Properties of single layer potentials for a pseudo- differential equation related to a linear transformation of a rotationally invariant stable stochastic process

Abstract

This article is aimed at determining existence conditions of single layer potentials for pseudo-differential equations related to some linear transformations of a rotationally invariant stable stochastic process in a multidimensional Euclidean space and investigating their properties as well. The carrier surface of the potential is smooth enough. In this article, we consider two main cases: the first, when this surface is bounded and closed; the second, when it is unbounded, but could be presented by an explicit equation in some coordinate system. The density of this potential is a continuous function. It is bounded with respect to the spatial variable and, probably, has an integrable singularity with respect to the time variable at zero. Classic properties of this potential, including a jump theorem of the action result of some operator (an analog of the co-normal differential) at its surface points, considered.

A rotationally invariant $\alpha$-stable stochastic process in $\mathbb{R}^d$ is a L\'{e}vy process with the characte\-ristic function of its value in the moment of time $t>0$ defined by the expression $\exp\{-tc|\xi|^\alpha\}$, $\xi\in\mathbb{R}^d$, where $\alpha\in(0,2]$, $c>0$ are some constants. If $\alpha=2$ and $c=1/2$, we get Brownian motion and classic theory of potential. There are many different results in this case. The situation of $\alpha\in(1,2)$ is considered in this paper. We study constant and invertible linear transformations of the rotationally invariant $\alpha$-stable stochastic process. The related pseudo-differential equation is the parabolic equation of the order $\alpha$ of the ``heat'' type in which the operator with respect to the spatial variable is the process generator. The single layer potential is constructed in the same way as the single layer potential for the heat equation in the classical theory of potentials. That is, we use the fundamental solution of the equation, which is the transition probability density of the related process. In our theory, the role of the gradient operator is performed by some vector pseudo-differential operator of the order $\alpha-1$. We have already studied the following main properties of the single layer potentials: the single layer potential is a solution of the relating equation outside of the carrier surface and the jump theorem is held. These properties can be useful to solving initial boundary value problems for the considered equations.