Schauder-type estimates for higher-order parabolic SPDEs

TL;DR

This paper establishes Schauder-type estimates for higher-order parabolic SPDEs, revealing that solution regularity depends on a novel coercivity condition that varies with the order's parity, and proves existence and uniqueness in stochastic Hölder spaces.

Contribution

It introduces a new coercivity condition for higher-order parabolic SPDEs, showing how solution regularity depends on parity and integrability, with proofs of existence and uniqueness.

Findings

Regularity of solutions depends on a parity-based coercivity condition.

Established Schauder estimates for solutions and derivatives up to order 2m.

Demonstrated the sharpness of the coercivity condition with an example.

Abstract

In this paper we consider the Cauchy problem for $2m$-order stochastic partial differential equations of parabolic type in a class of stochastic Hoelder spaces. The Hoelder estimates of solutions and their spatial derivatives up to order $2m$ are obtained, based on which the existence and uniqueness of solution is proved. An interesting finding of this paper is that the regularity of solutions relies on a coercivity condition that differs when $m$ is odd or even: the condition for odd $m$ coincides with the standard parabolicity condition in the literature for higher-order stochastic partial differential equations, while for even $m$ it depends on the integrability index $p$. The sharpness of the new-found coercivity condition is demonstrated by an example.

Equations

$$\text{$\mathrm{d}$}u=\bigg{[}(-1)^{m+1}\!\!\!\sum_{|\alpha|,\left|\beta\right|\leq m}A_{\alpha\beta}D^{\alpha+\beta}u+f\bigg{]}\mathrm{d}t+\sum_{k=1}^{\infty}\bigg{[}\sum_{|\alpha|\leq m}B_{\alpha}^{k}D^{\alpha}u+g^{k}\bigg{]}\mathrm{d}w_{t}^{k},$$

$$\|u\|_{\mathcal{C}_{p}^{\delta}}:=\bigg{[}\sup_{t,\,x}\mathbb{E}|u(x,t)|^{p}+\sup_{t,\,x\neq y}\frac{\mathbb{E}|u(x,t)-u(y,t)|^{p}}{|x-y|^{\delta p}}\bigg{]}^{\frac{1}{p}}<\infty$$

$$\sum _{|\alpha |=|\beta |=m}2A_{\alpha \beta }{\xi }_{\alpha }{\xi }_{\beta }-\lambda \sum _{|\alpha |=m}{\left|{\xi }_{\alpha }\right|}^2$$

$$\displaystyle=\left\{\begin{aligned} &\sum_{k=1}^{\infty}\bigg{|}\sum_{|\alpha|=m}B_{\alpha}^{k}\xi_{\alpha}\bigg{|}^{2}&\text{when }m\text{ is odd},\\ &(p-1)\sum_{k=1}^{\infty}\bigg{|}\sum_{|\alpha|=m}B_{\alpha}^{k}\xi_{\alpha}\bigg{|}^{2}&\text{when }m\text{ is even}.\end{aligned}\right.$$

$$\text{d}u=(-1{)}^{m+1}{D}^{2m}u\text{d}t+\mu D^{m}u\text{d}{w}_t$$

$$u(x,0)=\sum _{n\in \mathbb{Z}}{\mathrm{e}}^{-n^{2m}}\cdot {\mathrm{e}}^{\sqrt{-1}nx},x\in \mathbb{T}.$$

$$D^{\beta }f=\frac{{\partial }^{{\beta }_1}\cdots {\partial }^{{\beta }_n}}{\partial x_1^{{\beta }_1}\cdots \partial x_n^{{\beta }_n}}f,\left|\beta \right|={\beta }_1+\cdots +{\beta }_n.$$

$${\left|h\right|}_{k;\mathcal{O}}^E:=\underset{|\beta |\le k}{\max }\underset{x\in \mathcal{O}}{\sup }{\Vert D^{\beta }h(x)\Vert }_E,$$

$${\left[h\right]}_{k+\delta ;\mathcal{O}}^E:=\underset{|\beta |=k}{\max }\underset{x,y\in \mathcal{O},x\ne y}{\sup }\frac{{\Vert D^{\beta }h(x)-D^{\beta }h(y)\Vert }_E}{{\left|x-y\right|}^{\delta }},$$

$${\left|h\right|}_{k+\delta ;\mathcal{O}}^E:={\left|h\right|}_{k;\mathcal{O}}^E+{\left[h\right]}_{k+\delta ;\mathcal{O}}^E.$$

$${\left[u\right]}_{k+\delta ;Q}^E:=\underset{t\in I}{\sup }{\left[u(\cdot ,t)\right]}_{k+\delta ;\mathcal{O}}^E,$$

$${\left[u\right]}_{(k+\delta ,\delta /2m);Q}^E:=\underset{|\beta |=k}{\max }\underset{X,Y\in Q,X\ne Y}{\sup }\frac{{\Vert D^{\beta }u(X)-D^{\beta }u(Y)\Vert }_E}{{\left|X-Y\right|}_{\mathrm{p}}^{\delta }},$$

$${\left|u\right|}_{(k+\delta ,\delta /2m);Q}^E:={\left|u\right|}_{k;Q}^E+{\left[u\right]}_{(k+\delta ,\delta /2m);Q}^E.$$

$$\interleave\cdotp\interleave_{\ldots}\coloneqq\left|\cdotp\right|_{\ldots}^{L_{\omega}^{p}},\;\left\llbracket\cdotp\right\rrbracket_{\ldots}\coloneqq\left[\cdotp\right]_{\ldots}^{L_{\omega}^{p}}$$

$$u(x,T_1)-u(x,T_0)$$

$$B_r(x):=\{y\in {\mathbb{R}}^n:|y-x|<r\},Q_r(x,t):=B_r(x)\times (t-r^{2m},t]$$

$${\mathcal{Q}}_{r,T}(x):=B_r(x)\times (0,T],{\mathcal{Q}}_{r,T}:={\mathcal{Q}}_{r,T}(0),{\mathcal{Q}}_T:={\mathbb{R}}^n\times (0,T].$$

$$\text{interleave}u\text{interleave}{}_{(2m+\delta ,\delta /2m);{\mathcal{Q}}_T}\le C{e}^{CT}(\text{interleave}f{\text{interleave}}_{\delta ;{\mathcal{Q}}_{\mathcal{T}}}+\text{interleave}g{\text{interleave}}_{m+\delta ;{\mathcal{Q}}_T}).$$

$$\mathrm{d}u(x,t)=$$

$$\displaystyle+\sum_{k=1}^{\infty}\bigg{[}\sum_{|\alpha|=m}B_{\alpha}^{k}(t)D^{\alpha}u(x,t)+g^{k}(x,t)\bigg{]}\mathrm{d}w_{t}^{k}$$

$$L_{\omega }^p{L}_t^q{H}_x^m(Q):=L^p(\Omega ;L^q(I;H^m(\mathcal{O}))),$$

$$\Vert D^{\beta }u\Vert {}_{L_{\omega }^p{L}_t^{\infty }{L}_x^2}+\Vert D^{\beta }{D}^{m}u\Vert {}_{L_{\omega }^p{L}_t^2{L}_x^2}\le C(\Vert D^{\beta }f\Vert {}_{L_{\omega }^p{L}_t^2{H}_x^{-m}}+\Vert D^{\beta }g\Vert {}_{L_{\omega }^p{L}_t^2{L}_x^2})$$

$$\mathrm{d}\Vert u(\cdot ,t){\Vert }_{L_x^2}^2$$

$$\int_{\mathbb{R}^{n}}2u\Big{(}\sum_{|\alpha|=m}B_{\alpha}^{k}D^{\alpha}u\Big{)}\mathrm{d}x=0\quad\text{when }m\text{ is odd,}$$

$$\mathbb{E}\left[\bigg{(}\sup_{t\in[0,\tau]}\|u(t)\|_{L_{x}^{2}}^{2}+\int_{0}^{\tau}\|D^{m}u(t)\|_{L_{x}^{2}}^{2}\mathrm{d}t\bigg{)}^{\frac{p}{2}}\right]<+\infty.$$

$$\underset{t\in [0,\tau ]}{\sup }{\Vert u(t)\Vert }_{L_x^2}^2+{\int }_0^{\tau }{\Vert D^{m}u(t)\Vert }_{L_x^2}^2\mathrm{d}t$$

$$\mathbb{E}\underset{t\in [0,\tau ]}{\sup }\Vert u(t){\Vert }_{L_x^2}^p+\mathbb{E}{\left({\int }_0^{\tau }\Vert D^{m}u{\Vert }_{L_x^2}^2\mathrm{d}t\right)}^{\frac{p}{2}}\le C\mathbb{E}{\left|{\int }_0^{\tau }(\Vert f{\Vert }_{H_x^{-m}}^2+\Vert g{\Vert }_{L_x^2}^2)\mathrm{d}t\right|}^{\frac{p}{2}},$$

$$\mathrm{d}\Vert u(\cdot ,t){\Vert }_{L_x^2}^p$$

$$\displaystyle(p-2)\sum_{k}\bigg{[}\int_{\mathbb{R}^{n}}u\Big{(}\sum_{|\alpha|=m}B_{\alpha}^{k}D^{\alpha}u+g^{k}\Big{)}\mathrm{d}x\bigg{]}^{2}$$

$$\mathrm{d}\Vert u(\cdot ,t){\Vert }_{L_x^2}^p$$

$$\Vert u(s){\Vert }_{L_x^2}^p+\epsilon {\int }_0^s\Vert u{\Vert }_{L_x^2}^{p-2}\Vert D^{m}u{\Vert }_{L_x^2}^2\mathrm{d}t$$

$$\underset{t\in [0,T]}{\sup }E\Vert u(t\wedge \tau ){\Vert }_{L_x^2}^p+\mathbb{E}{\int }_0^{\tau }\Vert u{\Vert }_{L_x^2}^{p-2}\Vert D^{m}u{\Vert }_{L_x^2}^2\mathrm{d}t$$

Full text

1 Introduction

Let $(\varOmega,\mathcal{F},(\mathcal{F}_{t})_{t\geq 0},\mathbb{P})$ be a complete filtered probability space and $\{w_{\cdot}^{k}\}$ a sequence of independent standard Wiener processes adapted to the filtration $\mathcal{F}_{t}$. Consider the Cauchy problem for the following $2m$-order stochastic partial differential equations (SPDEs) of non-divergence form:

[TABLE]

where the coefficients, the free terms, and the unknown function are all random fields defined on $\mathbb{R}^{n}\times[0,\infty)\times\varOmega$ and adapted to $\mathcal{F}_{t}$. Typical examples of Equation (1.1) include the Zakai equation (see zakai1969optimal ; rozovskii1990stochastic for example), linearised stochastic Cahn–Hilliard equations (see da1996stochastic ; cardon2001cahn for example), and so on. General solvability theory for higher-order SPDEs of type (1.1) was first investigated in krylov1979stochastic under the framework of Hilbert spaces. This paper concerns the existence, uniqueness and regularity of solutions of (1.1) in some Hölder-type spaces that will be defined later. Regularity theory for linear equations often plays an important role in the study of nonlinear stochastic equations, see walsh1986introduction ; chow2014stochastic ; da2014stochastic and references therein.

The weak solution of Equation (1.1), which satisfies the equation in the (analytic) distribution sense, and its regularity in the framework of Sobolev spaces have been investigated by many researchers. Results for the second-order case (namely $m=1$) are numerous and fruitful; for instance, a complete $L^{p}$-theory $(p\geq 2)$ of second-order parabolic SPDEs has been developed, see for example pardoux1975equations ; krylov1977cauchy ; krylov1979stochastic ; rozovskii1990stochastic for $p=2$ and krylov1996l_p ; krylov1999analytic ; krylov2000spdes ; krylov200516 ; krylov2008brief for $p\geq 2$; degenerate equations were addressed in, for example, krylov1986characteristics ; and the Dirichlet problem were also extensively studied in many publications such as krylov1999sobolev ; kim2004p ; kim2004stochastic ; cioica2013lq ; kim2014weighted ; lindner2014singular ; kim2015bmo ; du2018w2 . For higher-order SPDEs, Krylov and Rozovskii krylov1979stochastic applied their abstract result to obtain the existence and uniqueness of solutions in the Sobolev space $W^{m}_{2}(\mathbb{R}^{n})$. Recently, van Neeven et al. van2012maximal and Portal and Veraar portal2019stochastic obtained some maximal $L^{p}$-regularity results for strong solutions of abstract stochastic parabolic time-dependent problems, which can also apply to higher-order SPDEs with proper conditions.

Another approach to the regularity problem of SPDEs is based on some Hölder spaces, corresponding to the celebrated Schauder theory for classical elliptic and parabolic PDEs (see gilbarg2015elliptic and references therein). This paper adopts this strategy to study Equation (1.1), stimulated by recent progress of the related research on second-order SPDEs. Actually, a $C^{2+\delta}$-theory for (1.1) with $m=1$ was once an open problem proposed by Krylov krylov1999analytic , which was partially addressed by Mikulevicius mikulevicius2000cauchy , and generally solved by Du and Liu du2019cauchy very recently. Introducing a Hölder-type space $\mathcal{C}_{p}^{\delta}$ containing all random fields $u$ satisfying

[TABLE]

with some constants $\delta\in(0,1)$ and $p\in[2,\infty)$, they proved that, under natural conditions on the coefficients, the solution $u$ and its derivatives $Du$ and $D^{2}u$ belong to $\mathcal{C}_{p}^{\delta}$, provided that $f$, $g$ and $Dg$ belong to the same space. In addition, Du and Liu du2019cauchy also obtained Hölder continuity in time of $D^{2}u$ with time irregular coefficients. A similar $C^{2+\delta}$-theory was also obtained recently for systems of second-order SPDEs in du2017stochastic .

This paper aims to prove a Schauder-type estimate for Equation (1.1) based on the space $\mathcal{C}_{p}^{\delta}$. To get more insight into such a kind of regularity of higher-order equations, let us recall some relevant work on deterministic PDEs. Boccia Boccia2013Schauder derived Schauder estimates for solutions of $2m$-order parabolic systems of non-divergence form in the classical $C_{x}^{2m+\delta}$-space provided that the free term $f$ (there are no terms like $g^{k}$ in deterministic equations) belongs to $C_{x}^{\delta}$, and for the divergence form Dong and Zhang dong2015schauder obtained $C_{x}^{m+\delta}$ regularity. Considering the feature of stochastic integral terms in SPDEs, a natural form of Schauder estimates for Equation (1.1) must be like this: the $\mathcal{C}_{p}^{\delta}$-norms of $u$ and its derivatives up to order $2m$ are dominated by the $\mathcal{C}_{p}^{\delta}$-norms of $f$ and $D^{\alpha}g$ with $|\alpha|\leq m$.

What surprises us during this work is not the above natural assertion but the structural condition that ensures the validity of this assertion. Let us give some explanation. It is well-known that the classical Schauder estimate for PDEs or PDE systems is based on certain coercivity conditions imposed on the leading coefficients and usually called strong ellipticity or strong parabolicity, and for second-order SPDEs either $L^{p}$-theory or $C^{2+\delta}$-theory requires a stochastic version of such conditions (see krylov1999analytic ; du2019cauchy for example). The solvability result of higher-order SPDEs in the space $W^{m}_{2}(\mathbb{R}^{n})$ obtained krylov1979stochastic relied on the following condition: there is a constant $\lambda>0$ such that for all $\xi_{\alpha}\in\mathbb{R}$,

[TABLE]

This is a natural condition as it can reduce to the standard ones for PDEs and for second-order SPDEs. However, things may change when one considers $L^{p}$-integrability ($p>2$) rather than square-integrability; more specifically, the coercivity condition (1.2) being adequate for $L^{2}$-theory seems not to be sufficient for $L^{p}$-integrability of solutions or their derivatives when $m\geq 2$. An indirect evidence is that, when the abstract maximal $L^{p}$-regularity results obtained in van2012maximal ; portal2019stochastic applied to higher-order SPDEs of type (1.1) the coefficients $B_{\alpha}$ with $|\alpha|=m$ were required to either be sufficiently small or have some additional analytic properties (see portal2019stochastic for details). Similar phenomena have been found also in complex valued SPDEs (see brzezniak2012stochastic ) and systems of second-order SPDEs (see kim2013note ; du2017stochastic ). This seems to be a unique feature of stochastic equations in contrast to deterministic PDEs.

A major contribution of this paper is the finding of a $p$-dependent coercivity condition that is just a small modification of (1.2) but perfectly works for the Schauder theory for Equation (1.1) based on $\mathcal{C}_{p}^{\delta}$. Let us state this condition as below: with some constants $\lambda>0$ and $p\geq 2$ it holds that

[TABLE]

Obviously, this condition is really $p$-dependent only when $m$ is even, and for odd $m$ it turns to be the same with (1.2). Though it might look strange at first glance, the following example demonstrates its sharpness to some extent.

Example 1

Given $\mu\in\mathbb{R}$, we consider the following equation on the torus $\mathbb{T}\coloneqq\mathbb{R}/(2\pi\mathbb{Z})$:

[TABLE]

with the initial condition

[TABLE]

If $\mu^{2}<2$, from Theorem 3.2.1 in krylov1979stochastic this equation admits a unique solution $u$ in $L^{2}(\Omega;C([0,T];H^{l}(\mathbb{T})))$ for any integer $l$. However, we have the following lemma.

Lemma 1

Let $m$ be even and $\mu^{2}<2$. If $p>1+2/\mu^{2}$, then $\mathbb{E}\left\|u(\cdot,t)\right\|_{L^{2}(\mathbb{T})}^{p}=+\infty$ for any $t>2/\varepsilon$ with $\varepsilon=(p-1)\mu^{2}-2$. Consequently, $\sup_{x\in\mathbb{T}}\mathbb{E}|u(x,t)|^{p}=+\infty$ for any $t>2/\varepsilon$.

The proof of Lemma 1 is presented in Section 6. This result indicates that the coefficient $p-1$ in the even case of the condition (1.3) couldn’t get any smaller if one wants to always ensure the finiteness of $\sup_{x}\mathbb{E}|u(x,t)|^{p}$, and this, of course, is a basic requirement in our theory.

Although our main result, Theorem 2.1 below, is stated (and also proved) only for linear equations of form (1.1), we point out that it is not difficult to extend it to the semilinear case where $f$ and $g$ depend on the unknown $u$ and are Lipschitz continuous with respect to all $D^{\alpha}u$ with $|\alpha|<2m$ and to all $D^{\beta}u$ with $|\beta|<m$, respectively. Besides, it is also interesting to ask if the coercivity condition (1.3) is sufficient or not to construct an $L^{p}$-theory for Equation (1.1).

Our approach to Schauder estimates, following the strategy used in du2019cauchy ; du2017stochastic , combines a perturbation scheme of Wang wang2006schauder with some integral-type estimates that were also used in trudinger1986new . The effect of the $p$-dependent condition (1.3) can be seen in the proof of the mixed norm estimates (Lemma 2); the latter leads to a local boundedness estimate that plays a key role in proving the fundamental interior Schauder estimate for the model equation (see (3.1) below).

This paper is organised as follows. In the next section we state our main theorem after introducing some notation and assumptions. Sections 3 and 4 are both devoted to the estimates for the model equation whose coefficients depend on $t$ and $\omega$ but not on $x$; we prove some auxiliary estimates in Section 3, and establish the interior Hölder estimate in Section 4. The proof of the main theorem is completed in Section 5. In the final section we prove Lemma 1.

2 Notation and main results

Before stating the main results, we introduce some notation and the working spaces. For a function $f$ of $x=(x_{1},\ldots,x_{n})\in\mathbb{R}^{n}$ and a multi-index $\beta=(\beta_{1},\ldots,\beta_{n})\in\mathbb{N}^{n}$, we define

[TABLE]

For $k\in\mathbb{N}=\{0,1,2,\dots\}$, $D^{k}f$ is regarded as the set of all $k$-order derivatives of $f$ and $\|D^{k}f\|_{E}={\sum_{|\alpha|=k}\|D^{\alpha}f\|_{E}}$ where $\|\cdot\|_{E}$ is the norm of a normed space $E$. All the derivatives of $E$-valued functions are defined with respect to the spatial variables in the strong sense as in phillips1957functional .

A Banach space-valued Hölder continuous function is a natural extension of the classical Hölder continuous function. Let $E$ be a Banach space, $\mathcal{O}$ be a domain in $\mathbb{R}^{n}$, $I\subset\mathbb{R}$ be an interval, and $Q\coloneqq\mathcal{O}\times I$. For a function $h:\,\mathcal{O}\rightarrow E$, we define

[TABLE]

with $k\in\mathbb{N}$ and $\alpha\in(0,1)$. For a function $u:\,Q\rightarrow E$, we define

[TABLE]

Moreover, we define the parabolic modulus $\left|X\right|_{\mathrm{p}}=\left|(x,t)\right|_{\mathrm{p}}\coloneqq\left|x\right|+\left|t\right|^{\frac{1}{2m}}$ and

[TABLE]

In this paper, $E$ is either i) $\mathbb{R}$, ii) $l^{2}$ or iii) $L_{\omega}^{p}\coloneqq L^{p}(\Omega)$. We omit the superscript in cases i) and ii), and in case iii) we denote

[TABLE]

for simplicity.

Definition 1

The Hölder-type spaces $C_{x}^{k+\delta}(Q;L_{\omega}^{p})$ and $C_{x,t}^{k+\delta,\delta/2m}(Q;L_{\omega}^{p})$ are defined as all predictable random fields $u$ defined on $Q\times\Omega$ and taking values in an Euclidean space or $l^{2}$ such that $u(\cdot,t)$ is an $L_{\omega}^{p}$-valued strongly continuous function for each $t$, and $\interleave u\interleave_{k+\delta;Q}$ and $\interleave u\interleave_{(k+\delta,\delta/2m);Q}$ are finite respectively.

Obviously, a function $u$ in $C_{x}^{k+\delta}(Q;L_{\omega}^{p})$ means that itself and its spatial derivatives up to order $k$ lie in the space $\mathcal{C}_{p}^{\delta}$ defined in the previous section.

In this paper we adopt a concept of quasi-classical solutions introduced in du2019cauchy .

Definition 2

A predictable random field u is called a *quasi-classical *solution of (1.1) if

(i) for each $t\in(0,\infty)$, $u(\cdot,t)$ is an $2m$ times strongly differentiable function from $\mathbb{R}^{n}$ to $L_{\omega}^{p}$ for some $p\geq 2$; and

(ii) for each $x\in\mathbb{R}^{n}$, the process $u(x,\cdot)$ is stochastically continuous and satisfies the integral equation

[TABLE]

almost surely (a.s.) for all $0\leq T_{0}<T_{1}<\infty$.

In particular, if $u(\cdot,t,\omega)\in C^{2m}(\mathbb{R}^{n})$ for each $(t,\omega)\in[0,\infty)\times\Omega$, then $u$ is a *classical *solution of (1.1).

Next we will introduce some notations for the domains:

[TABLE]

and simply write $B_{r}\coloneqq B_{r}(0),$ $Q_{r}\coloneqq Q_{r}(0,0)$. Also we denote

[TABLE]

Assumption 1

The following conditions hold throughout the paper unless otherwise stated:

The coercivity condition (1.3) is satisfied with some $\lambda>0$ and $p\geq 2$.

• 2)

The random fields $A_{\alpha\beta}$ and $f$ are real-valued, and $B_{\alpha}$ and $g$ are $l^{2}$-valued; all of them are predictable. The classical $C_{x}^{\delta}$-norms of $A_{\alpha\beta}(\cdot,t,\omega)$ and $C_{x}^{m+\delta}$-norms of $B_{\alpha}(\cdot,t,\omega)$ are all dominated by a constant $K>0$ uniformly in $(t,\omega)$.

• 3)

The free terms $f\in C_{x}^{\delta}(\mathcal{Q}_{T};L_{\omega}^{p})$ and $g\in C_{x}^{m+\delta}(\mathcal{Q}_{T};L_{\omega}^{p})$.

Now we are ready to state the main result in this paper which consists of the global Hölde estimate and the solvability.

Theorem 2.1

***Under Assumptions 1, there exists a unique quasi-classical solution $u\in C_{x,t}^{2m+\delta,\delta/2m}(\mathcal{Q}_{T};L_{\omega}^{p})$ to Equation (1.1) with the initial condition $u(\cdot,0)=0$. Moreover, there is a constant $C>0$ depending only on $n$, m, $\lambda$, p, $\delta$ and K such that

---

[TABLE]

In the proof of Theorem 2.1 the global Hölde estimate (2.2) is derived first, and then the existence and uniqueness of solutions of Equation (1.1) is obtained by the standard method of continuity.

We remark that the Cauchy problem with nonzero initial condition can be reduced into the case of zero initial condition by some simple calculation. Also, if $p$ is large enough one can obtain a modification of the solution that is Hölder continuous jointly in space and time by means of the Kolmogorov continuity theorem (see dalang2007hitting for example).

3 Auxillary estimates for the model equation

In Sections 3 and 4 we always assume that the coefficients $A_{\alpha\beta}$ and $B_{\alpha}$ with $|\alpha|=|\beta|=m$ are all bounded predictable processes (dominated by the constant $K$), independent of the spatial variable $x$, and satisfy the coercivity condition (1.3). Consider the following model equation

[TABLE]

with $(x,t)\in\mathbb{R}^{n}\times[-1,+\infty).$

Let $\mathcal{O\subset\mathbb{R}}^{n}$ and $H^{k}(\mathcal{O})=W^{k,2}(\mathcal{O})$ be the usual Sobolev spaces. Let $I\subset\mathbb{R}$ and $Q=\mathcal{O}\times I$. For $p,q\in[1,\infty]$, define

[TABLE]

and the domain $Q$ in the notation will be often omitted if there is no confusion.

Lemma 2

Let $\mathcal{Q}_{T}=\mathbb{R}^{n}\times[0,T]$, $p\geq 2$ and the integer $l\geq m$. Suppose $f\in L_{\omega}^{p}L_{t}^{2}H_{x}^{l-m}(\mathcal{Q}_{T})$ and $g\in L_{\omega}^{p}L_{t}^{2}H_{x}^{l}(\mathcal{Q}_{T})$. Then Equation (3.1) with zero initial value admits a unique weak solution $u\in L_{\omega}^{p}L_{t}^{\infty}H_{x}^{l}(\mathcal{Q}_{T})\cap L_{\omega}^{p}L_{t}^{2}H_{x}^{l+m}(\mathcal{Q}_{T})$. Moreover, for any multi-index $\beta$ such that $|\beta|\leq l$,

[TABLE]

where the constant $C$ depends only on $n$, $p$, $m$, $T$, $\lambda$, and $K$.

Proof

If $p=2$, the existence and uniqueness of the weak solution has already been obtained in krylov1979stochastic and rozovskii1990stochastic . So it remains to prove estimate (3.2) for general $p\geq 2.$ Since we can differentiate (3.1) with order $\beta$, it suffices to prove the estimate in the case $|\beta|=0$ .

By an Itô formula from (krylov1979stochastic, , Theorem 1.3.1), one can derive

[TABLE]

Note that in the last term one has

[TABLE]

but it is not true for even $m$.

Take a stopping time $\tau$ such that

[TABLE]

Let us consider two cases:

Case 1. $m$ is odd. Using the fact (3.4), and by Condition (1.3), the Sobolev–Gagliargo–Nirenberg inequality and Young’s inequality, we have

[TABLE]

Then computing $\mathbb{E}[\cdot]^{p/2}$ on both sides of the above inequality, and using the Burkholder–Davis–Gundy (BDG) inequality and Young’s inequality, one can obtain that

[TABLE]

where the constant $C$ depends only on $n$, $p$, $m$, $T$, and $\lambda$.

Case 2. $m$ is even. Applying Itô’s formula to $\|u(\cdot,t)\|_{L_{x}^{2}}^{p}$, one can derive

[TABLE]

With the help of Hölder inequality, one can obtain

[TABLE]

We choose $\epsilon>0$ so small that $(p-2)K\epsilon\leq\lambda/2$. Then combining with (1.3), (3) (3) and Sobolev-Gagliargo-Nirenberg inequality, we have

[TABLE]

For simplicity, we denote $\|D^{m}u\|_{L_{x}^{2}}=\sum_{|\alpha|=m}\|D^{\alpha}u\|_{L_{x}^{2}}$. Integrating with respect to $t$ on interval $[0,s]$ for any $s\in[0,T]$, we can obtain that

[TABLE]

where $\varepsilon=\varepsilon(m,\lambda)>0$. Choosing the stopping time $\tau$ as before and taking expectation on both sides of (3) and by Gronwall’s inequality, one can derive

[TABLE]

Then we can estimate $\mathbb{E}{\sup_{t\in[0,\tau]}\|u(t)\|_{L_{x}^{2}}^{p}}$ from (3) by the BDG inequality

[TABLE]

The last term of the above inequality is dominated by

[TABLE]

which along with (3) and (3) yields that

[TABLE]

Thus we obtain the estimate

[TABLE]

Next we need to estimate $\mathbb{E}(\int_{0}^{\tau}\|D^{m}u\|_{L_{x}^{2}}^{2}\mathrm{d}t)^{p/2}$. Back to (3) and integrating with respect to time, one can easily get that

[TABLE]

Computing $\mathbb{E}[\cdot]^{p/2}$ on both sides of the above inequality and by the Hölder’s inequality and BDG inequality, we derive that

[TABLE]

which along with (3.12) implies the estimate (3.5) in this case. Here the constant $C$ further depends on $K$.

Finally, we replace $\tau$ in (3.5) by the following sequence of stopping times

[TABLE]

and send $k$ to infinity. Then (3.5) yields the desired estimate for $l=0$ and the lemma is proved. ∎

Proposition 1

Let l be a positive integer, $l\geq m$, $p\geq 2,$ $r\in(0,1]$ and $\theta\in(0,1)$. Let $u\in L_{\omega}^{p}L_{t}^{2}H_{x}^{l+m}(Q_{r})$ solve (3.1) in $Q_{r}$ with free terms $f\in L_{\omega}^{p}L_{t}^{2}H_{x}^{l-m}(Q_{r})$ and $g\in L_{\omega}^{p}L_{t}^{2}H_{x}^{l}(Q_{r})$. Then there exists a constant $C$ depending only on $n$, $p$, $l$, $m$, $\lambda$, $K$, and $\theta$ such that

[TABLE]

Consequently, for $2(l-|\beta|)>n$,

[TABLE]

where the constant $C$ further depends on $|\beta|$.

Proof

By Sobolev’s embedding theorem, (3.14) can be derived directly from (3.13). Also we can reduce the problem for general $r>0$ to the case $r=1$ by rescaling. Indeed, for general $r>0$, we can apply the obtained estimates for $r=1$ to the rescaled function

[TABLE]

which solves the equation

[TABLE]

with free terms

[TABLE]

and obviously, $W^{k}$ are mutually independent Wiener processes. So it suffices to prove (3.13) for $r=1$. By induction, we shall only consider the case $l=m$.

For any $\theta\in(0,1)$, choose $m+1$ cut-off functions $\xi_{i}\in C_{0}^{\infty}(\mathbb{R}^{n+1})$ with $i=1,2,\cdots,m+1$, satisfying i) $0\leq\xi_{i}\leq 1,$ ii) $\xi_{i}=1$ in $Q_{\theta_{i}}$ and $\xi_{i}=0$ outside $Q_{\theta_{i+1}}$, where $\theta_{i}=\sqrt[i]{\theta}$. Let $v_{i}=\xi_{i}u$ which satisfy

[TABLE]

where

[TABLE]

where $C_{\gamma\eta}$ are the constants that can be derived from the Leibniz formula, depending only on $\gamma$, $\eta$, and $m$.

Applying Lemma 2 to (3) for $|\beta|=m+1-i$ with $i=1,2,\cdots,m+1,$ we have

[TABLE]

From the above inequalities, one can prove (3.13) for $l=m$. Higher-order estimates follow from induction. The proof is complete. ∎

Next we shall give an estimate for equation (3.1) with the Dirichlet boundary conditions

[TABLE]

Proposition 2

Let $f$ and $g$ be in $L_{\omega}^{p}L_{t}^{2}H_{x}^{l}(B_{r}\times(0,r^{2m}))$ for all $l\in\mathbb{N}$. Then the Dirichlet problem (3.1) with (3.16) admits a unique weak solution $u\in L_{\omega}^{2}L_{t}^{2}H_{x}^{m}(B_{r}\times(0,r^{2m}))$. For each $t\in(0,r^{2m})$, $u(\cdot,t)\in L^{p}(\Omega;C^{l}(B_{\varepsilon}))$ for all $l\geq 0$ and $\varepsilon\in(0,r)$. Moreover, there is a constant $C=C(n,p,\lambda,m,K)$ such that

[TABLE]

Proof

The existence and uniqueness of the weak solution of the Dirichlet problem (3.1) and (3.16) follow from (krylov1979stochastic, , Section 3.2). Then we choose a cut-off function $\varphi\in C_{0}^{\infty}(\mathbb{R}^{n})$ such that $\varphi(x)=1$ if $x\in B_{\varepsilon}$ and $\varphi(x)=0$ if $|x|>(r+\varepsilon)/2$ where $\varepsilon\in(0,r)$. Applying lemma 2 to $v\coloneqq\varphi D^{\alpha}u$ with Sobolev’s embedding theorem, the interior regularity can be obtained. We omit the proof of the estimate (2) because it’s analogous to the proof of (3.2) with the help of rescaling and Sobolev-Gagliargo-Nirenberg inequality. ∎

4 Interior Hölder estimates for the model equation

In this section we assume that $f\in C_{x}^{0}(\mathbb{R}^{n}\times[-1,\infty);L_{\omega}^{p})$ and $g\in C_{x}^{m}(\mathbb{R}^{n}\times[-1,\infty);L_{\omega}^{p})$, and $f(x,t)$ and $D^{m}g(x,t)$ are Dini continuous with respect to $x$ uniformly in $t$, namely, the modulus of continuity defined by

[TABLE]

satisfies that

[TABLE]

Theorem 4.1

Let u be a quasi-classical solution to (3.1) in $Q_{1}$. Under the above settings, there is a constant C depending only on n, $\lambda$, p, m and K, such that for any $X,\>Y\in Q_{1/16}$,

[TABLE]

where $\varrho=|X-Y|_{\textup{p}}$ and $M_{1}=\interleave u\interleave_{m-1;Q_{1}}+\interleave f\interleave_{0;Q_{1}}+\interleave g\interleave_{m;Q_{1}}$.

Proof

Firstly we mollify the functions $u$, $f$ and $g$ in the spatial variables. We choose a nonnegative and symmetric mollifier $\varphi:\,\mathbb{R}^{n}\rightarrow\mathbb{R}$ and define $\varphi^{\varepsilon}(x)=\varepsilon^{n}\varphi(x/\varepsilon)$, $u^{\varepsilon}=u\ast\varphi^{\varepsilon}$, $f^{\varepsilon}=f\ast\varphi^{\varepsilon}$ and $g^{\varepsilon}=g\ast\varphi^{\varepsilon}$. It is easy to check that $f^{\varepsilon}$ and $D^{m}g^{\varepsilon}$ are Dini continuous and have the same modulus of continuity $\varpi$ with $f$ and $D^{m}g$ and satisfy

[TABLE]

as $\varepsilon\rightarrow 0$. On the other hand, from Fubini’s theorem one can check that $u^{\varepsilon}$ satisfies the model equation (3.1) in the classical sense with free terms $f^{\varepsilon}$and $g^{\varepsilon}$. Therefore it suffices to prove the theorem for the mollified functions, and the general case is straightforward by passing the limits. The readers are referred to the appendix of du2019cauchy for more details. Then based on the smoothness of mollified functions, we can assume that $f$ and $g$ satisfy the following additional condition:

[TABLE]

From the definition of $\varpi$, one can see that for any $x,y\in\mathbb{R}^{n}$ and $t\in\mathbb{R}$,

[TABLE]

By translation we may suppose that $X=(0,0)$ and prove the theorem for any $Y\in Q_{1/8}$. Given $Y=(y,s)\in Q_{1/8},$ and $\tilde{\kappa}\in\mathbb{N}$ such that $\varrho\coloneqq|Y|_{\textup{p}}\in[\rho^{\tilde{\kappa}+2},\rho^{\tilde{\kappa}+1}).$ With $\rho=1/2$, we denote

[TABLE]

Let us introduce the following Dirichlet problems:

[TABLE]

where $\partial_{\textup{p}}Q^{\kappa}$ denotes the parabolic boundary of the cylinder $Q^{\kappa}$ for $\kappa=0,1,2,\dots$. Then the solvability and interior regularity of each $u^{\kappa}$ can be obtained by applying Proposition 2 to $u^{\kappa}-u$.

We have the following decomposition

[TABLE]

The next step is to estimate the three terms respectively. We split it into three lemmas.

Lemma 3

[TABLE]

Proof

Apply (3.14) to $u^{\kappa}-u^{\kappa+1}$ with $|\beta|=l,$ $r=\rho^{\kappa+1}$, $\theta=\frac{1}{2}$ to get

[TABLE]

In what follows, we define $\fint_{Q}=\frac{1}{|Q|}\int_{Q}$, where $|Q|$ is the Lebesgue measure of the set $Q\subset\mathbb{R}^{n+1}$.

On the other hand, from (2) one can obtain

[TABLE]

Combining the above we derive

[TABLE]

where $C$ is independent of $\kappa$. Choose $l=2m$, then

[TABLE]

which implies that $D^{2m}u^{\kappa}(0)$ converges in $L_{\omega}^{p}$ as $\kappa\rightarrow\infty$. Here [math] is the zero vector in $\mathbb{R}^{n+1}$. Next we shall prove that the limit is $D^{2m}u(0)$. It suffices to prove

[TABLE]

as $p\geq 2$. Applying (3.14) to $u^{\kappa}-u$ with $|\beta|=2m$, $l=n+2m$, $r=\rho^{\kappa}$, $\theta=1/2$ and $p=2$, we have

[TABLE]

where the last two terms tend to 0 as $\kappa\rightarrow\infty$. From (2) and (4.2) we have

[TABLE]

Therefore $D^{2m}u^{\kappa}(0)$ converges strongly to $D^{2m}u(0)$ in $L_{\omega}^{p}$. Moreover, we have

[TABLE]

where $C=C(n,m,\lambda,p,K)$. ∎

Lemma 4

[TABLE]

Proof

Define

[TABLE]

Then we decompose $K_{2}$ by

[TABLE]

As $D^{m+1}u^{0}$ satisfies the following homogeneous equation:

[TABLE]

in $Q_{3/4}.$ Using (3.14) to $D^{m+1}u^{0}$, one has

[TABLE]

Applying (3.13) to $u$, one can get

[TABLE]

Applying (3.13) and (2) to $u^{0}-u$ one can obtain

[TABLE]

Therefore,

[TABLE]

Hence, for $-8^{-2m}<s\leq t\leq 0$ and $x\in B_{1/8}$,

[TABLE]

Analogous to the above steps we can get

[TABLE]

Thus we get

[TABLE]

Note that $h^{\iota}$ satisfies

[TABLE]

in $Q^{\iota}$. By (4.4) we have

[TABLE]

Hence for $-\rho^{2m(\tilde{\kappa}+1)}\leq t\leq 0$ and $|x|\leq\rho^{\tilde{\kappa}+1}$,

[TABLE]

and

[TABLE]

Combining the last two estimates and (4.8), we can obtain

[TABLE]

The lemma is proved. ∎

Lemma 5

[TABLE]

Proof

We consider the following sequence of equations

[TABLE]

the equations associated with $\tilde{\kappa}$ and $\tilde{\kappa}+1$ are replaced by the following single equation

[TABLE]

As $|Y|_{\textup{p}}\in[\rho^{\tilde{\kappa}+2},\rho^{\tilde{\kappa}+1})$, it is easily seen that $Q^{\tilde{\kappa}+2}(Y)\subset Q^{\tilde{\kappa}}(0)$. So analogous to the proof of (4.6) we have

[TABLE]

where $C=C(n,m,\lambda,K,p)$. On the other hand, combining (3.14), (2) and (4.2), one can derive

[TABLE]

Then we have

[TABLE]

The lemma is proved. ∎

Now recalling (4.3) and combining Lemmas 3, 4 and 5, one has that

[TABLE]

The proof of Theorem 4.1 is complete. ∎

From the above theorem, one can easily derive the following interior Hölder estimate for (3.1), where we denote $\mathcal{Q}_{r,T}=B_{r}\times[0,T]$ for $r$, $T>0$.

Corollary 1

If u is a quasi-classical solution of (3.1) in $\mathbb{R}^{n}\times[0,\infty)$ with zero initial condition and $\delta\in(0,1)$. Then there is a positive constant C depending only on n, m, p, K, $\lambda$ and $\delta$, such that

[TABLE]

for any $T>0$, provided the right-hand side is finite.

Proof

Because of the zero initial condition, define $\tilde{u}(x,t)$, $\tilde{f}(x,t)$ and $\tilde{g}(x,t)$ to be zero whenever $t\in[-1,0)$, and be equal to $u(x,t)$, $f(x,t)$ and $g(x,t)$, respectively, whenever $t\geq 0$. Obviously, $\tilde{u}$ is a quasi-classical solution to (3.1) in $\mathbb{R}^{n}\times[-1,\infty)$. From (4.1) we have

[TABLE]

for any $X=\left(x,t\right)\in\mathbb{R}^{n}\times[0,\infty)$. Using the localization property of Hölder norms (see Lemma 4.1.1 in krylov1996l_p ), we obtain

[TABLE]

The proof is complete. ∎

5 Global Hölder estimates and the solvability

This section is devoted to the proof of Theorem 2.1. We need two technical lemmas; readers are referred to du2019cauchy for their proofs.

Lemma 6

Let $\varphi$ be a bounded nonnegative function from $[0,T]$ to $[0,\infty)$ satisfying

[TABLE]

for some nonnegative constants $\theta,\theta_{i}$ and $C_{i}$ (i=1,$\ldots$ ,k), where $\theta<1$. Then

[TABLE]

where $C$ depends only on $\theta_{1},\ldots\,,\theta_{k}$ and $\theta$.

Lemma 7

Let $B_{R}=\left\{x\in\mathbb{R}^{n}:|x|<R\right\}$ with $R>0$, $p\geq 1$, and $0\leq s<r$. There exists a positive constant $C$, depending only on $n$ and $p$, such that

[TABLE]

for any $u\in C^{r}(B_{R};L_{\omega}^{p})$ and $\varepsilon\in(0,R)$.

Now we are in a position to complete the proof of Theorem 2.1.

Proof (Proof of Theorem 2.1)

The proof is divided into two steps.

Step 1. Global Hölder estimate (2.2).

Suppose $u$ is the quasi-classical solution to (1.1) with zero initial condition. Let $\rho/2\leq r<R\leq\rho$ with $\rho\in(0,1/8)$ to be determined. Choose a nonnegative function $\zeta\in C_{0}^{\infty}(\mathbb{R}^{n})$ such that $\zeta(x)=1$ on $B_{r}$, $\zeta(x)=0$ outside $B_{R}$, and for $\delta>0$,

[TABLE]

Set $v=\zeta u$, and $A_{\alpha\beta,0}(t)=A_{\alpha\beta}(0,t)$, $B_{\alpha,0}^{k}(t)=B_{\alpha}^{k}(0,t)$. Then $v$ satisfies

[TABLE]

where

[TABLE]

We denote $\mathcal{Q}_{R,\tau}=B_{R}\times(0,\tau)$ for $\tau>0$ and define

[TABLE]

Then following from Lemma 7, we directly derive

[TABLE]

In the above two inequalities, $C_{1}=C_{1}(n,p,K,\varepsilon,\rho)$. Applying Corollary 1 and taking positive $\rho$, $\varepsilon$ so small that

[TABLE]

where $C$ is the constant in the corollary, then we get that

[TABLE]

Then by Lemma 6, we obtain

[TABLE]

Note that the above inequality is true for any point $x\in\mathbb{R}^{n}$ instead of [math]. Therefore, applying Lemma 7, we have

[TABLE]

The next step is to estimate $M_{\rho/2}^{\tau}(u)$. Applying Itô’s formula to $|u|^{p}$ and integrating in $\mathcal{Q}_{\rho/2,\tau}(x)\times\Omega$ with the use of Sobolev-Gagliargo-Nirenberg inequality, we get

[TABLE]

Taking $\tau=2\left(CC_{2}\right)^{-1},$ the above two inequalities yield

[TABLE]

Following from the localization property of Hölder norms, we get

[TABLE]

with $C_{\tau}=C_{\tau}(n,m,\delta,\lambda,K,p)\geq 1.$

Finally, we conclude the proof by induction. Assume that there is a constant $C_{S}\geq 1$ for some $S>0$ such that

[TABLE]

Then applying (5.1) to $v(x,t)\coloneqq u(x,t+S)-u(x,S)$ for $t\geq 0$, we can derive that

[TABLE]

where $\tilde{C}=\tilde{C}(m,K)\geq 1$. Thus we get

[TABLE]

which means $C_{S+\tau}\leq 3\tilde{C}C_{\tau}C_{S}$. As $\tau$ is fixed, by iteration we have $C_{S}\leq C\textup{e}^{CS}$ where $C=C(n,m,\delta,\lambda,p,K)$. This completes the proof of (2.2).

Step 2. The solvability.

For simplicity, we denote

[TABLE]

Define

[TABLE]

where $s\in[0,1]$ and $\Delta^{2m}\coloneqq\sum_{\left|\gamma\right|=2m}\delta_{\gamma}D^{\gamma}$ where

[TABLE]

Then consider the equation

[TABLE]

where $\mathcal{Q}\coloneqq\mathbb{R}^{n}\times[0,\infty)$. Evidently, the solutions of the above equations enjoy the estimate (2.2) with the same dominating constant $C$ (independent of $s$). So by the standard method of continuity (see (gilbarg2015elliptic, , Theorem 5.2)), it suffices to show the solvability of the following equation (the case $s=0$):

[TABLE]

Letting \text{\varphi$:$\mathbb{R}}^{n}\rightarrow\mathbb{R} be a nonnegative and symmetric mollifier (see Appendix in du2019cauchy ) and $\varphi^{\varepsilon}(x)=\varepsilon^{n}\varphi(x/\varepsilon)$, we define $f^{\varepsilon}=\varphi^{\varepsilon}\ast f$ and $g^{\varepsilon}=\varphi^{\varepsilon}\ast g$. From the results of Appendix in du2019cauchy , we obtain that $f^{\varepsilon}\in C_{x}^{\delta}(\mathcal{Q};L_{\omega}^{p})$ and $g^{\varepsilon}\in C_{x}^{m+\delta}(\mathcal{Q};L_{\omega}^{p})$ satisfying

[TABLE]

Moreover, $f^{\varepsilon}(x,t,\omega)$ and $g^{\varepsilon}(x,t,\omega)$ are smooth in $x$ for any $(t,\omega)$, and $f^{\varepsilon}$, $g^{\varepsilon}\in C^{k}(\mathcal{Q}_{T};L_{\omega}^{p})$ for all $k\in\mathbb{N}$. For any $k\in\mathbb{N}$ and $2r>n$, we have

[TABLE]

Considering the weighted Sobolev spaces and analogously to proving Lemma 2 but only with minor changes, we derive that (5.3) with free terms $f^{\varepsilon}$ and $g^{\varepsilon}$ admits a unique weak solution $u^{\varepsilon}$ satisfying

[TABLE]

for any large $r$. Following from Sobolev’s embedding theorem, $u^{\varepsilon}$ is smooth in $x$ and moreover,

[TABLE]

Then we have $\mathbb{E}\left|D^{k}u^{\varepsilon}(x,t)\right|^{p}<\infty$ for each $(x,t)\in\mathcal{Q}_{T}$ and $k\in\mathbb{N}.$ From global estimate (2.2) with $\delta/2$ instead of $\delta$ and (5.4), we obtain

[TABLE]

as $\varepsilon,\varepsilon^{\prime}\rightarrow 0$. Hence, $u^{\varepsilon}$ converges to a function $u\in C_{x,t}^{2m,0}(\mathcal{Q}_{T};L_{\omega}^{p})$ which is apparently a quasi-classical solution to (5.3). Then we can derive the uniqueness and regularity from the estimate (2.2). The solvability is proved.

To sum up, the proof of Theorem 2.1 is complete. ∎

6 Proof of Lemma 1

Recall Equation (1.4):

[TABLE]

with the initial condition with the initial condition

[TABLE]

Since $\mu^{2}<2$, this equation admits a unique solution $u\in L^{2}(\Omega;C([0,T];H^{l}(\mathbb{T})))$ for any integer $l$ (cf. krylov1979stochastic ). We shall prove that if $p>1+2/\mu^{2}$, then $\mathbb{E}\|u(\cdot,t)\|_{L^{2}(\mathbb{T})}^{p}=+\infty$ for any $t>2/\varepsilon$, where $\varepsilon=(p-1)\mu^{2}-2>0$.

Since $u\in L^{2}(\Omega;C([0,T];H^{l}(\mathbb{T})))$ for any integer $l$, one can express $u$ in the Fourier series

[TABLE]

where $u_{n}(t)$ satisfies

[TABLE]

Then we obtain that

[TABLE]

Set $f(t)\coloneqq 2+2t+(-1)^{m}\mu^{2}t$, then

[TABLE]

Using the condition that $m$ is even, one can obtain

[TABLE]

By Parseval’s identity,

[TABLE]

Therefore, we have

[TABLE]

Noticing the fact that $\mu(-1)^{m/2}y$ is positive on one of the intervals $(-\infty,0)$ and $(0,+\infty)$, one has

[TABLE]

Since

[TABLE]

one can further derive that

[TABLE]

Obviously, the last term is infinite if

[TABLE]

which is satisfied when $t>2/\varepsilon$, where $\varepsilon=(p-1)\mu^{2}-2>0$. The lemma is proved.

Remark 1

When $m$ is odd, it follows from (6.1) that

[TABLE]

where $f(t)=2+(2-\mu^{2})t$. Furthermore, one can obtain

[TABLE]

which means that the condition $\mu^{2}<2$ is sufficient to ensure $\mathbb{E}\left\|u(\cdot,t)\right\|_{L^{2}(\mathbb{T})}^{p}<+\infty$ for any $p\geq 2$.