# Viscous scalar conservation law with stochastic forcing: strong solution and invariant measure

## TL;DR

This paper studies one-dimensional viscous scalar conservation laws with stochastic forcing, proving existence and uniqueness of strong solutions and invariant measures under broad conditions, including degenerate noise.

## Contribution

It establishes the first rigorous results on strong solutions and invariant measures for viscous scalar conservation laws with degenerate stochastic forcing.

## Key findings

- Existence and uniqueness of strong solutions
- Existence and uniqueness of invariant measures
- Results hold under degenerate noise conditions

## Abstract

We are interested in viscous scalar conservation laws with a white-in-time but spatially correlated stochastic forcing. The equation is assumed to be one-dimensional and periodic in the space variable, and its flux function to be locally Lipschitz continuous and have at most polynomial growth. Neither the flux nor the noise need to be non-degenerate. In a first part, we show the existence and uniqueness of a global solution in a strong sense. In a second part, we establish the existence and uniqueness of an invariant measure for this strong solution.