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

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.