Large deviations for stochastic evolution equations beyond the coercive case

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 Itô--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.

Summary

This paper presents a novel framework for establishing large deviation principles (LDPs) for stochastic evolution equations in an L2 setting, expanding beyond traditional variational approaches that rely on strong coercivity conditions. By relaxing these conditions, the authors broaden the applicability of LDPs to a wider range of SPDEs, including those arising in reaction-diffusion systems and fluid dynamics.

Key Insights

• •The authors replace the coercivity condition with a set of more general assumptions, including a structural assumption, well-posedness of the SPDE, a priori estimates for the skeleton equation, and uniform boundedness in probability, allowing for the analysis of non-coercive SPDEs.
• •They introduce a modified criticality condition that weakens the growth conditions on the nonlinearities in the coefficients, accommodating more flexible growth rates compared to previous works and enabling the treatment of Stratonovich noise.
• •The paper provides a detailed analysis of the skeleton equation, proving local well-posedness and a blow-up criterion, relying on interpolation techniques to handle the more general nonlinearities.

Practical Implications

• •The relaxed conditions for establishing LDPs open up new possibilities for analyzing and understanding the long-time behavior of SPDEs arising in various fields, such as fluid dynamics, reaction-diffusion systems, and materials science.
• •The results have direct applications to specific models, including the 2D Allen-Cahn equation with transport noise and the stochastic 2D Burger's equation, providing new insights into their stochastic dynamics.
• •Future research can focus on further weakening the assumptions required for the LDP, developing more efficient methods for verifying these assumptions, and extending the framework to other classes of SPDEs, such as those with jumps or delays.

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