Malliavin Calculus and Stochastic Differential Equations

TL;DR

This paper explores the application of Malliavin calculus to stochastic differential equations with bounded Borel drift, establishing links between heat kernels and divergences, and providing explicit derivative estimates for solutions.

Contribution

It introduces a novel connection between heat kernels and Malliavin divergences, and derives explicit bounds for derivatives of SDE solutions with bounded Borel drift.

Findings

Established a link between heat kernels and iterated divergences in Malliavin calculus.

Derived explicit estimates for derivatives of SDE solutions in terms of the L-infinity norm of the drift.

Proved that the SDE generates a continuous flow of maps in Sobolev spaces.

Abstract

This paper is devoted to a study on SDEs with a bounded Borel drift b. We first remark that the original integration by parts formula due to P. Malliavin can be used to deal with derivatives with respect to space variables, then we obtain a link between the product of heat kernels and iterated divergences in Malliavin calculus. An explicit estimate for the derivative of solutions to SDE is obtained in terms of the L-infinity norm of b; as a result, we prove that the SDE defines a continuous flow of maps in Sobolev spaces.