Some of the Recent Topics on Stochastic Analysis

Abstract

First, we shall quickly explain why and how the space of generalized white noise functionals has been introduced. The space has big advantages to carry on the analysis of nonlinear functionals of white noise (or of Brownian motion) and to apply the theory to various fields. It should be noted that the introduction of generalized functionals was motivated by the Ito formula for Brownian functionals. Using this space we discuss the following two topics. 1. Path integrals. To formulate Lagrangian path integrals, we have to concretize the expressions of the Lagrangian in terms of paths. We propose that quantum mechanical paths (trajectories) are expressed as a sum of the classical paths and fluctuation which is taken to be a Brownian bridge. It is possible to give a plausible reason why a Brownian bridge is fitting in this case. With this choice of possible trajectories, there arises a difficulty that the kinetic energy becomes a generalized functional of a Brownian motion. It is now possible to overcome this difficulty to take our favorable space of generalized white noise functionals. Then follows the integration. Our method can be applied to a wider class of dynamics, for instance, to those cases with singular potentials and to some fields over non-euclidean space. 2. Infinite dimensional rotation group and unitary group. It is well known that the ”infinite dimensional rotation group” has naturally been introduced in connection with white noise, and the group describes certain invariance of the white noise measure. Hence, we may say that the white noise analysis should have an aspect of an infinite dimensional harmonic analysis. It seems natural, in fact by many reasons, to complexify the rotation group to have ”infinite dimensional unitary group”. Thus complexified group has various interesting applications to the analysis of nonlinear functionals of complex white noise. In addition, we can find good connections with Lie group theory and theory of quantum dynamics, to which we can give new interpretations.