Self-intersection local time for S '(R-d)-Wiener processes and related Ornstein-Uhlenbeck processes

Existence and continuity results are obtained for self-intersection local t ime of S'(R-d)-valued Ornstein-Uhlenbeck processes X-t = T-t'X-0 + integral (0)(t)T(t-s)' dW(s), where X-0 is Gaussian, W-t is an S'(R-d)-Wiener proces s (independent of X-0), and T-t' is the adjoint of a semigroup T-t on S(R-d ). Two types of covariance kernels for X-0 and for W are considered: square tempered kernels and homogeneous random field kernels. The case where Tt c orresponds to the spherically symmetric alpha-stable process in R-d, alpha is an element of (0, 2], is treated in detail. The method consists in provi ng first results for self-intersection local times of the ingredient proces ses: W-t, T-t'X-0 and integral(0)(t)T(t-s)' dW(s), from which the results f or X-t are derived. As a by-product, a class of non-finite tempered measure s on R-d whose Fourier transforms are functions is identified. The tools ar e mostly analytical.