## Local Times of the Solution to Stochastic Heat Equation with Fractional Noise

Wang Zhi,1, Yan Litan,2, Yu Xianye,3

 基金资助: 国家自然科学基金.  11701304国家自然科学基金.  11571071国家自然科学基金.  11701589浙江省自然科学基金.  LQ16A010006中央高校基本科研业务费.  171gpy17

 Fund supported: the NSFC.  11701304the NSFC.  11571071the NSFC.  11701589the Zhejiang Provincial Natural Science Foundation.  LQ16A010006the Fundamental Research Funds for the Central Universities.  171gpy17

Abstract

In this paper, we study the collision and intersection local times of the solution to stochastic heat equation with additive fractional noise. We mainly prove its existence and smoothness properties through local nondeterminism and chaos expansion.

Keywords： Stochastic heat equation ; Fractional noise ; Collision local time ; Intersection local time ; Chaos expansion

Wang Zhi, Yan Litan, Yu Xianye. Local Times of the Solution to Stochastic Heat Equation with Fractional Noise. Acta Mathematica Scientia[J], 2019, 39(3): 582-595 doi:

## 1 引言

$$$\label{1.5} \left\{\begin{array}{ll} \frac{\partial u}{\partial t}=\frac{1}{2}\Delta u+\dot{W}^{H}, \; &t\in[0, T], x\in\mathbb{R} ^{d}, \\u_{0, x}=0, &x\in\mathbb{R} ^{d}, \end{array}\right.$$$

$$$\label{1.4} E(W^{H}(t, x)W^{H}(s, y))=R_{H}(s, t)(x\wedge y),$$$

$H=\frac 12$时,噪声$(\dot{W}^{H}(t, x)_{t\in[0, T], x\in\mathbb{R} ^{d}})$是经典的时空白噪声.众所周知,下面的随机热方程

$$$\label{1.2}\left\{\begin{array}{ll} \frac{\partial u}{\partial t}=\frac{1}{2}\Delta u+\dot{W}, \; &t\in[0, T], x\in\mathbb{R} ^{d}, \\ u_{0, x}=0, &x\in\mathbb{R} ^{d} \end{array}\right.$$$

$$$\label{1.3} G(t, x)=\left\{\begin{array}{ll} (2\pi t)^{-d/2}\exp(-\frac{|x|^{2}}{2t}), \; &\mbox{if t>0, x\in\mathbb{R} ^{d}}, \\ 0, &\mbox{if t\leq0, x\in\mathbb{R} ^{d}}. \end{array}\right.$$$

$$$\label{1.6} U^{H}(t, x)=\int^{t}_{0}\int_{\mathbb{R} ^{d}}G(t-s, x-y)W^{H}({\rm d}s, {\rm d}y),$$$

$$$\label{02} F=\sum \limits_{n = 0}^\infty F_{n}$$$

$$$\label{3.1} \ell_{t}=\int^{t}_{0}\delta(F^{H_{1}}_{s}-F^{H_{2}}_{s}){\rm d}s,$$$

$$$\label{3.2} p_{\varepsilon}(x)=\frac{1}{\sqrt{2\pi\varepsilon}}{\rm e}^{-\frac{x^{2}}{2\varepsilon}} \equiv\frac{1}{2\pi}\int_{\mathbb{R} }{\rm e}^{{\rm i}x\xi}{\rm e}^{-\frac{1}{2}\varepsilon\xi^{2}}{\rm d}\xi,$$$

$\begin{eqnarray}\label{3.3} \ell_{\varepsilon, t}=\int^{t}_{0}p_{\varepsilon}(F^{H_{1}}_{s}-F^{H_{2}}_{s}){\rm d}s =\frac{1}{2\pi}\int^{t}_{0}\int_{\mathbb{R} }{\rm e}^{{\rm i}\xi(F^{H_{1}}_{s}-F^{H_{2}}_{s})}{\rm e}^{-\frac{1}{2}\varepsilon\xi^{2}}{\rm d}\xi {\rm d}s. \end{eqnarray}$

首先,证明对任意的$\varepsilon>0$,有$\ell_{\varepsilon, t}\in L^{2}(\Omega, {\cal F}, P)$.根据(3.3)式,有

$\begin{eqnarray}\label{3.4} E(\ell^{2}_{\varepsilon, t})&=&\frac{1}{4\pi^{2}}\int^{t}_{0}\int^{t}_{0}\int_{\mathbb{R} ^{2}}E{\rm e}^{{\rm i}\xi(F^{H_{1}}_{s}-F^{H_{2}}_{s})+{\rm i}\eta(F^{H_{1}}_{r}-F^{H_{2}}_{r})} {\rm e}^{-\frac{\varepsilon(\xi^{2}+\eta^{2})}{2}}{\rm d}\xi {\rm d}\eta {\rm d}r{\rm d}s\\ &=&\frac{1}{4\pi^{2}}\int^{t}_{0}\int^{t}_{0}\int_{\mathbb{R} ^{2}}{\rm e}^{-\frac{\sigma^{2}}{2}}{\rm e}^{-\frac{\varepsilon(\xi^{2}+\eta^{2})}{2}}{\rm d}\xi {\rm d}\eta {\rm d}r{\rm d}s, \end{eqnarray}$

$\begin{eqnarray}\label{3.5} \sigma^{2}&=&Var(\xi(F^{H_{1}}_{s}-F^{H_{2}}_{s})+\eta(F^{H_{1}}_{r}-F^{H_{2}}_{r}))\\ &=&Var(\xi(F^{H_{1}}_{s}-F^{H_{1}}_{r})-\xi(F^{H_{2}}_{s}-F^{H_{2}}_{r})+(\xi+\eta)(F^{H_{1}}_{r}-F^{H_{2}}_{r}))\\&\geq&k[\xi^{2}(|s-r|^{2H_{1}-\frac{d}{2}}+|s-r|^{2H_{2}-\frac{d}{2}})+(\xi+\eta)^{2}(r^{2H_{1}-\frac{d}{2}}+r^{2H_{2}-\frac{d}{2}})], \end{eqnarray}$

$$$\label{3.6} E|F^{H}_{t}-F^{H}_{s}|^{2}\asymp|t-s|^{2H-\frac{d}{2}}.$$$

## 4 碰撞局部时的光滑性

$H_{n}(x)$, $x\in \mathbb{R} $$n阶Hermite多项式,即 $$\label{4.1} H_{n}(x)=(-1)^{n}\frac{1}{n!}{\rm e}^{\frac{x^{2}}{2}}\frac{\partial^{n}}{\partial x^{n}}{\rm e}^{-\frac{x^{2}}{2}}.$$ 于是 $$\label{4.2} {\rm e}^{tx-\frac{t^{2}}{2}}=\sum\limits_{n = 0}^\infty t^{n}H_{n}(x),$$ 其中t>0$$x\in \mathbb{R}$.这意味着

$$$\label{4.3} \exp({\rm i}u\xi(F_{s}^{H_{1}}-F_{s}^{H_{2}})+\frac{1}{2}u^{2}\xi^{2}Var(F_{s}^{H_{1}}-F_{s}^{H_{2}}))=\sum\limits_{n=0}^{\infty}({\rm i}u)^{n}\sigma^{n}(s, \xi) H_{n}\left(\frac{\xi(F_{s}^{H_{1}}-F_{s}^{H_{2}})}{\sigma(s, \xi)}\right),$$$

$\exp({\rm i}u\xi(F_{s}^{H_{1}}-F_{s}^{H_{2}})+\frac{1}{2}u^{2}\xi^{2}Var(F_{s}^{H_{1}}-F_{s}^{H_{2}}))$$n阶混沌,其中s\in [0, T].下面我们给出命题4.1,这个命题是为了帮助证明定理4.1的. 命题 4.1 假设\frac12<H_i<1$$d<4H_i$,其中$i=1, 2$, $\lambda_{s}, \lambda_{r}$$\rho_{r, s}的定义如上.则\ell_{t}\in{\cal U}当且仅当 $$\label{4.4} \int^{t}_{0}\int^{t}_{0}\frac{\rho_{r, s}^{2}}{(\lambda_{s}\lambda_{r}-\rho_{r, s}^{2})^{\frac{3}{2}}}{\rm d}r{\rm d}s<\infty,$$ 其中t\in [0, T]. 对任意的\varepsilon>0$$t\in[0, T]$,有

$\begin{eqnarray}\label{4.6} Var(\xi F^{H}_{s}+\eta F^{H}_{r})&=&Var(\xi(F^{H}_{s}-F^{H}_{r})+(\xi+\eta)F^{H}_{r})\\ &\geq&k(\xi^{2}(s-r)^{2H-\frac{d}{2}}+(\xi+\eta)^{2}r^{2H-\frac{d}{2}}), \end{eqnarray}$

$$$\label{4.7} (\lambda_{s}-k(s-r)^{2H-\frac{d}{2}}-kr^{2H-\frac{d}{2}})\xi^{2}+2(\rho_{r, s}-kr^{2H-\frac{d}{2}})\xi\eta+(\lambda_{r}-kr^{2H-\frac{d}{2}})\eta^{2}\geq0.$$$

(4.7)式的判别式满足

## 5 碰撞局部时的正则性

对于$0\leq s\leq t\leq T$,根据(3.1)和(3.3)式,有

根据命题6.1,只需证明当$H<\frac{8+3d}{12}$时, (6.3)式是成立的.因此,只要证明对任意的$t\in[0, T]$$j=1, 2, 3时,下式成立 $$\label{6.5} \int_{{\cal T}_{j}}\frac{\mu^{2}}{(\lambda_{r, s}\lambda_{r', s'}-\mu^{2})^{\frac{3}{2}}}{\rm d}r{\rm d}s{\rm d}r'{\rm d}s'<\infty,$$ 其中 (r, s, r^{\prime}, s^{\prime})\in{\cal T}_{1}$${\cal T}_{2}$时,运用局部非确定性可得

$\begin{eqnarray}\label{6.6} &&Var(\xi(F^{H}_{s}-\tilde{F}^{H}_{r})+\eta(F^{H}_{s'}-\tilde{F}^{H}_{r'}))\\ &=&Var(\xi(F^{H}_{s}-F^{H}_{s'})-\xi(\tilde{F}^{H}_{r}-\tilde{F}^{H}_{r'})+(\xi+\eta)(F^{H}_{s'}-\tilde{F}^{H}_{r'}))\\ &\geq &k[\xi^{2}((s-s')^{2H-\frac{d}{2}}+(r-r')^{2H-\frac{d}{2}})+(\xi+\eta)^{2}(s'^{2H-\frac{d}{2}}+r'^{2H-\frac{d}{2}})], \end{eqnarray}$

$$$\label{6.7} Var(\xi(F^{H}_{s}-\tilde{F}^{H}_{r})+\eta(F^{H}_{s'}-\tilde{F}^{H}_{r'})) =\lambda_{r, s}\xi^{2}+2\mu\xi\eta+\lambda_{r', s'}\eta^{2}.$$$

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