Strong solutions to SDEs with singular drifts driven by fractional Brownian motions

TL;DR

This paper proves strong solutions and stochastic flows for SDEs with singular, integrable drifts driven by fractional Brownian motions with H<1/2, extending classical results to the fractional case.

Contribution

It establishes strong well-posedness and stochastic flow existence for SDEs with singular drifts driven by fractional Brownian motions, extending prior work to the fractional setting.

Findings

Proves strong solutions for SDEs with singular drifts driven by fractional Brownian motions.

Establishes existence of stochastic flows of Sobolev diffeomorphisms for these SDEs.

Extends classical results to the fractional Brownian motion case.

Abstract

In this paper, we establish the strong well-posedness of SDEs with merely integrable time-dependent drifts driven by fractional Brownian motions with Hurst parameter H<1/2. Our result holds over the entire subcritical regime and can be regarded as an extension of (Krylov and Rockner, Probab. Theory Relat. Fields, 131(2): 154-196 (2005)) to the fractional case. Furthermore, we prove the existence of stochastic flows of Sobolev diffeomorphisms for this class of SDEs, which generalizes a result in (Mohammed et al., Ann. Probab. 43, 1535-1576 (2015)). The approach adopted in our work is based on a compactness criterion for random fields in Wiener spaces.