Coupling of Brownian motions in Banach spaces

Consider a separable Banach space 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 W -valued Brownian motions B and (B) over tilde begun at starting points B(0) and (B) over tilde (0) if and only if the difference B(0) - (B) over tilde (0) of their initial positions belongs to the Cameron-Martin space H-mu of W corresponding to mu. For more general starting points, can there be a "coupling at time infinity", such that almost surely parallel to B(t) - (B) over tilde parallel to(W) -> 0 as t -> infinity? Such couplings exist if there exists a Schauder basis of W which is also a H-mu-orthonormal basis of 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 infinity is always possible" purely in terms of Banach space geometry?