The paper examines stochastic diffusion within an expanding space–time framework motivated by cosmological applications. Contrary to other results in the literature, for the considered general stochastic model, the expansion of space–time leads to a class of stochastic equations with non-constant coefficients that evolve with the expansion factor. 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 characterizes the extremal behaviour of the random solution by establishing upper bounds on the probabilities of large deviations. Numerical studies are carried out to illustrate the obtained theoretical results.

该论文研究了由宇宙学应用推动的扩展时空框架内的随机扩散。与文献中的其他结果相反，对于所考虑的一般随机模型，时空的扩展导致了一类具有随扩展因子演化的非常数系数的随机方程。提出并研究了具有随机初始条件的柯西问题。推导了膨胀球体上随机扩散方程的精确解。研究了解的各种概率性质，包括其依赖性结构、角功率谱的演化以及解的局部性质及其有限截断的近似。该论文还通过建立大偏差概率的上限来表征随机解的极值行为。进行数值研究来说明所获得的理论结果。