On a gateway between continuous and discrete Bessel and Laguerre processes

Miclo, Laurent; Patie, Pierre

Annales Henri Lebesgue, Tome 2 (2019), pp. 59-98.

En donnant des exemples d’approximation de diffusions linéaires par des processus de vie et de mort, Feller [Fel50] a mis en évidence un lien entre les mondes discrets et continus. En identifiant une relation d’entrelacement entre les processus de carrés de Bessel et certains processus de vie et de mort linéaires, nous montrons que ces liens sont encore plus profonds et vont également dans les deux sens. Cela nous permet d’exhiber des propriétés originales pour cette famille de processus de vie et de mort directement héritées des processus de carrés de Bessel. En particulier, nous obtenons une propriété d’auto-similarité discrète et un analogue discret à l’algèbre bêta-gamma. Nous expliquons ensuite comment ces propriétés s’étendent aux semi-groupes de Laguerre correspondants. Il apparaît à nouveau que les versions continues et discrètes sont plus proches qu’imaginé précédemment et qu’il est possible de transférer des informations d’un semi-groupe à l’autre.

By providing instances of approximation of linear diffusions by birth-death processes, Feller [Fel50] has offered an original path from the discrete world to the continuous one. In this paper, by identifying an intertwining relationship between squared Bessel processes and some linear birth-death processes, we show that this connection is in fact more intimate and goes in the two directions. As by-products, we identify some properties enjoyed by the birth-death family that are inherited from squared Bessel processes. For instance, these include a discrete self-similarity property and a discrete analogue of the beta-gamma algebra. We proceed by explaining that the same gateway identity also holds for the corresponding ergodic Laguerre semi-groups. It follows again that the continuous and discrete versions are more closely related than thought before, and this enables to pass information from one semi-group to the other one.

Classification : 53C28, 53C26, 32Q45
Mots clés : Squared Bessel processes, linear birth-death processes, intertwining, continuous and discrete scaling, spectral decomposition.

[AG81] Akhiezer, Naum Il’ich; Glazman, Izrail M. Theory of linear operators in Hilbert space. Vol. II, Monographs and Studies in Mathematics, 10, Pitman Advanced Publishing Program, 1981 | MR | Zbl

[Bak96] Bakry, Dominique Remarques sur les semigroupes de Jacobi, Astérisque, 236, Société Mathématique de France, 1996, pp. 23-39 | Numdam | Zbl

[BGL14] Bakry, Dominique; Gentil, Ivan; Ledoux, Michel Analysis and geometry of Markov diffusion operators, Grundlehren der Mathematischen Wissenschaften, 348, Springer, 2014 | MR | Zbl

[Bia95] Biane, Philippe Intertwining of Markov semi-groups, some examples, Séminaire de Probabilités XXIX (Lecture Notes in Mathematics), Volume 1613, Springer, 1995, pp. 30-36 | DOI | Numdam | MR | Zbl

[CC89] Craven, Thomas; Csordas, George Jensen polynomials and the Turán and Laguerre inequalities, Pac. J. Math., Volume 136 (1989) no. 2, pp. 241-260 | DOI | Zbl

[CP17] Choi, Michael C. H.; Patie, Pierre Analysis of non-reversible Markov chains via similarity orbit, 2017 (https://www.researchgate.net/publication/324518071_Analysis_of_non-reversible_Markov_chains_via_similarity_orbit)

[CP19] Choi, Michael C. H.; Patie, Pierre Skip-free Markov chains, Trans. Am. Math. Soc., Volume 371 (2019) no. 10, pp. 7301-7342 | DOI | MR | Zbl

[CPY98] Carmona, Philippe; Petit, Frédérique; Yor, Marc Beta-gamma random variables and intertwining relations between certain Markov processes, Rev. Mat. Iberoam., Volume 14 (1998) no. 2, pp. 311-368 | DOI | MR | Zbl

[Daw17] Dawson, Donald A. Introductory Lectures on Stochastic Population Systems (2017) (https://arxiv.org/abs/1705.03781)

[DF90] Diaconis, Persi; Fill, James Allen Strong stationary times via a new form of duality, Ann. Probab., Volume 18 (1990) no. 4, pp. 1483-1522 | DOI | MR | Zbl

[Dou69] Douglas, Ronald G. On the operator equation $S^{*}XT=X$ and related topics, Acta Sci. Math., Volume 30 (1969), pp. 19-32 | MR | Zbl

[Duf05] Dufresne, Daniel Bessel processes and Asian options, Numerical methods in finance (GERAD 25th Anniversary Series), Volume 9, Springer, 2005, pp. 35-57 | DOI | MR | Zbl

[Fel50] Feller, William Diffusion processes in genetics, Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, 1950, pp. 227-246 | Zbl

[GJY03] Göing-Jaeschke, Anja; Yor, Marc A survey and some generalizations of Bessel processes, Bernoulli, Volume 9 (2003) no. 2, pp. 313-349 | DOI | MR | Zbl

[Jen13] Jensen, Johan L. W. V. Recherches sur la théorie des équations, Acta Math., Volume 36 (1913), pp. 181-195 | DOI | Zbl

[KLS10] Koekoek, Roelof; Lesky, Peter A.; Swarttouw, René F. Hypergeometric orthogonal polynomials and their q-analogues, Springer Monographs in Mathematics, Springer, 2010 | Zbl

[KM58] Karlin, Samuel; McGregor, James Linear growth birth and death processes, J. Math. Mech., Volume 7 (1958), pp. 643-662 | MR | Zbl

[KT81] Karlin, Samuel; Taylor, Howard M. A second course in stochastic processes, Academic Press Inc., 1981 | Zbl

[Kub12] Kubrusly, Carlos S. Spectral theory of operators on Hilbert space, Birkhäuser, 2012 | Zbl

[Lam67] Lamperti, John Limiting distributions for branching processes, Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability (Berkeley, 1965/66), Vol. II: Contributions to Probability Theory, University of California Press, 1967, pp. 225-241 | Zbl

[MG10] Makarov, Roman N.; Glew, Devin Exact simulation of Bessel diffusions, Monte Carlo Methods Appl., Volume 16 (2010) no. 3-4, pp. 283-306 | MR | Zbl

[MP] Miclo, Laurent; Patie, Pierre On completely monotone intertwining relations and convergence to equilibrium (Work in progress)

[MP18] Miclo, Laurent; Patie, Pierre On a gateway between continuous and discrete Bessel and Laguerre processes (2018) (https://arxiv.org/abs/1807.09445v1) | Zbl

[MS65] Muckenhoupt, Benjamin; Stein, Elias M. Classical expansions and their relation to conjugate harmonic functions, Trans. Am. Math. Soc., Volume 118 (1965), pp. 17-92 | DOI | MR | Zbl

[PR81] Pitman, Jim; Rogers, L. C. G. Markov functions, Ann. Probab., Volume 9 (1981), pp. 573-582 | MR | Zbl

[PS18] Patie, Pierre; Savov, Mladen Spectral expansion of non-self-adjoint generalized Laguerre semigroups (2018) (to appear in Mem. Am. Math. Soc., https://www.researchgate.net/publication/277712416_Spectral_expansions_of_non-self-adjoint_generalized_Laguerre_semigroups)

[PY80] Pitman, Jim; Yor, Marc Bessel processes and infinitely divisible laws, Stochastic integrals. Proceedings of the LMS Durham Symposium (Durham, 1980) (Lecture Notes in Mathematics), Volume 851, Springer, 1980, pp. 285-370 | Zbl

[RS18] Redig, Frank; Sau, Federico Factorized duality, stationary product measures and generating functions, J. Stat. Phys., Volume 172 (2018) no. 4, pp. 980-1008 | DOI | MR | Zbl

[RY99] Revuz, Daniel; Yor, Marc Continuous martingales and Brownian motion, Grundlehren der Mathematischen Wissenschaften, 293, Springer, 1999 | MR | Zbl

[SK15] Slámová, Lenka; Klebanov, Lev B. Generalized definitions of discrete stability (2015) (https://arxiv.org/abs/1502.02588)

[SvH79] Steutel, Fred W.; van Harn, Klaas Discrete analogues of self-decomposability and stability, Ann. Probab., Volume 7 (1979) no. 5, pp. 893-899 | DOI | MR | Zbl

[Tit39] Titchmarsh, Edward C. The theory of functions, Oxford University Press, 1939 | Zbl