Uniqueness of stable processes with drift

Abstract:

Suppose that $d\geq 1$ and $\alpha \in (1, 2)$. Let $\mathcal {L}^b=-(-\Delta )^{\alpha /2} + b \cdot \nabla$, where $b$ is an $\mathbb {R}^d$-valued measurable function on $\mathbb {R}^d$ belonging to a certain Kato class of the rotationally symmetric $\alpha$-stable process $Y$ on $\mathbb {R}^d$. We show that the martingale problem for $(\mathcal {L}^b, C^\infty _c(\mathbb {R}^d))$ has a unique solution for every starting point $x\in \mathbb {R}^d$. Furthermore, we show that the stochastic differential equation $\mathrm {d} X_t=\mathrm {d} Y_t+b(X_t)\mathrm {d} t$ with $X_0=x$ has a unique weak solution for every $x\in \mathbb {R}^d$.