Electronic Communications in Probability

The frog model with drift on $\mathbb{R} $

Joshua Rosenberg

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Abstract

Consider a Poisson process on $\mathbb{R} $ with intensity $f$ where $0 \leq f(x)<\infty $ for ${x}\geq 0$ and ${f(x)}=0$ for $x<0$. The “points” of the process represent sleeping frogs. In addition, there is one active frog initially located at the origin. At time ${t}=0$ this frog begins performing Brownian motion with leftward drift $\lambda $ (i.e. its motion is a random process of the form ${B}_{t}-\lambda{t} $). Any time an active frog arrives at a point where a sleeping frog is residing, the sleeping frog becomes active and begins performing Brownian motion with leftward drift $\lambda $, independently of the motion of all of the other active frogs. This paper establishes sharp conditions on the intensity function $f$ that determine whether the model is transient (meaning the probability that infinitely many frogs return to the origin is 0), or non-transient (meaning this probability is greater than 0). A discrete model with $\text{Poiss} (f(n))$ sleeping frogs at positive integer points (and where activated frogs perform biased random walks on $\mathbb{Z} $) is also examined. In this case as well, we obtain a similar sharp condition on $f$ corresponding to transience of the model.

Article information

Source
Electron. Commun. Probab., Volume 22 (2017), paper no. 30, 14 pp.

Dates
Received: 27 June 2016
Accepted: 16 May 2017
First available in Project Euclid: 26 May 2017

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1495764227

Digital Object Identifier
doi:10.1214/17-ECP61

Mathematical Reviews number (MathSciNet)
MR3663101

Zentralblatt MATH identifier
1364.60098

Subjects
Primary: 60J65: Brownian motion [See also 58J65] 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
frog model transience birth-death process Poisson process Brownian motion

Rights
Creative Commons Attribution 4.0 International License.

Citation

Rosenberg, Joshua. The frog model with drift on $\mathbb{R} $. Electron. Commun. Probab. 22 (2017), paper no. 30, 14 pp. doi:10.1214/17-ECP61. https://projecteuclid.org/euclid.ecp/1495764227


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