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Heat kernel estimates for kinetic SDEs with drifts being unbounded and in Kato's class

Chongyang Ren, Xicheng Zhang

Abstract

In this paper we investigate the existence and uniqueness of weak solutions for kinetic stochastic differential equations with Hölder diffusion and unbounded singular drifts in Kato's class. Moreover, we also establish sharp two-sided estimates for the density of the solution. In particular, the drift $b$ can be in the mixed $L^q_tL^{p_1}_{x_1}L^{p_2}_{x_2}$ space with $\frac2q+\frac{d}{p_1}+\frac{3d}{p_2}<1$. As an application, we show the existence and uniqueness of weak solution to a second order singular interacting particle system in ${\mathbb R}^{d N}$.

Heat kernel estimates for kinetic SDEs with drifts being unbounded and in Kato's class

Abstract

In this paper we investigate the existence and uniqueness of weak solutions for kinetic stochastic differential equations with Hölder diffusion and unbounded singular drifts in Kato's class. Moreover, we also establish sharp two-sided estimates for the density of the solution. In particular, the drift can be in the mixed space with . As an application, we show the existence and uniqueness of weak solution to a second order singular interacting particle system in .
Paper Structure (7 sections, 21 theorems, 233 equations)

This paper contains 7 sections, 21 theorems, 233 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Solvability of kinetic PDEs with drifts in Kato class
  4. Well-posedness for SDE with singular drift
  5. Two-sided estimates of heat kernel
  6. Proof of the upper bound
  7. Proof of the lower bound

Key Result

Theorem 1.4

Under (H$^\sigma_b$), for each $s>0$ and ${\boldsymbol{x}}\in{\mathbb R}^{2d}$, there is a unique weak solution $({\mathbf X}_{t,s}({\boldsymbol{x}}))_{t\geqslant s}$ to SDE 1 starting from ${\boldsymbol{x}}$ at time $s$ in the sense of Definition weak. Moreover, if $|b_1|^\gamma\in {\mathbb K}_1$ where $\lambda_0,\lambda_1$ depend on $T, d, \gamma_0, \kappa_0, \kappa_1$, and $C_0,C_1$ depend on

Theorems & Definitions (45)

  • Definition 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Lemma 2.1
  • proof
  • Definition 2.2: Kato's class
  • Remark 2.3
  • Remark 2.4
  • ...and 35 more