In this seminar, I will briefly present the arguments of my PhD thesis: Weak regularization by degenerate Lévy noise and its applications.

After a general introduction on the regularization by noises phenomena and the motivations behind this work, I will start by establishing the Schauder estimates, a useful analytical tool often linked with the weak well-posedness 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 dynamics with non-linear drift, controlled by an α-stable operator acting only on the first component. To deal with the non-linear perturbation, I will also present some subtle duality results on Besov spaces. As an extension of the first one, I will show the Schauder estimates associated with a degenerate Ornstein-Uhlenbeck operator driven by a larger class of α-stable type 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. We will then move back to the stochastic framework. Exploiting a backward parametrix approach, I will finally prove the weak well-posedness 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. Moreover, I will show through suitable counter-examples that there exists an (almost) sharp threshold for the regularity exponents that ensure the weak well-posedness for the SDE. Time permitting, I will conclude with an application of the previous results for a class of degenerate diffusive Ornstein-Uhlenbeck operators. In particular, we will see that the constants appearing in some well-known controls involving this type of operators, such as the Schauder or the Sobolev estimates, are stable under perturbations of the diffusive matrix.