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Poisson process and sharp constants in Lp and Schauder estimates for a class of degenerate Kolmogorov operators
L. Marino</name> <arxiv:affiliation>LaMME</arxiv:affiliati · 2021-07-13 · via math.PR updates on arXiv.org

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].

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