The Annals of Probability

On transition semigroups of $(A,\Psi )$-superprocesses with immigration

Wilhelm Stannat

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We study the global properties of transition semigroups $(p_t^{\nu , \Psi , A})$ of $(A, \Psi )$-superprocesses over compact type spaces with possibly nonzero immigration $\nu$ in various function spaces. In particular, we compare the different rates of convergence of $(p_t^{\nu ,\Psi ,A})$ to equilibrium. Our analysis is based on an explicit formula for the Gateaux derivative of $p_t^{\nu ,\Psi , A} F$.

Article information

Ann. Probab., Volume 31, Number 3 (2003), 1377-1412.

First available in Project Euclid: 12 June 2003

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Zentralblatt MATH identifier

Primary: 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07]
Secondary: 47D07: Markov semigroups and applications to diffusion processes {For Markov processes, see 60Jxx} 60F10: Large deviations 60G57: Random measures 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 92D10: Genetics {For genetic algebras, see 17D92}

Superprocesses ergodic theorems gradient estimates Gamma processes short-time asymptotics.


Stannat, Wilhelm. On transition semigroups of $(A,\Psi )$-superprocesses with immigration. Ann. Probab. 31 (2003), no. 3, 1377--1412. doi:10.1214/aop/1055425784.

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  • [1] BAKRY, D. (1994). L'hy percontractivité et son utilisation en théorie des semigroupes. Ecole d'Ete de Probabilités de Saint Flour XXII. Lecture Notes in Math. 1581 1-114. Springer, Berlin.
  • [2] BELL, W. W. (1968). Special Functions. Van Nostrand, London.
  • [3] BOULEAU, N. and HIRSCH, F. (1991). Dirichlet Forms and Analy sis on Wiener Space. de Gruy ter, Berlin.
  • [4] CHERNOFF, P. R. (1968). Note on product formulas for operator semigroups. J. Funct. Anal. 2 238-242.
  • [5] DAWSON, D. (1993). Measure-valued Markov processes. Ecole d'Ete de Probabilités de Saint Flour XXI. Lecture Notes in Math. 1541 1-260. Springer, Berlin.
  • [6] ETHERIDGE, A. M. (2000). An Introduction to Superprocesses. Amer. Math. Soc., Providence, RI.
  • [7] ETHIER, S. N. and GRIFFITHS, R. C. (1993). The transition function of a measure-valued branching diffusion with immigration. In Stochastic Processes. A Festschrift in Honour of Gopinath Kallianpur (S. Cambanis, J. Ghosh, R. L. Karandikar and P. K. Sen, eds.) 71-79. Springer, New York.
  • [8] FELLER, W. (1951). Two singular diffusion problems. Ann. Math. 54 173-182.
  • [9] JACKA, S. and TRIBE, R. (2001). Comparison for measure-valued processes with interactions. Unpublished manuscript.
  • [10] RAMIREZ, J. A. (2001). Short-time asy mptotics in Dirichlet spaces. Comm. Pure Appl. Math. 54 259-293.
  • [11] SCHIED, A. (1997). Geometric aspects of Fleming-Viot and Dawson-Watanabe processes. Ann. Probab. 25 1160-1179.
  • [12] STANNAT, W. (2003). Spectral properties for a class of continuous state branching processes with immigration. J. Funct. Anal. To appear.
  • [13] SZEGÖ, G. (1967). Orthogonal Poly nomials. Amer. Math. Soc., Providence, RI.
  • [14] WATANABE, S. (1968). A limit theorem of branching processes and continuous state branching processes. J. Math. Ky oto Univ. 8 141-167.