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General solution of the poisson equation for quasi-birth-and-death processes

par Bini, Dario Andrea;Dendievel, Sarah ;Latouche, Guy ;Meini, Beatrice
Référence SIAM journal on applied mathematics, 76, 6, page (2397-2417)
Publication Publié, 2016

Article révisé par les pairs

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Résumé :

We consider the Poisson equation (I -P)u = g, where P is the transition matrix of a quasi-birth-and-death process with infinitely many levels, g is a given infinite dimensional vector, and u is the unknown. Our main result is to provide the general solution of this equation. To this purpose we use the block tridiagonal and block Toeplitz structure of the matrix P to obtain a set of matrix difference equations, which are solved by constructing suitable resolvent triples.