Arrival time for the fastest among $N$ switching stochastic particles

The first arrivals among $N$ Brownian particles is ubiquitous in the life sciences, as it often trigger cellular processes from the molecular level. We study here the case where stochastic particles, which represent molecules, proteins or molecules can switch between two states inside the non-negative real line. The switching process is modeled as a two-state Markov chain and particles can only escape in state 1. We estimate the fastest arrival time by solving asymptotically the Fokker-Planck equations for three different initial distributions: Dirac-delta, uniformly distributed and long-tail decay. The derived formulas reveal that the fastest particle avoid switching when the switching rates are much smaller than the diffusion time scale, but switches twice when the diffusion is state 2 is much faster than in state 1. The present results are compared to stochastic simulations revealing the range of validity of the derived formulas.