Avoiding-Probabilities For Brownian Snakes and Super-Brownian Motion

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


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 $\varepsilon$ from the origin. In particular we show that when $\varepsilon$ tends to 0, this probability respectively behaves (up to multiplicative constants) like $\varepsilon^4$, $\varepsilon^{2\sqrt{2}}$ and $\varepsilon^{(\sqrt {17}-1)/2}$, 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.

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Pages: 1-27

Publication Date: May 7, 1997

DOI: 10.1214/EJP.v2-17


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