Abstract
As an extension of the Doob-Meyer decomposition of a semimartingale and the Fukushima representation of a Dirichlet process, we introduce a general Doob decomposition in continuous time, where a square-integrable process is represented as the sum of a martingale and a process with “vanishing local risk”. For a probability measure Q on Wiener space, we discuss how entropy conditions on Q formulated with respect to Wiener measure P are connected with the Doob decomposition of the coordinate process W under Q. The situation is well understood if the relative entropy H(Q|P) is finite; in this case the decomposition is classical and yields an immediate proof of Talagrand’s transport inequality on Wiener space. To go beyond this restriction, we consider the specific relative entropy h(Q|P) on Wiener space that was introduced by Gantert in [11]. We discuss its interplay with the Doob decomposition of W under Q and a corresponding version of Talagrand’s inequality, with special emphasis on the case where W is a Dirichlet process under Q.
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Acknowledgements
It is a great pleasure to contribute to this volume in honor of Masatoshi Fukushima’s Beiju. His deep insight into the interplay between Markov processes and Dirichlet spaces has also inspired some of my own work, including the present paper. Since our first encounter during the 6th Berkeley Symposium, more than 50 years ago, our paths have crossed many times, including mutual visits in Hanover, Bonn, Osaka, Zürich and Berlin. I want to express my warmest thanks for the rewarding experience that each encounter has turned out to be, and my best wishes for the years to come.
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Föllmer, H. (2022). Doob Decomposition, Dirichlet Processes, and Entropies on Wiener Space. In: Chen, ZQ., Takeda, M., Uemura, T. (eds) Dirichlet Forms and Related Topics. IWDFRT 2022. Springer Proceedings in Mathematics & Statistics, vol 394. Springer, Singapore. https://doi.org/10.1007/978-981-19-4672-1_8
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DOI: https://doi.org/10.1007/978-981-19-4672-1_8
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