Electronic Journal of Probability

Conditional survival distributions of Brownian trajectories in a one dimensional Poissonian environment in the critical case

Martin Kolb and Mladen Savov

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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].

Article information

Electron. J. Probab., Volume 22 (2017), paper no. 14, 29 pp.

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

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Zentralblatt MATH identifier

Primary: 60G51: Processes with independent increments; Lévy processes 60G55: Point processes 60F17: Functional limit theorems; invariance principles

Brownian motion Poisson obstacles limit theorems for condititioned processes

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Kolb, Martin; Savov, Mladen. Conditional survival distributions of Brownian trajectories in a one dimensional Poissonian environment in the critical case. Electron. J. Probab. 22 (2017), paper no. 14, 29 pp. doi:10.1214/17-EJP4468. https://projecteuclid.org/euclid.ejp/1487127642

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