Stochastic diffusion within expanding space-time

Abstract

The paper examines stochastic diffusion within an expanding space-time framework. It starts with providing a rationale for the considered model and its motivation from cosmology where the expansion of space-time is used in modelling various phenomena. Contrary to other results in the literature, the considered in this paper general stochastic model takes into consideration the expansion of space-time. It leads to a stochastic diffusion equations with coefficients that are non-constant and evolve with the expansion factor. Then, the Cauchy problem with random initial conditions is posed and investigated. The exact solution to a stochastic diffusion equation on the expanding sphere is derived. Various probabilistic properties of the solution are studied, including its dependence structure, evolution of the angular power spectrum and local properties of the solution and its approximations by finite truncations. The paper also characterises the extremal behaviour of the random solution by establishing upper bounds on the probabilities of large deviations. Numerical studies are undertaken to illustrate the obtained theoretical results and demonstrate the evolution of the random solution.