| Avoiding-probabilities for Brownian snakes and super-Brownian motion (2007) | |||||||||||||||
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| We investigate the asymptotic behaviour of the probability that a d- dimensional Brownian snake conditioned to survive on a macroscopic scale, avoids 0 when starting at distance " from the origin. In particular we show that when " tends to 0, this probability respectively behaves (up to multiplicative constants) like " 4 , " 2 p 2 and " ( p 17\Gamma1)=2 , when d = 1, d = 2 and d = 3. Analogous results are derived for super-Brownian motion started from ffi x (conditioned to survive until some time) when the modulus of x tends to 0. AMS-classification: 60J25, 60J45 Key words: Brownian snakes, superprocesses, non-linear differential equations 1 Introduction The Brownian snake has been shown to be a powerful tool to study probabilistically some non-linear partial differential equations (see e.g. Le Gall [14, 15, 16], Dhersin-Le Gall [9]). It has also been studied on its own right (see e.g. [24], [25]), and because of its intimate connection with super-Brownian motion, it can be... | |||||||||||||||
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