Large deviations for stochastic evolution equations beyond the coercive case

TL;DR

This paper establishes a small-noise large deviation principle for stochastic evolution equations without requiring coercivity, broadening applicability to systems with nonlinearities and non-coercive coefficients.

Contribution

It introduces a framework for large deviations in non-coercive stochastic evolution equations, accommodating nonlinear drifts and multiple noise interpretations, extending beyond traditional variational methods.

Findings

Proves LDP for non-coercive stochastic evolution equations.

Applies to reaction-diffusion systems lacking coercivity.

Provides new LDP results for equations with critical nonlinearities.

Abstract

We prove the small-noise large deviation principle (LDP) for stochastic evolution equations in an $L^2$-setting. As the coefficients are allowed to be non-coercive, our framework encompasses a much broader scope than variational settings. To replace coercivity, we require only well-posedness of the stochastic evolution equation and two concrete, verifiable a priori estimates. Furthermore, we accommodate drift nonlinearities satisfying a modified criticality condition, and we allow for vanishing drift perturbations. The latter permits the inclusion of It\^o--Stratonovich correction terms, enabling the treatment of both noise interpretations. In another paper, our results have been applied to the 3D primitive equations with full transport noise. In the current paper, we give an application to a reaction-diffusion system which lacks coercivity, further demonstrating the versatility of the framework. Finally, we show that even in the coercive case, we obtain new LDP results for equations with critical nonlinearities that rely on our modified criticality condition, including the stochastic 2D Allen--Cahn equation in the weak setting.