Weak well-posedness for degenerate SDEs driven by L\'evy processes

TL;DR

This paper investigates the weak well-posedness of degenerate SDEs driven by Lévy processes, focusing on the propagation of noise through systems with Hölder continuous coefficients and establishing regularity conditions and density estimates.

Contribution

It provides new insights into the conditions ensuring weak well-posedness for degenerate Lévy-driven SDEs and characterizes sharp regularity exponents for specific dynamics.

Findings

Characterization of regularity exponents for weak well-posedness

Counterexamples illustrating limits of regularity conditions

Derivation of Krylov-type estimates for solution densities

Abstract

In this article, we study the effects of the propagation of a non-degenerate L\'evy noise through a chain of deterministic differential equations whose coefficients are H\"older continuous and satisfy a weak H\"ormander-like condition. In particular, we assume some non-degeneracy with respect to the components which transmit the noise. Moreover, we characterize, for some specific dynamics, through suitable counterexamples , the almost sharp regularity exponents that ensure the weak well-posedness for the associated SDE. As a by-product of our approach, we also derive some Krylov-type estimates for the density of the weak solutions of the considered SDE.