Fluctuations of random semi-linear advection equations

TL;DR

This paper studies the behavior of semi-linear advection equations with highly oscillatory Gaussian noise, revealing that nonlinearity acts as a random transformation on the linear homogenization limit.

Contribution

It demonstrates that nonlinearity in such equations functions as a random diffeomorphism, extending classical linear homogenization results to nonlinear settings.

Findings

Nonlinearity acts as a random diffeomorphism in the limit.

Point-wise distribution is obtained by applying the diffeomorphism to the linear limit.

Classical linear homogenization results are extended to nonlinear equations.

Abstract

We consider a semi-linear advection equation driven by a highly-oscillatory space-time Gaussian random field, with the randomness affecting both the drift and the nonlinearity. In the linear setting, classical results show that the characteristics converge in distribution to a homogenized Brownian motion, hence the point-wise law of the solution is close to a functional of the Brownian motion. Our main result is that the nonlinearity plays the role of a \emph{random diffeomorphism}, and the point-wise limiting distribution is obtained by applying the diffeomorphism to the limit in the linear setting.