Proof of the polar decomposition of the Wiener measure

TL;DR

This work proves a rigorous polar decomposition of the Wiener measure along the orbits of the diffeomorphism group $Diff^{1}_{+}([0,1])$, first in one dimension and then in a two-dimensional extension. It exploits the quasi-invariance of the Wiener measure, constructs explicit polar coordinates $x_{\rho,\varphi}(t)=\frac{\rho}{\sqrt{(\varphi^{-1}(t))'}}$, and derives a precise decomposition $w_{\sigma}(dx)=e^{-\frac{\sigma^{2}}{4\rho^{2}}}\ (\varphi'(0)\varphi'(1))^{\frac{3}{4}}\ \mu_{\frac{2\sigma}{\rho}}(d\varphi)\ d\rho$, with $\rho$ and $\varphi$ tied to $x$ by $\rho=\left(\int_{0}^{1} x(t)^{-2} dt\right)^{-1/2}$ and $\varphi^{-1}(t)=\rho^{2} \int_{0}^{t} x(\tau)^{-2} d\tau$. The proof proceeds through a chain of auxiliary results (Theorems and Lemmas) that relate path-space integrals to the $\rho,\varphi$-coordinates, compute Radon–Nikodym factors for diffeomorphisms, and establish Fourier-transform equality of two measure representations, yielding the claimed decomposition. The paper also outlines a complex, two-dimensional extension via a complex Wiener measure and a corresponding polar decomposition, with Theorem 4 asserting their equivalence. These results provide a rigorous bridge between Wiener path integrals, Schwarzian calculus, and conformal quantum mechanics.

Abstract

In the paper, we give the proof of the polar decomposition of the Wiener measure according to the orbits of the group of diffeomorphisms.