The frog model with drift on R

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.