Support theorem for a singular semilinear stochastic partial differential equation

TL;DR

This paper establishes a support theorem for the generalized parabolic Anderson equation in 2D, advancing understanding of its solution law within the framework of regularity structures and paracontrolled distributions.

Contribution

It provides a Stroock-Varadhan support theorem for gPAM, a singular SPDE, in Hölder-Besov spaces, linking recent solution theories to probabilistic support results.

Findings

Support theorem characterizes the law of gPAM solutions.

Connects regularity structures with probabilistic support analysis.

Advances the understanding of singular SPDEs in 2D.

Abstract

We consider the generalized parabolic Anderson equation (gPAM) in 2 dimensions with periodic boundary. This is an example of a singular semilinear stochastic partial differential equations, solutions of which require renormalization and have only be understood recently via Hairer's regularity structures and, in some cases equivalently, paracontrollled distributions due to Gubinelli, Imkeller and Perkowski. In the present paper we describe the law of gPAM, by establishing a Stroock{Varadhan type support theorem in suitable Holder{Besov spaces.