Stochastic Cauchy Initial Value Formulation Of The Heat Equation For Random Field Initial Data: Smoothing, Harnack-Type Bounds And p-Moments

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}-{\Delta}{x}\bigg)\widehat{u(x,t)}=0,~x\in$\mathbf{Q}$,t> 0 \end{align} \begin{align} \widehat{u(x,0)}=\phi(x)+\mathscr{J}(x),~x\in$\mathbf{Q}$,t=0 \end{align} where $\varphi (x)\in C^{\mathrm{\infty }}(\mathbf{Q})$, and $\mathcal{J}(x)$ is a classical Gaussian random scalar field with expectation $\mathbb{E}[\text{\hspace{0.17em}⁣}[\mathcal{J}(x)]\text{\hspace{0.17em}⁣}]=0$ and with a regulated covariance $\mathbb{E}[\text{\hspace{0.17em}⁣}[\mathcal{J}(x)\otimes \mathcal{J}(y)]\text{\hspace{0.17em}⁣}]=\zeta J(x,y;\mathrm{\ell })$, correlation length $\mathrm{\ell }$ and $$\mathbb{E}$[![ \mathscr{J}(x)\otimes\mathscr{J}(x)]!]=\zeta<\infty$. 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 $\mathbb{E}[\text{\hspace{0.17em}⁣}[\mathrm{\mid }\widehat{u(x,t)}{\mathrm{\mid }}^2]\text{\hspace{0.17em}⁣}]$ and p-moments $\mathbb{E}[\text{\hspace{0.17em}⁣}[\mathrm{\mid }\widehat{u(x,t)}{\mathrm{\mid }}^p]\text{\hspace{0.17em}⁣}]$ 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 $t\to \mathrm{\infty }$.