Distribution-flow dependent SDEs driven by (fractional) Brownian motion and Navier-Stokes equations

Abstract

Motivated by the probabilistic representation for solutions of the Navier-Stokes equations, we introduce a novel class of stochastic differential equations that depend on the entire flow of its time marginals. We establish the existence and uniqueness of both strong and weak solutions under one-sided Lipschitz conditions and for singular drifts. These newly proposed distribution-flow dependent stochastic differential equations are closely connected to quasilinear backward Kolmogorov equations and Fokker-Planck equations. Furthermore, we investigate a stochastic version of the 2D-Navier-Stokes equation associated with fractional Brownian noise. We demonstrate the global well-posedness and smoothness of solutions when the Hurst parameter $H$ lies in the range $(0, \frac12)$ and the initial vorticity is a finite signed measure.