SELF-INTERSECTION LOCAL TIME FOR ${\mathcal S}' ({\mathbb R}^d)$-WIENER PROCESSES AND RELATED ORNSTEIN–UHLENBECK PROCESSES

Abstract

Existence and continuity results are obtained for self-intersection local time of -valued Ornstein–Uhlenbeck processes , where X₀ is Gaussian, W_t is an -Wiener process (independent of X₀), and T'_t is the adjoint of a semigroup T_t on . Two types of covariance kernels for X₀ and for W are considered: square tempered kernels and homogeneous random field kernels. The case where T_t corresponds to the spherically symmetric α-stable process in ℝ^d, α∈(0,2], is treated in detail. The method consists in proving first results for self-intersection local times of the ingredient processes: W_t, T'_t X₀ and , from which the results for X_t are derived. As a by-product, a class of non-finite tempered measures on ℝ^d whose Fourier transforms are functions is identified. The tools are mostly analytical.