Kinetic SDEs with subcritical distributional drifts

In this paper we study the well-posedness of the kinetic stochastic differential equation (SDE) in ℝ^2d(d ≥ 2) driven by Brownian motion: dX_t = V_tdt, dV_t = b(t,X_t,V_t)dt+ ^√ 2dW_t, where the subcritical distribution-valued drift b belongs to the weighted anisotropic Hölder space L^q_T^bC_a^αb(p_k) with parameters α_b ∈ (−1,0), q_b ∈ ⁽₁₊²_αb,∞ ^] , k ∈ ^[0,1 + α_b⁾ and div_υb is bounded. We establish the well-posedness of weak solutions to the associated integral equation: t t X_t = X₀ + _∫ V_sds, V_t = V₀ + _nlim _→∞∫ b_n(s,X_s,V_s)ds+ ^√ 2W_t, 0 0 where b_n :=b∗Γ_n denotes the mollification of b and the limit is taken in the L²-sense. As an application, we discuss examples of binvolving Gaussian random fields.