A necessary and sufficient condition for the invertibility of adapted perturbations of identity on the Wiener space

TL;DR

This paper establishes a precise criterion linking the invertibility of adapted perturbations of the Wiener space to a balance between kinetic energy and relative entropy, providing a necessary and sufficient condition.

Contribution

It introduces a new necessary and sufficient condition for invertibility of adapted perturbations of identity on Wiener space based on energy-entropy equivalence.

Findings

Invertibility characterized by energy-entropy equality

Provides a clear criterion for adapted perturbations

Bridges stochastic analysis and information theory

Abstract

Let $(W,H,\mu)$ be the classical Wiener space, assume that $U=I_W+u$ is an adapted perturbation of identity satisfying the Girsanov identity. Then, $U$ is invertible if and only if the kinetic energy of $u$ is equal to the relative entropy of the measure induced with the action of $U$ on the Wiener measure $\mu$, in other words $U$ is invertible if and only if $$ \half \int_W|u|_H^2d\mu=\int_W \frac{dU\mu}{d\mu}\log\frac{dU\mu}{d\mu}d\mu . $$