Singular kinetic equations and applications

TL;DR

This paper develops a paracontrolled calculus framework for singular kinetic equations, establishing global well-posedness and solving related stochastic differential equations without renormalization.

Contribution

It introduces a kinetic paracontrolled calculus, proves well-posedness for linear and nonlinear singular kinetic equations, and addresses stochastic differential equations with singular drifts.

Findings

Global well-posedness for linear singular kinetic equations

No renormalization needed for Gaussian singular terms

Solution of martingale problem for distribution-dependent SDEs

Abstract

In this paper we study singular kinetic equations on $\mathbb{R}^{2d}$ by the paracontrolled distribution method introduced in \cite{GIP15}. We first develop paracontrolled calculus in the kinetic setting, and use it to establish the global well-posedness for the linear singular kinetic equations under the assumptions that the products of singular terms are well-defined. We also demonstrate how the required products can be defined in the case that singular term is a Gaussian random field by probabilistic calculation. Interestingly, although the terms in the zeroth Wiener chaos of regularization approximation are not zero, they converge in suitable weighted Besov spaces and no renormalization is required. As applications the global well-posedness for a nonlinear kinetic equation with singular coefficients is obtained by the entropy method. Moreover, we also solve the martingale problem for nonlinear kinetic distribution dependent stochastic differential equations with singular drifts.