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Theses

Weak regularization by degenerate Lévy noise and its applications

Abstract : After a general introduction about the regularization by noise phenomenon in the degenerate setting, the first part of this thesis focuses at establishing the Schauder estimates, a useful analytical tool to prove also the well-posedness of stochastic differential equations (SDEs), for two different classes of Kolmogorov equations under a weak Hörmander-like condition, whose coefficients lie in suitable anisotropic Hölder spaces with multi-indices of regularity.The first model considers a nonlinear system controlled by a symmetric ⍺-stable operator acting only on some components. Our method of proof relies on a perturbative approach based on forward parametrix expansions through Duhamel-type formulas. Due to the low regularising properties given by the degenerate setting, we also exploit some controls on Besov norms, in order to deal with the non-linear perturbation.As an extension of the first one, we also present Schauder estimates associated with a degenerate Ornstein-Uhlenbeck operator driven by a larger class of ⍺-stable-like operators, like the relativistic or the Lamperti stable one. The proof of this result relies instead on a precise analysis of the behaviour of the associated Markov semi-group between anisotropic Hölder spaces and some interpolation techniques.Exploiting a backward parametrix approach, the second part of this thesis aims at establishing the well-posedness in a weak sense of a degenerate chain of SDEs driven by the same class of ⍺-stable-like processes, under the assumptions of the minimal Hölder regularity on the coefficients. As a by-product of our method, we also presentKrylov-type estimates of independent interest for the associated canonical process.Finally, we emphasise through suitable counter-examples that there exists indeed an (almost) sharp threshold on the regularity exponents ensuring the weak well-posedness for the SDE.In connection with some mechanical applications for kinetic dynamics with friction, we conclude by investigating the stability of second-order perturbations for degenerate Kolmogorov operators in Lp and Hölder norms.
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Submitted on : Tuesday, January 25, 2022 - 5:38:19 PM
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Lorenzo Marino. Weak regularization by degenerate Lévy noise and its applications. General Mathematics [math.GM]. Université Paris-Saclay; Università degli studi di Milano - Bicocca, 2021. English. ⟨NNT : 2021UPASM031⟩. ⟨tel-03414121v2⟩

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