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On some perturbations of a symmetric stable process and the corresponding Cauchy problems
M. M. Osypchuk Vasyl Stefanyk Precarpathian National University
Аннотация:
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}+(a(\cdot),\mathbf{B}), $ where $\mathbf{A}$ is the generator of a symmetric stable process in $\mathbb{R}^d$ with the exponent $\alpha\in(1,2]$, $\mathbf{B}$ is the operator that is determined by the equality $\mathbf{A}=c\ \mathbf{div}(\mathbf{B})$ ($c>0$ is a given parameter), and a given $\mathbb{R}^d$-valued function $a\in L_p(\mathbb{R}^d)$ for some $p>d+\alpha$ (the case of $p=+\infty$ is not exclusion). 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. We construct a solution of the Cauchy problem for the parabolic equation $\frac{\partial u}{\partial t}=(\mathbf{A}+(a(\cdot),\mathbf{B}))u$.
Ключевые слова:
Markov process, Wiener process, symmetric stable process, perturbation, pseudo-differential operator, pseudo-differential equation, transition probability density.
Образец цитирования:
M. M. Osypchuk, “On some perturbations of a symmetric stable process and the corresponding Cauchy problems”, Theory Stoch. Process., 21(37):1 (2016), 64–72
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/thsp121 https://www.mathnet.ru/rus/thsp/v21/i1/p64
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Страница аннотации: | 145 | PDF полного текста: | 46 | Список литературы: | 26 |
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