Poisson process and sharp constants in $L^p$ and Schauder estimates for a class of degenerate Kolmogorov operators

Abstract

We consider a possibly degenerate Kolmogorov-Ornstein-Uhlenbeck operator of the form L = Tr(BD2) + 〈Az, D〉, where A, B are N × N matrices, z ∈ RN , N ≥ 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 + Tr(S(t)D2) where S(t) is a non-negative definite N × N matrix depending continuously on t ∈ [0, T ]. Our approach relies on the perturbative technique based on the Poisson process introduced in [15].