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 [N. V. Krylov and E. Priola, Arch. Ration. Mech. Anal. 225 (2017)].
Marino, L., Priola, E., Menozzi, S. (2022). Poisson process and sharp constants in Lp and Schauder estimates for a class of degenerate Kolmogorov operators. STUDIA MATHEMATICA, 267(3), 321-346 [10.4064/sm210819-13-4].
Poisson process and sharp constants in Lp and Schauder estimates for a class of degenerate Kolmogorov operators
Marino, L;
2022
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 [N. V. Krylov and E. Priola, Arch. Ration. Mech. Anal. 225 (2017)].
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