Coupling of Brownian motions in Banach spaces

Consider a separable Banach space W supporting a non-trivial Gaussian measure μ. The following is an immediate consequence of the theory of Gaussian measure on Banach spaces: there exist (almost surely) successful couplings of two W-valued Brownian motions B and B˜ begun at starting points B(0) and B˜(0) if and only if the difference B(0)−B˜(0) of their initial positions belongs to the Cameron-Martin space Hμ of W corresponding to μ. For more general starting points, can there be a “coupling at time ∞”, such that almost surely ∥B(t)−B˜(t)∥W→0 as t→∞? Such couplings exist if there exists a Schauder basis of W which is also a Hμ-orthonormal basis of Hμ. 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 ∞ is always possible” purely in terms of Banach space geometry