The self-intersections of a Gaussian random field

Let X ( t ) be an ( N , d , α ¯ ) non-deterministic Gaussian field. In this paper, the sufficient conditions for existence of the k -multiple points, the Hausdorff measure and the Hausdorff dimension for the k -multiple times set { ( t 1 , t 2 , … , t k ) : X ( t 1 ) = X ( t 2 ) = ⋯ = X ( t k ) for distinct t 1 , t 2 , … , t k } and the local times of the process Y ( T ) = { X ( t 2 ) − X ( t 1 ) , … , X ( t k ) − X ( t k − 1 ) } are evaluated.