Degenerate SDEs with Singular Drift and Applications to Heisenberg Groups

TL;DR

This paper develops new methods to prove existence and uniqueness of solutions for degenerate stochastic differential equations with singular drifts, applying these results to equations on generalized Heisenberg groups.

Contribution

It introduces a novel approach combining ultracontractivity and Krylov's estimate to handle degenerate SDEs with singular drifts, extending to Heisenberg groups.

Findings

Established Krylov's estimate for degenerate SDEs with singular drifts

Proved existence and pathwise uniqueness of solutions

Applied results to SDEs on generalized Heisenberg groups

Abstract

By using the ultracontractivity of a reference diffusion semigroup, Krylov's estimate is established for a class of degenerate SDEs with singular drifts, which leads to existence and pathwise uniqueness by means of Zvonkin's transformation. The main result is applied to singular SDEs on generalized Heisenberg groups.