Self-intersection local time for Gaussian '(d)-processes: Existence, path continuity and examples

In this paper we develop a criterion for existence or non-existence of self-intersection local time (SILT) for a wide class of Gaussian '(d)-valued processes, we show that quite generally the SILT process has continuous paths, and we give several examples which illustrate existence of SILT for different ranges of dimensions (e.g., d <= 3, d <= 7 and 5 <= d <= 11 in the Brownian case). Some of the examples involve branching and exhibit "dimension gaps". Our results generalize the work of Adler and coauthors, who studied the special case of "density processes" and proved that SILT paths are cadlag in the Brownian case making use of a "particle picture" approximation (this technique is not available for our general formulation).