The Annals of Probability

Path Properties of Superprocesses with a General Branching Mechanism

Jean-François Delmas

Full-text: Open access

Abstract

We first consider a super Brownian motion $X$ with a general branching mechanism. Using the Brownian snake representation with subordination, we get the Hausdorff dimension of supp $X_t$, the topological support of $X_t$ and, more generally, the Hausdorff dimension of $\Bigcup_{t/in B}\supp X _t$. We also provide estimations on the hitting probability of small balls for those random measures. We then deduce that the support is totally disconnected in high dimension. Eventually, considering a super $\alpha$-stable process with a general branching mechanism, we prove that in low dimension this random measure is absolutely continuous with respect to the Lebesgue measure.

Article information

Source
Ann. Probab., Volume 27, Number 3 (1999), 1099-1134.

Dates
First available in Project Euclid: 29 May 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1022677441

Digital Object Identifier
doi:10.1214/aop/1022677441

Mathematical Reviews number (MathSciNet)
MR1733142

Zentralblatt MATH identifier
0962.60033

Subjects
Primary: 60G57: Random measures 60J25: Continuous-time Markov processes on general state spaces 60J55 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords
Superprocesses measure valued processes Brownian snake exit mea-sure hitting probabilities Hausdorff dimension subordinator

Citation

Delmas, Jean-François. Path Properties of Superprocesses with a General Branching Mechanism. Ann. Probab. 27 (1999), no. 3, 1099--1134. doi:10.1214/aop/1022677441. https://projecteuclid.org/euclid.aop/1022677441


Export citation

References

  • [1] Abraham, R. (1995). On the connected components of super-Brownian motion and of its exit measure. Stochastic Process. Appl. 60 227-245.
  • [2] Bertoin, J. (1996). L´evy Processes. Cambridge Univ. Press.
  • [3] Bertoin, J., Le Gall, J.-F. and Le Jan, Y. (1997). Spatial branching processes and subordination. Canad. J. Math. 49 24-54.
  • [4] Bingham, N., Goldie, C. and Teugels, J. (1987). Regular Variation. Cambridge Univ. Press.
  • [5] Blumenthal, R. (1992). Excursions of Markov Processes. Birkh¨auser, Boston.
  • [6] Blumenthal, R. and Getoor, R. (1961). Sample functions of stochastic processes with stationary independent increments. J. Math. Mechanics 10 493-516.
  • [7] Dawson, D. A. (1993). Measure-valued Markov processes. ´Ecole d' ´Et´e de Probabilit´e de Saint-Flour 1991. Lecture Notes in Math. 1541 1-260. Springer, Berlin.
  • [8] Dawson, D. A., Iscoe, I. and Perkins, E. (1989). Super-Brownian motion: path properties and hitting probabilites. Probab. Theory Related Fields 83 135-205.
  • [9] Dawson, D. A. and Perkins, E. (1991). Historical Processes. Mem. Amer. Math. Soc. 93.
  • [10] Dynkin, E. (1991). Branching particle systems and superprocesses. Ann. Probab. 19 1157- 1194.
  • [11] Dynkin, E. (1991). A probabilistic approach to one class of nonlinear differential equations. Probab. Theory Related Fields 89 89-115.
  • [12] Dynkin, E. (1994). An Introduction to Branching Measure-valued Processes. Amer. Math. Soc., Providence, RI.
  • [13] Falconer, K. (1990). Fractal Geometry. Wiley, New York.
  • [14] Fitzsimmons, P. (1988). Construction and regularity of measure-valued Markov branching processes. Israel J. Math. 64 337-361.
  • [15] Fleischmann, K. (1988). Critical behavior of some measure-valued processes. Math. Nachr. 135 131-147.
  • [16] Fristedt, B. (1974). Sample functions of stochastic processes with stationary independent increments. In Advances in Probability 3 241-396. Dekker, New York.
  • [17] Le Gall, J.-F. (1993). A class of path-valued Markov processes and its applications to superprocesses. Probab. Theory Related Fields 95 25-46.
  • [18] Le Gall, J.-F. (1994). A path-valued Markov process and its connections with partial differential equations. In Proceedings in First European Congress of Mathematics 2 185-212. Birkh¨auser, Boston.
  • [19] Le Gall, J.-F. (1997). Brownian snakes, superprocesses and partial differential equations. Unpublished manuscript.
  • [20] Perkins, E. (1988). A space-time property of a class of measure-valued branching diffusions. Trans. Amer. Math. Soc. 305 743-795.
  • [21] Perkins, E. (1989). The Hausdorff measure of the closed support of super-Brownian motion. Ann. Inst. H. Poincar´e 25 205-224.
  • [22] Perkins, E. (1995). Measure-valued branching diffusions and interactions. In Proceedings of the International Congress of Mathematicians 1036-1045. Birkh¨auser, Boston.
  • [23] Revuz, D. and Yor, M. (1991). Continuous Martingales and Brownian Motion. Springer, Berlin.
  • [24] Rudin, W. (1986). Real and Complex Analysis, 3rd ed. McGraw-Hill, New York.
  • [25] Tribe, R. (1989). Path properties of superprocesses. Ph.D. thesis, Univ. British Columbia.
  • [26] Tribe, R. (1991). The connected components of the closed support of super-Brownian motion. Probab. Theory Related Fields 89 75-87.