On single-layer potentials for a class of pseudo-differential equations related to linear transformations of a symmetric $\alpha$-stable stochastic process

In this article an arbitrary invertible linear transformations of a symmetric $\alpha$-stable stochastic process in $d$-dimensional Euclidean space $\mathbb{R}^d$ are investigated. The result of such transformation is a Markov process in $\mathbb{R}^d$ whose generator is the pseudo-differential oper...

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Bibliographic Details
Main Authors: Kh.V. Mamalyha, M.M. Osypchuk
Format: Article
Language:English
Published: Vasyl Stefanyk Precarpathian National University 2019-12-01
Series:Karpatsʹkì Matematičnì Publìkacìï
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Online Access:https://journals.pnu.edu.ua/index.php/cmp/article/view/2114
Description
Summary:In this article an arbitrary invertible linear transformations of a symmetric $\alpha$-stable stochastic process in $d$-dimensional Euclidean space $\mathbb{R}^d$ are investigated. The result of such transformation is a Markov process in $\mathbb{R}^d$ whose generator is the pseudo-differential operator defined by its symbol $(-(Q\xi,\xi)^{\alpha/2})_{\xi\in\mathbb{R}^d}$ with some symmetric positive definite $d\times d$-matrix $Q$ and fixed exponent $\alpha\in(1,2)$. The transition probability density of this process is the fundamental solution of some parabolic pseudo-differential equation. The notion of a single-layer potential for that equation is introduced and its properties are investigated. In particular, an operator is constructed whose role in our consideration is analogous to that the gradient in the classical theory. An analogy to the classical theorem on the jump of the co-normal derivative of the single-layer potential is proved. This result can be applied for solving some boundary-value problems for the parabolic pseudo-differential equations under consideration. For $\alpha=2$, the process under consideration is a linear transformation of Brownian motion, and all the investigated properties of the single-layer potential are well known.
ISSN:2075-9827
2313-0210