This paper introduces new results on doubly stochastic Poisson processes, with log-Gaussian Hilbert-valued random intensity (LGHRI), defined from the Ornstein-Uhlenbeck process (O-U process) in Hilbert spaces. Sufficient conditions are derived for the existence of a counting measure on l2, for this type of doubly stochastic Poisson processes. Functional parameter estimation and prediction is achieved from the discrete-time approximation of the Hilbert-valued O-U process by an autoregressive Hilbertian process of order one (ARH(1) process). The results derived are applied to functional prediction of spatiotemporal log-Gaussian Cox processes, and an application to functional disease mapping is developed. The numerical results given, from the conditional simulation study undertaken, are compared to those ones obtained, when the random intensity is assumed to be a spatiotemporal long-range dependence (LRD) log-Gaussian process.
