Uniqueness of Stable Processes with Drift

Abstract

Suppose that $d\ge 1$ and $\alpha \in (1,2)$. Let $Y$ be a rotationally symmetric $\alpha$-stable process on ${\mathbb{R}}^d$ and $b$ a ${\mathbb{R}}^d$-valued measurable function on ${\mathbb{R}}^d$ belonging to a certain Kato class of $Y$. We show that $\rd X^b_t=\rd Y_t+b(X^b_t)\rd t$ with $X_0^b=x$ has a unique weak solution for every $x\in {\mathbb{R}}^d$. Let $\sL^b=-(-Δ)^{α/2} + b \cdot \nabla$, which is the infinitesimal generator of $X^b$. Denote by $C_c^{\infty }({\mathbb{R}}^d)$ the space of smooth functions on ${\mathbb{R}}^d$ with compact support. We further show that the martingale problem for $(\sL^b, C^\infty_c(\R^d))$ has a unique solution for each initial value $x\in {\mathbb{R}}^d$.

Cited by in corpus (3)

• Hölder estimates for nonlocal-diffusion equations with drifts
• On a Class of Stochastic Differential Equations With Jumps and Its Properties
• On Weak Solutions of SDEs with Singular Time-Dependent Drift and Driven by Stable Processes