A probabilistic representation for heat flow of harmonic map on manifolds with time-dependent Riemannian metric

TL;DR

This paper introduces a probabilistic approach using stochastic differential equations to analyze heat flow of harmonic maps on manifolds with evolving metrics, offering a new proof of solution existence.

Contribution

It develops a novel probabilistic representation for heat flow of harmonic maps on time-dependent Riemannian manifolds, and provides an alternative proof of local solution existence.

Findings

Probabilistic representation via forward-backward stochastic differential equations.

New stochastic method for proving existence of solutions.

Applicable to manifolds with evolving Riemannian metrics.

Abstract

In this paper we will give a probabilistic representation for the heat flow of harmonic map with time-dependent Riemannian metric via a forward-backward stochastic differential equation on manifolds. Moreover, we can provide an alternative stochastic method for the proof of existence of a unique local solution for heat flow of harmonic map with time-dependent Riemannian metric.