On Weak Solutions of SDEs with Singular Time-Dependent Drift and Driven by Stable Processes

TL;DR

This paper establishes the existence and uniqueness of weak solutions for a class of stochastic differential equations driven by stable processes with singular, time-dependent drift terms under certain integrability conditions.

Contribution

It introduces new conditions on the drift function that ensure weak solution uniqueness for SDEs driven by stable processes, extending previous results to more singular drifts.

Findings

Unique weak solutions exist under specified integrability conditions.

Conditions relate the drift's regularity to the stability index and dimension.

Results apply to non-degenerate stable processes with singular, time-dependent drifts.

Abstract

Let $d \ge 2$. In this paper, we study weak solutions for the following type of stochastic differential equation \[ dX_{t}=dS_{t}+b(s+t, X_{t})dt, \quad X_{0}=x, \] where $(s,x)\in \mathbb{R}_+ \times \mathbb{R}^{d}$ is the initial starting point, $b: \mathbb{R}_+ \times \mathbb{R}^{d} \to \mathbb{R}^{d}$ is measurable, and $S=(S_{t})_{t \ge 0}$ is a $d$-dimensional $\alpha$-stable process with index $\alpha \in (1,2)$. We show that if the $\alpha$-stable process $S$ is non-degenerate and $b \in L_{loc}^{\infty}(\mathbb{R}_{+};L^{\infty}(\mathbb{R}^{d}))+ L_{loc}^{q}(\mathbb{R}_{+};L^{p}(\mathbb{R}^{d}))$ for some $p,q>0$ with $d/ p+\alpha/q <\alpha-1$, then the above SDE has a unique weak solution for every starting point $(s,x)\in \mathbb{R}_+ \times \mathbb{R}^{d}$.