The frog model with drift on R

TL;DR

This paper analyzes a frog model with drift on the real line and a discrete version, establishing precise conditions on the initial distribution of sleeping frogs that determine whether the process is transient or recurrent.

Contribution

It provides sharp criteria for transience versus non-transience in both continuous and discrete frog models with drift, extending previous understanding of such processes.

Findings

Derived sharp conditions for transience based on the intensity function $f$.

Established equivalence of transience criteria between continuous and discrete models.

Identified thresholds for the number of frogs needed for recurrence or transience.

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.