Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Large deviations for stochastic evolution equations beyond the coercive case

Esmée Theewis

TL;DR

The paper extends large deviation principles for stochastic evolution equations beyond the coercive (variational) setting by introducing an $L^2$-framework with a flexible, non-coercive operator pair $(A,B)$. It develops a skeleton/perturbed-analysis approach under minimal a priori estimates, enabling LDPs for non-coercive SPDEs such as reaction-diffusion systems and 3D fluid models, as well as coercive cases with critical nonlinearities through an alpha-subcriticality condition. Central contributions include a robust non-coercive LDP via a weak convergence method, a perturbed-LDP framework to accommodate Itô–Stratonovich corrections, and multiple applications including the stochastic Brusselator, 2D Allen–Cahn with transport noise, and 2D Navier–Stokes with Stratonovich transport noise. The results significantly broaden the applicability of LDPs in SPDEs, capturing systems with weaker energy bounds and richer nonlinear structures. The framework provides practical criteria for well-posedness and a priori estimates, facilitating future work on complex SPDEs in physics and engineering.

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.

Large deviations for stochastic evolution equations beyond the coercive case

TL;DR

The paper extends large deviation principles for stochastic evolution equations beyond the coercive (variational) setting by introducing an -framework with a flexible, non-coercive operator pair . It develops a skeleton/perturbed-analysis approach under minimal a priori estimates, enabling LDPs for non-coercive SPDEs such as reaction-diffusion systems and 3D fluid models, as well as coercive cases with critical nonlinearities through an alpha-subcriticality condition. Central contributions include a robust non-coercive LDP via a weak convergence method, a perturbed-LDP framework to accommodate Itô–Stratonovich corrections, and multiple applications including the stochastic Brusselator, 2D Allen–Cahn with transport noise, and 2D Navier–Stokes with Stratonovich transport noise. The results significantly broaden the applicability of LDPs in SPDEs, capturing systems with weaker energy bounds and richer nonlinear structures. The framework provides practical criteria for well-posedness and a priori estimates, facilitating future work on complex SPDEs in physics and engineering.

Abstract

We prove the small-noise large deviation principle (LDP) for stochastic evolution equations in an -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.
Paper Structure (17 sections, 23 theorems, 140 equations)

This paper contains 17 sections, 23 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. The non-coercive case
  4. The coercive case
  5. The skeleton equation with flexible nonlinearities
  6. Linearized problem
  7. Nonlinear problem
  8. Proof of Theorem \ref{['th:LDP general']}
  9. Perturbed LDP results
  10. The non-coercive perturbed case
  11. The coercive perturbed case
  12. Proof of Theorem \ref{['th:LDP perturbed2']}
  13. Applications
  14. The stochastic 2D Brusselator
  15. Coercive applications
  16. ...and 2 more sections

Key Result

Theorem 2.9

Let Assumption ass:coer replace hold and let $T>0$. Then the family $(Y^\varepsilon)_{\varepsilon\in(0,\varepsilon_0)}$ of solutions to eq:SPDE satisfies the large deviation principle on $L^2(0,T;V)\cap C([0,T];H)$ with rate function $I\colon L^2(0,T;V)\cap C([0,T];H)\to [0,+\infty]$ defined by where $\inf\varnothing\coloneqq +\infty$ and $u^\psi$ is the unique strong solution to eq:skeleton.

Theorems & Definitions (58)

  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Remark 2.8
  • Theorem 2.9: LDP
  • Example 2.10
  • Lemma 2.11
  • proof
  • ...and 48 more