Geodesic
On some perturbations of a symmetric stable process and the corresponding Cauchy problems
Teoriâ slučajnyh processov, Tome 21 (2016) no. 1, pp. 64-72.

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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 $\frac{\partial u}{\partial t}=(\mathbf{A}+(a(\cdot),\mathbf{B}))u$.
Keywords: Markov process, Wiener process, symmetric stable process, pseudo-differential operator, pseudo-differential equation, transition probability density.
Mots-clés : perturbation 
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M. M. Osypchuk. On some perturbations of a symmetric stable process and the corresponding Cauchy problems. Teoriâ slučajnyh processov, Tome 21 (2016) no. 1, pp. 64-72. http://geodesic.mathdoc.fr/item/THSP_2016_21_1_a6/

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