Perturbation of an isotropic $\alpha$-stable stochastic process by a pseudo-gradient with a generalized coefficient

Abstract

The article is devoted to the perturbation of an isotropic $\alpha$-stable stochastic process in a finite-dimensional Euclidean space by a pseudo-gradient operator multiplied by a delta-function on a hypersurface. This is analogous to the construction of some membrane in the phase space. Semigroup of operators on the space of continuous bounded functions is constructed. It has the infinitesimal generator (in some generalized sense) $c\Delta_\alpha+(q\delta_S\nu,\nabla_\beta)$, where $c$ is some positive constant, $\Delta_\alpha$ is the fractional Laplacian of the order $\alpha$, $\delta_S$ is the delta-function on the hypersurface $S$, which has a normal vector $\nu$, $q$ is some continuous bounded function, $\nabla_\beta$ is a fractional gradient (pseudo-gradient), that is the pseudo-differentional operator defined by the symbol $i\lambda|\lambda|^{\beta-1}$. The order of the pseudo-gradient is less than $\alpha-1$. Some properties of the obtained semigroup are investigated. This semigroup defines a pseudo-process.