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

TL;DR

The paper addresses strong well-posedness of the SDE $\mathrm{d}X_t=b(t,X_t)\mathrm{d}t+\mathrm{d}B_t^H$ with a singular drift $b\in L^p_xL^q_t$ under the subcritical condition $\frac{1}{q}+\frac{Hd}{p}<1-H$ for a fractional Brownian driver with $H<\tfrac12$. It introduces a density-based Malliavin calculus framework and a compactness criterion in Wiener space, combined with a Girsanov transform for fBm, to construct strong solutions as limits of smooth approximations and to establish pathwise uniqueness. It also proves the existence of a stochastic flow of Sobolev diffeomorphisms associated with the SDE, showing Sobolev regularity of the flow and its inverse. The results extend Krylov–Röckner-type strong well-posedness to the fractional Brownian setting in the subcritical regime and provide new insights into regularization by fractional noise for non-martingale drivens, with potential PDE applications in transport and fluid dynamics.

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.

Theorems & Definitions (39)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Remark 1.4
• Remark 1.5
• Definition 1.6
• Definition 1.7
• Definition 1.8
• Proposition 2.1
• Proposition 2.2