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Conservative stochastic PDE and fluctuations of the symmetric simple exclusion process

Nicolas Dirr, Benjamin Fehrman, Benjamin Gess

Abstract

In this paper, we provide a continuum model for the fluctuations of the symmetric simple exclusion process about its hydrodynamic limit. The model is based on an approximating sequence of stochastic PDEs with nonlinear, conservative noise. In the small-noise limit, we show that the fluctuations of the solutions are to first-order the same as the fluctuations of the particle system. Furthermore, the SPDEs correctly simulate the rare events in the particle process. We prove that the solutions satisfy a zero-noise large deviations principle with rate function equal to the rate function describing the deviations of the symmetric simple exclusion process from its hydrodynamic limit.

Conservative stochastic PDE and fluctuations of the symmetric simple exclusion process

Abstract

In this paper, we provide a continuum model for the fluctuations of the symmetric simple exclusion process about its hydrodynamic limit. The model is based on an approximating sequence of stochastic PDEs with nonlinear, conservative noise. In the small-noise limit, we show that the fluctuations of the solutions are to first-order the same as the fluctuations of the particle system. Furthermore, the SPDEs correctly simulate the rare events in the particle process. We prove that the solutions satisfy a zero-noise large deviations principle with rate function equal to the rate function describing the deviations of the symmetric simple exclusion process from its hydrodynamic limit.

Paper Structure

This paper contains 15 sections, 27 theorems, 227 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Comments on Applications and Numerics
  3. Comments on the literature
  4. Organization of the paper
  5. The well-posedness of the SPDE
  6. The definition of the noise
  7. Uniqueness of stochastic kinetic solutions
  8. Existence of renormalized kinetic solutions
  9. The central limit theorem
  10. A quantitative LLN for the approximating SPDE
  11. The CLT for the approximating SPDE
  12. The CLT for the SPDE with singular coefficients
  13. The large deviations principle
  14. The skeleton equation
  15. The large deviations principle

Key Result

Theorem 1

Let $T\in(0,\infty)$, let $\varepsilon\in(0,1)$, let the noise $\{\xi^K\}_{K\in\mathbb{N}}$ satisfy Assumption def_noise_trig (see also Assumption def_noise) with respect to a filtration $(\mathcal{F}_t)_{t\in[0,\infty)}$ on some probability space $(\Omega,\mathcal{F},\mathbb{P})$, and let $\rho_0\i

Figures (2)

  • Figure 1: Particles (left), linear SPDE (centre), SPDE \ref{['intro_fd_eq']} (right). First row smoothed output, second row smoothed fluctuations, third row averaged fluctuations.
  • Figure 2: $x-$axis: $N=1/\epsilon.$$y$-axis: fluctuations integrated against the specific test function \ref{['eq:testfct']}. Red particles, black SPDE \ref{['intro_fd_eq']} (multiplicative noise), green SPDE \ref{['intro_clt']} (additive noise, theoretical value).

Theorems & Definitions (59)

  • Theorem : cf. Theorems \ref{['thm_rks_unique']}, \ref{['thm_rks_exist']}
  • Theorem : cf. Theorem \ref{['prop_est_infty']}
  • Theorem : cf. Theorem \ref{['thm_clt_prob']}, Corollary \ref{['cor_clt_prob']}
  • Theorem : cf. Theorem \ref{['thm_new']}
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 2.5
  • Theorem 2.6
  • proof
  • ...and 49 more