Stochastic evolution equations with singular drift and gradient noise via curvature and commutation conditions

TL;DR

This paper establishes the existence and uniqueness of solutions for a class of nonlinear stochastic evolution equations with singular drifts and gradient noise on a torus, using advanced mathematical techniques under specific geometric and commutation conditions.

Contribution

It introduces a novel approach to handle singular drifts and gradient noise in stochastic evolution equations through curvature and commutation conditions, expanding the theoretical understanding of such equations.

Findings

Proved well-posedness of the stochastic evolution equations under certain conditions.

Developed a framework using resolvent, Dirichlet forms, and an approximative Itô formula.

Extended the theory to include singular p-Laplace and total variation flows.

Abstract

We prove existence and uniqueness of solutions to a nonlinear stochastic evolution equation on the $d$-dimensional torus with singular $p$-Laplace-type or total variation flow-type drift with general sublinear doubling nonlinearities and Gaussian gradient Stratonovich noise with divergence-free coefficients. Assuming a weak defective commutator bound and a curvature-dimension condition, the well-posedness result is obtained in a stochastic variational inequality setup by using resolvent and Dirichlet form methods and an approximative It\^{o}-formula.