Well-posedness of the martingale problem for non-local perturbations of Lévy-type generators (1708.04467v1)

Abstract: 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}\backslash{0}}[f(x+y)-f(x)-\mathbf{1}{\alpha>1}\mathbf{1}{{|y|\le1}}y\cdot\nabla f(x)]M(t,x,dy), ] where $M(t,x,\cdot)$ is a L\'evy 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|^{\beta}M(t,x,dy)<\infty ] for some $0<\beta<\alpha$, then the martingale problem for $\mathcal{L}_{t}$ is well-posed.