Gradient walk and $p$-harmonic functions

TL;DR

This paper introduces stochastic processes that approximate $p$-harmonic functions, providing explicit convergence rates and a diffusion representation, addressing challenges related to the zero gradient set.

Contribution

It develops new stochastic approximation methods for $p$-harmonic functions with proven uniform convergence and explicit rates, including a continuous-time diffusion representation.

Findings

Stochastic processes converge uniformly to $p$-harmonic functions

Explicit convergence rates are established

A diffusion representation in continuous time is derived

Abstract

We consider a class of stochastic processes and establish its connection to $p$-harmonic functions. In particular, we obtain stochastic approximations that converge uniformly to a $p$-harmonic function, with an explicit convergence rate, and also obtain a precise diffusion representation in continuous time. The main difficulty is how to deal with the zero set of the gradient of the underlying function.