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Extra resources for Eigenvalues, Inequalities, and Ergodic Theory (Probability and its Applications)
Additional discrete examples are included in Appendix B. To conclude this subsection, let us consider the Ornstein–Uhlenbeck process in Rd . 30 (4), we have Lρ(x, y) −ρ(x, y), and so x,y E ρ(Xt , Yt ) ρ(x, y)e−t . 42 with the help of a localizing procedure, this gives us λ1 1, which is indeed exact! Ergodicity Coupling methods are often used to study the ergodicity of Markov processes. 6) where π is the stationary distribution of the process. 6) simply means that the process is exponentially ergodic with respect to W .
4). Then the ρ-optimal solution c(x, y) is given as follows: (1) If d = 1, then c(x, y) = − Lf (|x − y|) = a1 (x)a2 (y), and moreover, 1 2 + a1 (x) + a2 (y) 2 f (|x − y|) (x − y)(b1 (x) − b2 (y)) f (|x − y|). 3 Optimality with respect to closed functions 31 Next, suppose that ak = σk2 (k = 1, 2) is nondegenerate and write c(x, y) = σ1 (x)H ∗ (x, y)σ2 (y). (2) If f (r) < 0 for all r > 0, then H(x, y) = U (γ)−1 U (γ)U (γ)∗ γ =1 − |x−y|f (|x−y|) , f (|x − y|) 1/2 , where U (γ) = σ1 (x)(I −γ u ¯u ¯∗ )σ2 (y).
1) g ∈ Dw (L), (2) supx=y |g(x) − g(y)| < ∞. supx=y Ex,y T Then for every coupling Px,y , we have λ −1 . Proof. Set f (x, y) = g(x) − g(y). By the martingale formulation as in the last proof, we have t∧T f (x, y) = Ex,y f Xt∧T , Yt∧T − Ex,y Lf Xs , Ys ds 0 t∧T = Ex,y f Xt∧T , Yt∧T + λEx,y f Xs , Xs ds. 0 Hence t∧T |g(x) − g(y)| Ex,y g Xt∧T − g Yt∧T g Xs − g Ys ds. + λEx,y 0 Assume supx=y Ex,y T < ∞, and so Px,y [T < ∞] = 1. Letting t ↑ ∞, we obtain T |g(x) − g(y)| λEx,y g Xs − g Ys ds. 4 Applications of coupling methods 37 Choose xn and yn such that lim |g(xn ) − g(yn )| = sup |g(x) − g(y)|.