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Optimal Couplings on Wiener Space and An Extension of Talagrand’s Transport Inequality

In: Stochastic Analysis, Filtering, and Stochastic Optimization

Author

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  • Hans Föllmer

    (Humboldt-Universität zu Berlin, Institut fùr Mathematik, Unter den Linden 6, 10099)

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

Suggested Citation

  • Hans Föllmer, 2022. "Optimal Couplings on Wiener Space and An Extension of Talagrand’s Transport Inequality," Springer Books, in: George Yin & Thaleia Zariphopoulou (ed.), Stochastic Analysis, Filtering, and Stochastic Optimization, pages 147-175, Springer.
    • Handle: RePEc:spr:sprchp:978-3-030-98519-6_7
    DOI: 10.1007/978-3-030-98519-6_7
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