Electronic Communications in Probability

Coupling of Brownian motions in Banach spaces

Elisabetta Candellero and Wilfrid S. Kendall

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Abstract

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?

Article information

Source
Electron. Commun. Probab., Volume 23 (2018), paper no. 9, 13 pp.

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

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1519182084

Digital Object Identifier
doi:10.1214/18-ECP109

Mathematical Reviews number (MathSciNet)
MR3771767

Zentralblatt MATH identifier
1390.60294

Subjects
Primary: 60J65: Brownian motion [See also 58J65] 60H99: None of the above, but in this section 28C20: Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) [See also 46G12, 58C35, 58D20, 60B11]

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

Rights
Creative Commons Attribution 4.0 International License.

Citation

Candellero, Elisabetta; Kendall, Wilfrid S. Coupling of Brownian motions in Banach spaces. Electron. Commun. Probab. 23 (2018), paper no. 9, 13 pp. doi:10.1214/18-ECP109. https://projecteuclid.org/euclid.ecp/1519182084


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