Please use this identifier to cite or link to this item: http://hdl.handle.net/123456789/5874
Title: One type of sіngular perturbatіons of a multіdіmensіonal stable process
Authors: Osypchuk, Mykhailo
Portenko, Mykola
Keywords: Markov process
Wiener process
symmetric stable process
singular perturbation
pseudo-differential operator
pseudo-differential equation
semigroup of operators
transition probability density
Issue Date: Dec-2014
Citation: Osypchuk M.M. One type of sіngular perturbatіons of a multіdіmensіonal stable process/ M.M. Osypchuk, M.І. Portenko// Theory Stoch. Process. -2014. -V. 19(35), 2. -P. 42-51.
Abstract: A semigroup of linear operators on the space of all continuous bounded functions given on a d-dimensional Euclidean space R d is constructed such that its generator can be written in the following form A + q(x)δ S (x)B ν , where A is the generator of a symmetric stable process in R d (that is, a pseudo-differential operator whose symbol is given by (−c|ξ| α ) ξ∈R d , parameters c > 0 and α ∈ (1, 2] are fixed); B ν is the operator with the symbol (2ic|ξ| α−2 (ξ, ν)) ξ∈R d (i =√−1 and ν ∈ R d is a fixed unit vector); S is a hyperplane in R d that is orthogonal to ν; (δ S (x)) x∈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∈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 R d corresponding to this semigroup because it does not preserve the property of a function to take on only non-negative values.
URI: http://hdl.handle.net/123456789/5874
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