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.
Categories: math.PR