  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?
