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

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 Riemannian 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.