Local time as an element of the Sobolev space

For a centered Gaussian random field taking its values in d, we investigate the existence of a local time as a generalized functional, i.e an element of some Sobolev space. We give the sufficient condition for such an existence in terms of the field covariation and apply it in several examples: the self-intersection local time for a fractional Brownian motion and the intersection local time for two Brownian motions.