An extended variational setting for critical SPDEs with Lévy noise
Sebastian Bechtel, Fabian Germ, Mark Veraar
TL;DR
This work extends the critical variational setting for stochastic evolution equations to incorporate a flexible left-norm via $\alpha\in[0,\tfrac{1}{2}]$, a singular-in-time drift component, and Lévy noise. By decomposing the drift as $A=a_L+A_S$ and establishing a robust coercivity framework, the authors obtain a priori estimates and stochastic maximal $L^2$-regularity for the linear problem, which feed into a local well-posedness theory and a global well-posedness result under (nonlinear) coercivity conditions. The framework unlocks new well-posedness results for equations previously out of reach, including the Allen–Cahn equation in 2D weak setting and various fluid- and pattern-forming models under Lévy noise, on bounded or unbounded domains. The paper also provides blow-up criteria and continuous dependence results, with detailed applications to reaction–diffusion, stochastic fluid dynamics, Kuramoto–Sivashinsky, and Swift–Hohenberg equations, highlighting both theoretical and practical significance for SPDEs with non-Gaussian 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 Lévy noise for which the critical setting was not available yet.