Malliavin Calculus and Stochastic Differential Equations

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.