Abstract
For a probability measure Q on Wiener space, Talagrand’s transport inequality takes the formWϰ (Q,P)2 ≤2H(Q|P), where theWasserstein distanceWϰ is defined in terms of the Cameron-Martin norm, and where H(Q|P) denotes the relative entropy with respect to Wiener measure P. Talagrand’s original proof takes a bottom-up approach, using finite-dimensional approximations. As shown by Feyel and Üstünel in [3] and Lehec in [10], the inequality can also be proved directly on Wiener space, using a suitable coupling of Q and P. We show how this top-down approach can be extended beyond the absolutely continuous case Q<<P. Here the Wasserstein distance is defined in terms of quadratic variation, and H(Q|P) is replaced by the specific relative entropy h(Q|P) on Wiener space that was introduced by N. Gantert in [7].
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21 June 2022
The original version of the book was inadvertently published with incorrect abstracts in the chapters. This has now been amended.
In addition to this, the affiliation of author Dr. Bertram Tschiderer has been changed to Faculty of Mathematics, University of Vienna in the online version of Chapter 10.
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Föllmer, H. (2022). Optimal Couplings on Wiener Space and An Extension of Talagrand’s Transport Inequality. In: Yin, G., Zariphopoulou, T. (eds) Stochastic Analysis, Filtering, and Stochastic Optimization. Springer, Cham. https://doi.org/10.1007/978-3-030-98519-6_7
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DOI: https://doi.org/10.1007/978-3-030-98519-6_7
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Publisher Name: Springer, Cham
Print ISBN: 978-3-030-98518-9
Online ISBN: 978-3-030-98519-6
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