Geometric analysis for symmetric Fleming-Viot operators: Rademacher's theorem and exponential families

Potential Analysis

Volumn 17, Issue 4, 2002, Pages 351-374

  Schied, Alexander ^a  

^a TECHNISCHE UNIVERSITÄT BERLIN   (Germany)

Author keywords

Bhattacharya distance; Dirichlet form; Dirichlet point process; Fleming Viot process; Rademacher's theorem; Weak differentiability

Indexed keywords

EID: 0141680538     PISSN: 09262601     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1016360318443     Document Type: Article

References (30)

1. Albeverio, S., Kondratiev, Yu.G. and Röckner, M.: 'Analysis and geometry on configuration spaces', J. Funct. Anal. 154(2) (1998), 444-500.
2. Albeverio, S., Kondratiev, Yu.G. and Röckner, M.: 'Analysis and geometry on configuration spaces: The Gibbsian case', J. Funct. Anal. 157(1) (1998), 242-291.
3. Amari, S.: Differential-Geometrical Methods in Statistics, Lecture Notes in Statist. 28, Springer-Verlag, Berlin, 1985.
4. Bogachev, V.I. and Mayer-Wolf, E.: 'Some remarks on Rademacher's theorem in infinite dimensions', Potential Anal. 5 (1996), 23-30.
5. Cheeger, J.: 'Differentiability of Lipschitz functions on metric measure spaces', GAFA, Geom. Funct. Anal. 9 (1999), 418-517.
6. Dawson, D.A.: 'Measure-valued Markov processes', in Ecole d'Eté de Probabilités de Saint-Flour XXI, Lecture Notes in Math. 1541, Springer, Berlin, 1993, pp. 1-260.
7. Enchev, O. and Stroock, D.: 'Rademacher's theorem for Wiener functionals', Ann. Probab. 21 (1993), 25-33.
8. Ethier, S. and Kurtz, T: 'Fleming-Viot processes in population genetics', SIAM J. Control Optim. 31(2) (1993), 345-386.
9. Ethier, S. and Kurtz, T.: Markov Processes. Characterization and Convergence, Wiley Series in Probability and Mathematical Statistics, Wiley, New York, 1986.
10. Friedrich, T.: 'Die Fisher-Information und symplektische Strukturen', Math. Nachr 153 (1991), 273-296.
11. Handa, K.: Quasi-invariance and reversibility in the Fleming-Viot process, Preprint, Saga University, 1999; to appear in Probab. Theory Related Fields.
12. Hille, E. and Phillips, R.: Functional Analysis and Semi-Groups, Amer. Math. Soc., Providence, 1957.
13. Kusuoka, S.: 'Dirichlet forms and diffusion processes on Banach spaces', J. Fac. Sci., Univ. Tokyo, Sect. I A 29 (1982), 79-95.
14. Li, Z., Shiga, T. and Yao, L.: 'A reversibility problem for Fleming Viot processes', Elect. Comm. in Probab. 4 (1999), 65-76.
15. Ma, Z.M. and Röckner, M.: Introduction to the Theory of (Nonsymmetric) Dirichlet Forms, Springer-Verlag, Berlin, 1992.
16. McShane, E.: 'Extension of range of functions', Bull. Amer. Math. Psyche. 40 (1934), 837-842.
17. Overbeck, L. and Röckner, M.: 'Geometric aspects of finite and infinite-dimensional Fleming-Viot processes', Random Oper Stoch. Equation 5(1) (1997), 35-58.
18. Overbeck, L., Röckner, M. and Schmuland, B.: 'An analytic approach to the Fleming-Viot processes with interactive selection', Ann. Probab. 23(1) (1995), 1-36.
19. Rademacher, H.: 'Über partielle und totale Differenzierbarkeit von Funktionen mehrerer Variabeln und über die Transformation der Doppelintegrale', Math. Ann. 79 (1919), 340-359.
20. Reed, M. and Simon, B.: Methods of Modern Mathematical Physics II: Fourier Analysis, Self-adjointness, Academic Press, San Diego, 1975.
21. Riesz, F. and Nagy, S.B.: Vorlesungen über Funktionalanalysis, VEB Deutscher Verlag der Wissenschaften, Berlin, 1956.
22. Röckner, M. and Schied, A.: 'Rademacher's theorem on configuration spaces and applications', J. Funct. Anal. 169(2) (1999), 325-356.
23. Schied, A.: 'Geometric aspects of Fleming-Viot and Dawson-Watanabe processes', Ann. Probab. 25(3) (1997), 1160-1179.
24. Shiga, T.: 'A stochastic equation based on a Poisson system for a class of measure-valued diffusion processes', J. Math. Kyoto Univ. 30 (1990), 245-279.
25. Stannat, W.: 'On the validity of the log-Sobolev inequality for symmetric Fleming-Viot operators', Ann. Probab. 28(2) (2000), 667-684.
26. Sturm, K.-T.: 'The geometric aspect of Dirichlet forms', in J. Jost et al. (eds), New Directions in Dirichlet Forms, AMS/IP Stud. Adv. Math. 8, Amer. Math. Soc., Providence, RI, 1998, pp. 233-277.
27. Sturm, K.-T.: 'On the geometry defined by Dirichlet forms', in E. Bolthausen et al. (eds), Seminar on Stochastic Analysis, Random Fields and Applications. Proceedings of a Seminar Held at the Centro Stefano Franscini, Ascona, Switzerland, 7-12 June, 1993, Prog. Probab. 36, Birkhäuser, Basel, 1995, pp. 231-242.
28. p-Liouville properties', J. Reine Angew. Math. 456 (1994), 173-186.
29. Tsilevich, N. and Vershik, A.: 'Quasi-invariance of the gamma process and multiplicative properties of the Poisson-Dirichlet measures', C. R. Acad. Sci. Paris, Ser I Math. 329(2) (1999), 163-168.
30. Tsilevich, N., Vershik, A. and Yor, M.: Distinguished properties of the gamma process and related topics, Preprint, Sankt Petersburg, 2000.