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

Volume 267 / 2022

L. Marino, S. Menozzi, E. Priola Studia Mathematica 267 (2022), 321-346 MSC: Primary 35K10; Secondary 35K15, 60J76. DOI: 10.4064/sm210819-13-4 Published online: 8 September 2022


We consider a possibly degenerate Kolmogorov Ornstein–Uhlenbeck operator of the form $L={\rm 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+{\rm 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 (2017)].


  • L. MarinoInstitute of Mathematics
    Polish Academy of Sciences
    Śniadeckich 8
    00-656 Warszawa, Poland
  • S. MenozziDipartimento di Matematica
    Università di Pavia
    Via Adolfo Ferrata 5
    27100 Pavia, Italy
  • E. PriolaLaboratoire de Modélisation Mathématique
    d’Evry (LaMME), UMR CNRS 8071
    Université Paris-Saclay
    Université d’Evry Val d’Essonne
    23 Boulevard de France
    91037 Evry, France
    Laboratory of Stochastic Analysis
    Higher School of Economics (HSE)
    Pokrovsky Boulevard, 11
    Moscow, Russian Federation

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