Super-Brownian motion: Path properties and hitting probabilities. (English) Zbl 0692.60063

The model under consideration is a super-Brownian motion \(X=\{X_ t;t\geq 0\}\) in \({\mathbb{R}}^ d\), i.e. a measure-valued diffusion process related to the equation \(\partial u/\partial t=2^{-1}(\Delta u-u^ 2)\) via the Laplace transition functional. The basic goal is the derivation of path properties of \(\{S(X_ t)\); \(t\geq 0\}\) where \(S(X_ t)\) is the closed support of the measure state \(X_ t\) at time t. This includes a modulus of continuity (d\(\geq 1)\), a study of “polar sets” and “multiple points”, and the derivation of an exact Hausdorff measure function for the “range” of this support process (d\(\geq 4).\)
The super-Brownian motion exhibits a richer family of behaviors than Brownian motion. For instance, points are hit in dimensions \(d\leq 3\), in \(d=4\) multiple points of all orders exist, and for \(d\leq 7\) (and only for these dimensions) there are double points. Proofs are much more complicated as for analogous Brownian motion sample path properties. They include analytic estimates for the hitting of balls by the X process. Also nonstandard analysis arguments are involved.
Reviewer: K.Fleischmann


60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
Full Text: DOI


[1] Albeverio, S., Fenstad, J.E., Hoegh-Krohn, R., Lindstrom, T.: Nonstandard methods in stochastic analysis and mathematical physics. New York: Academic Press 1986
[2] Anderson, R.M., Rashid, S.: A nonstandard characterization of weak convergence. Proc. Am. Math. Soc. 69, 327-332 (1978) · Zbl 0393.03047
[3] Ciesielski, Z., Taylor, S.J.: First passage times and sojourn times for Brownian motion in space and the exact Hausdorff measure of the sample path. Trans. Am. Math. Soc. 103, 434-450 (1962) · Zbl 0121.13003
[4] Cutland, N.: Nonstandard measure theory and its applications. Bull. London Math. Soc. 15, 529-589 (1983) · Zbl 0529.28009
[5] Cutler, C.: Some measure-theoretical and topological results for measure-valued and set-valued stochastic processes, Ph.D. thesis, Carleton University 1984
[6] Dawson, D.A.: The critical measure diffusion process. Z. Wahrscheinlichkeitstheor. Verw. Geb. 40, 125-145 (1977) · Zbl 0343.60001
[7] Dawson, D.A.: Limit theorems for interaction free geostochastic systems. Colloq. Math. Soc. Janos Bolyai 24, 27-47 (1978)
[8] Dawson, D.A., Hochberg, K.J.: The carrying dimension of a stochastic measure diffusion. Ann. Probab. 7, 693-703 (1979) · Zbl 0411.60084
[9] Dawson, D.A., Iscoe, I., Perkins, E.A.: Sample path properties of the support process of super-Brownian motion. C.R. Math. Rep. Acad. Sci. Canada 10, 83-88 (1988) · Zbl 0644.60089
[10] Dugundji, J.: Topology. Boston: Allyn and Bacon 1966 · Zbl 0144.21501
[11] Dynkin, E.B.: Representation for functionals of superprocesses by multiple stochastic integral, with applications to self-intersection local times. Astérisque 157-158, 147-171 (1988)
[12] Ethier, S.N., Kurtz, T.G.: Markov Processes: Characterization and Convergence. New York: Wiley (1986) · Zbl 0592.60049
[13] Gmira, A., Veron, L.: Large time behavior of the solutions of a semilinear parabolic equation in ?N. J. Differ. Equations 53, 258-276 (1984) · Zbl 0529.35041
[14] Harris, T.E.: The Theory of Branching Processes. Berlin Heidelberg New York: Springer 1963 · Zbl 0117.13002
[15] Hoover, D.N., Perkins, E.: Nonstandard construction of the stochastic integral and applications to stochastic differential equations. I. Trans. Am. Math. Soc. 275, 1-58 (1983) · Zbl 0533.60063
[16] Hurd, A.E., Loeb, P.A.: An introduction to nonstandard real analysis. New York: Academic Press 1985 · Zbl 0583.26006
[17] Iscoe, I.: A weighted occupation time for a class of measure-valued critical branching Brownian motion. Probab. Theory Rel. Fields 71, 85-116 (1986a) · Zbl 0555.60034
[18] Iscoe, I.: Ergodic theory and a local occupation time for measure-valued branching processes. Stochastics 18, 197-243 (1986b) · Zbl 0608.60077
[19] Iscoe, I.: On the supports of measure-valued critical branching Brownian motion. Ann. Probab. 16, 200-221 (1988) · Zbl 0635.60094
[20] Lady?enskaya, O.A., Sollonnikov, V.A., Ural’ceva, N.N.: Linear and quasilinear equations of parabolic type. (Am. Math. Soc. Monogr. vol. 23) 1968
[21] Le Gall, J.-F.: Exact Hausdorff measure of Brownian multiple points. In: Cinlar, E., Chung, K.L., Getoor, R.K. (eds.). Seminar on Stochastic Processes, 1986. Boston: Birkhäuser 1987 · Zbl 0609.60079
[22] Loeb, P.A.: Conversion from nonstandard to standard measure spaces and applications in probability theory. Trans. Am. Math. Soc. 211, 113-122 (1975) · Zbl 0312.28004
[23] Loeb, P.A.: An introduction to nonstandard analysis and hyperfinite probability theory. In: Reid-Bharuch, A. (ed.). (Probabilistic analysis and related topics, vol. 2). New York: Academic Press 1979 · Zbl 0441.03027
[24] Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, Berlin Heidelberg New York: 1983 · Zbl 0516.47023
[25] Perkins, E.A.: A space-time property of a class of measure-valued branching diffusions. Trans. Am. Math. Soc. 305, 743-795 (1988a) · Zbl 0641.60060
[26] Perkins, E.A.: Polar sets and multiple points for super Brownian motion. Lower Bounds (1988b) · Zbl 0721.60046
[27] Perkins, E.A.: The Hausdorff measure of the closed support of super-Brownian motion. Ann. Inst. H. Poincaré 25, 205-224 (1989a) · Zbl 0679.60053
[28] Perkins, E.A.: Unpublished lecture notes (1989b)
[29] Ray, D.: Sojourn times and the exact Hausdorff measure of the sample path for planar Brownian motion. Trans. Am. Math. Soc. 106, 436-444 (1963) · Zbl 0119.14602
[30] Roelly-Coppoletta, S.: A criterion of convergence of measure-valued processes: application to measure branching processes. Stochastics 17, 43-65 (1986) · Zbl 0598.60088
[31] Rogers, C.A.: Hausdorff Measures. Cambridge: Cambridge University Press 1970 · Zbl 0204.37601
[32] Rogers, C.A., Taylor, S.J.: Functions continuous and singular with respect to a Hausdorff measure. Mathematika 8, 1-31 (1961) · Zbl 0145.28701
[33] Sawyer, S., Fleischman, J.: Maximum geographic range of a mutant allele considered as a subtype of Brownian branching random field. Proc. Natl. Acad. Sci. USA 76, 872-875 (1979) · Zbl 0404.92011
[34] Stroock, D.W.: An introduction to the theory of large deviations. Berlin Heidelberg New York Springer: 1984 · Zbl 0552.60022
[35] Stroock, D.W., Varadhan, S.R.S.: Multidimensional diffusion processes. Berlin Heidelberg New York: Springer 1979 · Zbl 0426.60069
[36] Sugitani, S.: Some properties for the measure-valued branching diffusion processes (preprint 1987) · Zbl 0634.60071
[37] Taylor, S.J.: On the connection between generalized capacities and Hausdorff measures. Proc. Cambridge Philos. Soc. 57, 524-531 (1961) · Zbl 0106.26802
[38] Taylor, S.J.: The exact Hausdorff measure of the sample path for planar Brownian motion. Proc. Cambridge Philos. Soc. 60, 253-258 (1964) · Zbl 0149.13104
[39] Taylor, S.J.: Multiple points for the sample paths of the symmetric stable process. Z. Wahrscheinlichkeitsth. Verw. Geb. 5, 247-264 (1966) · Zbl 0146.37905
[40] Walsh, J.B.: An Introduction to stochastic partial differential equations. (Lect. Notes in Math., vol. 1180). Berlin Heidelberg New York: Springer 1986 · Zbl 0608.60060
[41] Watanabe, S.: A limit theorem of branching processes and continuous state branching processes. J. Math. Kyoto Univ. 8, 141-167 (1968) · Zbl 0159.46201
[42] Zähle, U.: The fractal character of localizable measure-valued processes. III. Fractal carrying sets of branching diffusions. Math. Nachr. 138, 293-311 (1988) · Zbl 0692.60062
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.