Stochastic modeling of rain processes: A diffusion rate model

Real data obtained from experiments performed at several regions on the globe show that rain rate, averaged over a large area, is approximately lognormally distributed. A theoretical justification for this fact is provided under some conditions. Rain rate is modeled as a temporally homogeneous diffusion process on the nonnegative semiaxis with appropriate boundary conditions, drift and diffusion coefficients. Assuming that the process is conservative, the asymptotic probability distribution is obtained. The limit is a parametric family of probability measures over the nonnegative semiaxis. The class of lognormal distributions is obtained as a special case.