Strong Feller Regularisation of 1-d Nonlinear Transport by Reflected Ornstein-Uhlenbeck Noise

TL;DR

This paper introduces a stochastic perturbation to nonlinear transport equations on the circle, demonstrating that the resulting measure-valued process exhibits strong Feller regularity, thus providing a noise-induced regularization effect.

Contribution

The paper presents a novel stochastic regularization method for nonlinear transport equations using reflected Ornstein-Uhlenbeck noise, establishing strong Feller properties of the induced process.

Findings

The stochastic dynamics form a measure-valued Markov process.

The process is shown to be strong Feller in the relevant topology.

Regularization by noise phenomenon is demonstrated.

Abstract

We consider equations of nonlinear transport on the circle with regular self interactions appearing in aggregation models and deterministic mean field dynamics. We introduce a random perturbation of such systems through a stochastic orientation preserving flow, which is given as an integrated infinite dimensional periodic Ornstein- Uhlenbeck process with reflection. As our main result we show that the induced stochastic dynamics yields a measure valued Markov process on a class of regular measures. Moreover, we show that this process is strong Feller in the corresponding topology. This is interpreted as a qualitative regularisation by noise phenomenon.