×

Large deviations from the hydrodynamical limit for a system of independent Brownian particles. (English) Zbl 0734.60099

Author’s introduction: The problem of hydrodynamical limit for infinite particle systems consists in proving that as a parameter tends to zero some macroscopic field converges in probability to the solution of a deterministic PDE. It is therefore a natural question to ask how large is the probability to observe a different density profile than the one predicted by hydrodynamics.
The present paper is an attempt to use the (by now standard) recipe: “Find the right perturbation of your system and you get the large deviation functional easily” for infinite particle systems in the hydrodynamical scaling.
It turns out that when the density field is autonomous, which covers the case of independent or weakly dependent (mean field interaction) the perturbations obtained by adding a microscopically small drift (but macroscopically non-trivial) are enough to make this approach work. In order not to blur the main ideas we have chosen to present our results for independent Brownian motions on the d-dimensional torus \(\Pi^ d\) rather than in the most general case.
An analogous result for the McKean-Vlasov limit has been obtained by D. A. Dawson and J. Gärtner [Stochastics 20, 247-308 (1987; Zbl 0613.60021)] using a kind of contraction principle.
Our results give the usual upper (lower) bounds for closed (open) sets with an action functional which is given in a variational way but which can be expressed when it is finite as the sum of two quantities: the initial entropy of the state with respect to the slowly varying Poisson initial state plus a dynamical action functional.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60F10 Large deviations

Citations:

Zbl 0613.60021
PDF BibTeX XML Cite
Full Text: DOI