Non-local quasilinear singular SPDEs

TL;DR

This paper develops a local-in-time solution theory for a nonlocal, quasilinear singular stochastic PDE on the 2D torus, extending the generalized parabolic Anderson model using paracontrolled calculus.

Contribution

It introduces a novel approach to handle nonlocal coefficients in quasilinear singular SPDEs within the paracontrolled calculus framework.

Findings

Established local existence and uniqueness of solutions.

Extended the framework of gPAM to nonlocal coefficient settings.

Provided conditions under which the solution theory applies.

Abstract

We study in this short note a counterpart to the quasilinear generalized parabolic Anderson model (gPAM) on the 2-dimensional torus where the coefficients are nonlocal functionals of the solution. Under a positivity assumption on the diffusion coefficient we give a local in time solution theory within the framework of paracontrolled calculus.