Regularization effects of a noise propagating through a chain of differential equations: an almost sharp result

TL;DR

This paper analyzes how Brownian noise propagates through a chain of rough differential equations with weak Hörmander conditions, establishing near-optimal regularity criteria for well-posedness and deriving density estimates for solutions.

Contribution

It provides almost sharp regularity exponents ensuring weak well-posedness for SDEs with noise propagation in rough coefficient settings.

Findings

Characterized near-optimal regularity conditions for well-posedness.

Constructed counterexamples demonstrating sharpness of conditions.

Derived Krylov-type density estimates for solutions.

Abstract

We investigate the effects of the propagation of a non-degenerate Brownian noise through a chain of deterministic differential equations whose coefficients are rough and satisfy a weak like H{\"o}rmander structure (i.e. a non-degeneracy condition w.r.t. the components which transmit the noise). In particular we characterize, through suitable counterexamples , almost sharp regularity exponents that ensure that weak well posedness holds for the associated SDE. As a by-product of our approach, we also derive some density estimates of Krylov type for the weak solutions of the considered SDEs.