# Hydrodynamic limit large deviation from nonlinear heat equation given by stochastic Carleman particles, a Hamilton-Jacobi approach

### Mathematical Physics and Probability Seminar

**Hydrodynamic limit large deviation from nonlinear heat equation given by stochastic Carleman particles, a Hamilton-Jacobi approach**

**Location:**MATH 402

**Presenter:**Jin Feng, University of Kansas

The deterministic Carleman equation can be considered as an one dimensional two speed fictitious gas model. Its associated hydrodynamic limit gives a nonlinear heat equation. The first rigorous derivation of such limit was given by Kurtz in 1973. In this talk, starting from a more refined stochastic model giving the Carleman equation as the mean field, we derive a macroscopic fluctuation structure associated with the hydrodynamic limit.

The large deviation result is established through an abstract Hamilton-Jacobi method applied to this specific setting. The principal idea is to identify a two scale averaging structure in the context of Hamiltonian convergence in the space of probability measures. This is achieved through a change of coordinate to the density-flux description of the problem. We also extend a method in the weak KAM theory to the infinite particle context for explicitly identifying the effective Hamiltonian. In the end, we conclude by establishing a comparison principle for a set of Hamilton-Jacobi equation in the space of measures.

I will present some subtle issues involved and put the method in perspective regarding challenges we face when applying the method to other hydrodynamic limit issues.

This is a joint work with Toshio Mikami and Johannes Zimmer.