega;\mathfrak{H})}\mathrm{d}y\mathrm{d}s$
		$\displaystyle\leq Cn^{-1}\sup_{t,x}\\|Du(t,x)\\|_{L^{4}(\Omega;\mathfrak{H})}\leq Cn^{-1},$

where (4.2) and Lemma 4.2 were used in the second line. Similarly, by the boundedness of $f^{\prime}$ , Lemma 2.3, Lemma 4.2, and (3.9),

	$\displaystyle\\|J^{n,4}_{t,x}\\|_{L^{2}(\Omega;\mathfrak{H})}$	$\displaystyle\leq\int_{0}^{t}\int_{\mathcal{O}}\|\Delta_{n}G^{n}_{t-s}(x,y)-\Delta G_{t-s}(x,y)\|\\|f^{\prime}(u(s,y))Du(s,y)\\|_{L^{2}(\Omega;\mathfrak{H})}\mathrm{d}y\mathrm{d}s$
		$\displaystyle\leq C\int_{0}^{t}\int_{\mathcal{O}}\|\Delta_{n}G^{n}_{t-s}(x,y)-\Delta G_{t-s}(x,y)\|\mathrm{d}y\mathrm{d}s\leq Cn^{-1}.$

Since $\sigma$ is bounded and Lipschitz continuous, it follows from the Minkowski inequality, (4.2), (3.10), and (2.1) that for $p\geq 2$ ,

	$\displaystyle\\|I_{t,x}^{n}\\|_{L^{p}(\Omega;\mathfrak{H})}^{2}$	$\displaystyle\leq\int_{0}^{t}\int_{\mathcal{O}}\\|G^{n}_{t-r}(x,z)\sigma(u^{n}(r,\kappa_{n}(z)))-G_{t-r}(x,z)\sigma(u(r,z))\\|_{p}^{2}\mathrm{d}z\mathrm{d}r$
		$\displaystyle\leq 2\int_{0}^{t}\int_{\mathcal{O}}\|G^{n}_{t-r}(x,z)-G_{t-r}(x,z)\|^{2}\\|\sigma(u^{n}(r,\kappa_{n}(z)))\\|_{p}^{2}\mathrm{d}z\mathrm{d}r$
		$\displaystyle\quad+2\int_{0}^{t}\int_{\mathcal{O}}\|G_{t-r}(x,z)\|^{2}\\|\sigma(u^{n}(r,\kappa_{n}(z)))-\sigma(u(r,z))\\|_{p}^{2}\mathrm{d}z\mathrm{d}r\leq Cn^{-2}.$

Replacing $\sigma$ by $\sigma^{\prime}$ in the above inequality, we also have

\displaystyle\int_{0}^{t}\int_{\mathcal{O}}\|G^{n}_{t-s}(x,y)\sigma^{\prime}(u^{n}(s,\kappa_{n}(y)))-G_{t-s}(x,y)\sigma^{\prime}(u(s,y))\|_{4}^{2}\mathrm{d}y\mathrm{d}s\leq Cn^{-2},

which along with the Burkholder inequality for Hilbert space valued martingales (see e.g. [2, (4.18)]), the Hölder inequality and Lemma 4.2 indicates

	$\displaystyle\\|K^{n,3}_{t,x}\\|_{L^{2}(\Omega;\mathfrak{H})}^{2}$	$\displaystyle\leq C\int_{0}^{t}\int_{\mathcal{O}}\left\\|\left(G^{n}_{t-s}(x,y)\sigma^{\prime}(u^{n}(s,\kappa_{n}(y)))-G_{t-s}(x,y)\sigma^{\prime}(u(s,y))\right)Du(s,y)\right\\|^{2}_{L^{2}(\Omega;\mathfrak{H})}\mathrm{d}y\mathrm{d}s$
		$\displaystyle\leq Cn^{-2}\sup_{t,x}\\|Du(t,x)\\|_{L^{4}(\Omega;\mathfrak{H})}^{2}\leq Cn^{-2}.$

In order to estimate $I^{n,2}_{t,x}$ and $K^{n,2}_{t,x}$ , we claim that for $p\geq 2$ , there exists some constant $C=C(p,T)$ such that for any $x_{1},x_{2}\in\mathcal{O}$ and $t\in(0,T]$ ,

\displaystyle\|Du(t,x_{1})-Du(t,x_{2})\|_{L^{p}(\Omega;\mathfrak{H})}\leq C|x_{1}-x_{2}|.

(4.7)

Indeed, from (4.2), we have that for $r\leq t$ ,

	$\displaystyle D_{r,z}u(t,x_{1})-D_{r,z}u(t,x_{2})$	$\displaystyle=\left[G_{t-r}(x_{1},z)-G_{t-r}(x_{2},z)\right]\sigma(u(r,z))$
		$\displaystyle\quad+\int_{r}^{t}\int_{\mathcal{O}}\left[\Delta G_{t-s}(x_{1},y)-\Delta G_{t-s}(x_{2},y)\right]f^{\prime}(u(s,y))D_{r,z}u(s,y)\mathrm{d}y\mathrm{d}s$
		$\displaystyle\quad+\int_{r}^{t}\int_{\mathcal{O}}\left[G_{t-s}(x_{1},y)-G_{t-s}(x_{2},y)\right]\sigma^{\prime}(u(s,y))D_{r,z}u(s,y)W(\mathrm{d}s,\mathrm{d}y)$
		$\displaystyle=:L_{1}(r,z)+L_{2}(r,z)+L_{3}(r,z).$

For $r>t$ , let $L_{i}(r,z)=0$ , $i=1,2,3$ . The boundedness of $\sigma$ and Lemma 2.1 indicate

\|L_{1}\|^{2}_{L^{p}(\Omega;\mathfrak{H})}\leq C\int_{0}^{t}\int_{\mathcal{O}}|G_{t-r}(x_{1},z)-G_{t-r}(x_{2},z)|^{2}\mathrm{d}z\mathrm{d}r\leq C|x_{1}-x_{2}|^{2}.

Since $f^{\prime}$ is bounded, it follows from (2) and Lemma 4.2 that

\displaystyle\|L_{2}\|_{L^{p}(\Omega;\mathfrak{H})}

\displaystyle\leq C\int_{0}^{t}\int_{\mathcal{O}}\left|\Delta G_{t-s}(x_{1},y)-\Delta G_{t-s}(x_{2},y)\right|\|Du(s,y)\|_{L^{p}(\Omega;\mathfrak{H})}\mathrm{d}y\mathrm{d}s\leq C|x_{1}-x_{2}|.

Similarly, it follows from the Burkholder inequality, Lemmas 2.1 and 4.2 that $\|L_{3}\|_{L^{p}(\Omega;\mathfrak{H})}\leq C|x_{1}-x_{2}|.$ Gathering the above estimates of $L_{1}$ , $L_{2}$ and $L_{3}$ , we obtain (4.7). By means of (4.7) and (3.21), it can be verified that $\|J^{n,2}_{t,x}\|_{L^{2}(\Omega;\mathfrak{H})}+\|K^{n,2}_{t,x}\|_{L^{2}(\Omega;\mathfrak{H})}\leq Cn^{-1}.$ Substituting the above estimates of $\|I^{n}_{t,x}\|_{L^{2}(\Omega;\mathfrak{H})}$ , $\|J^{n,i}_{t,x}\|_{L^{2}(\Omega;\mathfrak{H})}$ , $i=2,3,4$ , and $\|K^{n,j}_{t,x}\|_{L^{2}(\Omega;\mathfrak{H})}$ , $j=2,3$ , into (4.5), we deduce that for $0<\epsilon\ll 1$ ,

		$\displaystyle\quad\\|Du^{n}(t,x)-Du(t,x)\\|^{2}_{L^{2}(\Omega;\mathfrak{H})}$
		$\displaystyle\leq Cn^{-2}+C\left\|\int_{0}^{t}\int_{\mathcal{O}}\|\Delta_{n}G^{n}_{t-s}(x,y)\|\\|Du^{n}(s,\kappa_{n}(y))-Du(s,\kappa_{n}(y))\\|_{L^{2}(\Omega;\mathfrak{H})}\mathrm{d}y\mathrm{d}s\right\|^{2}$
		$\displaystyle\quad+C\int_{0}^{t}\int_{\mathcal{O}}\|G^{n}_{t-s}(x,y)\|^{2}\\|Du^{n}(s,\kappa_{n}(y))-Du(s,\kappa_{n}(y))\\|^{2}_{L^{2}(\Omega;\mathfrak{H})}\mathrm{d}y\mathrm{d}s$
		$\displaystyle\leq Cn^{-2}+C\int_{0}^{t}\int_{\mathcal{O}}(t-s)^{-\frac{3}{4}-\epsilon}\\|Du^{n}(s,\kappa_{n}(y))-Du(s,\kappa_{n}(y))\\|^{2}_{L^{2}(\Omega;\mathfrak{H})}\mathrm{d}y\mathrm{d}s,$		(4.8)

where the last step used the Hölder inequality and (3.21). Similar to (3.22), by taking the supremum over $x\in\mathcal{O}$ on both sides of (4.2) and applying the Gronwall lemma with weak singularities (see e.g., [15, Lemma 3.4]), we complete the proof. ∎

4.3. Convergence of density

In this part, we present the convergence of density of the numerical solution $\{u^{n}(T,kh)\}_{n\geq 2}$ for $k\in\mathbb{Z}_{n}$ . In order to apply Proposition 4.1 with $X_{\infty}=u(T,x)$ , we impose Assumption 4 and investigate the negative moment estimate of $Du(t,x)$ .

Assumption 4.

There exists some $\sigma_{0}>0$ such that $|\sigma(x)|>\sigma_{0}$ , for any $x\in\mathbb{R}$ .

Lemma 4.4.

Let $x\in\mathcal{O}$ and Assumptions 1 and 4 hold. Then there is $\rho\in(0,1]$ such that

\displaystyle\mathbb{E}\big{[}\|Du(T,x)\|^{-2\rho}_{\mathfrak{H}}\big{]}\leq C(\rho,T).

(4.9)

Proof.

To prove (4.9), we need to use [8, Proposition 3.2], which is summarized as follows: under Assumption 4, if $x_{i}\in\mathcal{O}$ , $i=1,\ldots,d$ , are distinct points, then for some $p_{0}>0$ , there exists $\varepsilon_{0}=\varepsilon_{0}(p_{0})$ such that for all $\varepsilon\in(0,\varepsilon_{0})$ ,

\displaystyle\sup_{\xi\in\mathbb{R}^{d},\|\xi\|=1}\mathbb{P}\left(\xi^{\top}\mathbb{C}(t)\xi\leq\varepsilon\right)\leq\varepsilon^{p_{0}},

(4.10)

where $\mathbb{C}(t):=(\langle Du(t,x_{i}),Du(t,x_{j})\rangle_{\mathfrak{H}})_{1\leq i,j\leq d}$ denotes the Malliavin covariance matrix of $(u(t,x_{1}),\ldots,u(t,x_{d}))$ (the notation $u$ corresponds to $X_{R}$ in [8]).

As a consequence of (4.10) with $d=1$ and $t=T$ , we have that for all $0<\varepsilon\leq\varepsilon_{0}$ , $\mathbb{P}(\|Du(T,x)\|^{2}_{\mathfrak{H}}\leq\varepsilon)\leq\varepsilon^{p_{0}},$ which implies that for any $\rho<p_{0}$ ,

\displaystyle\sum_{n=1}^{\infty}n^{\rho-1}\mathbb{P}\left(\|Du(T,x)\|^{-2}_{\mathfrak{H}}\geq n\right)\leq\sum_{n=1}^{\lfloor\varepsilon_{0}^{-1}\rfloor}n^{\rho-1}+\sum_{n=\lfloor\varepsilon_{0}^{-1}\rfloor+1}^{\infty}n^{\rho-1}n^{-p_{0}}\leq C(\rho,\varepsilon_{0}).

Then we have that for $0<\rho<\min\{p_{0},1\}$ and $Z:=\|Du(T,x)\|^{-2}_{\mathfrak{H}}$ ,

	$\displaystyle\mathbb{E}\left[Z^{\rho}\right]$	$\displaystyle\leq 1+\sum_{n=1}^{\infty}(n+1)^{\rho}\mathbb{P}(n\leq Z<n+1)\leq 2+\sum_{n=1}^{\infty}\left((n+1)^{\rho}-n^{\rho}\right)\mathbb{P}(Z\geq n)$
		$\displaystyle\leq 2+\rho\sum_{n=1}^{\infty}n^{\rho-1}\mathbb{P}(Z\geq n)\leq C(\rho,\varepsilon_{0}),$

which implies (4.9). The proof is completed. ∎

We are ready to give the main result of this section, which states that for $k\in\mathbb{Z}_{n}$ , the density of the numerical solution $u^{n}(T,kh)$ exists and converges in $L^{1}(\mathbb{R})$ to the density of the exact solution. The readers are referred to [4, 8] for the existence of the density $p_{u(t,x)}$ of the exact solution $u(t,x)$ for any $(t,x)\in[0,T]\times\mathcal{O}$ .

Theorem 4.5.

Suppose that Assumptions 1, 3, and 4 hold for $k=2$ , and $u_{0}\in\mathcal{C}^{3}(\mathcal{O})$ . Then for any $k\in\mathbb{Z}_{n}$ , $u^{n}(T,kh)$ admits a density $p_{u^{n}(T,kh)}$ , and moreover

\lim_{n\rightarrow\infty}\|p_{u(T,kh)}-p_{u^{n}(T,kh)}\|_{L^{1}(\mathbb{R})}=0.

Proof.

By [23, Theorem 2.3.3] and (3.2), we obtain that under Assumption 4, for any $t\in(0,T]$ , the law of $U(t)$ is absolutely continuous with respect to the Lebesgue measure on $\mathbb{R}^{n-1}$ . Thus, $\{u^{n}(T,kh)\}_{k\in\mathbb{Z}_{n}}$ admits a density. Theorem 3.3, Lemma 4.2, Proposition 4.3, and Lemma 4.4 indicate that the conditions of Proposition 4.1 are fulfilled for $\alpha=2\rho$ , $X_{n}=u^{n}(T,kh)$ and $X_{\infty}=u(T,kh)$ . As a result,

\lim_{n\rightarrow\infty}\mathrm{d}_{\mathrm{TV}}(u(T,kh),u^{n}(T,kh))=0,

which together to (4.1) completes the proof. ∎

5. Full discretization

For the purpose of effective computation, we combine the spatial FDM with a temporal exponential Euler method to obtain the full discretization of Eq. (1.1), and give the strong convergence rate of the fully discrete numerical solution in this section.

Let $\{t_{i}=i\tau,i=0,1,\ldots,m\}$ (with $m\geq 2$ ) be a uniform partition of $[0,T]$ , where $\tau:=T/m$ is the uniform time stepsize. Denote by $\eta_{m}(s)=\tau\lfloor s/\tau\rfloor$ the largest time grid point smaller than $s$ . By replacing $s$ in (3) by $\eta_{m}(s)$ , we obtain the full discretization $u^{m,n}=\{u^{m,n}(t,x);(t,x)\in[0,T]\times\mathcal{O}\}$ given by

$\displaystyle u^{m,n}(t,x)$	$\displaystyle=\int_{\mathcal{O}}G^{n}_{t}(x,y)u_{0}(\kappa_{n}(y))\mathrm{d}y$
	$\displaystyle\quad+\int_{0}^{t}\int_{\mathcal{O}}\Delta_{n}G^{n}_{t-\eta_{m}(s)}(x,y)f(u^{m,n}(\eta_{m}(s),\kappa_{n}(y)))\mathrm{d}y\mathrm{d}s$
	$\displaystyle\quad+\int_{0}^{t}\int_{\mathcal{O}}G^{n}_{t-\eta_{m}(s)}(x,y)\sigma(u^{m,n}(\eta_{m}(s),\kappa_{n}(y)))W(\mathrm{d}s,\mathrm{d}y).$	(5.1)

The discrete Green function $G^{n}$ satisfies the following estimates.

Lemma 5.1.

Let $\gamma\in(0,\frac{3}{8})$ . Then for any $x,y\in\mathcal{O}$ and $s,t\in[0,T]$ with $s<t$ ,

	$\displaystyle\int_{0}^{t}\int_{\mathcal{O}}\|G^{n}_{t-r}(x,z)-G^{n}_{t-r}(y,z)\|^{2}\mathrm{d}z\mathrm{d}r\leq C\|x-y\|^{2},$
	$\displaystyle\int_{0}^{s}\int_{\mathcal{O}}\|G^{n}_{t-r}(x,z)-G^{n}_{s-r}(x,z)\|^{2}\mathrm{d}z\mathrm{d}r+\int_{s}^{t}\int_{\mathcal{O}}\|G^{n}_{t-r}(x,z)\|^{2}\mathrm{d}z\mathrm{d}r\leq C_{\gamma}\|t-s\|^{2\gamma}.$

Proof.

The proof is similar to that of [4, Lemma 1.8]. It can be verified that

|\phi_{j,n}(x)-\phi_{j,n}(y)|\leq\sqrt{2\pi^{-1}}j|x-y|.

(5.2)

A combination of (3.5) and (5.2) implies

\displaystyle\int_{0}^{t}\int_{\mathcal{O}}|G^{n}_{t-r}(x,z)-G^{n}_{t-r}(y,z)|^{2}\mathrm{d}z\mathrm{d}r\leq

\displaystyle\sum_{j=1}^{n-1}\frac{1}{2\lambda_{j,n}^{2}}|\phi_{j,n}(x)-\phi_{j,n}(y)|^{2}\leq C|x-y|^{2}.

By the uniform boundedness of $\phi_{j,n}$ and (3.5),

\displaystyle\int_{\mathcal{O}}|G^{n}_{t}(x,z)|^{2}\mathrm{d}z=\sum_{j=1}^{n-1}\exp(-2\lambda_{j,n}^{2}t)|\phi_{j,n}(x)|^{2}\leq Ct^{-\frac{1}{4}}\int_{0}^{\infty}e^{-z^{4}}\mathrm{d}z\leq Ct^{-\frac{1}{4}},

which indicates $\int_{s}^{t}\int_{\mathcal{O}}|G^{n}_{t-r}(x,z)|^{2}\mathrm{d}z\mathrm{d}r\leq C|t-s|^{\frac{3}{4}}$ . Using (3.5) and (2.20), we obtain

	$\displaystyle\int_{0}^{s}\int_{\mathcal{O}}\|G^{n}_{t-r}(x,z)-G^{n}_{s-r}(x,z)\|^{2}\mathrm{d}z\mathrm{d}r$	$\displaystyle=\sum_{j=1}^{n-1}\|1-\exp(-\lambda_{j,n}^{2}(t-s))\|^{2}\frac{1-\exp(-2\lambda_{j,n}^{2}s)}{2\lambda_{j,n}^{2}}\|\phi_{j,n}(x)\|^{2}$
		$\displaystyle\leq C\sum_{j=1}^{n-1}\lambda_{j,n}^{4\gamma-2}(t-s)^{2\gamma}\leq C(t-s)^{2\gamma},$		(5.3)

where $\gamma<\frac{3}{8}$ . The proof is completed. ∎

In a similar way, one can prove the following lemma.

Lemma 5.2.

Let $\alpha\in(0,1)$ . Then for any $x,y\in\mathcal{O}$ and $s,t\in[0,T]$ with $s<t$ ,

	$\displaystyle\int_{s}^{t}\int_{\mathcal{O}}\|\Delta_{n}G^{n}_{t-r}(x,z)\|\mathrm{d}z\mathrm{d}r\leq C\|t-s\|^{\frac{3\alpha}{8}},$		(5.4)
	$\displaystyle\int_{0}^{s}\int_{\mathcal{O}}\|\Delta_{n}G^{n}_{t-r}(x,z)-\Delta_{n}G^{n}_{s-r}(y,z)\|\mathrm{d}z\mathrm{d}r\leq C(\alpha)(\|x-y\|+\|t-s\|^{\frac{3\alpha}{8}}).$		(5.5)

Proof.

By using the orthogonality of $\{\phi_{j}\circ\kappa_{n}\}_{j\in\mathbb{Z}_{n}}$ and the Cauchy–Schwarz inequality, (5.4) and (5.5) with $x=y$ and (5.5) with $t=s$ can be obtained similarly as in (2), (2) and (2), respectively. ∎

Proposition 5.3.

Let Assumption 1 hold and $u_{0}\in\mathcal{C}^{2}(\mathcal{O})$ . Then for any $\alpha\in(0,1)$ and $p\geq 1$ , there exists $C=C(p,T,\alpha)$ such that for any $(t,x)\in[0,T]\times\mathcal{O}$ ,

\displaystyle\|u^{n}(t,x)-u^{n}(s,y)\|_{p}\leq C(|t-s|^{\frac{3\alpha}{8}}+C|x-y|).

Proof.

As a result of Theorem 3.3 and Lemma 2.3, for $n\geq 2$ and $(t,x)\in[0,T]\times\mathcal{O}$ ,

\displaystyle\|u^{n}(t,x)\|_{p}\leq\|u^{n}(t,x)-u(t,x)\|_{p}+\|u(t,x)\|_{p}\leq C(p,T,K).

(5.6)

Recall that $\tilde{u}^{n}_{1}(t,x):=\int_{\mathcal{O}}G^{n}_{t}(x,y)u_{0}(\kappa_{n}(y))\mathrm{d}y$ . It follows from (3) and (3.18) that

\displaystyle|\tilde{u}^{n}_{1}(t,x)-\tilde{u}^{n}_{1}(t,y)|

\displaystyle\leq C|x-y|+\int_{0}^{t}\int_{\mathcal{O}}|\Delta_{n}G^{n}_{r}(x,z)-\Delta_{n}G^{n}_{r}(y,z)||\Delta_{n}u_{0}(z)|\mathrm{d}z\mathrm{d}r.

Since $u_{0}\in\mathcal{C}^{2}(\mathcal{O})$ , $|\Delta_{n}u_{0}(z)|\leq C$ for $z\in\mathcal{O}$ , and hence (5.5) implies $|\tilde{u}^{n}_{1}(t,x)-\tilde{u}^{n}_{1}(t,y)|\leq C|x-y|.$ Similarly, based on (3), (5.4), and (5.5), one also has that for any $\alpha\in(0,1)$ , $|\tilde{u}^{n}_{1}(t,x)-\tilde{u}^{n}_{1}(s,x)|\leq C|t-s|^{\frac{3\alpha}{8}}.$ Based on a standard argument as in the proof of Lemma 2.2, it follows from (5.6) and Lemmas 5.1 and 5.2 that $\|u^{n}(t,x)-\tilde{u}^{n}_{1}(t,x)-u^{n}(s,y)+\tilde{u}^{n}_{1}(s,y)\|_{p}\leq C(|t-s|^{\frac{3\alpha}{8}}+C|x-y|)$ . The proof is completed. ∎

Theorem 5.4.

Suppose that Assumption 1 holds and $u_{0}\in\mathcal{C}^{3}(\mathcal{O})$ . Then for every $p\geq 1$ and $0<\epsilon\ll 1$ , there exists some constant $C=C(p,T,K,\epsilon)$ such that for any $(t,x)\in[0,T]\times\mathcal{O}$ ,

\displaystyle\|u^{m,n}(t,x)-u(t,x)\|_{p}\leq C(n^{-1}+m^{-\frac{3-\epsilon}{8}}).

Proof.

Let $(t,x)\in[0,T]\times\mathcal{O}$ and $0<\epsilon\ll 1$ . By virtue of Theorem 3.3, it remains to show

\displaystyle\|u^{m,n}(t,x)-u^{n}(t,x)\|_{p}\leq Cm^{-\frac{3-\epsilon}{8}}.

By (3), (5), the Minkowski inequality, the Burkholder inequality, and the Lipschitz continuity of $\sigma$ and $b$ , we obtain that for any $p\geq 2$ ,

	$\displaystyle\\|u^{m,n}(t,x)-u^{n}(t,x)\\|^{2}_{p}\leq CH_{1}^{m,n}+CH_{2}^{m,n}+CQ_{1}^{m,n}+CQ_{2}^{m,n}$
	$\displaystyle+C\int_{0}^{t}\int_{\mathcal{O}}\|\Delta_{n}G^{n}_{t-\eta_{m}(s)}(x,y)\|\\|u^{m,n}(\eta_{m}(s),\kappa_{n}(y))-u^{n}(\eta_{m}(s),\kappa_{n}(y))\\|^{2}_{p}\mathrm{d}y\mathrm{d}s$
	$\displaystyle+C\int_{0}^{t}\int_{\mathcal{O}}\|G^{n}_{t-\eta_{m}(s)}(x,y)\|^{2}\\|u^{m,n}(\eta_{m}(s),\kappa_{n}(y))-u^{n}(\eta_{m}(s),\kappa_{n}(y))\\|_{p}^{2}\mathrm{d}y\mathrm{d}s,$

where

	$\displaystyle H_{1}^{m,n}:$	$\displaystyle=\Big{\|}\int_{0}^{t}\int_{\mathcal{O}}\big{\|}\Delta_{n}G^{n}_{t-\eta_{m}(s)}(x,y)-\Delta_{n}G^{n}_{t-s}(x,y)\big{\|}\\|f(u^{n}(\eta_{m}(s),\kappa_{n}(y)))\\|_{p}\mathrm{d}y\mathrm{d}s\Big{\|}^{2},$
	$\displaystyle H_{2}^{m,n}:$	$\displaystyle=\Big{\|}\int_{0}^{t}\int_{\mathcal{O}}\|\Delta_{n}G^{n}_{t-s}(x,y)\|\\|f(u^{n}(\eta_{m}(s),\kappa_{n}(y)))-f(u^{n}(s,\kappa_{n}(y)))\\|_{p}\mathrm{d}y\mathrm{d}s\Big{\|}^{2},$
	$\displaystyle Q_{1}^{m,n}:$	$\displaystyle=\int_{0}^{t}\int_{\mathcal{O}}\|G^{n}_{t-\eta_{m}(s)}(x,y)-G^{n}_{t-s}(x,y)\|^{2}\\|\sigma(u^{n}(\eta_{m}(s),\kappa_{n}(y)))\\|_{p}^{2}\mathrm{d}y\mathrm{d}s,$
	$\displaystyle Q_{2}^{m,n}:$	$\displaystyle=\int_{0}^{t}\int_{\mathcal{O}}\|G^{n}_{t-s}(x,y)\|^{2}\\|\sigma(u^{n}(\eta_{m}(s),\kappa_{n}(y)))-\sigma(u^{n}(s,\kappa_{n}(y)))\\|_{p}^{2}\mathrm{d}y\mathrm{d}s.$

Taking advantage of Corollary 5.3 and (3.21), we obtain

H_{2}^{m,n}+Q_{2}^{m,n}\leq C_{\epsilon}\sup_{t}|\eta_{m}(t)-t|^{\frac{3-\epsilon}{4}}\leq C_{\epsilon}m^{-\frac{3-\epsilon}{4}}.

Similar to the proof of (5), one has that for $\alpha<\frac{3}{8}$ ,

\displaystyle\int_{0}^{t}\int_{\mathcal{O}}|G^{n}_{t-\eta_{m}(s)}(x,y)-G^{n}_{t-s}(x,y)|^{2}\mathrm{d}y\mathrm{d}s\leq C_{\epsilon}\sup_{t}|\eta_{m}(t)-t|^{2\alpha},

which along with the boundedness of $\sigma$ shows that $Q_{1}^{m,n}\leq Cm^{-\frac{3(1-\epsilon)}{4}}$ . Similar to (5.5) with $x=y$ , we also have that for $\alpha<\frac{3}{8}$ ,

\displaystyle\int_{0}^{t}\int_{\mathcal{O}}\big{|}\Delta_{n}G^{n}_{t-\eta_{m}(s)}(x,y)-\Delta_{n}G^{n}_{t-s}(x,y)\big{|}\mathrm{d}y\mathrm{d}s\leq C\sup_{t}|\eta_{m}(t)-t|^{\alpha},

which along with (5.6) reveals that $H_{1}^{m,n}\leq C_{\epsilon}m^{-\frac{3-\epsilon}{4}}$ . Gathering the above estimates together yields that for any $t\in[0,T]$ ,

	$\displaystyle\quad\sup_{x}\\|u^{m,n}(t,x)-u^{n}(t,x)\\|^{2}_{p}$
	$\displaystyle\leq C_{\epsilon}m^{-\frac{3-\epsilon}{4}}+C\int_{0}^{t}\int_{\mathcal{O}}\|\Delta_{n}G^{n}_{t-\eta_{m}(s)}(x,y)\|\sup_{x}\\|u^{m,n}(\eta_{m}(s),x)-u^{n}(\eta_{m}(s),x)\\|^{2}_{p}\mathrm{d}y\mathrm{d}s$
	$\displaystyle\quad+C\int_{0}^{t}\int_{\mathcal{O}}\|G^{n}_{t-\eta_{m}(s)}(x,y)\|^{2}\sup_{x}\\|u^{m,n}(\eta_{m}(s),x)-u^{n}(\eta_{m}(s),x)\\|^{2}_{p}\mathrm{d}y\mathrm{d}s.$

Letting $a(t):=\sup_{x}\|u^{m,n}(t,x)-u^{n}(t,x)\|^{2}_{p}$ , $t\in[0,T]$ and using (3.21), we obtain

\displaystyle a(t)\leq C_{\epsilon}m^{-\frac{3-\epsilon}{4}}+C\int_{0}^{t}(t-\eta_{m}(s))^{-\frac{3}{4}-\epsilon}a(\eta_{m}(s))\mathrm{d}s,\quad t\in[0,T].

(5.7)

Hence, for any $k=1,2,\ldots,m$ ,

\displaystyle a(t_{k})

\displaystyle\leq C_{\epsilon}m^{-\frac{3-\epsilon}{4}}+C\int_{0}^{t_{k}}(t_{k}-\eta_{m}(s))^{-\frac{3}{4}-\epsilon}a(\eta_{m}(s))\mathrm{d}s=C_{\epsilon}m^{-\frac{3-\epsilon}{4}}+C\tau\sum_{i=0}^{k-1}t_{k-i}^{-\frac{3}{4}-\epsilon}a(t_{i}),

which together with the discrete Gronwall lemma (see e.g. [20, Lemma A.4]) implies that $\sup_{0\leq k\leq m}a(t_{k})\leq C_{\epsilon}m^{-\frac{3-\epsilon}{4}}.$ Finally, taking (5.7) into account gives

\displaystyle a(t)

\displaystyle\leq C_{\epsilon}m^{-\frac{3-\epsilon}{4}}+C_{\epsilon}m^{-\frac{3-\epsilon}{4}}\int_{0}^{t}(t-\eta_{m}(s))^{-\frac{3}{4}-\epsilon}\mathrm{d}s\leq C_{\epsilon}m^{-\frac{3-\epsilon}{4}},\quad t\in[0,T].

Thus the proof is complete. ∎

Remark 5.5.

The application of the orthogonality of $\{\phi_{j}\circ\kappa_{n}\}_{j\in\mathbb{Z}_{n}}$ plays a key role to obtain the temporal convergence order nearly $\frac{3}{8}$ of the full discretization in Theorem 5.4. For example, if the left hand of (5.4) is estimated in the following way

	$\displaystyle\quad\int_{0}^{s}\int_{\mathcal{O}}\|\Delta_{n}G^{n}_{t-r}(x,z)-\Delta_{n}G^{n}_{s-r}(x,z)\|\mathrm{d}z\mathrm{d}r$
	$\displaystyle\leq C\int_{0}^{s}\sum_{j=0}^{n-1}\|\lambda_{j,n}\|\exp(-\lambda_{j,n}^{2}(s-r))[1-\exp(-\lambda_{j,n}^{2}(t-s))]\mathrm{d}r$
	$\displaystyle\leq C\sum_{j=1}^{n-1}\frac{\lambda_{j,n}^{2\alpha}(t-s)^{\alpha}}{-\lambda_{j,n}}\leq C\sum_{j=1}^{\infty}j^{4\alpha-2}(t-s)^{\alpha}\leq C_{\alpha}(t-s)^{\alpha},$

with $\alpha\in(0,\frac{1}{4})$ , then we can only obtain $\int_{0}^{s}\int_{\mathcal{O}}|\Delta_{n}G^{n}_{t-r}(x,z)-\Delta_{n}G^{n}_{s-r}(x,z)|\mathrm{d}z\mathrm{d}r\leq C(\alpha)|t-s|^{\frac{1}{4}-\epsilon}$ with $0<\epsilon\ll 1$ . As a result, the temporal Hölder continuity exponent of $u^{n}$ is only nearly ${\frac{1}{4}}$ , which leads to that the temporal convergence order of the exponential Euler method is only nearly $\frac{1}{4}$ .

Combining Theorem 5.4 with a localized argument, we show an $L^{p}(\Omega;\mathbb{R})$ convergence order localized on a set of arbitrarily large probability for Eq. (1.1) with polynomial nonlinearity.

Corollary 5.6.

Suppose that Assumption 2 hold and $u_{0}\in\mathcal{C}^{3}(\mathcal{O})$ . Then for any $R\geq 1$ , $0<\epsilon\ll 1$ and $p\geq 1$ , there exists $C=C(R,T,p,\epsilon)$ such that

\displaystyle\mathbb{E}\left[\mathbf{1}_{\Omega_{R}\cap\Omega_{R}^{m,n}}|u^{m,n}(t,x)-u(t,x)|^{p}\right]\leq C\big{(}n^{-1}+m^{-\frac{3-\epsilon}{8}}\big{)}.

Proof.

For $R\geq 1$ , denote $\Omega_{R}:=\big{\{}\omega\in\Omega:\sup_{t,x}|u(t,x,\omega)|\leq R\big{\}}.$ Set $f_{R}=K_{R}f$ with $K_{R}$ defined by (2.3). Consider the localized Cahn-Hilliard equation

\displaystyle\partial_{t}u_{R}+\Delta^{2}u_{R}=\Delta f_{R}(u_{R})+\sigma(u_{R})\dot{W},\quad R\geq 1

(5.8)

with $u_{R}(0,\cdot)=u_{0}$ and DBCs. Then the local property of stochastic integrals shows $u=u_{R}$ (i.e., for any $(t,x)\in[0,T]\times\mathcal{O}$ , $u(t,x)=u_{R}(t,x)$ ) on $\Omega_{R}$ a.s. Consider the numerical solution $u_{R}^{m,n}$ of (5.8) based on the FDM in space and the exponential Euler method in time, i.e.,

	$\displaystyle u_{R}^{m,n}(t,x)=$	$\displaystyle\int_{\mathcal{O}}G^{n}_{t}(x,y)u_{0}(\kappa_{n}(y))\mathrm{d}y+\int_{0}^{t}\int_{\mathcal{O}}\Delta_{n}G^{n}_{t-\eta_{m}(s)}(x,y)f_{R}(u_{R}^{m,n}(\eta_{m}(s),\kappa_{n}(y)))\mathrm{d}y\mathrm{d}s$
		$\displaystyle+\int_{0}^{t}\int_{\mathcal{O}}G^{n}_{t-\eta_{m}(s)}(x,y)\sigma(u_{R}^{m,n}(\eta_{m}(s),\kappa_{n}(y)))W(\mathrm{d}s,\mathrm{d}y),$		(5.9)

for $n,m\geq 2$ and $(t,x)\in[0,T]\times\mathcal{O}$ . Seting $\Omega_{R}^{m,n}:=\big{\{}\omega\in\Omega:\sup_{t,x}|u^{m,n}(t,x,\omega)|\leq R\big{\}}$ , and comparing (5) with (5), it follows from the local property of stochastic integrals that $u_{R}^{m,n}=u^{m,n}$ on $\Omega_{R}^{m,n}$ a.s. For fixed $R\geq 1$ , since $f_{R}$ satisfies Assumption 1, Theorem 5.4 indicates that there exists some constant $C=C(R,T,p,\epsilon)$ such that for $0<\epsilon\ll 1$ and $p\geq 1$ ,

\mathbb{E}\left[|u^{m,n}_{R}(t,x)-u_{R}(t,x)|^{p}\right]\leq C\big{(}n^{-1}+m^{-\frac{3-\epsilon}{8}}\big{)}.

Since $u^{m,n}(t,x)$ and $u(t,x)$ have almost surely continuous trajectories, we have $\lim_{R\rightarrow\infty}\mathbb{P}(\Omega_{R})=\lim_{R\rightarrow\infty}\mathbb{P}(\Omega_{R}^{m,n})=1$ , which implies $\lim_{R\rightarrow\infty}\mathbb{P}(\Omega_{R}\cap\Omega_{R}^{m,n})=1$ . By $u=u_{R}$ on $\Omega_{R}$ a.s., and $u_{R}^{m,n}=u^{m,n}$ on $\Omega_{R}^{m,n}$ a.s., we obtain

\displaystyle\mathbb{E}\left[\mathbf{1}_{\Omega_{R}\cap\Omega_{R}^{m,n}}|u^{m,n}(t,x)-u(t,x)|^{p}\right]\leq\mathbb{E}\left[|u^{m,n}_{R}(t,x)-u_{R}(t,x)|^{p}\right]\leq C\big{(}n^{-1}+m^{-\frac{3-\epsilon}{8}}\big{)}.

The proof is completed. ∎

Remark 5.7.

Theorems 3.3 and 4.5 indicate that when applying the spatial FDM to the localized Cahn–Hilliard equation (1.2), the associated numerical solution is strongly convergent and the density of the numerical solution converges in $L^{1}(\mathbb{R})$ . In addition, Section 2 gives the uniform moment estimate and Hölder continuity of the exact solution for Eq. (1.1) with $f$ being a polynomial of degree $3$ with a positive dominant coefficient. We expect to combine the above results with the localization technique to study the strong convergence of the spatial FDM and the density convergence of the associated numerical solution for the stochastic Cahn–Hilliard equation with polynomial nonlinearity and multiplicative noise in the future.

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		$\displaystyle\quad\int_{0}^{s}\int_{\mathcal{O}}\|\Delta G_{t-r}(x,z)-\Delta G_{s-r}(x,z)\|\mathrm{d}z\mathrm{d}r$
		$\displaystyle\leq\sqrt{\pi}\int_{0}^{s}\left(\int_{\mathcal{O}}\|\Delta G_{t-r}(x,z)-\Delta G_{s-r}(x,z)\|^{2}\mathrm{d}z\right)^{\frac{1}{2}}\mathrm{d}r$
		$\displaystyle\leq C\int_{0}^{s}\Big{(}\sum_{j=1}^{\infty}j^{4}e^{-2j^{4}(s-r)}\|1-e^{-j^{4}(t-s)}\|^{2}\Big{)}^{\frac{1}{2}}\mathrm{d}r$
		$\displaystyle\leq C\Big{(}\sum_{j=1}^{\infty}j^{4-4\rho+8\alpha_{3}}\Big{)}^{\frac{1}{2}}\int_{0}^{s}(s-r)^{-\frac{\rho}{2}}\mathrm{d}r(t-s)^{\alpha_{3}}\leq C(t-s)^{\alpha_{3}}.$		(2.21)

	$\displaystyle\|\tilde{u}^{n}_{1}(t,x)-u_{1}(t,x)\|$	$\displaystyle\leq\frac{C}{n}+\int_{0}^{t}\int_{\mathcal{O}}\|\Delta_{n}G^{n}_{r}(x,z)-\Delta G_{r}(x,z)\|\mathrm{d}z\mathrm{d}r$
		$\displaystyle\quad+\int_{0}^{t}\int_{\mathcal{O}}\|\Delta G_{r}(x,z)\|\|u^{\prime\prime}_{0}(z)-\Delta_{n}u_{0}(z)\|\mathrm{d}z\mathrm{d}r\leq Cn^{-1}.$		(3.19)

		$\displaystyle\quad\\|\tilde{u}^{n}_{3}(t,x)-u_{3}(t,x)\\|_{p}^{2}$
		$\displaystyle\leq\int_{0}^{t}\int_{\mathcal{O}}\|G_{t-s}^{n}(x,y)-G_{t-s}(x,y)\|^{2}\mathrm{d}y\mathrm{d}s+C\int_{0}^{t}\int_{\mathcal{O}}G^{2}_{t-s}(x,y)\\|u(s,\kappa_{n}(y))-u(s,y)\\|_{p}^{2}\mathrm{d}y\mathrm{d}s$
		$\displaystyle\leq Cn^{-2}+C\int_{0}^{t}(t-s)^{-\frac{1}{2}}\sup_{y\in\mathcal{O}}\\|u(s,\kappa_{n}(y))-u(s,y)\\|_{p}^{2}\mathrm{d}s\leq Cn^{-2}.$		(3.20)

	$\displaystyle\\|e^{n}(t,x)\\|_{p}^{2}$	$\displaystyle\leq Cn^{-2}+C\int_{0}^{t}\int_{\mathcal{O}}\|\Delta_{n}G^{n}_{t-s}(x,y)\|\\|u^{n}(s,\kappa_{n}(y))-u(s,\kappa_{n}(y))\\|_{p}^{2}\mathrm{d}y\mathrm{d}s$
		$\displaystyle\quad+C\int_{0}^{t}\int_{\mathcal{O}}\|G^{n}_{t-s}(x,y)\|^{2}\\|u^{n}(s,\kappa_{n}(y))-u(s,\kappa_{n}(y))\\|_{p}^{2}\mathrm{d}y\mathrm{d}s$
		$\displaystyle\leq Cn^{-2}+C_{\epsilon}\int_{0}^{t}\int_{\mathcal{O}}(t-s)^{-\frac{3}{4}-\epsilon}\\|u^{n}(s,\kappa_{n}(y))-u(s,\kappa_{n}(y))\\|_{p}^{2}\mathrm{d}y\mathrm{d}s,$

	$\displaystyle\\|I_{t,x}^{n}\\|_{L^{p}(\Omega;\mathfrak{H})}^{2}$	$\displaystyle\leq\int_{0}^{t}\int_{\mathcal{O}}\\|G^{n}_{t-r}(x,z)\sigma(u^{n}(r,\kappa_{n}(z)))-G_{t-r}(x,z)\sigma(u(r,z))\\|_{p}^{2}\mathrm{d}z\mathrm{d}r$
		$\displaystyle\leq 2\int_{0}^{t}\int_{\mathcal{O}}\|G^{n}_{t-r}(x,z)-G_{t-r}(x,z)\|^{2}\\|\sigma(u^{n}(r,\kappa_{n}(z)))\\|_{p}^{2}\mathrm{d}z\mathrm{d}r$
		$\displaystyle\quad+2\int_{0}^{t}\int_{\mathcal{O}}\|G_{t-r}(x,z)\|^{2}\\|\sigma(u^{n}(r,\kappa_{n}(z)))-\sigma(u(r,z))\\|_{p}^{2}\mathrm{d}z\mathrm{d}r\leq Cn^{-2}.$

Convergence analysis of a finite difference method for stochastic Cahn–Hilliard equation

Abstract.

Key words and phrases:

2010 Mathematics Subject Classification:

1. Introduction

2. Preliminaries

Assumption 1.

Assumption 2.

Lemma 2.1.

Lemma 2.2.

Proof.

Lemma 2.3.

Proof.

3. Finite difference method

Lemma 3.1.

Proof.

Lemma 3.2.

Proof.

Theorem 3.3.

Proof.

4. Convergence of density

4.1. Malliavin calculus

Proposition 4.1.

4.2. Convergence in 𝔻1,2\mathbb{D}^{1,2}

Assumption 3.

Lemma 4.2.

Proof.

Proposition 4.3.

Proof.

4.3. Convergence of density

Assumption 4.

Lemma 4.4.

Proof.

Theorem 4.5.

Proof.

5. Full discretization

Lemma 5.1.

Proof.

Lemma 5.2.

Proof.

Proposition 5.3.

Proof.

Theorem 5.4.

Proof.

Remark 5.5.

Corollary 5.6.

Proof.

Remark 5.7.

References

4.2. Convergence in $\mathbb{D}^{1,2}$