Asymptotics of PDE in random environment by paracontrolled calculus

TL;DR

This paper uses paracontrolled calculus to analyze the asymptotic behavior of a quasilinear PDE with noise, proving convergence, well-posedness, and comparison results for the associated stochastic PDE without renormalization.

Contribution

It introduces a novel application of paracontrolled calculus to establish convergence and well-posedness of a stochastic PDE derived from a particle system in a random environment.

Findings

Proves convergence of the PDE to a stochastic limit

Establishes local in time well-posedness of the limit SPDE

Shows the limit SPDE does not require renormalization

Abstract

We apply the paracontrolled calculus to study the asymptotic behavior of a certain quasilinear PDE with smeared mild noise, which originally appears as the space-time scaling limit of a particle system in random environment on one dimensional discrete lattice. We establish the convergence result and show a local in time well-posedness of the limit stochastic PDE with spatial white noise. It turns out that our limit stochastic PDE does not require any renormalization. We also show a comparison theorem for the limit equation.