SDEs with subcritical Lebesgue--H\"{o}lder drift and driven by $\alpha$-stable processes

TL;DR

This paper establishes the existence and uniqueness of solutions for certain stochastic differential equations with irregular drift driven by stable processes, expanding understanding of their well-posedness under various conditions.

Contribution

It provides the first comprehensive analysis of SDEs with subcritical Lebesgue--Hölder drift driven by α-stable processes, including weak and strong solution criteria.

Findings

Weak well-posedness for specific parameter ranges.

Pathwise and Davie's type uniqueness results.

Counterexample illustrating limits of uniqueness.

Abstract

We obtain the unique weak and strong solvability for time inhomogeneous stochastic differential equations with the drift in subcritical Lebesgue--H\"{o}lder spaces $L^p([0,T];{\mathcal C}_b^{\beta}({\mathbb R}^d;{\mathbb R}^d))$ and driven by $\alpha$-stable processes for $\alpha\in (0,2)$. The weak well-posedness is derived for $\beta\in (0,1)$, $\alpha+\beta>1$ and $p>\alpha/(\alpha+\beta-1)$ through Prohorov's theorem, Skorohod's representation and the regularity estimates of solutions for a class of fractional parabolic partial differential equations. The pathwise uniqueness and Davie's type uniqueness are proved for $\beta>1-\alpha/2$ by using It\^{o}--Tanaka's trick. Moreover, we give a counterexample to the pathwise uniqueness for the supercritical Lebesgue--H\"{o}lder drifts to explain the present result is sharp.