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2017 Conditional survival distributions of Brownian trajectories in a one dimensional Poissonian environment in the critical case
Martin Kolb, Mladen Savov
Electron. J. Probab. 22: 1-29 (2017). DOI: 10.1214/17-EJP4468

Abstract

In this work we consider a one-dimensional Brownian motion with constant drift moving among a Poissonian cloud of obstacles. Our main result proves convergence of the law of processes conditional on survival up to time $t$ as $t$ converges to infinity in the critical case where the drift coincides with the intensity of the Poisson process. This complements a previous result of T. Povel, who considered the same question in the case where the drift is strictly smaller than the intensity. We also show that the end point of the process conditioned on survival up to time $t$ rescaled by $\sqrt{t} $ converges in distribution to a non-trivial random variable, as $t$ tends to infinity, which is in fact invariant with respect to the drift $h>0$. We thus prove that it is sub-ballistic and estimate the speed of escape. The latter is in a sharp contrast with discrete models of dimension larger or equal to $2$ when the behaviour at criticality is ballistic, see [7], and even to many one dimensional models which exhibit ballistic behaviour at criticality, see [8].

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Martin Kolb. Mladen Savov. "Conditional survival distributions of Brownian trajectories in a one dimensional Poissonian environment in the critical case." Electron. J. Probab. 22 1 - 29, 2017. https://doi.org/10.1214/17-EJP4468

Information

Received: 6 August 2015; Accepted: 7 December 2016; Published: 2017
First available in Project Euclid: 15 February 2017

zbMATH: 1359.60062
MathSciNet: MR3622884
Digital Object Identifier: 10.1214/17-EJP4468

Subjects:
Primary: 60F17, 60G51, 60G55

JOURNAL ARTICLE
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