Electronic Communications in Probability

Coupling of Brownian motions in Banach spaces

Elisabetta Candellero and Wilfrid S. Kendall

Full-text: Open access


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?

Article information

Electron. Commun. Probab., Volume 23 (2018), paper no. 9, 13 pp.

Received: 6 June 2017
Accepted: 16 January 2018
First available in Project Euclid: 21 February 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J65: Brownian motion [See also 58J65] 60H99: None of the above, but in this section 28C20: Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) [See also 46G12, 58C35, 58D20, 60B11]

Banach space Brownian motion Cameron-Martin space coupling coupling at time $\infty $ Gaussian measure Hilbert spaces Markushevich basis (M-basis) reflection coupling Schauder basis

Creative Commons Attribution 4.0 International License.


Candellero, Elisabetta; Kendall, Wilfrid S. Coupling of Brownian motions in Banach spaces. Electron. Commun. Probab. 23 (2018), paper no. 9, 13 pp. doi:10.1214/18-ECP109. https://projecteuclid.org/euclid.ecp/1519182084

Export citation


  • [1] David J. Aldous, Random walks on finite groups and rapidly mixing Markov chains, Séminaire de probabilités de Strasbourg 17 (1983), 243–297.
  • [2] Graeme K. Ambler and Bernard W. Silverman, Perfect simulation using dominated coupling from the past with application to area-interaction point processes and wavelet thresholding, Probability and Mathematical Genetics (Nicholas H Bingham and Charles M Goldie, eds.), Cambridge University Press, 2010, pp. 64–90.
  • [3] Stefan Banach, Théorie des opérations linéaires, Monografie Matematyczne, Warzawa, 1932.
  • [4] Sayan Banerjee and Wilfrid S. Kendall, Rigidity for Markovian maximal couplings of elliptic diffusions, Probability Theory and Related Fields 168 (2016), no. 1, 55–112.
  • [5] Gérard Ben Arous, Michael Cranston, and Wilfrid Stephen Kendall, Coupling constructions for hypoelliptic diffusions: Two examples, Stochastic Analysis (Michael Cranston and Mark Pinsky, eds.), vol. 57, AMS, 1995, pp. 193–212.
  • [6] R. H. Cameron and W. T. Martin, Transformations of Wiener Integrals Under Translations, Annals of Mathematics 45 (1944), no. 2, 386–396.
  • [7] Peter G. Casazza, Approximation Properties, Handbook of the Geometry of Banach Spaces Volume I (Gerard Meurant, ed.), Elsevier Science B.V., 2001, pp. 271–316.
  • [8] Michael Cranston, Gradient estimates on manifolds using coupling, J. Funct. Analysis 99 (1991), no. 1, 110–124.
  • [9] William J. Davis and Joram Lindenstrauss, On total nonnorming subspaces, Proc. Amer. Math. Soc. 31 (1972), 109–111.
  • [10] Aryeh Dvoretzky, Some results on convex bodies and Banach spaces, Proc. Internat. Sympos. Linear Spaces (Jerusalem, 1960), Jerusalem Academic Press, 1961, pp. 123–160.
  • [11] Nate Eldredge, Analysis and Probability on Infinite-Dimensional Spaces, arXiv 1607.03591 (2016), 68pp.
  • [12] Per Enflo, A counterexample to the approximation problem in Banach spaces, Acta Math. 130 (1973), no. 1, 309–317.
  • [13] Vladimir P. Fonf, Operator bases and generalized summation bases., Dokl. Akad. Nauk Ukr. SSR, Ser. A 1986 (1986), no. 11, 16–18 (Russian).
  • [14] Alison L. Gibbs, Convergence in the Wasserstein metric for Markov chain Monte Carlo algorithms with applications to image restoration, Stochastic Models 20 (2004), no. 4, 473–492.
  • [15] M. Hairer, Exponential mixing properties of stochastic PDEs through asymptotic coupling, Probability Theory and Related Fields 124 (2002), no. 3, 345–380.
  • [16] M. Hairer, J. C. Mattingly, and M. Scheutzow, Asymptotic coupling and a general form of Harris’ theorem with applications to stochastic delay equations, Probability Theory and Related Fields 149 (2011), no. 1, 223–259.
  • [17] Martin Hairer, An Introduction to Stochastic PDEs, arXiv 0907.4178 (2009), 78pp.
  • [18] Petr Hájek, Václav Zizler, Vicente Montesinos Santalucía, and Jon Vanderwerff, Biorthogonal systems in Banach spaces, CMS Books in Mathematics, Springer, 2007.
  • [19] Wilfrid Stephen Kendall, Nonnegative Ricci curvature and the Brownian coupling property, Stochastics 19 (1986), no. 1–2, 111–129.
  • [20] Wilfrid Stephen Kendall, Stochastic differential geometry, a coupling property, and harmonic maps, J. London Math. Soc.33 (1986), 554–566.
  • [21] Wilfrid Stephen Kendall, From Stochastic Parallel Transport to Harmonic Maps, New Directions in Dirichlet Forms (J. Jost and W.S. Kendall and U. Mosco and M. Roeckner and K. Sturm, ed.), Studies in Advanced Mathematics, AMS, 1998, pp. 49–115.
  • [22] Wilfrid Stephen Kendall, Coupling all the Lévy stochastic areas of multidimensional Brownian motion, Ann. Probab. 35 (2007), no. 3, 935–953.
  • [23] Wilfrid Stephen Kendall, Coupling time distribution asymptotics for some couplings of the Lévy stochastic area, Probability and Mathematical Genetics (Nicholas H Bingham and Charles M Goldie, eds.), London Math. Soc. Lecture Notes, Cambridge University Press, 2010, pp. 446–463.
  • [24] Wilfrid Stephen Kendall, Coupling, local times, immersions, Bernoulli 21 (2015), no. 2, 1014–1046.
  • [25] Wilfrid Stephen Kendall and Roland G. Wilson, Ising models and multiresolution quad-trees, Adv. Appl. Prob. 35 (2003), no. 1, 96–122.
  • [26] H. H. Kuo, Gaussian measures in Banach spaces, LNMath, no. 463, Springer-Verlag, 1975 (en).
  • [27] Torgny Lindvall, On Coupling of Brownian Motions, Tech report 1982:23, Univ. Göteborg, 1982.
  • [28] Torgny Lindvall and L. C. G. Rogers, Coupling of multidimensional diffusions by reflection, Ann. Probab. 14 (1986), no. 3, 860–872.
  • [29] C. W. McArthur, Developments in Schauder basis theory, Bull. AMS 78 (1972), no. 6, 877–908 (EN).
  • [30] R. I. Ovsepian and A. Pelczynski, On the existence of a fundamental total and bounded biorthogonal sequence in every separable Banach space, and related constructions of uniformly bounded orthonormal systems in $L^2$, Stud. Math. 44 (1975), 151–159.
  • [31] Ju. I. Petunin, Conjugate Banach spaces containing subspaces of characteristic zero, Dokl. Akad. Nauk SSSR 154 (1964), 527–529.
  • [32] Gilles Pisier, Martingales in Banach Spaces, Cambridge University Press, 2016 (English).
  • [33] I. Singer, Basic sequences and reflexivity of Banach spaces, Studia Math. 21 (1961/1962), 351–369.
  • [34] D. W. Stroock, Probability theory: an analytic view, Cambridge University press, Cambridge, 2011 (English).
  • [35] Daniel W. Stroock, Abstract Wiener space, revisited, Comm. Stoch. Analysis 2 (2008), no. 1, 145–141.
  • [36] Paolo Terenzi, Every separable Banach space has a bounded strong norming biorthogonal sequence which is also a Steinitz basis, Stud. Math. 111 (1994), no. 3, 207–222.
  • [37] Max-K. Von Renesse, Intrinsic Coupling on Riemannian Manifolds and Polyhedra, Elect. J. Prob. 9 (2004), 411–435.