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**Extra resources for Eigenvalues, Inequalities, and Ergodic Theory (Probability and its Applications)**

**Example text**

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)|.