The wave maps equation and Brownian paths

TL;DR

This paper establishes probabilistic local well-posedness for the wave maps equation with initial data given by Brownian paths on a Riemannian manifold, integrating analytic, geometric, and probabilistic techniques.

Contribution

It introduces a novel probabilistic approach to the wave maps equation using Brownian paths as initial data, advancing understanding of well-posedness in this context.

Findings

Proves local well-posedness with Brownian initial data

Develops methods combining analysis, geometry, and probability

Provides new insights into wave maps with stochastic initial conditions

Abstract

We discuss the $(1+1)$-dimensional wave maps equation with values in a compact Riemannian manifold $\mathcal{M}$. Motivated by the Gibbs measure problem, we consider Brownian paths on the manifold $\mathcal{M}$ 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.