Poisson process and sharp constants in $L^p$ and Schauder estimates for a class of degenerate Kolmogorov operators
Volume 267 / 2022
Abstract
We consider a possibly degenerate Kolmogorov Ornstein–Uhlenbeck operator of the form $L={\rm Tr}(BD^2)+\langle Az,D\rangle $, where $A$, $B $ are $N\times N $ matrices, $z \in \mathbb R^N$, $N\ge 1 $, which satisfy the Kalman condition which is equivalent to the hypoellipticity condition. We prove the following stability result: the Schauder and Sobolev estimates associated with the corresponding parabolic Cauchy problem remain valid, with the same constant, for the parabolic Cauchy problem associated with a second order perturbation of $L$, namely for $L+{\rm Tr}(S(t) D^2) $ where $S(t)$ is a non-negative definite $N\times N $ matrix depending continuously on $t \in [0,T]$. Our approach relies on the perturbative technique based on the Poisson process introduced in [N. V. Krylov and E. Priola, Arch. Ration. Mech. Anal. 225 (2017)].
