Diffusing Diffusivity: Survival in a Crowded Rearranging and Bounded Domain

We consider a particle diffusing in a bounded, crowded, rearranging medium. The rearrangement happens on a time scale longer than the typical time scale of diffusion of the article; as a result, effectively, the diffusion coefficient of the particle varies as a stochastic function of time. What is the probability that the particle will survive within the bounded region; given that it is absorbed the first time it hits the boundary of the region in which it difffises? This question is of great interest in a variety of chemical and biological problems: If; the `diffilision `coefficient is a constant, then analytical solutions for a variety of cases are available in the literature. However, there is no solution available for the case in which the difftision coefficient is a random function of time. We discuss a class of models for which' it is possible to find, analytical solutions to the prOblem. We illu-Strate the method for a circular, two-dimensional region, but our methods are easy, to apply to diffusion in arbitrary dimensions and spherical/rectangular regions. Our solution shows that if the dimension of the region is large, then only the average value of the diffusion coefficient determines the survival probability. However, for smaller -sized regions, one would be able to `see the effects of the stochasticity of the generalizations: of the results to N dimensions. diffusion coefficient. We also give generalizations: of the results to N dimensions