Uniqueness in law for stable-like processes of variable order

Abstract

Let $d\ge1$. Consider a stable-like operator of variable order \begin{align*} \mathcal{A}f(x) & =\int_{\mathbb{R}^{d} \backslash\{0\}} \left[f(x+h) -f(x) -\mathbf{1}_{\{|h|\le1\}}h \cdot\nabla f(x)\right]\frac{n(x,h)}{|h|^{d+\alpha(x)}} \mathrm{d}h, \end{align*} where $0<\inf_{x}\alpha(x) \le \sup_{x}\alpha(x)<2$ and $n(x,h)$ satisfies \[ n(x,h)=n(x,-h),\quad0<\kappa_{1}\le n(x,h)\le\kappa_{2},\quad\forall x,h\in \mathbb{R}^{d}, \] 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 $\mathcal{A}$.