The Wave Maps Equation and Brownian Paths

Abstract

We discuss the ( 1 + 1 ) -dimensional wave maps equation with values in a compact Riemannian manifold . Motivated by the Gibbs measure problem, we consider Brownian paths on the manifold as initial data. Our main theorem is the probabilistic local well-posedness of the associated initial value problem. The analysis in this setting combines analytic, geometric, and probabilistic methods.