Large Deviations for $(1+1)$-dimensional Stochastic Geometric Wave Equation

TL;DR

This paper proves a large deviations principle for solutions of a stochastic wave map equation on the real line with values in a compact Riemannian manifold, establishing global well-posedness and analyzing the effects of vanishing noise.

Contribution

It introduces the first large deviations analysis for the stochastic geometric wave equation with solutions in Sobolev spaces, demonstrating global existence and uniqueness.

Findings

Established global, strong solutions in Sobolev spaces

Proved large deviations principle for vanishing noise

Ensured well-posedness of the stochastic wave map equation

Abstract

We consider stochastic wave map equation on real line with solutions taking values in a $d$-dimensional compact Riemannian manifold. We show first that this equation has unique, global, strong in PDE sense, solution in local Sobolev spaces. The main result of the paper is a proof of the Large Deviations Principle for solutions in the case of vanishing noise.