The self-intersections of a Gaussian random field

Let X(t)X(t) be an (N,d,α¯) non-deterministic Gaussian field. In this paper, the sufficient conditions for existence of the kk-multiple points, the Hausdorff measure and the Hausdorff dimension for the kk-multiple times set {(t1,t2,…,tk):X(t1)=X(t2)=⋯=X(tk){(t1,t2,…,tk):X(t1)=X(t2)=⋯=X(tk) for distinct t1,t2,…,tk}t1,t2,…,tk} and the local times of the process Y(T)={X(t2)−X(t1),…,X(tk)−X(tk−1)}Y(T)={X(t2)−X(t1),…,X(tk)−X(tk−1)} are evaluated.