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Theory of Stochastic Processes, 2014, Volume 19(35), Issue 2, Pages 42–51 (Mi thsp12)  

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
Full-text PDF (305 kB) Citations (1)
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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.
Bibliographic databases:
Document Type: Article
MSC: Primary 47D06, 47G30; Secondary 60E07, 60G52
Language: English
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
Citation in format AMSBIB
\Bibitem{OsyPor14}
\by M.~M.~Osypchuk, M.~I.~Portenko
\paper One type of singular perturbations of a~multidimensional stable process
\jour Theory Stoch. Process.
\yr 2014
\vol 19(35)
\issue 2
\pages 42--51
\mathnet{http://mi.mathnet.ru/thsp12}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=3405382}
\zmath{https://zbmath.org/?q=an:1340.47089}
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  • https://www.mathnet.ru/eng/thsp12
  • https://www.mathnet.ru/eng/thsp/v19/i2/p42
  • This publication is cited in the following 1 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Theory of Stochastic Processes
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    References:89
     
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