Kinetic SDEs with subcritical distributional drifts

Abstract In this paper we study the well-posedness of the kinetic stochastic differential equation (SDE) in R 2 d ( d ⩾ 2 ) ${\mathbb{R}}^{2d}\left(d{\geqslant}2\right)$ driven by Brownian motion: d X t = V t d t , d V t = b ( t , X t , V t ) d t + 2 d W t , $$\mathrm{d}{X}_{t}={V}_{t}\mathrm{d}t, \mathrm{d}{V}_{t}=b\left(t,{X}_{t},{V}_{t}\right)\mathrm{d}t+\sqrt{2}\mathrm{d}{W}_{t},$$ where the subcritical distribution-valued drift b belongs to the weighted anisotropic Hölder space L T q b C a α b ( ρ κ ) ${\mathbb{L}}_{T}^{{q}_{b}}{\mathbf{C}}_{\boldsymbol{a}}^{{\alpha }_{b}}\left({\rho }_{\kappa }\right)$ with parameters α b ∈ (−1, 0), q b ∈ 2 1 + α b