Maximal Coupling of Euclidean Brownian Motions
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Maximal Coupling of Euclidean Brownian Motions Elton P. Hsu · Karl-Theodor Sturm
Received: 1 March 2013 / Accepted: 6 March 2013 / Published online: 27 March 2013 © School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag Berlin Heidelberg 2013
Abstract We prove that the mirror coupling is the unique maximal Markovian coupling of two Euclidean Brownian motions starting from single points and discuss the connection between the uniqueness of maximal Markovian coupling of Brownian motions and certain mass transportation problems. Keywords Euclidean Brownian motion · Mirror coupling · Maximal coupling Mathematics Subject Classification (2010) 60J65
1 Introduction Let (E1 , B1 , μ1 ) and (E2 , B2 , μ2 ) be two probability spaces. A coupling of the probability measures μ1 and μ2 is a probability measure μ on the product measurable space (E1 × E2 , B1 × B1 ) whose marginal probabilities are μ1 and μ2 , respectively. We denote the set of coupling of μ1 and μ2 by C (μ1 , μ2 ). Accordingly, a coupling of two Euclidean Brownian motions on Rn starting from x1 and x2 , respectively, is a C(R+ , Rn × Rn )-valued random variable (X1 , X2 ) on a probability space (Ω, F , P) such that the components X1 and X2 have the law of Brownian motion starting from x1 and x2 , respectively. In this case, we say simply that (X1 , X2 ) is a coupling of Brownian motions from (x1 , x2 ). In the present work we discuss the uniqueness problem of maximal couplings of Euclidean Brownian motion. As usual, the maximality of a coupling is defined as a E.P. Hsu () Department of Mathematics, Northwestern University, Evanston, IL 60208, USA e-mail: [email protected] K.-T. Sturm Institute for Applied Mathematics, University of Bonn, Endenicher Allee, 53115 Bonn, Germany e-mail: [email protected]
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E.P. Hsu, K.-T. Sturm
coupling for which the coupling inequality (see below) becomes an equality. It is well known that the mirror coupling is a maximal coupling. We show by an example that in general a maximal coupling need not be unique. To prove a uniqueness result, we consider a more restricted class of couplings, that of Markovian couplings. In this class we show that the mirror coupling is the unique maximal coupling. This will be done by two methods. The first method is a martingale argument. This method, although the simpler of the two, depends on the linear structure of the Euclidean state space, thus has a rather limited scope of application to other more general settings. In the second method, we use the Markovian hypothesis to reduce the problem to a mass transport problem on the state space. In the Euclidean case under consideration this mass transportation problem has a well-known solution. This second method demonstrate an interesting connection between maximal Markov coupling and mass transportation with a cost function defined by the transition density function (the heat kernel), and has the potential of generalization to more general settings (e.g., Brownian motion on a Riemannian manifold
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