Kinetic SDEs with subcritical distributional drifts

Abstract

In this paper we study the well-posedness of the kinetic stochastic differential equation (SDE) in ${\mathbb{R}}^{2d}(d\ge 2)$ driven by Brownian motion: $$\mathrm{d}{X}_t=V_t\mathrm{d}t,\mathrm{d}{V}_t=b(t,X_t,V_t)\mathrm{d}t+\sqrt{2}\mathrm{d}{W}_t,$$ where the subcritical distribution-valued drift $b$ belongs to the weighted anisotropic Hölder space ${\mathbb{L}}_T^{q_b}{\mathbf{C}}_{\mathbf{a}}^{{\alpha }_b}({\rho }_{\kappa })$ with parameters ${\alpha }_b\in (-1,0)$, $q_b\in (\frac{2}{1+{\alpha }_b},\infty ]$, $\kappa \in [0,1+{\alpha }_b)$ and ${÷}_{v}b$ is bounded. We establish the well-posedness of weak solutions to the associated integral equation: $$X_t=X_0+{\int }_0^t{V}_s\mathrm{d}s,V_t=V_0+\underset{n\to \infty }{\lim }{\int }_0^t{b}_n(s,X_s,V_s)\mathrm{d}+\sqrt{2}{W}_t,$$ where $b_n:=b\ast {\Gamma }_n$ denotes the mollification of $b$ and the limit is taken in the $L^2$-sense. As an application, we discuss examples of $b$ involving Gaussian random fields.