The wave maps equation and Brownian paths

Bjoern Bringmann, Jonas Luhrmann, Gigliola Staffilani

Published 2021-11-14, updated 2023-11-11Version 2

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.