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Coupling of Brownian motions in Banach spaces. (English) Zbl 1390.60294

Summary: 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 \(\|\mathbf{B}(t)-\widetilde{\mathbf{B}}(t)\|_{\mathcal{W}} \rightarrow 0\) as \(t\rightarrow \infty\)? 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?

MSC:

60J65 Brownian motion
60H99 Stochastic analysis
28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
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