Poisson process and sharp constants in Lp and Schauder estimates for a class of degenerate Kolmogorov operators

TL;DR

This paper establishes the stability of Schauder and Sobolev estimates for a class of degenerate Kolmogorov operators under second-order perturbations, using a Poisson process-based perturbative approach.

Contribution

It proves that key regularity estimates remain valid for perturbed degenerate operators, extending previous results to a broader class of operators.

Findings

Schauder and Sobolev estimates are stable under certain perturbations.

The approach uses a Poisson process-based perturbative technique.

Results apply to operators satisfying the Kalman hypoellipticity condition.

Abstract

We consider a possibly degenerate Kolmogorov-Ornstein-Uhlenbeck operator of the form L = Tr(BD 2) + Az, D , where A, B are N x N matrices, z $\in$ 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 + Tr(S(t)D 2) where S(t) is a non-negative N x N matrix depending continuously on t $\ge$ 0. Our approach relies on the perturbative technique based on the Poisson process introduced in [15].