Abstract
This whole chapter is devoted to Markov processes that are reversible. The emphasis is on the restoration of quantitative results that were lost in Chap. 4 when Doeblin’s condition was not present. The chapter concludes with the application of these results to the simulated annealing algorithm.
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Notes
- 1.
The reader who did Exercise 2.4.1 should recognize that the condition below is precisely the same as the statement that P=P ⊤. In particular, if one knows the conclusion of that exercise, then one has no need for the discussion which follows.
- 2.
A Hilbert space is a vector space equipped with an inner product which determines a norm for which the associated metric is complete.
- 3.
If one knows spectral theory, especially Stone’s theorem, the rather cumbersome argument that follows can be avoided.
- 4.
If that is not already so, we can make it so by choosing k 0 to be a point at which H takes its minimum value and replacing H by H−H(k 0). Such a replacement leaves both γ(β) and Q(β) as well as the quantity on the right hand side of (6.4.10) unchanged.
- 5.
Actually, there is good reason to doubt that monotonically increasing β may not be the best strategy. Indeed, the name “simulated annealing” derives from the idea that what one wants to do is simulate the annealing process familiar to chemists, material scientists, skilled carpenters, and followers of Metropolis. Namely, what these people do is alternately heat and cool to achieve their goal, and there is reason to believe we should be following their example. However, I have chosen not to follow them on the unforgivable, but understandable, grounds that my analysis is capable of handling only the monotone case.
References
Diaconis, P., Stroock, D.: Geometric bounds for eigenvalues of Markov chains. Ann. Appl. Probab. 1(1), 36–61 (1991)
Holley, R., Stroock, D.: Simulated annealing via Sobolev inequalities. Commun. Math. Phys. 115(4), 553–569 (1988)
Riesz, F., Sz.-Nagy, B.: Functional Analysis. Dover, New York (1990). Translated from the French edition by F. Boron, reprint of 1955 original
Stroock, D.: Mathematics of Probability. GSM, vol. 149. AMS, Providence (2013)
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Stroock, D.W. (2014). Reversible Markov Processes. In: An Introduction to Markov Processes. Graduate Texts in Mathematics, vol 230. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40523-5_6
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