Exact solution to the first-passage problem for a particle with a dichotomous diffusion coefficient

We consider the problem of first-passage time for reaching a boundary of a particle which diffuses in one dimension and is confined to the region x∈(0,L), with a diffusion coefficient that switches randomly between two states, having diffusivities that are different. Exact analytical expressions are found for the survival probability of the particle as a function of time. The survival probability has a multiexponential decay, and to characterize it, we use the average rate constant k, as well as the instantaneous rate r(t). Our approach can easily be extended to the case where the diffusion coefficient takes n different values. The model should be of interest to biological processes, in which a reactant searches for a target in a heterogeneous environment, making the diffusion coefficient a random function of time. The best example for this is a protein searching for a target site on the DNA.