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Large Deviations for Stochastic Evolution Equations in the Critical Variational Setting

Esmée Theewis, Mark Veraar

TL;DR

This work establishes a large deviation principle for small-noise stochastic evolution equations in the critical variational setting AV22variational. It develops a robust weak convergence framework, proving global well-posedness for skeleton equations via maximal regularity and fixed-point arguments, and proving compactness of rate-function sublevel sets without requiring monotonicity or compact embeddings. The results apply to a broad class of semilinear and quasilinear SPDEs, including gradient-noise variants of the 2D Navier–Stokes and Boussinesq equations on unbounded domains, thereby resolving open problems in the literature. The theory connects the stochastic dynamics to a Laplace principle, enabling precise probabilistic descriptions of rare events in fluid dynamics and related PDEs under gradient-type noise.

Abstract

Using the weak convergence approach, we prove the large deviation principle (LDP) for solutions to quasilinear stochastic evolution equations with small Gaussian noise in the critical variational setting, a recently developed general variational framework. No additional assumptions are made apart from those required for well-posedness. In particular, no monotonicity is required, nor a compact embedding in the Gelfand triple. Moreover, we allow for flexible growth of the diffusion coefficient, including gradient noise. This leads to numerous applications for which the LDP was not established yet, in particular equations on unbounded domains with gradient noise. Since our framework includes the 2D Navier-Stokes and Boussinesq equations with gradient noise and unbounded domains, our results resolve an open problem that has remained unsolved for over 15 years.

Large Deviations for Stochastic Evolution Equations in the Critical Variational Setting

TL;DR

This work establishes a large deviation principle for small-noise stochastic evolution equations in the critical variational setting AV22variational. It develops a robust weak convergence framework, proving global well-posedness for skeleton equations via maximal regularity and fixed-point arguments, and proving compactness of rate-function sublevel sets without requiring monotonicity or compact embeddings. The results apply to a broad class of semilinear and quasilinear SPDEs, including gradient-noise variants of the 2D Navier–Stokes and Boussinesq equations on unbounded domains, thereby resolving open problems in the literature. The theory connects the stochastic dynamics to a Laplace principle, enabling precise probabilistic descriptions of rare events in fluid dynamics and related PDEs under gradient-type noise.

Abstract

Using the weak convergence approach, we prove the large deviation principle (LDP) for solutions to quasilinear stochastic evolution equations with small Gaussian noise in the critical variational setting, a recently developed general variational framework. No additional assumptions are made apart from those required for well-posedness. In particular, no monotonicity is required, nor a compact embedding in the Gelfand triple. Moreover, we allow for flexible growth of the diffusion coefficient, including gradient noise. This leads to numerous applications for which the LDP was not established yet, in particular equations on unbounded domains with gradient noise. Since our framework includes the 2D Navier-Stokes and Boussinesq equations with gradient noise and unbounded domains, our results resolve an open problem that has remained unsolved for over 15 years.
Paper Structure (16 sections, 25 theorems, 200 equations)

This paper contains 16 sections, 25 theorems, 200 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. The critical variational setting
  4. Statement of the main result
  5. Well-posedness of the skeleton equation
  6. Linearized skeleton equation
  7. Local well-posedness
  8. Global well-posedness
  9. Proof of the large deviation principle
  10. Weak convergence approach
  11. Weakly continuous dependence in the skeleton equation
  12. Stochastic continuity criterion
  13. Proof of Theorem \ref{['th: main LDP theorem']}
  14. Application to fluid dynamics
  15. Abstract model
  16. ...and 1 more sections

Key Result

Theorem 2.3

Let $(A,B)$ satisfy Assumption ass: crit var set and suppose that $(A,B)$ is coercive in the following sense: for all $T>0$, there exist $\theta, M>0$ and $\phi\in L^2(0,T)$ such that for all $v\in V$ and $t\in[0,T]$, Then, for any $x\in H$ and $T>0$, there exists a unique strong solution $u$ to eq: original stoch ev eq on $[0,T]$.

Theorems & Definitions (58)

  • Definition 2.1
  • Theorem 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6
  • Definition 3.1
  • Proposition 3.2
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • ...and 48 more