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Let ( W , H , μ ) 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 μ , in other words U is invertible if and only if 1 2 ∫ W | u | H 2 d μ = ∫ W d U μ d μ log d U μ d μ d μ . To cite this article: A.S. Üstünel, C. R. Acad. Sci. Paris, Ser. I 346 (2008). Résumé Soit ( W , H , μ ) l'espace de Wiener classique, et soit U = I W + u une perturbation d'identité adaptée satisfaisant á l'identité de Girsanov. Alors U est inversible si et seulement si l'énergie cinétique de u est égale à l'entropie relative de la mesure induite par l'action de U sur la mesure de Wiener μ ; en d'autres termes U est inversible si et seulement si 1 2 ∫ W | u | H 2 d μ = ∫ W d U μ d μ log d U μ d μ d μ . Pour citer cet article : A.S. Üstünel, C. R. Acad. Sci. Paris, Ser. I 346 (2008). References [1] D. Feyel A.S. Üstünel M. Zakai Realization of positive random variables via absolutely continuous transformations of measure on Wiener space Probability Surveys 3 2006 170 205 (electronic) [2] M. Fujisaki G. Kallianpur H. Kunita Stochastic differential equations for the non linear filtering problem Osaka J. Math. 9 1972 19 40 [3] N. Ikeda S. Watanabe Stochastic Differential Equations and Diffusion Processes 1981 North-Holland Amsterdam Kodansha Ltd., Tokyo [4] A.S. Üstünel M. Zakai Transformation of Measure on Wiener Space 1999 Springer-Verlag [5] A.S. Üstünel M. Zakai The invertibility of adapted perturbations of identity on the Wiener space C. R. Acad. Sci. Paris, Ser. I 342 2006 689 692 [6] A.S. Üstünel M. Zakai Sufficient conditions for the invertibility of adapted perturbations of identity on the Wiener space Probab. Theory Relat. Fields 139 2007 207 234

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