Turnover in the mean survival time for a particle moving between two traps

There is presently no solution to the problem of characterizing the statistics of trapping of a one-dimensional Brownian particle that moves between two traps, when velocity is relevant. We analyze a simplified version of this problem in which the continuous velocity is replaced by three velocities, ±v and 0, allowing the particle to make random transitions between these states. This model is easily solved and exhibits turnover behavior as a function of the noise intensity (i.e., the jump rate).