Self-intersection local time for some S '(R-d)-Ornstein-Uhlenbeck processes related to inhomogeneous fields

We study existence and path continuity of the self-intersection local time (SILT) for some S' (R-d)-valued Ornstein-Uhlenbeck processes. Examples of s uch processes arise in particular as fluctuation limits of particle systems . We analyze the effect that different types of measures involved in the co variances of the processes have on existence and continuity of SILT. These measures do not necessarily have the regularity and homogeneity proper-ties of those that have been considered before; this precludes some key ingredi ents of the previous techniques. We develop new technical tools and prove s ufficient conditions and some necessary conditions for existence of SILT, a nd sufficient conditions for path continuity of SILT. The questions of exis tence and continuity involve problems of existence of integrals and singula r integrals. We give examples which illustrate how different types of measu res (e.g., atomic or with L-2-density) may produce different critical dimen sions for existence of SILT. One of our motivations is the desire for a bet ter understanding of what SILT represents for S'(R-d)-valued processes.