Self-intersection local time of an S′(Rd)-valued process involving motions of two types

We study existence and continuity of self-intersection local time (SILT) for a Gaussian S′(Rd)-valued process which arises as a high-density fluctuation limit of a particle system in Rd, where the particle motion switches back and forth between symmetric stable processes of indices α1 and α2 at exponential time intervals. We prove that SILT exists if and only if d<2 min{α1,α2}. This means that existence of SILT is determined by the “most mobile” of the two types, and we interpret this result in terms of the particle picture. In contrast with the single-type case, there are technical difficulties due to the lack of self-similarity of the particle paths.