Entropy, Invertibility and Variational Calculus of the Adapted Shifts on Wiener space

Abstract

In this work we study the necessary and sufficient conditions for a positive random variable whose expectation under the Wiener measure is one, to be represented as the Radon-Nikodym derivative of the image of the Wiener measure under an adapted perturbation of identity with the help of the associated innovation process. We prove that the innovation conjecture holds if and only if the original process is almost surely invertible. We also give variational characterizations of the invertibility of the perturbations of identity and the representability of a positive random variable whose total mass is equal to unity. We prove in particular that an adapted perturbation of identity $U=I_W+u$ satisfying the Girsanov theorem, is invertible if and only if the kinetic energy of $u$ is equal to the entropy of the measure induced with the action of $U$ on the Wiener measure $μ$, in other words $U$ is invertible iff $$ \half \int_W|u|_H^2dμ=\int_W \frac{dUμ}{dμ}\log\frac{dUμ}{dμ}dμ>. $$ otherwise the l.h.s. is always strictly greater than the r.h.s. The relations with the Monge-Kantorovitch measure transportation are also studied. An application of these results to a variational problem related to large deviations is also given.