Optimal strategy to capture a skittish lamb wandering near a precipice

TL;DR

This paper analyzes the probabilities of capturing a skittish lamb near a precipice using a model of Brownian motion with moving boundaries, revealing optimal strategies depending on boundary speeds and initial conditions.

Contribution

It provides new analytical expressions for splitting probabilities in a dynamic boundary setting, highlighting optimal boundary movement strategies for capture.

Findings

Capture probability is maximized at a non-zero optimal shepherd speed.

Boundary motion significantly influences splitting probabilities.

Multiple behavioral regimes depend on boundary speeds and initial positions.

Abstract

We study the splitting probabilities for a one-dimensional Brownian motion in a cage whose two boundaries move at constant speeds $c_1$ and $c_2$. This configuration corresponds to the capture of a diffusing, but skittish lamb, with an approaching shepherd on the left and a precipice on the right. We derive compact expressions for these splitting probabilities when the cage is expanding. We also obtain the time-dependent first-passage probability to the left boundary, as well as the splitting probability to this boundary, when the cage is either expanding or contracting. The boundary motions have a non-trivial impact on the splitting probabilities, leading to multiple regimes of behavior that depend on the expansion or contraction speed of the cage. In particular, the probability to capture the lamb is maximized when the shepherd moves at a non-zero optimal speed if the initial lamb position and the ratio between the two boundary speeds satisfy certain conditions.