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Stochastic equations with singular drift driven by fractional Brownian motion

Oleg Butkovsky, Khoa Lê and Leonid Mytnik

Vol. 6 (2025), No. 3, 857–912
Abstract

We consider the stochastic differential equation

dXt = b(Xt)dt + dWtH,

where the drift b is either a measure or an integrable function, and WH is a d-dimensional fractional Brownian motion with Hurst parameter H (0,1), d . For the case where b Lp(d), p [1,], we show weak existence of solutions to this equation under the condition

d p < 1 H 1,

which is an extension of the Krylov–Röckner condition (2005) to the fractional case. We construct a counterexample showing optimality of this condition. If b is a Radon measure, particularly the delta measure, we prove weak existence of solutions to this equation under the optimal condition H < 1 d+1. We also show strong well-posedness of solutions to this equation under certain conditions. To establish these results, we utilize the stochastic sewing technique and develop a new version of the stochastic sewing lemma.

Keywords
regularization by noise, fractional Brownian motion, stochastic sewing, weak existence, local times
Mathematical Subject Classification
Primary: 60G22, 60H10, 60H50
Milestones
Received: 29 March 2023
Revised: 24 April 2025
Accepted: 21 May 2025
Published: 21 June 2025
Authors
Oleg Butkovsky
Weierstrass Institute
Berlin
Germany
Institut für Mathematik
Humboldt-Universität zu Berlin
Berlin
Germany
Khoa Lê
School of Mathematics
University of Leeds
United Kingdom
Leonid Mytnik
Faculty of Data and Decision Sciences
Technion — Israel Institute of Technology
Haifa
Israel