Home | Contents | Submissions, editors, etc. | Login | Search | ECP
 Electronic Journal of Probability > Vol. 2 (1997) > Paper 3 open journal systems 


Avoiding-Probabilities For Brownian Snakes and Super-Brownian Motion

Romain Abraham, Université René Descartes (Paris 5)
Wendelin Werner, Université Paris-Sud and IUF


Abstract
We investigate the asymptotic behaviour of the probability that a normalized $d$-dimensional Brownian snake (for instance when the life-time process is an excursion of height 1) avoids 0 when starting at distance $eps$ from the origin. In particular we show that when $eps$ tends to 0, this probability respectively behaves (up to multiplicative constants) like $eps^4$, $eps^{2sqrt{2}}$ and $eps^{(sqrt {17}-1)/2}$, when $d=1$, $d=2$ and $d=3$. Analogous results are derived for super-Brownian motion started from $delta_x$ (conditioned to survive until some time) when the modulus of $x$ tends to 0.


Full text: PDF

Pages: 1-27

Published on: May 7, 1997


Bibliography

M. Abramowitz, I.A. Stegun (Eds.): Handbook of mathematical functions, Dover, New-York, 1965. Math. Review link 94b:00012

P. Biane, M. Yor: Quelques précisions sur le méandre Brownien, Bull. Sci. Math. 111, 101-109 (1988) Math. Review link 89i:60156

D.A. Dawson: Measure-valued Markov processes, Ecole d'été de St-Flour 1991, Lecture Notes in Math. 1541, Springer, Berlin, 1993. Math. Review link 94m:60101

D.A. Dawson, I. Iscoe, E.A. Perkins: Super-Brownian motion: Path properties and hitting probabilities, Probab. Theor. Rel. Fields 83, 135-205 (1989) Math. Review link 90k:60073

D.A. Dawson, E.A. Perkins: Historical superprocesses, Memoirs Amer. Math. Soc. 454, 1991. Math. Review link 92a:60145

A. Dembo, O. Zeitouni: Large deviations for random distribution of mass, Proceedings of the IMA workshop on random discrete structures (Ed. D.J. Aldous, R. Pemantle), IMA vol. 76, Springer, 45-53 (1994) Math. Review link 97d:60051

J.-S. Dhersin: Super-mouvement brownien, serpent brownien et équations aux dérivées partielles, Thèse de doctorat de l'université Paris 6, 1997.

J.-S. Dhersin, J.-F. Le Gall: Wiener's test for super-Brownian motion and the Brownian snake, Probab. Theor. Rel. Fields, to appear.

E.B. Dynkin: A probabilistic approach to one class of nonlinear differential equations, Probab. Theor. Rel. Fields 89, 89-115 (1991) Math. Review link 92d:35090

E.B. Dynkin: An introduction to branching measure-valued processes, CRM Monograph Series Vol.6, Amer. Math. Soc., Providence, 1994. Math. Review link 96f:60145

E.B. Dynkin, S.E. Kuznetsov: Markov snakes and superprocesses, Probab. Theor. Rel. Fields 103, 433-473 (1995) Math. Review link 96k:60188

J.-P. Imhof: Density factorizations for Brownian motion and the three-dimensional Bessel processes and applications, J. Appl. Prob. 21, 500-510 (1984) Math. Review link 85j:60152

J.-F. Le Gall: A class of path-valued Markov processes and its applications to superprocesses, Probab. Th. Rel. Fields 95, 25-46 (1993) Math. Review link 94f:60093

J.-F. Le Gall: A path-valued Markov process and its connections with partial differential equations, Proceedings 1st European Congress of Mathematics, Vol. II, pp. 185-212, Birkhäuser, Boston, 1994. Math. Review link 96m:60169

J.-F. Le Gall: The Brownian snake and solutions of $Delta u = u^2$ in a domain, Probab. Th. Rel. Fields 104, 393-432 Math. Review link 96c:60098

J.-F. Le Gall: A probabilistic Poisson representation for positive solutions of $Delta u = u^2$ in a domain, Comm. Pure Appl. Math. 50, 69-103 (1997)

J.-F. Le Gall: Superprocesses, Brownian snakes and partial differential equations, Lecture Notes from the 11th winter school on Stochastic processes, Sigmundsburg, March 1996, Prépublication 337 du Laboratoire de Probabilités, Université Paris VI (1996).

J.-F. Le Gall, E.A. Perkins: The Hausdorff measure of the support of two-dimensional super-Brownian motion, Ann. Probab. 23, 1719-1747 (1995) Math. Review link 96m:60114

S.C. Port, C.J. Stone: Brownian motion and classical potential theory, Academic Press, New-York, 1978. Math. Review link 58#11459

D. Revuz, M. Yor: Continuous martingales and Brownian motion, Springer, Berlin, 1991. Math. Review link 92d:60053

L. Serlet: Some dimension results for super-Brownian motion, Probab. Theor. Rel. Fields 101, 371-391 (1995) Math. Review link 96m:60115

L. Serlet: On the Hausdorff measure of multiple points and collision points of super-Brownian motion, Stochastics Stoch. Rep. 54, 169-198 (1995)

L. Serlet: The occupation measure of super-Brownian motion conditioned on non-extinction, J. Theor. Prob. 9, 561-578 (1996)

L. Serlet: Large deviation principle for the Brownian snake, Stoch. Proc. Appl., to appear.

J. Verzani: Cone paths for the planar Brownian snake, Probab. Theor. Rel. Fields, to appear

M. Yor: Some aspects of Brownian motion, Part I: Some special functionals, Lectures in Mathematics, ETH Zürich, Birkhäuser, 1992 Math. Review link 93i:60155

M. Yor: Generalized meanders as limits of weighted Bessel processes, and an elementary proof of Spitzer's asymptotic result on Brownian windings, Stud. Sci. Math. Hung., to appear.















Research
Support Tool
Capture Cite
View Metadata
Printer Friendly
Context
Author Address
Action
Email Author
Email Others


Home | Contents | Submissions, editors, etc. | Login | Search | ECP

Electronic Journal of Probability. ISSN: 1083-6489