Small fluctuations for time-dependent spherical random fields

We consider time-dependent space isotropic and time stationary spherical Gaussian random fields where the covariance functions is separable in space and time. We establish Chung’s law of the iterated logarithm and solve the small ball probabilities problem. Our results depend on the high-frequency behaviour of the angular power spectrum: the speed of decay of the small ball probability is faster as the memory parameter or the space-parameter decreases.