The Annals of Probability

Geometric aspects of Fleming-Viot and Dawson-Watanabe processes

Alexander Schied

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Abstract

This paper is concerned with the intrinsic metrics of the two main classes of superprocesses. For the Fleming-Viot process, we identify it as the Bhattacharya distance, and for Dawson-Watanabe processes, we find the Kakutani-Hellinger metric. The corresponding geometries are studied in some detail. In particular, representation formulas for geodesics and arc length functionals are obtained. The relations between the two metrics yield a geometric interpretation of the identification of the Fleming-Viot process as a Dawson-Watanabe superprocess conditioned to have total mass 1. As an application, a functional limit theorem for super-Brownian motion conditioned on local extinction is proved.

Article information

Source
Ann. Probab., Volume 25, Number 3 (1997), 1160-1179.

Dates
First available in Project Euclid: 18 June 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1024404509

Digital Object Identifier
doi:10.1214/aop/1024404509

Mathematical Reviews number (MathSciNet)
MR1457615

Zentralblatt MATH identifier
0895.60082

Subjects
Primary: 60J60: Diffusion processes [See also 58J65] 60G57: Random measures 58G32 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords
Intrinsic metric Fleming-Viot process Dawson-Watanabe superprocess Kakutani-Hellinger distance Bhattacharya metric

Citation

Schied, Alexander. Geometric aspects of Fleming-Viot and Dawson-Watanabe processes. Ann. Probab. 25 (1997), no. 3, 1160--1179. doi:10.1214/aop/1024404509. https://projecteuclid.org/euclid.aop/1024404509


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References

  • AMARI, S. 1985. Differential-geometrical methods in statistics. Lecture Notes in Statist. 28. Springer, Berlin. Z.
  • BALLMANN, W. 1990. Singular spaces of non-positive curvature. In Sur les groupes Hyperboliques d'apres Mikhael Gromov 189 202. Birkhauser, Boston. ´ ¨
  • CARLEN, E. A., KUSUOKA, S. and STROOK, D. W. 1987. Upper bounds for symmetric Markov transition functions. Ann. Inst. H. Poincare 2 245 287. ´ Z.
  • DAVIES, E. B. 1989. Heat Kernels and Spectral Theory. Cambridge Univ. Press. Z.
  • DAWSON, D. A. 1993. Measure-valued Markov processes. Ecole d'Ete de Probabilites de Saint´ ´ Flour XXI. Lecture Notes in Math. 1541 1 260. Springer, Berlin. Z.
  • DUNFORD, N. and SCHWARTZ, J. 1967. Linear Operators. Interscience, New York. Z.
  • ETHERIDGE, A. and MARCH, P. 1991. A note on superprocesses. Probab. Theory Related Fields 89 141 147. Z.
  • ETHIER, S. N and KURTZ, T. G. 1986. Markov Processes Characterization and Convergence. Wiley, New York. Z.
  • HILLE, E. and PHILLIPS, R. 1957. Functional Analysis and Semi-groups. Amer. Math. Soc., Providence, RI. Z.
  • IKEDA, N. and WATANABE, S. 1989. Stochastic Differential Equations and Diffusion Processes, 2nd ed. North-Holland, Amsterdam. Z.
  • JACOD, J. and SHIRYAEV, A. N. 1987. Limit Theorems for Stochastic Processes. Springer, Berlin. Z.
  • KONNO, N. and SHIGA, T. 1988. Stochastic partial differential equations for some measurevalued diffusions. Probab. Theory Related Fields 79 201 225. Z.
  • KUWAE, K. and UEMURA, T. 1995. Weak convergence of symmetric diffusion processes. Probab. Theory Related Fields. To appear. Z.
  • LYNCH, J. and SETHURAMAN, J. 1987. Large deviations for processes with independent increments. Ann. Probab. 15 610 627. Z.
  • OVERBECK, L. and ROCKNER, M. 1996. Geometric aspects of finite and infinite dimensional ¨ Fleming Viot processes. In Random Operators and Stochastic Equations. To appear. Z.
  • PARTHASARATHY, K. R. 1967. Probability Measures on Metric Spaces. Academic Press, New York. Z.
  • PERKINS, E. 1991. Conditional Dawson Watanabe processes and Fleming Viot processes. In Seminar on Stochastic Processes Progr. Probab. 29 143 156. Birkhauser, Basel. ¨ Z.
  • SCHIED, A. 1996. Sample path large deviations for super-Brownian motion. Probab. Theory Related Fields 104 319 348. Z.
  • SHIGA, T. 1990. A stochastic equation based on a Poisson system for a class of measure-valued diffusion processes. J. Math. Kyoto Univ. 30 245 279. Z.
  • STURM, K.-TH. 1994. Analysis on local Dirichlet spaces I. Recurrence, conservativeness and L p-Liouville properties. J. Reine Angew. Math. 456 173 196. Z.
  • STURM, K.-TH. 1995. On the geometry defined by Dirichlet forms. In Seminar on Stochastic Analysis, Random Fields and Applications Prog. Probab. 36 231 242. Birkauser, ¨ Basel.