2025/11/25 by Mayorcas, Avi, Mądry, Łukasz
Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Fractional Differential Equations Solutions #Navier-Stokes equation solutions #Probability (math.PR) #Stochastic processes and financial applications
paper · doi:10.48550/arxiv.2511.20556
openalex publication_date 2025/11/25 · openalex created_date 2025/11/28 · openalex updated_date 2026/07/28
We show existence and uniqueness of invariant measures for SDE of the form dXt = g(Xt)dt + u(Xt)dt + dWHt where WH is a fractional Brownian motion (fBm) with Hurst parameter H∈ (0,(1)/(2)), u is a linearly dispersive term and g is any Bα∞,∞(ℝd) distribution in the class treated by Catellier--Gubinelli `16, i.e. α>1-(1)/(2H). The significant challenge is to combine the regularizing effect of the fBm with an ergodic theory suited to non-Markovian SDE. Concerning the latter our first main contribution is to construct a bona fide stochastic dynamical system (SDS) (Hairer `05 and Hairer--Ohashi `07) associated to the equation above. Since the solution map is only continuous in the support of the stationary noise process we weaken the definitions introduced by Hairer `05 and Hairer--Ohashi `07 but manage to retain the Doob--K'hashminksii provided by Hairer--Ohashi `07. Our second innovation is to introduce a family of flexible local-global stochastic sewing lemmas, in the vein of Lê `20, which allows us to efficiently treat small and large scales simultaneously. By tuning the local scale as a function of ‖g‖_Bα∞,∞ we are able to obtain the necessary continuity of the semi-group and stability estimates to show unique ergodicity for all g∈ Bα∞,∞(ℝd). We believe that these local-global sewing lemmas may be of independent interest.