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Local Invertibility of Adapted Shifts on Wiener Space and Related Topics

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 34)

Abstract

In this article we show that the invertibility of an adapted shift on the Brownian sheet is a local property in the usual sense of stochastic calculus. Thanks to this result we give a short proof of the invertibility for some processes which occur in free euclidean quantum mechanics and we relate this result to optimal transport. We also investigate some applications to information theory of a recent criterion which relates the invertibility of a shift to an equality between the energy of the signal and the relative entropy of the measure it induces. In particular, thanks to a change of measure, we interpret Shannon’s inequality as a consequence of information loss in Gaussian channels and we extend it to any abstract Wiener space. Finally, we extend the criterion of invertibility to the case of some stochastic differential equations with dispersion.

Keywords

Strong solutions Entropy Invertibility Shannon’s inequality 

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.LTCI CNRS Dépt. Infres, Institut Telecom, Telecom ParisTechParisFrance

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