A note on weak existence for singular SDEs

TL;DR

This paper extends the weak existence results for singular stochastic differential equations (SDEs) with integrable drifts by introducing a new technique based on a partial Zvonkin transform, accommodating more general growth conditions.

Contribution

It provides a variant of Krylov's weak existence result for SDEs, using an alternative proof method that handles drifts with growth at infinity and in local Lebesgue spaces.

Findings

Establishes weak existence of solutions under broader conditions.

Introduces a partial Zvonkin transform technique.

Allows for drifts with growth at infinity.

Abstract

Recently Krylov established weak existence of solutions to SDEs for integrable drifts in mixed Lebesgue spaces, whose exponents satisfy the condition $1/q+d/p\leq 1$, thus going below the celebrated Ladyzhenskaya-Prodi-Serrin condition. We present here a variant of such result, whose proof relies on an alternative technique, based on a partial Zvonkin transform; this allows for drifts with growth at infinity and/or in uniformly local Lebesgue spaces.