SDEs with subcritical Lebesgue--Hölder drift and driven by α-stable processes

2025/02/06 by Tian, Rongrong, Wei, Jinlong
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2502.03712

Abstract

We obtain the unique weak and strong solvability for time inhomogeneous stochastic differential equations with the drift in subcritical Lebesgue--Hölder spaces L^p([0,T];\mathcal C_b^β(\mathbb R^d;\mathbb R^d)) and driven by α-stable processes for α∈ (0,2). The weak well-posedness is derived for β∈ (0,1), α+β>1 and p>α/(α+β-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 β>1-α/2 by using Itô--Tanaka's trick. Moreover, we give a counterexample to the pathwise uniqueness for the supercritical Lebesgue--Hölder drifts to explain the present result is sharp.

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