2024/05/29 by Zimo Hao, Michael Röckner, Hao, Zimo +3
Economics, Econometrics and Finance · Engineering · #FOS: Mathematics #Fluid Dynamics and Turbulent Flows #Probability (math.PR) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2405.19034
openalex publication_date 2024/05/29 · openalex created_date 2024/05/31 · openalex updated_date 2026/07/28
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.