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This article is cited in 1 scientific paper (total in 1 paper)
One type of singular perturbations of a multidimensional stable process
M. M. Osypchuka, M. I. Portenkob a Vasyl Stefanyk Precarpathian National University
b Institute of Mathematics of Ukrainian National Academy of Sciences
Abstract:
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.
Keywords:
Markov process, Wiener process, symmetric stable process, singular perturbation, pseudo-differential operator, pseudo-differential equation, semigroup of operators, transition probability density.
Citation:
M. M. Osypchuk, M. I. Portenko, “One type of singular perturbations of a multidimensional stable process”, Theory Stoch. Process., 19(35):2 (2014), 42–51
Linking options:
https://www.mathnet.ru/eng/thsp12 https://www.mathnet.ru/eng/thsp/v19/i2/p42
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