Analytic continuation of time in Brownian motion. Stochastic distributions approach

With the use of Hida's white noise space theory and spaces of stochastic distributions, we present a detailed analytic continuation theory for classes of Gaussian processes, with focus here on Brownian motion. For the latter, we prove and make use of bounds in the complex plane for the Hermite functions; as well as a new approach to stochastic distributions. This in turn allows us to present (in Section 6) an explicit formula for an analytically continued white noise process, realized this way in the complex domain. With the use of the Wick product, we then apply our complex white noise analysis in Section 7 in a derivation of a new realization of Hilbert space-valued stochastic integrals.