Well-posedness of the martingale problem for non-local perturbations of L\'evy-type generators

Let $L$ be a L'evy-type generator whose L'evy measure is controlled from below by that of a non-degenerate $\alpha$-stable ($0<\alpha <2$) process. In this paper, we study the martingale problem for the operator ${\mathcal{L}}_t=L+K_t$, with $K_t$ being a time-dependent non-local operator defined by $$K_{t}f(x):={\int }_{{\mathbb{R}}^d\mathrm{\backslash }\{0\}}[f(x+y)-f(x)-{\mathbf{1}}_{\alpha >1}{\mathbf{1}}_{\{\mathrm{\mid }y\mathrm{\mid }\le 1\}}y\cdot \mathrm{\nabla }f(x)]M(t,x,dy),$$ where $M(t,x,\cdot )$ is a L'evy measure on ${\mathbb{R}}^d\mathrm{\backslash }\{0\}$ for each $(t,x)\in $\mathbb{R}$_+ \times $\mathbb{R}$^{d}$. We show that if $$\underset{t\ge 0,x\in {\mathbb{R}}^d}{\sup }{\int }_{{\mathbb{R}}^d\mathrm{\backslash }\{0\}}1\wedge \mathrm{\mid }y{\mathrm{\mid }}^{\beta }M(t,x,dy)<\mathrm{\infty }$$ for some $0<\beta <\alpha$, then the martingale problem for ${\mathcal{L}}_t$ is well-posed.