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

TL;DR

This paper develops a stochastic formulation of the heat equation with random initial data, deriving bounds, inequalities, and stability results for the solution's moments and regularity properties.

Contribution

It introduces a stochastic convolution approach to extend classical heat equation results to random initial conditions with Gaussian fields, including Harnack inequalities and decay estimates.

Findings

Derived a stochastic Li-Yau differential Harnack inequality.

Established decay estimates for solution volatility over time.

Proved stability of the solution as randomness dissipates with time.

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 $t\rightarrow\infty$.