Propagation of microlocal singularities for stochastic partial differential equations

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.