Time changed spherical Brownian motions with longitudinal drifts

Abstract

We consider a time change of a drifted Brownian motion on the two-dimensional unit sphere. Precisely, we find strong solutions to the related time-nonlocal Kolmogorov equation under suitably regular initial data and we determine the spectral decomposition of its probability density function. Moreover, we study the speed of convergence to the stationary state, proving a non-exponential rate to the equilibrium. Finally, we provide very weak solutions of the same time-nonlocal Kolmogorov equation with general initial data. These results improve the known ones in terms of both the presence of a perturbation and the generality of the initial data.