Strong Feller Regularisation of 1-d Nonlinear Transport by Reflected Ornstein-Uhlenbeck Noise
Max-K. von Renesse, Feng-Yu Wang, Alexander Weiß
TL;DR
The paper studies regularisation by noise for a 1-d nonlinear measure-valued transport model on the circle, perturbing the dynamics by a reflected integrated periodic Ornstein–Uhlenbeck flow. By translating the nonlinear transport into an inverse-CDF/Lagrangian framework and differentiating, it derives a coupled SPDE for the derivative $g_t$ and a scalar moment $M_t$ with a reflection term to preserve positivity, yielding a well-posed, measure-valued Markov diffusion on $\mathcal{P}_2^1(\mathbb{R})$. Under global Lipschitz/bounded regularity assumptions on the self-interaction $b$ (Assumption (A1)), the authors establish existence and uniqueness, prove the Markov property, and derive quantitative entropy bounds that imply a strong Feller property for the associated semigroup. The results demonstrate a qualitative regularisation by noise in a convex, infinite-dimensional state space setting and connect to McKean–Vlasov-type interactions; an explicit example includes $b(u,\mu)=\int h(u-v)\,\mu(dv)$ with $h\in C_c^{\infty}$. Overall, the work combines a reflection-based positivity enforcement, a detailed Lagrangian/Differential-operator formulation, and a coupling/Girsanov-based entropy analysis to show how noise enhances regularity in measure-valued transport dynamics.
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.