Entropy, invertibility and variational calculus of adapted shifts on Wiener space
Author
Üstünel, A.S.
Date
2009Source Title
Journal of Functional Analysis
Print ISSN
0022-1236
Electronic ISSN
1096-0783
Volume
257
Issue
11
Pages
3655 - 3689
Language
English
Type
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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 = IW + 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 ifffrac(1, 2) under(∫, W) | u |H 2 d μ = under(∫, W) frac(d U μ, d μ) log frac(d U μ, d μ) d μ . 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. © 2009 Elsevier Inc. All rights reserved.
Keywords
Calculus of variationsEntropy
Invertibility
Large deviations
Malliavin calculus
Monge transportation