Poisson process and sharp constants in \(L^p\) and Schauder estimates for a class of degenerate Kolmogorov operators. (English) Zbl 1498.35121
Summary: We consider a possibly degenerate Kolmogorov Ornstein-Uhlenbeck operator of the form \(L=\mathrm{Tr}(BD^2)+\langle Az,D \rangle \), 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 \(L+\mathrm{Tr}(S(t) D^2)\) 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, No. 3, 1089–1126 (2017; Zbl 1375.35551)].
MSC:
| 35B45 | A priori estimates in context of PDEs |
| 35H10 | Hypoelliptic equations |
| 35K15 | Initial value problems for second-order parabolic equations |
| 35K65 | Degenerate parabolic equations |
| 35K70 | Ultraparabolic equations, pseudoparabolic equations, etc. |
| 60J76 | Jump processes on general state spaces |
Keywords:
degenerate Ornstein-Uhlenbeck operators; multidimensional parabolic equations; \(L^p\) and Sobolev-space estimates; Schauder estimates; Poisson processCitations:
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