  After a general introduction about the regularization by noise phenomenon in
the degenerate setting, the first part of this PhD 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 H\"ormander-like condition, whose
coefficients lie in suitable anisotropic H\"older spaces with multi-indices of
regularity. The first class 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 regularizing 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. Exploiting a backward parametrix
approach, the second part of this work aims at establishing the weak
well-posedness for a degenerate chain of SDEs driven by the same class of
stable-like processes, under the assumptions of the minimal H\"older regularity
on the coefficients. As a by-product of our method, we also present Krylov-type
estimates of independent interest for the associated canonical process.
Finally, we show through suitable counter-examples the existence of an (almost)
sharp threshold on the regularity exponents ensuring the weak well-posedness
for the SDE. In connection with some possible applications to kinetic dynamics
with friction, we conclude by investigating the stability of second-order
perturbations for degenerate Kolmogorov operators in Lp and H\"older norms.
