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**Example text**

Let ak (x) = σk (x)σk (x)∗ , k = 1, 2. Take c(x, y) = σ1 (x)σ2 (y)∗ . The two choices given in the next example are due to T. G. F. Li (1989), respectively. 17 (Coupling by reﬂection). Let L1= L2 and a(x) = σ(x)σ(x)∗. We have two choices: c(x, y) = σ(x) σ(y)∗ − 2 σ(y)−1 u ¯u ¯∗ , |σ(y)−1 u ¯|2 c(x, y) = σ(x) I − 2¯ uu¯∗ σ(y)∗ , det σ(y) = 0, x = y, x = y, where u ¯ = (x − y)/|x − y|. S. Kendall (1986) and M. Cranston (1991). In the case that x = y, the ﬁrst and the third couplings here are deﬁned to be the same as the second one.

We are now going to prove the variational formula for the lower bounds (cf. 3) 44 3 New Variational Formulas for the First Eigenvalue where W = {w : w0 = 0, wi ↑↑}, i 0, w ¯i = wi − π(w), Ii (w) = W = {w : wi ↑↑, π(w) 0}, ∞ 1 µi bi (wi+1 − wi ) j=i+1 µj wj , i w∈W, 0, and “↑↑” means strictly increasing. 3), since {w ¯:w∈W}⊂W. (a) First, we prove that Ii (w) > 0 for each w ∈ W and all i 1. ∞ µ w > 0 for all i 0. Otherwise, let i satisfy Equivalently, j j 0 j=i+1 ∞ 0. Then, since wj is strictly increasing, it follows that j=i0 +1 µj wj wi0 < 0, and furthermore, ∞ µj wj = 0 ∞ i0 j=0 i0 µj wj + µj wj µj wj j=i0 +1 j=0 i0 wi0 j=0 µj < 0.

24) for details. This example illustrates the ﬂexibility in the application of couplings. The details of this chapter, except for diﬀusions, are included in Chapter 5 of the second edition of Chen (1992a). Finally, we mention that the coupling methods are also powerful for timeinhomogeneous Markov processes, not touched on in this book. 14 is valid for Markov jump processes valued in Polish spaces [cf. L. Zheng (1993)]. I. L. I. Zeifman (1997). 2. 2. Then, three sections are used to explain the ideas in detail for the proof in the geometric case.