The Annals of Probability

Shy couplings, $\operatorname{CAT} ({0})$ spaces, and the lion and man

Maury Bramson, Krzysztof Burdzy, and Wilfrid Kendall

Full-text: Open access


Two random processes $X$ and $Y$ on a metric space are said to be $\varepsilon$-shy coupled if there is positive probability of them staying at least a positive distance $\varepsilon$ apart from each other forever. Interest in the literature centres on nonexistence results subject to topological and geometric conditions; motivation arises from the desire to gain a better understanding of probabilistic coupling. Previous nonexistence results for co-adapted shy coupling of reflected Brownian motion required convexity conditions; we remove these conditions by showing the nonexistence of shy co-adapted couplings of reflecting Brownian motion in any bounded $\operatorname{CAT} ({0})$ domain with boundary satisfying uniform exterior sphere and interior cone conditions, for example, simply-connected bounded planar domains with $C^{2}$ boundary.

The proof uses a Cameron–Martin–Girsanov argument, together with a continuity property of the Skorokhod transformation and properties of the intrinsic metric of the domain. To this end, a generalization of Gauss’ lemma is established that shows differentiability of the intrinsic distance function for closures of $\operatorname{CAT} ({0})$ domains with boundaries satisfying uniform exterior sphere and interior cone conditions. By this means, the shy coupling question is converted into a Lion and Man pursuit–evasion problem.

Article information

Ann. Probab., Volume 41, Number 2 (2013), 744-784.

First available in Project Euclid: 8 March 2013

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J65: Brownian motion [See also 58J65]

CAT(0) CAT($\kappa$) co-adapted coupling coupling eikonal equation first geodesic variation Gauss’ lemma greedy strategy intrinsic metric Lion and Man problem Lipschitz domain Markovian coupling pursuit–evasion problem reflected Brownian motion Reshetnyak majorization shy coupling Skorokhod transformation total curvature uniform exterior sphere condition uniform interior cone condition


Bramson, Maury; Burdzy, Krzysztof; Kendall, Wilfrid. Shy couplings, $\operatorname{CAT} ({0})$ spaces, and the lion and man. Ann. Probab. 41 (2013), no. 2, 744--784. doi:10.1214/11-AOP723.

Export citation


  • Alexander, S., Bishop, R. L. and Ghrist, R. (2006). Pursuit and evasion in non-convex domains of arbitrary dimension. In Proceedings of Robotics: Science and Systems, Philadelphia, USA.
  • Alexander, S., Bishop, R. and Ghrist, R. (2010). Total curvature and simple pursuit on domains of curvature bounded above. Geom. Dedicata 149 275–290.
  • Benjamini, I., Burdzy, K. and Chen, Z.-Q. (2007). Shy couplings. Probab. Theory Related Fields 137 345–377.
  • Berestovskij, V. N. and Nikolaev, I. G. (1993). Multidimensional generalized Riemannian spaces. In Geometry, IV. Encyclopaedia Math. Sci. 70 165–243, 245–250. Springer, Berlin.
  • Bieske, T. (2010). The Carnot–Carathéodory distance vis-à-vis the eikonal equation and the infinite Laplacian. Bull. Lond. Math. Soc. 42 395–404.
  • Bishop, R. L. (2008). The intrinsic geometry of a Jordan domain. Int. Electron. J. Geom. 1 33–39.
  • Bollobas, B., Leader, I. and Walters, M. (2012). Lion and man—can both win? Israel J. Mathematics. To appear.
  • Bramson, M., Burdzy, K. and Kendall, W. S. (2011). Rubber bands, pursuit games and shy couplings. Unpublished manuscript.
  • Bridson, M. R. and Haefliger, A. (1999). Metric Spaces of Non-Positive Curvature. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 319. Springer, Berlin.
  • Burago, D., Burago, Y. and Ivanov, S. (2001). A Course in Metric Geometry. Graduate Studies in Mathematics 33. Amer. Math. Soc., Providence, RI.
  • Cheeger, J. and Ebin, D. G. (2008). Comparison Theorems in Riemannian Geometry. Amer. Math. Soc., Providence, RI.
  • Croft, H. T. (1964). “Lion and man”: A postscript. J. London Math. Soc. 39 385–390.
  • Émery, M. (2005). On certain almost Brownian filtrations. Ann. Inst. Henri Poincaré Probab. Stat. 41 285–305.
  • Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
  • Isaacs, R. (1965). Differential Games. A Mathematical Theory with Applications to Warfare and Pursuit, Control and Optimization. Wiley, New York.
  • Jun, C. (2011). Pursuit–evasion and time-dependent gradient flow in singular spaces. Ph.D. Thesis, Univ. Illinois, Urbana-Champaign.
  • Karuwannapatana, W. and Maneesawarng, C. (2007). The lower semi-continuity of total curvature in spaces of curvature bounded above. East–West J. Math. 9 1–9.
  • Kendall, W. S. (2001). Symbolic Itô calculus in AXIOM: An ongoing story. Stat. Comput. 11 25–35.
  • Kendall, W. S. (2009). Brownian couplings, convexity, and shy-ness. Electron. Commun. Probab. 14 66–80.
  • Lions, P. L. and Sznitman, A. S. (1984). Stochastic differential equations with reflecting boundary conditions. Comm. Pure Appl. Math. 37 511–537.
  • Littlewood, J. E. (1986). Littlewood’s Miscellany. Cambridge Univ. Press, Cambridge.
  • Nahin, P. J. (2007). Chases and Escapes: The Mathematics of Pursuit and Evasion. Princeton Univ. Press, Princeton, NJ.
  • Rešetnjak, J. G. (1968). Non-expansive maps in a space of curvature no greater than $K$. Sibirsk. Mat. Ž. 9 918–927.
  • Saisho, Y. (1987). Stochastic differential equations for multidimensional domain with reflecting boundary. Probab. Theory Related Fields 74 455–477.
  • Stroock, D. W. and Varadhan, S. R. S. (1979). Multidimensional Diffusion Processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 233. Springer, Berlin.