Levy solutions of a randomly forced Burgers equation

TL;DR

This paper investigates Levy process solutions to a one-dimensional Burgers equation driven by spatial Brownian and temporal white noise, deriving the evolution of the characteristic exponent and providing explicit solutions for specific initial conditions.

Contribution

It introduces Levy process solutions to a stochastic Burgers equation with explicit evolution equations and solutions, expanding understanding of stochastic PDEs with Levy noise.

Findings

Existence of Levy process solutions for the stochastic Burgers equation.

Derivation of the evolution equation for the characteristic exponent.

Explicit solution for the case with zero initial condition.

Abstract

We consider the one dimensional Burgers equation forced by a brownian in space and white noise in time process $\partial_t u + u \partial_x u = f(x,t)$, with $2E(f(x,t)f(y,s)) = (|x|+|y|-|x-y|)\delta(t-s)$ and we show that there are Levy processes solutions, for which we give the evolution equation of the characteristic exponent. In particular we give the explicit solution in the case $u_0(x)=0$.