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 ∂tu+v·∇xu+a(t,x,v):∇v2u +b(t,x,v)·∇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.
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BibTeX
@misc{hao2026harnack,
title = {{Harnack estimates for nondivergence kinetic equations}},
author = {Zimo Hao and Xicheng Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15534},
eprint = {2609.15534},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15534},
note = {CC0},
}
RIS
TY - UNPB
TI - Harnack estimates for nondivergence kinetic equations
AU - Zimo Hao
AU - Xicheng Zhang
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15534
UR - https://arxiv.org/abs/2609.15534
AB - We establish a weak Harnack inequality for nonnegative strong supersolutions and a Harnack inequality for nonnegative strong solutions of the nondivergence-form kinetic equation ∂_tu+v·∇_xu+a(t,x,v):∇_v^2u +b(t,x,v)·∇_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.
ER -