Propagation of microlocal singularities for stochastic partial differential equations

TL;DR

This paper extends microlocal analysis to stochastic PDEs, demonstrating that the propagation of singularities theorem applies to hyperbolic SPDEs driven by Brownian motion, revealing invariance of wave front sets under stochastic flows.

Contribution

It introduces a novel application of microlocal analysis to hyperbolic SPDEs, establishing the propagation of singularities theorem in this stochastic context.

Findings

Wave front set invariance under stochastic Hamiltonian flow

Extension of H"ormander's theorem to hyperbolic SPDEs

Introduction of a class of random pseudodifferential operators

Abstract

Microlocal analysis techniques are extended and applied to stochastic partial differential equations (SPDEs). In particular, the H\"ormander propagation of singularities theorem is shown to be valid for hyperbolic SPDEs driven by a standard Brownian motion. In this case the wave front set of the solution is invariant under the stochastic Hamiltonian flow associated to the principal symbol of the SPDE. This study leads to the introduction of a class of random pseudodifferential operators.