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The large deviation principle for the stochastic 3D primitive equations with transport noise

Antonio Agresti, Esmée Theewis

TL;DR

The paper develops a rigorous small-noise large deviation principle for the three-dimensional primitive equations with transport noise, treating both Itô and Stratonovich forms. Central to the analysis is handling transport noise acting directly on the full horizontal velocity, which introduces a noncoercive setting and turbulent pressure effects; the authors overcome this with the T25L2LDP framework and a detailed barotropic/baroclinic decomposition to obtain essential a priori estimates. The skeleton equation and tilted SPDE are controlled to produce MR(0,T) bounds, enabling the LDP with a rate function given by the minimal control energy required to realize a path. This work extends the mathematical understanding of rare events in geophysical fluid dynamics models with physically meaningful transport noise and provides a pathway for analyzing climate-relevant scenarios via large deviations.

Abstract

We prove the small-noise large deviation principle for the three-dimensional primitive equations with transport noise and turbulent pressure. Transport noise is important for geophysical fluid dynamics applications, as it takes into account the effect of small scales on the large scale dynamics. The main mathematical challenge is that we allow for the transport noise to act on the full horizontal velocity, therefore leading to a non-trivial turbulent pressure, which requires an involved analysis to obtain the necessary energy bounds. Both Stratonovich and Itô noise are treated.

The large deviation principle for the stochastic 3D primitive equations with transport noise

TL;DR

The paper develops a rigorous small-noise large deviation principle for the three-dimensional primitive equations with transport noise, treating both Itô and Stratonovich forms. Central to the analysis is handling transport noise acting directly on the full horizontal velocity, which introduces a noncoercive setting and turbulent pressure effects; the authors overcome this with the T25L2LDP framework and a detailed barotropic/baroclinic decomposition to obtain essential a priori estimates. The skeleton equation and tilted SPDE are controlled to produce MR(0,T) bounds, enabling the LDP with a rate function given by the minimal control energy required to realize a path. This work extends the mathematical understanding of rare events in geophysical fluid dynamics models with physically meaningful transport noise and provides a pathway for analyzing climate-relevant scenarios via large deviations.

Abstract

We prove the small-noise large deviation principle for the three-dimensional primitive equations with transport noise and turbulent pressure. Transport noise is important for geophysical fluid dynamics applications, as it takes into account the effect of small scales on the large scale dynamics. The main mathematical challenge is that we allow for the transport noise to act on the full horizontal velocity, therefore leading to a non-trivial turbulent pressure, which requires an involved analysis to obtain the necessary energy bounds. Both Stratonovich and Itô noise are treated.
Paper Structure (20 sections, 10 theorems, 179 equations)

This paper contains 20 sections, 10 theorems, 179 equations.

Table of Contents

  1. Introduction
  2. Motivation and LDP
  3. Related literature
  4. Strategy and novelties
  5. Preliminaries, assumptions and main result
  6. Notation
  7. Preliminaries
  8. Primitive equations reformulation and solution notion
  9. The case of Stratonovich-type transport noise
  10. Proof of the LDP -- case of Itô noise
  11. Skeleton equation
  12. $L^2$-estimates for the horizontal skeleton velocity and temperature
  13. Intermediate estimates for skeleton barotropic and baroclinic modes
  14. $H^1$-estimate for the horizontal skeleton velocity
  15. Controlled stochastic equation
  16. ...and 5 more sections

Key Result

Theorem 1.1

Suppose that $(\phi_n)_{n\geq 1},(\psi_n)_{n\geq 1}\in C^{1+\alpha}(\mathcal{O};\ell^2)$ for some $\alpha>0$, and $\nabla \cdot \psi_n=0$ in distribution on $\mathcal{O}$ for all $n\geq 1$. Moreover, letting $\phi^j_n$ and $\psi^j_n$ be the components of $\phi_n$ and $\psi_n$ for $j\in \{1,2,3\}$, r Then, the solutions (Definition def:sol_strong_weak) to the stochastic primitive equations eq:primi

Theorems & Definitions (24)

  • Theorem 1.1: LDP for the primitive equations with Stratonovich transport noise
  • Definition 2.1
  • Remark 2.3
  • Definition 2.4: $L^2$-strong-weak solutions
  • Theorem 2.6
  • proof
  • Theorem 2.7
  • Theorem 2.9
  • Remark 3.1
  • Lemma 3.2: Coercivity of the linear part
  • ...and 14 more