Excursions of diffusion processes and continued fractions

Comtet, Alain; Tourigny, Yves

Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 3, pp. 850-874.

Il est bien connu que les excursions d'un processus de diffusion peuvent être étudiées en considérant une certaine équation de Riccati associée au processus. On montre que, dans beaucoup de cas intéressants, certaines solutions de cette équation de Riccati peuvent être développées en fraction continue. On examine le contenu probabiliste de ce développement. Ces résultats sont illustrés par quelques exemples de diffusions en milieux aléatoires et déterministes.

It is well-known that the excursions of a one-dimensional diffusion process can be studied by considering a certain Riccati equation associated with the process. We show that, in many cases of interest, the Riccati equation can be solved in terms of an infinite continued fraction. We examine the probabilistic significance of the expansion. To illustrate our results, we discuss some examples of diffusions in deterministic and in random environments.

[1] N. I. Akhiezer. The Classical Moment Problem and Some Related Questions in Analysis. Fitzmatgiz, Moscow, 1961; English transl., Oliver and Boyd, Edinburgh, 1965. | MR | Zbl

[2] D. Applebaum. Lévy Processes and Stochastic Calculus. Cambridge Univ. Press, Cambridge, 2004. | MR | Zbl

[3] C. Aslangul, N. Pottier and D. Saint-James. Random walk in a one-dimensional random medium. Phys. A 164 (1990) 52-80. | MR

[4] C. Bender and S. A. Orszag. Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill, New York, 1978. | MR | Zbl

[5] J. Bernasconi, S. Alexander and R. Orbach. Classical diffusion in one-dimensional disordered lattice. Phys. Rev. Lett. 41 (1978) 185-187.

[6] J. Bertoin. Lévy Processes. Cambridge Univ. Press, Cambridge, 1996. | MR | Zbl

[7] P. Biane. Comparaison entre temps d'atteinte et temps de séjour de certaines diffusions réelles. In Sém. Probabilités Strasbourg, XIX 291-296. Lecture Notes in Math. 1123. Springer, Berlin, 1985. | Numdam | MR | Zbl

[8] P. Biane and M. Yor. Variations sur une formule de Paul Lévy. Ann. Inst. H. Poincaré Probab. Statist. 23 (1987) 359-377. | Numdam | MR | Zbl

[9] G. Bordes and B. Roehner. Application of Stieltjes theory for S-fractions to birth and death processes. Adv. in Appl. Probab. 15 (1983) 507-530. | MR | Zbl

[10] A. N. Borodin and P. Salminen. Handbook of Brownian Motion - Facts and Formulae. Birkhäuser, Basel, 1996. | MR | Zbl

[11] J. P. Bouchaud, A. Comtet, A. Georges and P. Le Doussal. Classical diffusion of a particle in a one-dimensional random force field. Ann. Phys. 201 (1990) 285-341. | MR

[12] M. M. Crum. Associated Sturm-Liouville systems. Quart. J. Math. Oxford (2) 6 (1955) 121-127. | MR | Zbl

[13] A. K. Common and D. E. Roberts. Solutions of the Riccati equation and their relation to the Toda lattice. J. Phys. A Math. Gen. 19 (1986) 1889-1898. | MR | Zbl

[14] Z. Ciesielski and S. J. Taylor. First passage times and Sojourn times for Brownian motion in space and the exact Hausdorff measure of the sample path. Trans. Amer. Math. Soc. 103 (1962) 434-450. | MR | Zbl

[15] M. G. Darboux. Sur une proposition relative aux équations linéaires. C. R. Acad. Sci. Paris 94 (1882) 1456-1459. | JFM

[16] F. Den Hollander. Large Deviations. Amer. Math. Soc., Providence, RI, 2000. | MR | Zbl

[17] H. Dette, J. A. Fill, J. Pitman and W. J. Studden. Wall and Siegmund duality relations for birth and death chains with reflecting barrier. J. Theoret. Probab. 10 (1997) 349-374. | MR | Zbl

[18] C. Donati-Martin and M. Yor. Some explicit Krein representations of certain subordinators, including the Gamma process. Publ. Res. Inst. Math. Sci. 42 (2006) 879-895. | MR | Zbl

[19] H. Dym and H. P. Mckean. Gaussian Processes, Function Theory, and the Inverse Spectral Problem. Academic Press, New York, 1976. | MR | Zbl

[20] L. Euler. De fractionibus continuis dissertatio. Comm. Acad. Sci. Petropol. 9 (1744) 98-137; English transl.: M. Wyman and B. Wyman. An essay on continued fractions. Math. Systems Theory 18 (1985) 295-328. | MR

[21] P. Flajolet and F. Guillemin. The formal theory of birth-and-death processes, lattice path combinatorics and continued fractions. Adv. in Appl. Probab. 32 (2000) 750-778. | MR | Zbl

[22] H. L. Frisch and S. P. Lloyd. Electron levels in a one-dimensional lattice. Phys. Rev. 120 (1960) 1175-1189. | Zbl

[23] F. Guillemin and D. Pinchon. Excursions of birth and death processes, orthogonal polynomials, and continued fractions. J. Appl. Probab. 36 (1999) 752-770. | MR | Zbl

[24] M. E. H. Ismail and D. H. Kelker. Special functions, Stieltjes transforms and infinite divisibility. SIAM J. Math. Anal. 10 (1979) 884-901. | MR | Zbl

[25] K. Itô and H. P. Mckean. Diffusion Processes and Their Sample Paths. Springer, New York, 1974. | MR | Zbl

[26] K. M. Jansons. Excursions into a new duality relation for diffusion processes. Elect. Comm. Probab. 1 (1996) 65-69. | MR | Zbl

[27] S. Karlin and J. L. Mcgregor. The differential equations of birth-and-death processes, and the Stieltjes moment problem. Trans. Amer. Math. Soc. 85 (1957) 489-546. | MR | Zbl

[28] F. B. Knight. Characterisation of the Lévy measures of inverse local times of gap diffusion. In Seminar on Stochastic Processes 53-78. Birkhäuser, Basel, 1981. | MR | Zbl

[29] F. B. Knight. Essentials of Brownian Motion and Diffusion. Amer. Math. Soc., Providence, RI, 1981. | MR | Zbl

[30] S. Kotani. On asymptotic behaviour of the spectra of a one-dimensional Hamiltonian with a certain random coefficient. Publ. RIMS Kyoto Univ. 12 (1976) 447-492. | MR | Zbl

[31] K. Kawazu and H. Tanaka. A diffusion process in a Brownian environment with drift. J. Math. Soc. Japan 49 (1997) 189-211. | MR | Zbl

[32] S. Kotani and S. Watanabe. Krein's spectral theory of strings and generalized diffusion processes. In Functional Analysis in Markov Processes 235-259. Springer, New York, 1982. | MR | Zbl

[33] G. Letac and W. Seshadri. A characterisation of the generalised inverse Gaussian distribution by continued fractions. Z. Wahrsch. Werw. Gebiete 62 (1983) 485-489. | MR | Zbl

[34] S. N. Majumdar and A. Comtet. Exact asymptotic results for persistence in the Sinai problem with arbitrary drift. Phys. Rev. E 66 (2002) 061105-061116. | MR

[35] J. Marklof, Y. Tourigny and L. Wolowski. Explicit invariant measures for products of random matrices. Trans. Amer. Math. Soc. 360 (2008) 3391-3427. | MR | Zbl

[36] J. Marklof, Y. Tourigny and L. Wolowski. Padé approximants of random Stieltjes functions. Proc. Roy. Soc. A 463 (2007) 2813-2832. | MR | Zbl

[37] E. M. Nikishin and W. N. Sorokin. Rational Approximation and Orthogonality. Nauk, Moscow, 1988; English transl., Amer. Math. Soc., Providence, RI, 1991. | MR | Zbl

[38] B. Øksendal. Stochastic Differential Equations. Springer, Berlin, 1998.

[39] J. Pitman and M. Yor. Bessel processes and infinitely divisible laws. In Stochastic Integrals 285-370. Lecture Notes in Math. 851. Springer, Berlin, 1981. | MR | Zbl

[40] J. Pitman and M. Yor. Hitting, occupation and inverse local times of one-dimensional diffusions: Martingale and excursion approaches. Bernoulli 9 (2003) 1-24. | MR | Zbl

[41] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion. Springer, Berlin, 1999. | MR | Zbl

[42] P. Salminen. One-dimensional diffusions and their exit spaces. Math. Scand. 54 (1984) 209-220. | MR | Zbl

[43] F. Soucaliuc. Réflection entre deux diffusions conjuguées. C. R. Acad. Sci. Paris Ser. I 334 (2002) 1119-1124. | MR | Zbl

[44] T. J. Stieltjes. Recherches sur les fractions continues. Ann. Fac. Sci. Toulouse 8 (1894) 1-122. | JFM | Numdam | MR

[45] B. Tóth. Generalized Ray-Knight theory and limit theorems for self-interacting random walks on ℤ1. Ann. Probab. 24 (1996) 1324-1367. | MR | Zbl