Compound Poisson Processes and Lévy Processes in Groups and Symmetric Spaces

Journal of Theoretical Probability

Volumn 13, Issue 2, 2000, Pages 383-425

  Applebaum, David ^a  

^a NOTTINGHAM TRENT UNIVERSITY   (United Kingdom)

Author keywords

Brownian motion; Cartan decomposition; Compound poisson process; Hunt generator; Interlacing with jumps; L vy process; Lie algebra; Lie group; Poisson random measure; Spherical function; Spherical measure; Stochastic differential equation

Indexed keywords

EID: 0034563514     PISSN: 08949840     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1007845508326     Document Type: Article

References (44)

1. Albevero, S., Høegh-Krohn, R., Marion, J., Testard, D., and Torrésani, B. (1993). Non-commutative Distributions, Marcel Dekker, Inc.
2. Applebaum, D. (1995). A horizontal Levy process on the bundle of orthonormal frames over a complete Riemannian manifold. In J. Azéma, M. Emery, P. A. Meyer, and M. Yor (eds.), Séminaire de Probabilités XXIX, Lecture Notes in Math., Vol. 1603, Springer-Verlag, Berlin/Heidelberg, pp. 166-181.
3. Applebaum, D. (1996). Covariant Poisson fields in Fock space. In R. Bhatia (ed.), Analysis, Geometry and Probability, Essays in Honor of K.R. Parthasarathy, Hindustan Book Agency, pp. 1-16.
4. Applebaum, D., and Brooks, M. (1996). Infinite series of quantum spectral stochastic integrals. J. Oper. Th. 36, 295-316.
5. Applebaum, D., and Kunita, H. (1993). Lévy flows on manifolds and Lévy processes on Lie groups. J. Math. Kyoto Univ. 33, 1103-1123.
6. Araki, H. (1970). Factorizable representation of current algebra. Publ. Res. Inst. Math. Sci. (Kyoto Univ.) 5, 361-422.
7. Bertoin, J. (1996). Lévy Processes, Cambridge University Press.
8. Bretagnolle, J. L. (1973). Processus a accroissments independants. In Lecture Notes Math., Vol. 307, Springer-Verlag, Berlin/Heidelberg, pp. 1-26.
9. Cohen, S. (1995). Some Markov properties of stochastic differential equations with jumps. In J. Azéma, M. Emery, P. A. Meyer, and M. Yor (eds.), Séminaire de Probabilités XXIX, Lecture Notes in Math., Vol. 603, Springer-Verlag, Berlin/Heidelberg, pp. 181-194.
10. Elworthy, K. D. (1982). Stochastic Differential Equations on Manifolds, Cambridge University Press.
11. Elworthy, K. D. (1988). Geometric aspects of diffusions on manifolds in école d'été de probabilités de Saint Flour XVII. In P. L. Hennequin (ed.), Lecture Notes in Math., Vol. 1362, Springer-Verlag, Berlin/Heidelberg, pp. 276-425.
12. Estrade, A. (1990). Calcul Stochastique Discontinu sur les Groups des Lie, Université d'Orleans, Ph. D. thesis.
13. Estrade, A. (1992). Exponentielle stochastique et intégrale multiplicatives discontinu. Ann. Inst. Henri Poincaré (Prob. Stat.) 28, 107-129.
14. Feinsilver, P. (1978). Processes with independent increments on a Lie group. Trans. Amer. Math. Soc. 242, 73-121.
15. Fristedt, B., and Gray, L. (19997). A Modern Approach to Probability Theory, Birkhäuser, Boston.
16. Fujiwara, T. (1991). Stochastic differential equations of jump type on manifolds and Lévy flows. J. Math. Kyoto Univ. 31, 99.
17. Galmarino, A. R. (1963). Representation of an isotropic diffusion as a skew product. Z. Wahrsch. Verw. Geb. 1, 359-378.
18. Gangolli, R. (1964). Isotropic infinitely divisible measures on symmetric spaces. Acta Math. 111, 213-246.
19. Gangolli, R. (1965). Sample functions of certain differential processes on symmetric spaces. Pacific J. Math. 15, 477-496.
20. Getoor, R. K. (1961). Infinite divisible probabilities on the hyperbolic plane. Pacific J. Math. 11, 1387-1308.
21. Guichardet, A. (1972). Symmetric Hilbert Spaces and Related Topic, Lecture Notes Math., Vol. 261, Springer-Verlag, Berlin/Heidelberg.
22. Helgason, S. (1978). Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press.
23. Helgason, S. ( 1984). Groups and Geometric Analysis, Academic Press.
24. Heyer, H. (1977). Probability Measures on Locally Compact Groups, Springer-Verlag, Berlin/Heidelberg.
25. Hudson, R. L., and Parthasarathy, K. R. (1984). Quantum Itô's formula and stochastic evolution. Commun. Math. Phys. 93, 301-323.
26. Hunt, G.A. (1956). Semigroups of measures on Lie groups. Trans. Amer. Math. Soc. 81, 264-293.
27. Itô, K. (1950). Brownian motions on a Lie group. Proc. Japan Acad. 26, 4-10.
28. Knapp, A. W. (1996). Lie Groups Beyond An Introduction, Birkhäuser, Boston.
29. Kobayishi, S., and Nomizu, K. (1963). Foundations of Differential Geometry, Vol. 1, Wiley, Interscience, New York.
30. Kunita, H. (1994). Stable Lévy processes on nilpotent Lie groups. In Stochastic Analysis in Infinite-Dimensional Spaces, Pitman Res. Notes on Math. Ser. A, pp. 167-182.
31. Liao, M. (1994). The Brownian motion and the canonical stochastic flow on a symmetric space. Trans. Amer. Math. Soc. 341, 253-274.
32. McKean, H. P. (1969). Stochastic Integrals, Academic Press.
33. Marcus, S. I. (1981). Modelling and approximation of stochastic differential equations driven by semimartingales. Stochastics 4, 223-245.
34. Malliavin, M. P., and Malliavin, P. (1974). Factorisations et lois limites de la diffusion horizontale au-dessus d'un espaces Riemmanian symetrique. In Théorie du Potentiel et Analyse Harmonique, Lecture Notes Math., Vol. 404, Springer-Verlag, Berlin, pp. 164-217.
35. Nelson, E. (1959). Analytic vectors. Ann. Math. 70, 572-615.
36. Newman, C. M. (1972). Ultralocal quantum field theory in terms of currents. Commun. Math. Phys. 26, 169-204.
37. Nomizu, K. (1956). Lie groups and differential geometry. Publ. Math. Soc. Japan 2.
38. Parthasarathy, K. R., and Schmidt, K. (1972). Positive definite kernels, continuous tensor products and central limit theorems of probability theory, Lecture Notes Math., Vol. 272, Springer-Verlag, Berlin/Heidelberg.
39. Parthasarathy, K. R. (1992). An Introduction to Quantum Stochastic Calculus, Birkhäuser, Verlag, Basel.
40. Parthasarathy, K. R. (1986). One parameter semigroups of completely positive maps on groups arising from quantum stochastic differential equations. Boll. Un. Math. Ital. 5A, 391-397.
41. Ramaswami, S. (1974). Semigroups of measures on Lie groups. J. Indian Math. Soc. 38, 175-189.
42. Streater, R. F. (1969). Current commutation relations, continuous tensor products and infinitely divisible group representations. In R. Jost (ed.), Local Quantum Theory, Academic Press.
43. Taylor, J. C. (1988). The Iwasawa decomposition and the limiting behavior of Brownian motion on a symmetric space of non-compact type. Contemp. Math. 73, 303-332.
44. Terras, A. (1985, 1988). Harmonic analysis on Symmetric Spaces and Applications, Vois. 1 and 2, Springer-Verlag, New York.