Transformations of Wiener Measure and Orthogonal Expansions

Abstract

In this paper we study the structure of square integrable functionals measurable with respect to coalescing stochastic flows. The case of $L^2$ space generated by the process $\eta(\cdot)=w(\min(\tau,\cdot)),$ where $w$ is a Brownian motion and $\tau$ is the first moment when $w$ hits the given continuous function $g$ is considered. We present a new construction of multiple stochastic integrals with respect to the process $\eta.$ Our approach is based on the change of measure technique. The analogue of the It\^o-Wiener expansion for the space $L^2(\eta)$ is constructed.