$p$-Brownian motion and the $p$-Laplacian

TL;DR

This paper introduces a nonlinear Markov process associated with the parabolic p-Laplace equation, extending the classical connection between Brownian motion and the heat equation to a nonlinear setting.

Contribution

It constructs a new stochastic process linked to the p-Laplace equation, providing a probabilistic interpretation similar to Brownian motion for the classical heat equation.

Findings

Established a nonlinear Markov process related to the p-Laplace equation

Extended the classical stochastic process- PDE relationship to nonlinear PDEs

Provided a foundation for further probabilistic analysis of nonlinear PDEs

Abstract

In this paper we construct a stochastic process, more precisely, a (nonlinear) Markov process, which is related to the parabolic $p$-Laplace equation in the same way as Brownian motion is to the classical heat equation given by the (2-) Laplacian.