Brownian survival among Poissonian traps with random shapes at critical intensity

Berg, van den Maarten; Bolthausen, Erwin; Hollander, den WThF Frank

doi:10.1007/s00440-004-0393-4

Cited by 14 publications

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“…The model of Brownian motion in a static random medium has been thoroughly investigated. For the directed polymer with immobile Poissonian traps or catalysts, we refer to Sznitman's book [39] for general collection (up to the year 1998), [30] for a survey and [2], [3], [28], [36] for specific topics. For the results motivated by the parabolic Anderson models with timeindependent random potentials, we cite [7], [8], [14], [16], [20], [26], [27], [38] as a partial list of the publications on this subject.…”

Section: Introductionmentioning

confidence: 99%

Annealed Asymptotics for Brownian Motion of Renormalized Potential in Mobile Random Medium

Chen

Xiong

2014

J Theor Probab

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Motivated by the study of the directed polymer model with mobile Poissonian traps or catalysts and the stochastic parabolic Anderson model with time dependent potential, we investigate the asymptotic behavior ofwhere θ > 0 is a constant, V is the renormalized Poisson potential of the formand ω s is the measure-valued process consisting of independent Brownian particles whose initial positions form a Poisson random measure on R d with Lebesgue measure as its intensity. Different scaling limits are obtained according to the parameter p and dimension d. For the logarithm of the negative exponential moment, the range of d 2 < p < d is divided into 5 regions with various scaling rates of the orders t d/p , t 3/2 , t (4−d−2p)/2 , t log t and t, respectively. For the positive exponential moment, the limiting behavior is studied according to the parameters p and d in three regions. In the sub-critical region (p < 2), the double logarithm of the exponential moment has a rate of t. In the critical region (p = 2), it has different behavior over two parts decided according to the comparison of θ with the best constant in the Hardy inequality. In the super-critical region (p > 2), the exponential moments become infinite for all t > 0.

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