Well-posedness of the martingale problem for non-local perturbations of Lévy-type generators

Abstract

Let $L$ be a Lévy-type generator whose Lévy 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 \[ K_{t}f(x):=\int_{\mathbb{R}^{d}\backslash\{0\}}[f(x+y)-f(x)-\mathbf{1}_{α>1}\mathbf{1}_{\{|y|\le1\}}y\cdot\nabla f(x)]M(t,x,dy), \] where $M(t,x,\cdot )$ is a Lévy measure on ${\mathbb{R}}^d\backslash \{0\}$ for each $(t,x)\in {\mathbb{R}}_{+}\times {\mathbb{R}}^d$. We show that if \[ \sup_{t\geq0,x\in\mathbb{R}^{d}}\int_{\mathbb{R}^{d}\backslash\{0\}}1\wedge|y|^βM(t,x,dy)<\infty \] for some $0<\beta <\alpha$, then the martingale problem for ${\mathcal{L}}_t$ is well-posed.

25 pages