One type of singular perturbations of a multidimensional stable process

M. M. Osypchuk, M. I. Portenko
Theory of Stochastic Processes
Vol.19 (35), no.2, 2014, pp.42-51
A semigroup of linear operators on the space of all continuous bounded functions given on a $d$-dimensional Euclidean space $\mathbb{R}^d$ is constructed such that its generator can be written in the following form $\mathbf{A}+q(x)\delta_S(x)\mathbf{B}_\nu,$ where $\mathbf{A}$ is the generator of a symmetric stable process in $\mathbb{R}^d$ (that is, a pseudo-differential operator whose symbol is given by $(-c|\xi|^\alpha)_{\xi\in\mathbb{R}^d}$, parameters $c>0$ and $\alpha\in(1,2]$ are fixed); $\mathbf{B}_\nu$ is the operator with the symbol $(2ic|\xi|^{\alpha-2}(\xi,\nu))_{\xi\in\mathbb{R}^d}$ ($i=\sqrt{-1}$ and $\nu\in\mathbb{R}^d$ is a fixed unit vector); $S$ is a hyperplane in $\mathbb{R}^d$ that is orthogonal to $\nu$; $(\delta_S(x))_{x\in\mathbb{R}^d}$ is a generalized function whose action on a test function consists in integrating the latter one over $S$ (with respect to Lebesgue measure on $S$); and $(q(x))_{x\in S}$ is a given bounded continuous function with real values. This semigroup is generated by some kernel that can be given by an explicit formula. However, there is no Markov process in $\mathbb{R}^d$ corresponding to this semigroup because it does not preserve the property of a function to take on only non-negative values.