Kinetic SDEs with subcritical distributional drifts

In this paper we study the well-posedness of the kinetic stochastic differential equation (SDE) in R²ᵈ(d≥2) driven by Brownian motion: d Xₜ=Vₜ d t,\ d Vₜ=b(t,Xₜ,Vₜ) d t+√2 d Wₜ, where the subcritical distribution-valued drift b belongs to the weighted anisotropic Hölder space LTqb Cₐαb(ρ_κ) with parameters αb∈(-1,0), qb∈(2/1+αb,∞], κ∈[0,1+αb) and ÷ᵥ b is bounded. We establish the well-posedness of weak solutions to the associated integral equation: Xₜ=X₀+∫₀ᵗ Vₛ d s,\ Vₜ=V₀+limn→∞∫₀ᵗ bₙ(s,Xₛ,Vₛ) d+√2Wₜ, where bₙ:=b*Γₙ denotes the mollification of b and the limit is taken in the L²-sense. As an application, we discuss examples of b involving Gaussian random fields.