Uniqueness in Law for Stable-Like Processes of Variable Order

Abstract

Let $$d\ge 1$$ . Consider a stable-like operator of variable order $$\begin{aligned} {\mathcal {A}}f(x)&=\int _{{\mathbb {R}}^{d}\backslash \{0\}} \left[ f(x+h)-f(x)-\mathbf {1}_{\{|h|\le 1\}}h\cdot \nabla f(x)\right] n(x,h)|h|^{-d-\alpha (x)}\mathrm{d}h, \end{aligned}$$ where $$0<\inf _{x}\alpha (x)\le \sup _{x}\alpha (x)<2$$ and n(x, h) satisfies $$\begin{aligned} n(x,h)=n(x,-h),\quad 0<\kappa _{1}\le n(x,h)\le \kappa _{2}, \quad \forall x,h\in {\mathbb {R}}^{d}, \end{aligned}$$ with $$\kappa _{1}$$ and $$\kappa _{2}$$ being some positive constants. Under some further mild conditions on the functions n(x, h) and $$\alpha (x)$$ , we show the uniqueness of solutions to the martingale problem for $${\mathcal {A}}$$ .