Malliavin Matrix of Degenerate SDE and Gradient Estimate

TL;DR

This paper establishes the p-integrability of the inverse Malliavin matrix for certain degenerate SDEs under relaxed conditions, providing gradient estimates and proving the strong Feller property of the associated semigroup.

Contribution

It introduces new integrability results for the Malliavin matrix in degenerate SDEs without requiring full smoothness of coefficients.

Findings

Inverse Malliavin matrix is p-integrable under specified conditions

Uniform estimates for the Malliavin matrix are obtained

The semigroup is shown to be strong Feller

Abstract

In this paper, we prove that the inverse of Malliavin matrix is p integrable for a kind of degenerate stochastic differential equation under some conditions, which like to Hormander condition, but don't need all the coefficients of the SDE are smooth. Furthermore, we obtain a uniform estimation for Malliavin matrix, a gradient estimate, and prove that the semigroup generated by the SDE is strong Feller. Also some examples are given.