A stochastic flow approach to De Giorgi-Nash-Moser estimates for SPDEs with smooth transport noise

TL;DR

This paper proves H"older continuity for solutions of certain parabolic SPDEs with transport noise using stochastic flow techniques, extending classical regularity results to stochastic settings.

Contribution

It introduces a stochastic flow method to establish De Giorgi-Nash-Moser estimates for SPDEs with regular transport noise coefficients, a significant advancement in stochastic regularity theory.

Findings

Established H"older continuity for solutions to specific SPDEs with transport noise.

Developed new a-priori estimates for stochastic flow inverses.

Proved existence of global regular solutions to quasilinear SPDEs with transport noise.

Abstract

The celebrated De Giorgi-Nash-Moser theory ensures that solutions to uniformly elliptic or parabolic PDEs are bounded and H\"older continuous, even with merely bounded measurable coefficients. For parabolic SPDEs with transport noise, boundedness has recently been established, but H\"older continuity remains a key open problem in the regularity theory of parabolic SPDEs. In this work, we resolve this question under the assumption that the noise coefficients are sufficiently regular in space. Our approach relies on Kunita's stochastic method of characteristics, which allows us to transform the original SPDE-via a stochastic flow of diffeomorphisms-into a random PDE to which the classical De Giorgi-Nash-Moser estimates apply. This program is accomplished through new a-priori estimates for the inverse of stochastic flows of diffeomorphisms, and a novel version of the It\^o-Wentzell formula adapted to rough random fields. To demonstrate the applicability of our results, we establish the existence of global, regular solutions to quasilinear SPDEs with transport noise.