Harnack estimates for nondivergence kinetic equations

Abstract

We establish a weak Harnack inequality for nonnegative strong supersolutions and a Harnack inequality for nonnegative strong solutions of the nondivergence-form kinetic equation \[ \partial_tu+v\cdot\nabla_xu+a(t,x,v):\nabla_v^2u +b(t,x,v)\cdot\nabla_vu+c(t,x,v)u=0, \] where the coefficients $a$, $b$, and $c$ are merely Borel measurable, the matrix $a$ is uniformly elliptic, and $b$ and $c$ are bounded. As a consequence, we derive interior kinetic Hölder estimates for strong solutions. Our approach develops a probabilistic kinetic analogue of the Krylov--Safonov method. The main new ingredient is a quantitative version of Krylov's estimate for kinetic Itô processes with progressively measurable coefficients, which in turn yields the global existence of weak solutions to the associated kinetic SDE with bounded measurable coefficients. These results appear to be the first Harnack and Hölder regularity theory for kinetic equations with merely measurable coefficients.