(Homogeneous) markovian bridges

Vigon, Vincent

Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 3, pp. 875-916.

Les ponts markoviens (homogènes) sont des chaines de Markov (homogènes) qui démarrent à un point donné et meurent à un point donné. Pour préserver l'homogénéité, une telle chaine de Markov a nécessairement une durée de vie aléatoire. Nous étudions les ponts pour eux mêmes et pour leur utilité à décrire les transformations d'une chaine de Markov : restriction à un intervalle aléatoire, renversement temporel, changement de temps, conditionnements variés : notamment le confinement dans une partie de l'espace d'état. Ces ponts nous conduisent à considérer les chaines de Markov d'un point de vue inhabituel : nous ne travaillons plus avec une seule matrice de transition comme à l'accoutumée, mais avec une classe de matrices qui se déduisent les unes des autres par transformation de Doob. Cette méthode a l'avantage de mieux décrire les symétries passé ↔ futur : symétrie de l'indépendance conditionnelle (bien connue) et symétrie de l'homogénéité (moins bien connue).

(Homogeneous) Markov bridges are (time homogeneous) Markov chains which begin at a given point and end at a given point. The price to pay for preserving the homogeneity is to work with processes with a random life-span. Bridges are studied both for themselves and for their use in describing the transformations of Markov chains: restriction on a random interval, time reversal, time change, various conditionings comprising the confinement in some part of the state space. These bridges lead us to look at Markov chains from an unusual point of view: we will work, no longer with only one transition matrix, but with a class of matrices which can be deduced one from the other by Doob transformations. This way of proceeding has the advantage of better describing the “past ↔ future symmetries”: The symmetry of conditional independence (well known) and the symmetry of homogeneity (less well known).

Classification : 60J10, 60J45, 47A68, 15A23, 60J50
Mots clés : Markov chains, random walks, LU-factorization, path-decomposition, fluctuation theory, probabilistic potential theory, infinite matrices, Martin boundary

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