Probabilistic representation of solutions to the parabolic $p$-Laplace equation

TL;DR

This paper establishes a probabilistic representation for solutions to the parabolic p-Laplace equation using McKean-Vlasov SDEs, under specific conditions on initial data and for p ≥ 4.

Contribution

It introduces a novel second order regularity result for weak solutions and links them to McKean-Vlasov SDEs for p ≥ 4.

Findings

Solutions can be represented as laws of stochastic processes for p ≥ 4.

New regularity results enable probabilistic representation of solutions.

Representation holds for initial data with specific regularity and support conditions.

Abstract

This work is concerned with the probabilistic representation of solutions to the $p$-Laplace evolution equation $\frac{\partial u}{\partial t}={\rm div}(|\nabla u|^{p-2}\nabla u)$ in $(0,\infty)\times\mathbb{R}^d$, $u(0,x)=u_0(x),$ $x\in\mathbb{R}^d$. One proves that, if $p\geq 4$, and if $u_0$ is a probability density with compact support and $u_0\in L^2$, $|\nabla u_0|\in L^\infty$, then $u$ can be represented as $u(t,x)dx=\mathscr L_{X(t)}(dx)$, where $\mathscr L_{X(t)}$ denotes the time marginal law of $X$ at time $t$ with $X$ being a probabilistically weak solution to a corresponding McKean-Vlasov stochastic differential equation. This result is based on a new second order global regularity result for the weak solutions to the parabolic $p$-Laplace equation.