(Homogeneous) markovian bridges
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).

DOI : 10.1214/10-AIHP391
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 
@article{AIHPB_2011__47_3_875_0,
     author = {Vigon, Vincent},
     title = {(Homogeneous) markovian bridges},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {875--916},
     publisher = {Gauthier-Villars},
     volume = {47},
     number = {3},
     year = {2011},
     doi = {10.1214/10-AIHP391},
     mrnumber = {2841078},
     zbl = {1267.60080},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1214/10-AIHP391/}
}
TY  - JOUR
AU  - Vigon, Vincent
TI  - (Homogeneous) markovian bridges
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2011
SP  - 875
EP  - 916
VL  - 47
IS  - 3
PB  - Gauthier-Villars
UR  - http://archive.numdam.org/articles/10.1214/10-AIHP391/
DO  - 10.1214/10-AIHP391
LA  - en
ID  - AIHPB_2011__47_3_875_0
ER  - 
%0 Journal Article
%A Vigon, Vincent
%T (Homogeneous) markovian bridges
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2011
%P 875-916
%V 47
%N 3
%I Gauthier-Villars
%U http://archive.numdam.org/articles/10.1214/10-AIHP391/
%R 10.1214/10-AIHP391
%G en
%F AIHPB_2011__47_3_875_0
Vigon, Vincent. (Homogeneous) markovian bridges. Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 3, pp. 875-916. doi : 10.1214/10-AIHP391. http://archive.numdam.org/articles/10.1214/10-AIHP391/

[1] J. Bertoin. Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge Univ. Press, Cambridge, 1998. | MR | Zbl

[2] J. D. Biggins. Random walk conditioned to stay positive. J. London Math. Soc. (2) 67 (2003) 259-272. | MR | Zbl

[3] R. M. Blumenthal and R. K. Getoor. Markov Processes and Potential Theory. Pure and Applied Mathematics 29. Academic Press, New York-London, 1968. | MR | Zbl

[4] C. Dellacherie and P.-A. Meyer. Probabilités et potentiel. Chapitres IX à XI: Théorie discrète du potentiel. Publications de l'Institut de Mathématique de l'Université de Strasbourg XVIII. Actualités Scientifiques et Industrielles 1410. Hermann, Paris, 1983. | MR | Zbl

[5] J. L. Doob. Conditional Brownian motion and the boundary limits of harmonic functions. Bull. Soc. Math. France 85 (1957) 431-458. | Numdam | MR | Zbl

[6] E. B. Dynkin. Boundary theory of Markov processes (the discrete case). Russian Math. Surveys 24 (1969) 1-42. | MR | Zbl

[7] W. Feller. An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edition. Wiley, 1966. | MR | Zbl

[8] P. J. Fitzsimmons. On the excursions of Markov processes in classical duality. Probab. Theory Related Fields 75 (1987) 159-178. | MR | Zbl

[9] P. Fitzsimmons. Markov processes with identical bridges. Electron. J. Probab. 3 (1998). | MR | Zbl

[10] P. Fitzsimmons, J. Pitman and M. Yor. Markovian bridges: Construction, Palm interpretation, and splicing. In Seminar on Stochastic Processes 101-134. Progress in Probability 33. Birkhäuser Boston, Boston, 1992. | MR | Zbl

[11] S. Fourati. Vervaat et Lévy. Ann. Inst. H. Poincaré Probab. Statist. 41 (2005) 461-478. | Numdam | MR | Zbl

[12] R. K. Getoor and M. J. Sharpe. The Markov property at co-optional times. Z. Wahrsch. Verw. Gebiete 48 (1979) 201-211. | MR | Zbl

[13] R. K. Getoor and M. J. Sharpe. Excursions of dual processes. Adv. Math. 45 (1982) 259-309. | MR | Zbl

[14] J. Hoffmann-Jørgensen. Markov sets. Math. Scand. 24 (1969) 145-166. | MR | Zbl

[15] G. A. Hunt. Markoff processes and potentials. Illinois J. Math. 2 (1958) 151-213. | MR | Zbl

[16] G. A. Hunt. Markoff chains and Martin boundaries. Illinois J. Math. 4 (1960) 316-340. | MR | Zbl

[17] K. Itô. Poisson point processes attached to Markov processes. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Vol. III: Probability Theory 225-239. Univ. California Press, Berkeley, 1972. | MR | Zbl

[18] M. Jacobsen. Splitting times for Markov processes and a generalised Markov property for diffusions. Z. Wahrsch. Verw. Gebiete 30 (1974) 27-43. | MR | Zbl

[19] M. Jacobsen and J. M. Pitman. Birth, death and conditioning of Markov chains. Ann. Probab. 5 (1977) 430-450. | MR | Zbl

[20] O. Kallenberg. Foundations of Modern Probability, 2nd edition. Probability and Its Applications (New York). Springer, New York, 2002. | MR | Zbl

[21] A. N. Kolmogorov. Zur Theorie der Markoffschen Ketten. Math. Ann. 112 (1936) 155-160. | MR | Zbl

[22] B. Maisonneuve. Sytèmes régénératifs. Astérique 15. Société mathématique de France, 1974. | Numdam | MR | Zbl

[23] P. A. Meyer, R. T. Smythe and J. B. Walsh. Birth and death of Markov processes. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Vol. III: Probability Theory 295-305. Univ. California Press, Berkeley, 1972. | MR | Zbl

[24] P. W. Millar. Zero-one laws and the minimum of a Markov process. Trans. Amer. Math. Soc. 226 (1977) 365-391. | MR | Zbl

[25] M. Nagasawa. Time reversions of Markov processes. Nagoya Math. J. 24 (1964) 177-204. | MR | Zbl

[26] J. Pitman and M. Yor. Itô's excursion theory and its applications. Japan J. Math. 2 (2007) 83-96. | MR | Zbl

[27] A. O. Pittenger and C. T. Shih. Coterminal families and the strong Markov property. Trans. Amer. Math. Soc. 182 (1973) 1-42. | MR | Zbl

[28] H. Rost. Markoff-Ketten bei sich füllenden Löchern im Zustandsraum. Ann. Inst. Fourier (Grenoble) 21 (1971) 253-270. | Numdam | MR | Zbl

[29] E. Seneta. Non-Negative Matrices. An Introduction to Theory and Applications. Allen & Unwin, London, 1973. | MR | Zbl

[30] T. Simon. Subordination in the wide sense for Lévy processes. Probab. Theory Related Fields 115 (1999) 445-477. | MR | Zbl

[31] H. Thorisson. Coupling, Stationarity, and Regeneration. Probability and Its Applications (New York). Springer, New York, 2000. | MR | Zbl

[32] W. Vervaat. A relation between Brownian bridge and Brownian excursion. Ann. Probab. 7 (1979) 141-149. | MR | Zbl

[33] V. Vigon. Simplifiez vos Lévy en titillant la factorisation de Wiener-Hopf. Editions Universitaires Europeennes, also disposable on HAL and on my web page, 2002.

[34] D. Williams. Decomposing the Brownian path. Bull. Amer. Math. Soc. 76 (1970) 871-873. | MR | Zbl

[35] W. Woess. Random Walks on Infinite Graphs and Groups. Cambridge Tracts in Mathematics 138. Cambridge Univ. Press, Cambridge, 2000. | MR | Zbl

Cité par Sources :