# Density large deviations for multidimensional stochastic hyperbolic conservation laws

## TL;DR

This paper studies the probability of rare density fluctuations in multidimensional hyperbolic conservation laws, deriving explicit large deviation functions and analyzing the structure of optimal currents, especially when conductivity and diffusivity are not proportional.

## Contribution

It provides explicit calculations of the large deviation function for step-like profiles and explores the structure of optimal currents without the proportionality assumption.

## Key findings

- Explicit large deviation function for step-like density profiles.
- Optimal current exhibits non-trivial structure when conductivity and diffusivity are not proportional.
- Lower bound established for the large deviation function in general cases.

## Abstract

We investigate the density large deviation function for a multidimensional conservation law in the vanishing viscosity limit, when the probability concentrates on weak solutions of a hyperbolic conservation law conservation law. When the conductivity and dif-fusivity matrices are proportional, i.e. an Einstein-like relation is satisfied, the problem has been solved in [4]. When this proportionality does not hold, we compute explicitly the large deviation function for a step-like density profile, and we show that the associated optimal current has a non trivial structure. We also derive a lower bound for the large deviation function, valid for a general weak solution, and leave the general large deviation function upper bound as a conjecture.