Energy solutions of singular SPDEs on Hilbert spaces with applications to domains with boundary conditions

TL;DR

This paper advances the theory of energy solutions for singular stochastic partial differential equations (SPDEs) driven by irregular noise, using Hilbert space analysis and invariant measures to handle boundary conditions.

Contribution

It introduces a new framework employing Gelfand triples and infinite-dimensional analysis, expanding the applicability of energy solutions to more complex SPDEs with boundary conditions.

Findings

Broadened the class of SPDEs solvable with energy solutions.

Eliminated reliance on Fourier series and chaos expansions.

Applied the framework to scaling limits of interacting particle systems.

Abstract

In this paper we extend the theory of energy solutions for singular SPDEs, focusing on equations driven by highly irregular noise with bilinear nonlinearities, including scaling critical examples. By introducing Gelfand triples and leveraging infinite-dimensional analysis in Hilbert spaces together with an integration by parts formula under the invariant measure, we largely eliminate the need for Fourier series and chaos expansions. This approach broadens the applicability of energy solutions to a wider class of SPDEs, offering a unified treatment of various domains and boundary conditions. Our examples are motivated by recent work on scaling limits of interacting particle systems.