We consider a possibly degenerate Kolmogorov Ornstein–Uhlenbeck operator of the form L = Tr(BD 2 ) + A z \cdot D, where A, B are N × N matrices, z ∈ R^N , 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)D 2 ) 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 [N. V. Krylov and E. Priola, Arch. Ration. Mech. Anal. 225 (2017)].
Poisson process and sharp constants in Lp and Schauder estimates for a class of degenerate Kolmogorov operators
Enrico Priola
Membro del Collaboration Group
2022-01-01
Abstract
We consider a possibly degenerate Kolmogorov Ornstein–Uhlenbeck operator of the form L = Tr(BD 2 ) + A z \cdot D, where A, B are N × N matrices, z ∈ R^N , 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)D 2 ) 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 [N. V. Krylov and E. Priola, Arch. Ration. Mech. Anal. 225 (2017)].I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
