Weak existence for SDEs with singular drifts and fractional Brownian or Levy noise beyond the subcritical regime

TL;DR

This paper establishes weak existence of solutions for multidimensional SDEs with singular drifts driven by fractional Brownian or Levy noise, extending previous results beyond the subcritical regime and demonstrating optimality of conditions.

Contribution

It extends weak existence results for SDEs with singular drifts driven by fractional Brownian or Levy noise beyond the subcritical regime, and constructs a counterexample for optimality.

Findings

Weak solutions exist under optimal integrability conditions.

Counterexample shows the conditions are sharp.

Strong solutions exist in one dimension under the same conditions.

Abstract

We study a multidimensional stochastic differential equation with additive noise: \[ d X_t=b(t, X_t) dt +d \xi_t, \] where the drift $b$ is integrable in space and time, and $\xi$ is either a fractional Brownian motion or a L\'evy process. We show weak existence of solutions to this equation under the optimal condition on integrability indices of $b$, going beyond the subcritical Krylov--R\"ockner (Prodi--Serrin--Ladyzhenskaya) regime. This extends the recent results of Krylov (2020) to the fractional Brownian and L\'evy cases. We also construct a counterexample to demonstrate the optimality of this condition. In the one-dimensional case, we show the existence of a strong solution under the same condition. Our methods are built upon a version of the stochastic sewing lemma of L\^e and the John--Nirenberg inequality.