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Transactions of the American Mathematical Society

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Absolute continuity results for superprocesses with some applications

Authors: Steven N. Evans and Edwin Perkins
Journal: Trans. Amer. Math. Soc. 325 (1991), 661-681
MSC: Primary 60G30; Secondary 60J80
MathSciNet review: 1012522
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Abstract: Let $ {X^1}$ and $ {X^2}$ be instances of a measure-valued Dawson-Watanabe $ \xi $-super process where the underlying spatial motions are given by a Borel right process, $ \xi $, and where the branching mechanism has finite variance. A necessary and sufficient condition on $ X_0^1$ and $ X_0^2$ is found for the law of $ X_s^1$ to be absolutely continuous with respect to the law of $ X_t^2$. The conditions are the natural absolute continuity conditions on $ \xi $, but some care must be taken with the set of times $ s$, $ t$ being considered. The result is used to study the closed support of super-Brownian motion and give sufficient conditions for the existence of a nontrivial "collision measure" for a pair of independent super-Lévy processes or, more generally, for a super-Lévy process and a fixed measure. The collision measure gauges the extent of overlap of the two measures. As a final application, we give an elementary proof of the instantaneous propagation of a super-Lévy process to all points to which the underlying Lévy process can jump. This result is then extended to a much larger class of superprocesses using different techniques.

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  • [1] A. Benveniste and J. Jacod, Systèmes de Lévy des processus de Markov, Invent. Math. 21 (1973), 183-198. MR 0343375 (49:8117)
  • [2] R. M. Blumenthal and R. K. Getoor, Markov processes and potential theory, Academic Press, New York and London, 1968. MR 0264757 (41:9348)
  • [3] D. A. Dawson, The critical measure diffusion process, Z. Wahrsch. Verw. Gebiete 40 (1977), 125-145. MR 0478374 (57:17857)
  • [4] D. A. Dawson, I. Iscoe and E. Perkins, Super-Brownian motion: path properties and hitting probabilities, Probab. Theory Related Fields 83 (1989), 135-206. MR 1012498 (90k:60073)
  • [5] C. Dellacherie and P. A. Meyer, Probability and potential, North-Holland Math. Stud., no. 29, North-Holland, Amsterdam, 1978.
  • [6] E. B. Dynkin, Representation for functionals of superprocesses by multiple stochastic integrals, with applications to self-intersection local times, In: Colloque Paul Lévy sur les processus stochastiques, Astérisque 157-158 (Société Mathématique de France) (1988). MR 976217 (90b:60103)
  • [7] N. El Karoui and S. Roelly-Coppoletta, Study of a general class of measure-valued branching processes: A Lévy-Hincin representation, preprint.
  • [8] S. N. Ethier and T. G. Kurtz, Markov processes: Characterization and convergence, Wiley, New York, 1986. MR 838085 (88a:60130)
  • [9] P. J. Fitzsimmons, Construction and regularity of measure-valued Markov branching processes, Israel J. Math 64 (1988), 337-361. MR 995575 (90f:60147)
  • [10] I. I. Gihman and A. V. Skorohod, The theory of stochastic processes II, Springer-Verlag, New York, Heidelberg, and Berlin, 1975. MR 0375463 (51:11656)
  • [11] I. Iscoe, A weighted occupation time for a class of mesaure-valued branching processes, Probab. Theory Related Fields 71 (1986), 85-116. MR 814663 (87c:60070)
  • [12] O. Kallenberg, Random measures, 3rd ed., Akademie-Verlag, Berlin; Academic Press, New York, 1983. MR 818219 (87g:60048)
  • [13] F. B. Knight, Essentials of Brownian motion and diffusion, Math. Surveys Monogr., no. 18, Amer. Math. Soc., Providence, R.I., 1981. MR 613983 (82m:60098)
  • [14] P. A. Meyer, Intégrales stochastiques, Séminaire de Probabilités I, Lecture Notes in Math., Vol. 39, Springer-Verlag, New York, Heidelberg, and Berlin, 1967.
  • [15] E. Perkins, Polar sets and multiple point for super-Brownian motion, Ann. Probab. 18 (1990), 453-491. MR 1055416 (91i:60109)
  • [16] -, The Hausdorff measure of the closed support of super-Brownian motion, Ann. Inst. H. Poincaré25 (1989), 205-224. MR 1001027 (90k:60074)
  • [17] -, A space-time property of a class of measure-valued diffusions, Trans. Amer. Math. Soc. 305 (1988), 743-795. MR 924777 (89c:60064)
  • [18] R. Tribe, Ph.D. thesis, Univ. of British Columbia.
  • [19] S. Watanabe, A limit theorem of branching processes and continuous state branching processes, J. Math. Kyoto Univ. 8 (1968), 141-167. MR 0237008 (38:5301)
  • [20] M. Yor, Continuité des temps locaux, Temps Locaux, Astérisque, Vols. 52-53, Soc. Math. France, Paris, 1978.

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Keywords: Superprocesses, absolute continuity, measure-valued diffusion, measure-valued branching process, random measure
Article copyright: © Copyright 1991 American Mathematical Society

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