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

Abstract: 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 $\phi(x)\in C^{\infty}({\mathbf{Q}})$, and $\mathscr{J}(x)$ is a classical Gaussian random scalar field with expectation $\mathbb{E}[![\mathscr{J}(x)]!]=0 $ and with a regulated covariance $\mathbb{E}[![ \mathscr{J}(x)\otimes\mathscr{J}(y)]!]=\zeta J(x,y;\ell)$, correlation length $\ell$ 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 $\mathbb{E}[![|\widehat{u(x,t)}|^{2}]!]$ 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 $t\rightarrow\infty$.