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.

LDP and CLT for SPDEs with transport noise

“…All of the above works however do not include the case of multiplicative transport noise; in this sense, a much closer paper to our setting is the one by Slavík [82], proving an LDP, MDP and CLT for stochastic 3D viscous primitive equations. At the same time the equation considered in [82] is truly parabolic, while our setting is of hyperbolic nature with conservative noise, which makes it closer to the results obtained in [25] (resp. [53]), treating conservative local (resp.…”

“…nonlocal) SPDEs. Like here, [82] and [25] establish strong convergence for the fluctuations, while [53] even provides a rate of convergence which is shown to be optimal. An alternative pathwise approach to CLTs for SPDEs is being developed in [54], in the context of Landau-Lifschitz equation.…”

Well-posedness of the Dean--Kawasaki and the nonlinear Dawson--Watanabe equation with correlated noise

“…The asymmetric simple exclusion process corresponds to Φ(ρ) = ρ, ν(ρ) = a 2 (ρ) = ρ(1 − ρ). In this case, the exclusion rule prevents concentration of mass, which allows a much simpler treatment of the stochastic PDE, see [71] and [25]. However, prior to this work, even for this case it remained necessary to introduce an approximation of the square root ρ(1 − ρ) in order to obtain the wellposedness of the equation.…”

The Dean-Kawasaki equation and the structure of density fluctuations in systems of diffusing particles

“…In [11], they have proven a large deviation principle for regularised variants of (6); in a suitable limit, the rate functional of their large deviations principle and the corresponding one of the interacting particle system are shown to approach each other. In a paper written independently of -and simultaneously to -the present manuscript, Dirr, Fehrman, and Gess [6] have given a rigorous justification of the fluctuating hydrodynamics SPDE associated with the symmetric simple exclusion process…”