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2018 Coupling of Brownian motions in Banach spaces
Elisabetta Candellero, Wilfrid S. Kendall
Electron. Commun. Probab. 23: 1-13 (2018). DOI: 10.1214/18-ECP109


Consider a separable Banach space $\mathcal{W} $ supporting a non-trivial Gaussian measure $\mu $. The following is an immediate consequence of the theory of Gaussian measure on Banach spaces: there exist (almost surely) successful couplings of two $\mathcal{W} $-valued Brownian motions $\mathbf{B} $ and $\widetilde{\mathbf {B}} $ begun at starting points $\mathbf{B} (0)$ and $\widetilde{\mathbf {B}} (0)$ if and only if the difference $\mathbf{B} (0)-\widetilde{\mathbf {B}} (0)$ of their initial positions belongs to the Cameron-Martin space $\mathcal{H} _\mu $ of $\mathcal{W} $ corresponding to $\mu $. For more general starting points, can there be a “coupling at time $\infty $”, such that almost surely $\|{\mathbf {B}(t)-\widetilde {\mathbf {B}}(t)}\|_{\mathcal{W} } \to 0$ as $t\to \infty $? Such couplings exist if there exists a Schauder basis of $\mathcal{W} $ which is also a $\mathcal{H} _\mu $-orthonormal basis of $\mathcal{H} _\mu $. We propose (and discuss some partial answers to) the question, to what extent can one express the probabilistic Banach space property “Brownian coupling at time $\infty $ is always possible” purely in terms of Banach space geometry?


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Elisabetta Candellero. Wilfrid S. Kendall. "Coupling of Brownian motions in Banach spaces." Electron. Commun. Probab. 23 1 - 13, 2018.


Received: 6 June 2017; Accepted: 16 January 2018; Published: 2018
First available in Project Euclid: 21 February 2018

zbMATH: 1390.60294
MathSciNet: MR3771767
Digital Object Identifier: 10.1214/18-ECP109

Primary: 28C20 , 60H99 , 60J65

Keywords: Banach space , Brownian motion , Cameron-Martin space , coupling , coupling at time $\infty $ , Gaussian measure , ‎Hilbert spaces , Markushevich basis (M-basis) , Reflection coupling , Schauder basis


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