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Small noise fluctuations and large deviations of conservative SPDEs with Dirichlet boundary conditions

Shyam Popat

TL;DR

The paper studies small-noise fluctuations and rare events for the generalised Dean–Kawasaki equation on bounded $C^2$ domains with Dirichlet boundary data, connecting stochastic PDE fluctuations to those of interacting particle systems like the zero-range process. It develops a comprehensive framework that combines a quantitative law of large numbers and a central limit theorem for regularised and singular SPDEs with a uniform large deviations principle, using a skeleton equation and variational representations. A key feature is the treatment of Dirichlet boundaries, which correspond to particle absorption/injection, and the rigorous handling of boundary data within the kinetic formulation and LDP arguments. The results hinge on a joint scaling regime $\epsilon\to0$, $K\to\infty$ and provide explicit rates and functionals that match the large-deviation behavior of the underlying particle systems, offering a rigorous link between microscopic dynamics and macroscopic fluctuation theory in bounded domains.

Abstract

We establish a central limit theorem and large deviations principle that characterises small noise fluctuations of the generalised Dean--Kawasaki stochastic PDE. The fluctuations agree to first order with fluctuations of certain interacting particle systems, such as the zero range process, about their hydrodynamic limits. Our main contribution is that we are able to consider stochastic PDEs on general $C^2$ bounded domains with Dirichlet boundary conditions. On the level of particles, the boundary condition corresponds to absorption or injection of particles at the boundary.

Small noise fluctuations and large deviations of conservative SPDEs with Dirichlet boundary conditions

TL;DR

The paper studies small-noise fluctuations and rare events for the generalised Dean–Kawasaki equation on bounded domains with Dirichlet boundary data, connecting stochastic PDE fluctuations to those of interacting particle systems like the zero-range process. It develops a comprehensive framework that combines a quantitative law of large numbers and a central limit theorem for regularised and singular SPDEs with a uniform large deviations principle, using a skeleton equation and variational representations. A key feature is the treatment of Dirichlet boundaries, which correspond to particle absorption/injection, and the rigorous handling of boundary data within the kinetic formulation and LDP arguments. The results hinge on a joint scaling regime , and provide explicit rates and functionals that match the large-deviation behavior of the underlying particle systems, offering a rigorous link between microscopic dynamics and macroscopic fluctuation theory in bounded domains.

Abstract

We establish a central limit theorem and large deviations principle that characterises small noise fluctuations of the generalised Dean--Kawasaki stochastic PDE. The fluctuations agree to first order with fluctuations of certain interacting particle systems, such as the zero range process, about their hydrodynamic limits. Our main contribution is that we are able to consider stochastic PDEs on general bounded domains with Dirichlet boundary conditions. On the level of particles, the boundary condition corresponds to absorption or injection of particles at the boundary.

Paper Structure

This paper contains 19 sections, 30 theorems, 281 equations.

Table of Contents

  1. Introduction
  2. Application to particle systems
  3. Background literature
  4. Motivation of boundary data
  5. Organisation of the paper
  6. Setup
  7. Quantitative law of large numbers and central limit
  8. Central limit theorem for approximate SPDE
  9. Central limit theorem for the stochastic PDE with singular coefficients
  10. The large deviations principle
  11. Existence of solution maps
  12. Lack of uniformity of solution map with respect to initial condition
  13. Energy estimates for the regularised controlled SPDE
  14. The uniform large deviations principle
  15. Appendix
  16. ...and 4 more sections

Key Result

Proposition 1

Let $\rho^{n,\epsilon,K}$ be the weak solution of the regularised equation equation for rho n epsilon k, and let $\bar{\rho}$ denote solution to the regularised hydrodynamic limit equation eq: regularised hydrodynamic limit. Under Assumptions assumption on noise, assumption of constant boundary data and the $p$-dependent estimate where $F_i^K$, $i=1,2,3$ relate to the spatial component of the noi

Theorems & Definitions (88)

  • Proposition 1: Propositions \ref{['ppn: energy estimate for difference of regularised equation and hydrodynamic limit']}, \ref{['prop: second energy estimate for rho n epsilon k']} below
  • Theorem 1: Theorem \ref{['theorem CLT for approximating equation']} below
  • Theorem 2: Theorem \ref{['thm: l infinity estimate for singular equation']} below
  • Proposition 2: Propositions \ref{['prop: existence of solution map for skeleton equation']},\ref{['prop: existence of solution map for controlled SPDE']} below
  • Theorem 3: Theorem \ref{['thm: convergence of solutions when initial data and controls converge']} below
  • Theorem 4: Theorem \ref{['thm: LDP theorem from BDM']} below
  • Definition 1: Truncated noise
  • Remark 1
  • Remark 2
  • Remark 3
  • ...and 78 more