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IBM, SIBM and IBS. (English) Zbl 1044.60077

Summary: We construct a super iterated Brownian motion (SIBM) from a historical version of iterated Brownian motion (IBM) using an iterated Brownian snake (IBS). It is shown that the range of super iterated Brownian motion is qualitatively quite different from that of super Brownian motion in that there are points with explosions in the branching. However, at a fixed time the support of SIBM has an exact Hausdorff measure function that is the same (up to a constant) as that of super Brownian motion at a fixed time.

MSC:

60J65 Brownian motion
60G17 Sample path properties
60G57 Random measures
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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References:

[1] Blumenthal, R. (1992). Excursions of Markov Processes. Birkhäuser, Boston. · Zbl 0983.60504
[2] Burdzy K. (1993). Some path properties of iterated Brownian motion. In Seminar on Stochastic Processes (E. \?Cinlar, K. L. Chung and M. J. Sharpe, eds.) 67-87 Birkhäuser, Boston. · Zbl 0789.60060
[3] Burdzy, K. and Khoshnevisan (1998). Brownian motion in a Brownian crack. Ann. Appl. Probab. 8 708-748. · Zbl 0937.60081
[4] Dawson, D., Iscoe, I. and Perkins E. (1989) Super-Brownian motion: path properties and hitting probabilities. Probab. Theory Related Fields 83 135-205. · Zbl 0692.60063
[5] Dawson, D. and Perkins, E. (1991) Historical Processes. Mem. Amer. Math Soc. 93 · Zbl 0754.60062
[6] Dynkin, E. B. and Kuznetsov, S. E. (1995) Markov snakes and superprocesses. Probab. Theory Related Fields 103 433-473. · Zbl 0862.60072
[7] Fitzsimmons P. J. (1988) Construction and regularity of measure-valued branching processes. Israel J. Math. 64 337-361. · Zbl 0673.60089
[8] Karatzas I. and Shreve S. (1991) Brownian Motion and Stochastic Calculus. Springer, Berlin. · Zbl 0734.60060
[9] Le Gall, J. F. (1999). Spatial Branching Processes, Random Snakes and Partial Differential Equations. Birkha üser, Boston. · Zbl 0938.60003
[10] Le Gall, J. F. (1993) A class of path-valued Markov processes and its applications to superprocesses. Probab. Theory Related Fields 95 25-46. · Zbl 0794.60076
[11] Le Gall, J. F. (1994) A path-valued Markov process and its connection with partial differential equations. In Proceedings First European Congress of Mathematics (A. Joseph, F. Mignot, F. Murat, B. Prum, R. Rentschler, eds.) 2 185-212. Birka üser, Boston. · Zbl 0812.60058
[12] Le Gall J. F. and Perkins E. A. (1995) The Hausdorff measure of the support of twodimensional super-Brownian motion. Ann. Probab. 23 1719-1747 · Zbl 0856.60055
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