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

Abstract

Consider a Poisson process on R with intensity f where 0 ≤ f (x) < ∞ for x ≥ 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 λ (i.e. its motion is a random process of the form Bt − λt). 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 λ, 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 Poiss(f (n)) sleeping frogs at positive integer points (and where activated frogs perform biased random walks on Z) is also examined. In this case as well, we obtain a similar sharp condition on f corresponding to transience of the model.

The critical density for the frog model is the degree of the tree

“…Recurrence versus transience is perhaps the most basic question for the frog model. It has been studied on Z d under a variety of initial conditions and frog paths [TW99, Pop01, GS09, DP14, KZ16,Ros16]. In [HJJ16b,HJJ16a,JJ16], we address the question on d-ary trees.…”

The frog model on trees with drift

“…studied the range of the frog model in the transient case [GNR17]. Similar observations were made by Rosenberg when the frog paths are Brownian motions in R [Ros17a,Ros17b]. The question is more subtle and challenging in higher dimensions.…”

The nonhomogeneous frog model on ℤ

“…and which, for the particular case where X j = 1, builds on the result from [1] by giving a sharp result that supersedes the soft condition in (2) and, for the case where p j = 1 2 + C log j (for all but finitely many j), implies the existence of a phase transition at C = π 2 24 . Finally, 1.1 will also be employed to obtain a formula that builds on the result from [5] In order to move towards a proof of Theorem 1.1, we begin by defining the process {M j } where, for each j ≥ 1, M j represents the number of frogs originating in {0, 1, . .…”

“…In this section we'll address the final model discussed in the introduction (see [5]), establishing sharp conditions for the case where the drift values of individual frogs are dependent on where they originate. Our result is as follows.…”

“…This time the number of sleeping frogs X j at x = j (for j ≥ 1) has Poisson distribution Poi(η j ), where the X j are mutually independent and {η j } is an increasing sequence. At each step activated frogs go left with probability p (for 1 2 < p < 1) and right with probability 1 − p. This model was introduced in [5], where it was established that the model is nontransient if and only if…”