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

Abstract

This work is concerned with the probabilistic representation of solutions to the $p$-Laplace evolution equation $\frac{\partial u}{\partial t}=\mathrm{div}(|\nabla u{|}^{p-2}\nabla u)$ in $(0,\infty )\times {\mathbb{R}}^d$, $u(0,x)=u_0(x),$ $x\in {\mathbb{R}}^d$. One proves that, if $p\ge 4$, 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 $u(t,x)dx={\mathcal{L}}_{X(t)}(dx)$, where ${\mathcal{L}}_{X(t)}$ 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.