On some perturbations of a stable process and solutions to the Cauchy problem for a class of pseudo-differential equations


  • M.M. Osypchuk Vasyl Stefanyk Precarpathian National University, 57 Shevchenka str., 76018, Ivano-Frankivsk, Ukraine https://orcid.org/0000-0001-6100-1654


stable process, Cauchy problem, pseudo-differential equation, transition probability density
Published online: 2015-07-03


A fundamental solution for some class of pseudo-differential equations is constructed by the method based on the theory of perturbations. We consider a symmetric $\alpha$-stable process in multidimensional Euclidean space. Its generator $\mathbf{A}$ is a pseudo-differential operator whose symbol is given by $-c|\lambda|^\alpha$, were the constants $\alpha\in(1,2)$ and $c>0$ are fixed. The vector-valued operator $\mathbf{B}$ has the symbol $2ic|\lambda|^{\alpha-2}\lambda$. We construct a fundamental solution of the equation $u_t=(\mathbf{A}+(a(\cdot),\mathbf{B}))u$ with a continuous bounded vector-valued function $a$.

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How to Cite
Osypchuk, M. On Some Perturbations of a Stable Process and Solutions to the Cauchy Problem for a Class of Pseudo-Differential Equations. Carpathian Math. Publ. 2015, 7, 101-107.