An extended variational setting for critical SPDEs with L\'evy noise

TL;DR

This paper extends the critical variational framework for SPDEs to include more general nonlinearities, singular drifts, and Lévye noise, broadening its applicability to complex stochastic systems.

Contribution

It introduces a flexible range space for nonlinear drifts, allows singular time-dependent drifts, and develops the theory for SPDEs driven by Lévye noise.

Findings

Extended the variational setting to include borderline cases like the 2D Allen-Cahn equation.

Allowed singular in time drifts, relevant for large deviation skeleton equations.

Established the critical framework for SPDEs with Lévye noise.

Abstract

The critical variational setting was recently introduced and shown to be applicable to many important SPDEs not covered by the classical variational setting. In this paper, we extend the critical variational setting in several ways. We introduce a flexibility in the range space for the nonlinear drift term, due to which certain borderline cases can now also be included. An example of this is the Allen-Cahn equation in dimension two in the weak setting. In addition to this, we allow the drift to be singular in time, which is something that naturally arises in the study of the skeleton equations for large deviation principles for SPDEs. Last but not least, we present the theory in the case of L\'evy noise for which the critical setting was not available yet.