Kinetic SDEs with subcritical distributional drifts

Abstract

In this paper we study the well-posedness of the kinetic stochastic differential equation (SDE) in $\mathbb R^{2d}(d\geq2)$ driven by Brownian motion: $$\mathord{{\rm d}} X_t=V_t\mathord{{\rm d}} t,\ \mathord{{\rm d}} V_t=b(t,X_t,V_t)\mathord{{\rm d}} t+\sqrt{2}\mathord{{\rm d}} W_t,$$ where the subcritical distribution-valued drift $b$ belongs to the weighted anisotropic H\"{o}lder space $\mathbb L_T^{q_b}\mathbf C_{\boldsymbol{a}}^{\alpha_b}(\rho_\kappa)$ with parameters $\alpha_b\in(-1,0)$, $q_b\in(\frac{2}{1+\alpha_b},\infty]$, $\kappa\in[0,1+\alpha_b)$ and $\div_v b$ is bounded. We establish the well-posedness of weak solutions to the associated integral equation: $$X_t=X_0+\int_0^t V_s\mathord{{\rm d}} s,\ V_t=V_0+\lim_{n\to\infty}\int_0^t b_n(s,X_s,V_s)\mathord{{\rm d}}+\sqrt{2}W_t,$$ where $b_n:=b*\Gamma_n$ 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.