Asymptotic expansion for the distribution of a Markovian random motion

Pogorui, A. A.

doi:10.1515/rose.2009.013

Cited by 3 publications

(1 citation statement)

References 10 publications

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“…However, the resulting Markov process has a complicated continuous phase space. Upon observing that the projective and potential operators for this Markov process were obtained in [4], we apply the method of asymptotic expansion for Markov random evolutions [7]- [9] to find a recursive solution.…”

Section: Discussionmentioning

confidence: 99%

Asymptotic expansion for transport processes in semi-Markov media

Pogorui

Rodríguez‐Dagnino

2011

Theor. Probability and Math. Statist.

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Abstract. In this paper we study asymptotic expansions for a solution of the singularly perturbed equation for a functional of a semi-Markov random evolution on the line. By using the method for solutions of singularly perturbed equations, we obtain the solution in the form of a series of regular and singular terms. The first regular term satisfies a diffusion-type equation, and the first singular term is a semigroup with the infinitesimal operator of the respective related bivariate process. Each regular and singular term can be calculated recursively. IntroductionAsymptotic expansions for perturbed equations of Markov and semi-Markov random evolution have generated a great deal of research; see [7]-[10] and others. In this paper we investigate solutions of singularly perturbed equations of semi-Markov random evolutions by reducing a semi-Markov process to a Markov process with a more complicated phase space.Let {ξ(t), t ≥ 0} be a semi-Markov process on the phase space {E, F} with the semi-where P (x, B) are the transition probabilities of the embedded Markov chain {ξ n , n ≥ 0}, and G x (t) is the cumulative distribution function (cdf) of a sojourn time (holding time) of ξ(t) in x ∈ E. Now, let us assume a function C(u, x), u ∈ R, x ∈ E such that it satisfies the uniquevalued solvability condition of the following evolution equation:In addition, we assume that the derivative ∂ C(u, x)/∂u is bounded. For the fixed parameter ε > 0 consider the following random transport process u ε (t) in the scaled semi-Markov medium {ξ(t/ε 2 )}, as follows [1, 2]:(1) du ε (t) dt = 1 ε C u ε (t), ξ t/ε 2 , u ε (0) = u 0 .2010 Mathematics Subject Classification. Primary 60J25; Secondary 35C20.

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Section: Discussionmentioning

confidence: 99%