Summary
The Dawson-Watanabe super-Brownian motion has been intensively studied in the last few years. In particular, there has been much work concerning the Hausdorff dimension of certain remarkable sets related to super-Brownian motion. We contribute to this study in the following way. Let (Y t)t≧0 be a super-Brownian motion on ℝd(d≥2) andH be a Borel subset of ℝd. We determine the Hausdorff Dimension of {t≧0; SuppY t∩H≠Ø}, improving and generalizing a result of Krone. We also obtain a new proof of a result of Tribe which gives, whend≧4, the Hausdorff dimension of\( \cup _{t \in {\rm B}}\) SuppY t as a function of the dimension ofB.
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Dawson, D.A., Iscoe, I., Perkins, E.A.: Super-Brownian motion: path properties and hitting probabilities. Probab. Theory Relat. Fields83, 135–205 (1989)
Dawson, D.A., Perkins, E.A.: Historical processes. Mem. Am. Math. Soc.454, (1991)
Dynkin, E.B.: Additive functionals of several time reversible Markov processes. J. Funct. Anal.42, 64–101 (1981)
Dynkin, E.B.: A probabilistic approach to one class of non linear differential equations. Probab. Theory Relat. Fields89, 89–115 (1991)
Evans, S.N., Perkins, E.A.: Absolute continuity results for superprocesses with some applications. Trans. Am. Math. Soc.325(2), 661–681 (1991)
Kaufman, R.: Une propriété métrique du mouvement brownien. C. R. Acad. Sci. Paris Sér. A tome268, 727–728 (1969)
Kaufman, R.: Measures of Hausdorff-type, and Brownian motion. Mathematika19, 115–119 (1972)
Krone, S.M.: Local times for super-diffusions. Ann Probab. (to appear)
Le Gall, J.F.: A class of path-valued Markov processes and its applications to superprocesses. Probab. Theory Relat. Fields95, 25–46 (1993)
Le Gall, J.F.: A path-valued Markov process and its connections with partial differential equations. Proc. First European Congress of Math. Birkhaüser (to appear)
Le Gall, J.F.: A lemma on super-Brownian motion with some applications. Festchrift in honor of E.B. Dynkin, Birkhaüser (to appear)
Perkins, E.A.: The Hausdorff measure of the closed support of super-Brownian motion. Ann. Inst. H. Poincaré (Probabilités-Statistiques)25, 205–224 (1989)
Revuz, D., Yor, M.: Continuous martingales and Brownian motion. Berlin Heidelberg New York: Springer 1991
Serlet, L.: Quelques propriétés du super-mouvement brownien. Thèse de Doctorat de l'Université Paris VI (1993)
Serlet, L.: On the Hausdorff measure of multiple points and collision points of super-Brownian motion. Stochastics. (to appear)
Tribe, R.: Path properties of superprocesses. Ph.D. Thesis, University of British Colombia (1989)