Coupling of Brownian motions in Banach spaces

2017/05/23 by Candellero, Elisabetta, Kendall, Wilfrid S.
#FOS: Mathematics #Functional Analysis (math.FA) #Probability (math.PR)

paper · doi:10.48550/arxiv.1705.08300

Abstract

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 \widetildeB begun at starting points B(0) and \widetildeB(0) if and only if the difference B(0)-\widetildeB(0) of their initial positions belongs to the Cameron-Martin space H_μ of W corresponding to μ. For more general starting points, can there be a "coupling at time ∞", such that almost surely ‖B(t)-\widetildeB(t)‖_W → 0 as t→∞? Such couplings exist if there exists a Schauder basis of W which is also a H_μ -orthonormal basis of H_μ . 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?

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