Joint continuity of renormalized intersection local times

We study k-fold self-intersection local times alpha(x(2), ..., x(k); t), and k-fold renormalized self-intersection local times gamma(x(2), ...,x(k); t) for Levy processes. Our main result says that the k-fold renormalized self-intersection local time gamma(x(2), ..., x(k); t) for the symmetric stable process of order beta in R(2) is jointly continuous almost surely if and only if (2k - 1)(2 - beta) < 2.