Hypoelliptic diffusions with singular drifts

TL;DR

This paper proves the existence and uniqueness of solutions for a class of stochastic differential equations with degenerate, hypoelliptic diffusions and singular drifts on Carnot groups, extending classical results.

Contribution

It establishes well-posedness for hypoelliptic SDEs with singular drifts on Carnot groups, broadening the scope beyond classical non-degenerate cases.

Findings

Unique strong solutions exist for a large class of singular drifts.

Generalizes classical well-posedness results to hypoelliptic diffusions.

Bridges the gap between ODE theory and existing stochastic results.

Abstract

We establish the well-posedness of stochastic differential equations possessing degenerate diffusions and singular drifts. We prove that SDEs defined on the homogeneous Carnot group, whose hypoelliptic diffusion part is given by the horizontal Brownian motion, admit a unique strong solution for a large class of singular drifts. It considerably generalizes the classical well-posedness results of singular SDEs with non-degenerate diffusions. It also provides an intermediate result between the Cauchy-Lipschitz theorem in ordinary differential equations and the result proved by Krylov and R\"ockner [38], which states the well-posedness of SDEs with the additive noise and singular drifts.