Existence and Smoothness of Local Time and Self-intersection Local Time for Spherical Gaussian Random Fields

Abstract

In this paper, we study properties of local time for spherical Gaussian random fields T on $${\mathbb{S}^2}$$ . For 2 < α < 4, we give a formally expression of local time for spherical Gaussian random fields, and obtain that the existence of local time in L2 if and only if (α − 2)d < 4 and the smoothness in the sense of Meyer–Watanabe if and only if (α − 2)(d + 2) < 4, respectively. Finally, the existence and smoothness of self-intersection local time of T are considered.