Some new examples of Markov processes which enjoy the time-inversion property. (English) Zbl 1087.60058
S. Watanabe [Z. Wahrscheinlichkeitstheor. Verw. Geb. 31, 115–124 (1975; Zbl 0286.60035)] established the problem of finding time-depending homeomorphisms \(g(t,.)\) for the state spaces \(E\) and \(E'\) of 2 conservative diffusions \((X(t),P)\) and \((X'(t),P')\) in such a way that \((g(t,X(1/t)),P)\) and \((X'(t),P')\) are equivalent. As an important tool to find some answers Watanabe used M. G. Krein’s spectral theory of the string.
The present authors have modified the time-inversion problem from above to the following one: For a given homogeneous Markov process \((X(t),P.)\) in \(\mathbb{R}^n\) find conditions for the associated Markov semigroup under which the process \((tX(1/t),P.)\) is also a homogeneous Markov process.
Using quite different methods some answers are given. The impressive set of very different examples contains, e.g., the examples of Bessel processes considered by Watanabe and several new ones as Dunkl processes and their radial parts.
The present authors have modified the time-inversion problem from above to the following one: For a given homogeneous Markov process \((X(t),P.)\) in \(\mathbb{R}^n\) find conditions for the associated Markov semigroup under which the process \((tX(1/t),P.)\) is also a homogeneous Markov process.
Using quite different methods some answers are given. The impressive set of very different examples contains, e.g., the examples of Bessel processes considered by Watanabe and several new ones as Dunkl processes and their radial parts.
Reviewer: Wilfried Schenk (Dresden)
Keywords:
Homogeneous Markov processes; Semi-stable processes; Time inversion property; “Opérateur carré du champ”; Bessel processes; Wishart processes; Dunkl processes; Radial Dunkl processes; Dunkl processes with drift; Brownian motion in a Weyl chamberCitations:
Zbl 0286.60035References:
| [1] | Biane, P., Bougerol, P., O’Connell, N.: Littlemann paths and Brownian paths. Preprint (March 2004) · Zbl 1161.60330 |
| [2] | Bru, R. Acad. Sci. Paris, Série I, t., 308, 29 (1989) · Zbl 0666.60049 |
| [3] | Bru, Multivariate Anal., 29, 127 (1989) · Zbl 0687.62048 · doi:10.1016/0047-259X(89)90080-8 |
| [4] | Bru, J. Theor. Prob.,, 4, 725 (1991) · Zbl 0737.60067 |
| [5] | Chaumont, L., Yor, M.: Exercises in Probability. A guided Tour from Measure Theory to Random Processes, via Conditioning. Cambridge University Press, 2003 · Zbl 1065.60001 |
| [6] | Dellacherie, C., Maisonneuve, B., Meyer, P.A.: Probabilités et Potentiels, Chap. XVII à XXIV. Hermann, Paris 1992 |
| [7] | Donati-Martin, C., Doumerc, Y., Matsumoto, H., Yor, M.: Some properties of the Wishart processes and a matrix extension of the Hartman-Watson laws. To appear in Publ. RIMS, Kyoto, 2004 · Zbl 1076.60067 |
| [8] | Dunkl, Trans. Amer. Math. Soc., 311, 167 (1) · Zbl 0652.33004 |
| [9] | Dunkl, Contemp. Math.,, 138, 123 (1992) |
| [10] | Dunkl, C., Xu, Y.: Orthogonal Polynomials of Several Variables. Cambridge University Press, 2001 · Zbl 0964.33001 |
| [11] | Gallardo, L., Yor, M.: Some remarkable properties of the Dunkl martingales. to appear in Sém. Probas. XXXVIII, dedicated to Paul André Meyer. Springer (Springer LNM 2005) |
| [12] | Godefroy, L.: Frontière de Martin sur les hypergroupes et principe d’invariance relatif au processus de Dunkl. Thèse Université de Tours (Déc. 2003) |
| [13] | Graversen, Prob. Math. Stat., 20, 63 (2000) · Zbl 0988.60079 |
| [14] | Hirsch, F.: L’opérateur carré du champ, d’après J.P. Roth. Séminaire Bourbaki, 29 ième année, exposé 501, (juin 1977) |
| [15] | Kunita, Nagoya Math. J., 36, 1 (1969) · Zbl 0186.51203 |
| [16] | Lamperti, Zeit. für Wahr., 22, 205 (1972) · Zbl 0274.60052 |
| [17] | Pitman, J., Yor, M.: Bessel processes and infinitely divisible laws. In : D. Williams (ed) Stochastic Integrals. Lect. Notes in Math. 851 (1981) · Zbl 0469.60076 |
| [18] | Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Springer. Third Edition, corrected second printing, 2001 · Zbl 0917.60006 |
| [19] | Rösler, Commun. Math. Phys., 192, 519 (1998) · Zbl 0908.33005 · doi:10.1007/s002200050307 |
| [20] | Rösler, Adv. App. Math., 21, 575 (4) · Zbl 0919.60072 · doi:10.1006/aama.1998.0609 |
| [21] | Roth, Ann. Inst. Fourier,, 26, 1 (4) · Zbl 0331.47021 |
| [22] | Watanabe, Zeit. für Wahr., 31, 115 (1975) · Zbl 0286.60035 |
| [23] | Watanabe, In : J. Korean Math. Soc., 35, 659 (3) |
| [24] | Watanabe, S.: Bilateral Bessel diffusion processes with drifts and time-inversion. Preprint (1997-1998) |
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