- Date: 22 Jul. (Thu.)
- Place: On the Web
- Time: 16:30 – 18:00
- Speaker: Lorenzo Marino (Université d’Evry Val d’Essonne, University of Pavia)
- Title: Weak Regularization by Degenerate Lévy noise and Applications
- Abstract:
In this talk, we briefly present the arguments of the PhD thesis: “Weak Regularization by Degenerate Lévy noise and Applications”. After a general introduction on the regularization by noises phenomena and the motivations behind this work, we start by showing the Schauder estimates, a useful analytical tool for the wellposedness of SDEs, for two different classes of integro-differential equations whose coefficients lie in suitable anisotropic Hölder spaces with multi-indices of regularity. The first one focuses on non-linear dynamics controlled by an α-stable operator acting only on the first component. To deal with the non-linear perturbation, we also need some subtle controls on Besov norms. As an extension of the first one, we also present the Schauder estimates associated with a degenerate Ornstein-Uhlenbeck operator driven by a larger class of α-stabletype operators, like the relativistic or Lamperti stable ones. The proof of this result relies instead on a precise analysis of the behaviour of the associated Markov semigroup between anisotropic Hölder spaces and some interpolation techniques. Exploiting a backward parametrix approach, we finally prove the weak wellposedness of the associated degenerate chain of SDEs. As a by-product of our method, Krylov-type estimates on the canonical solution process are also presented. Time permitting, we conclude by showing through suitable counter-examples that there exists an (almost) sharp threshold for the regularity exponents that ensure the weak well-posedness for the SDE.
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