Fiber Brownian motion and the "hot spots" problem

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Efficient Markovian couplings: examples and counterexamples (2000, methodology): “Coupling techniques can be applied to obtain various estima tes in probability and analysis, both in purely theoretical research and in situat ions directly related to applications ([32] provides a good introduction; see also [24 , 5]).”
Geometrical Structure of Laplacian Eigenfunctions (2012, background): “This conjecture was proved by Banuelos and Burdzy for a class of planar domains [40], but in general the statement is wrong, as shown by several counterexamples [54, 101, 104, 257].”