Flow of diffeomorphisms for SDEs with unbounded H\"older continuous drift

TL;DR

This paper proves the existence of a global flow of diffeomorphisms for certain SDEs with unbounded H"older continuous drift, using a novel transformation and advanced elliptic estimates, and derives a derivative formula for the diffusion semigroup.

Contribution

It introduces a new method to establish stochastic flows for SDEs with unbounded H"older drift using a special drift transformation and elliptic estimates.

Findings

Existence of a global flow of diffeomorphisms for the SDE.

Development of a Bismut-Elworthy-Li type formula for derivatives.

Application of non-standard elliptic estimates in H"older spaces.

Abstract

We consider a SDE with a smooth multiplicative non-degenerate noise and a possibly unbounded Holder continuous drift term. We prove existence of a global flow of diffeomorphisms by means of a special transformation of the drift of Ito-Tanaka type. The proof requires non-standard elliptic estimates in Holder spaces. As an application of the stochastic flow, we obtain a Bismut-Elworthy-Li type formula for the first derivatives of the associated diffusion semigroup.