## 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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