Stochastic equations with singular drift driven by fractional Brownian motion
Oleg Butkovsky, Khoa Lê, Leonid Mytnik
TL;DR
This work analyzes stochastic differential equations driven by fractional Brownian motion with singular drifts, establishing weak existence under a fractional Krylov–Röckner-type condition and proving optimality via counterexamples. It extends the SDE framework to drift terms that are measures or Besov-space functions, linking regularized solutions to measure-valued formulations through local time and occupation densities. A novel stochastic sewing approach, including a Rosenthal-type refinement and random controls, enables sharp bounds on drift integrals and local time construction, avoiding Girsanov-type methods. The results uncover regularization-by-noise phenomena for fractional noise, provide strong well-posedness in 1D under improved thresholds, and offer a robust toolkit potentially applicable to SPDEs and other rough-noise settings.
Abstract
We consider stochastic differential equation $$ d X_t=b(X_t) dt +d W_t^H, $$ where the drift $b$ is either a measure or an integrable function, and $W^H$ is a $d$-dimensional fractional Brownian motion with Hurst parameter $H\in(0,1)$, $d\in\mathbb{N}$. For the case where $b\in L_p(\mathbb{R}^d)$, $p\in[1,\infty]$ we show weak existence of solutions to this equation under the condition $$ \frac{d}p<\frac1H-1, $$ which is an extension of the Krylov-Röckner condition (2005) to the fractional case. We construct a counter-example 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 $H<\frac1{d+1}$. 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.