随机场初始数据的Heat Equation的随机Cauchy初值公式：光滑、Harnack型界和p-矩
Stochastic Cauchy Initial Value Formulation Of The Heat Equation For Random Field Initial Data: Smoothing, Harnack-Type Bounds And p-Moments
摘要：
研究了区域$ \mathbf{Q}\subset{\mathbf{R}}^{n}$上抛物型热方程在随机场上的随机Cauchy初值问题. \begin{align} &{\square}\widehat{u(x,t)} \equiv \bigg(\frac{\partial}{\partial t}-Δ_{x}\bigg)\widehat{u(x,t)}=0,~x\in\mathbf{Q},t> 0 \end{align} \begin{align} \widehat{u(x,0)}=ϕ(x)+\mathscr{J}(x),~x\in\mathbf{Q},t=0 \end{align}是具有期望$\mathbb{E}[\![\mathscr{J}(x)]\!]=0 $和具有调节协方差$\mathbb{E}[\![ \mathscr{J}(x)\otimes\mathscr{J}(y)]\!]=ζJ(x,y;\ell)$、相关长度$\ell$和$\mathbb{E}[\![ \mathscr{J}(x)\otimes\mathscr{J}(x)]\!]=ζ<\infty$的经典高斯随机标量场。随机扰动解$\widehat{u(x,t)}$是一个随机卷积积分。这导致随机扩展和版本的一些经典结果的热方程，特别是一个李-丘微分Harnack不等式\begin{align} \mathbb{E}\left[\!\!\left[\frac{|\nabla\widehat{u(x,t)}|^{2}}{|\widehat{u(x,t)}|^{2}} \right]\!\!\right]-\mathbb{E}\left[\!\!\left[\frac{\tfrac{\partial}{\partial t}\widehat{u(x,t)}}{\widehat{u(x,t)}} \right]\!\!\right]\le \frac{1}{2}n\frac{1}{t} \end{align}和抛物Harnack不等式。推导出波动率$\mathbb{E}[\![|\widehat{u(x,t)}|^{2}]\!]$和p矩$\mathbb{E}[\![|\widehat{u(x,t)}|^{p}]\!] $的衰减估计和界限。由于$\lim_{t\uparrow \infty}\mathbb{E}[\![|\widehat{u(x,t)}|^{p}]\!]=0 $，随机扰动解的柯西演化是稳定的，因为热方程平滑或消散了由初始数据随机性引起的波动，如$t\rightarrow\infty$。
英文摘要：
The following stochastic Cauchy initial-value problem is studied for the parabolic heat equation on a domain $ \mathbf{Q}\subset{\mathbf{R}}^{n}$ with random field initial data. \begin{align} &{\square}\widehat{u(x,t)} \equiv \bigg(\frac{\partial}{\partial t}-Δ_{x}\bigg)\widehat{u(x,t)}=0,~x\in\mathbf{Q},t> 0 \end{align} \begin{align} \widehat{u(x,0)}=ϕ(x)+\mathscr{J}(x),~x\in\mathbf{Q},t=0 \end{align} where $ϕ(x)\in C^{\infty}({\mathbf{Q}})$, and $\mathscr{J}(x)$ is a classical Gaussian random scalar field with expectation $\mathbb{E}[\![\mathscr{J}(x)]\!]=0 $ and with a regulated covariance $\mathbb{E}[\![ \mathscr{J}(x)\otimes\mathscr{J}(y)]\!]=ζJ(x,y;\ell)$, correlation length $\ell$ and $\mathbb{E}[\![ \mathscr{J}(x)\otimes\mathscr{J}(x)]\!]=ζ<\infty$. The randomly perturbed solution $\widehat{u(x,t)}$ is a stochastic convolution integral. This leads to stochastic extensions and versions of some classical results for the heat equation; in particular, a Li-Yau differential Harnack inequality \begin{align} \mathbb{E}\left[\!\!\left[\frac{|\nabla\widehat{u(x,t)}|^{2}}{|\widehat{u(x,t)}|^{2}} \right]\!\!\right]-\mathbb{E}\left[\!\!\left[\frac{\tfrac{\partial}{\partial t}\widehat{u(x,t)}}{\widehat{u(x,t)}} \right]\!\!\right]\le \frac{1}{2}n\frac{1}{t} \end{align} and a parabolic Harnack inequality. Decay estimates and bounds for the volatility $\mathbb{E}[\![|\widehat{u(x,t)}|^{2}]\!]$ and p-moments $\mathbb{E}[\![|\widehat{u(x,t)}|^{p}]\!] $ are derived. Since $\lim_{t\uparrow \infty}\mathbb{E}[\![|\widehat{u(x,t)}|^{p}]\!]=0 $, the Cauchy evolution of the randomly perturbed solution is stable since the heat equation smooths out or dissipates volatility induced by initial data randomness as $t\rightarrow\infty$.
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