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Search: All articles in the CJM digital archive with keyword coupling

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1. CJM 2013 (vol 66 pp. 1050)

Holmes, Mark; Salisbury, Thomas S.
 Random Walks in Degenerate Random Environments We study the asymptotic behaviour of random walks in i.i.d. random environments on $\mathbb{Z}^d$. The environments need not be elliptic, so some steps may not be available to the random walker. We prove a monotonicity result for the velocity (when it exists) for any 2-valued environment, and show that this does not hold for 3-valued environments without additional assumptions. We give a proof of directional transience and the existence of positive speeds under strong, but non-trivial conditions on the distribution of the environment. Our results include generalisations (to the non-elliptic setting) of 0-1 laws for directional transience, and in 2-dimensions the existence of a deterministic limiting velocity. Keywords:random walk, non-elliptic random environment, zero-one law, couplingCategory:60K37

2. CJM 1999 (vol 51 pp. 673)

Barlow, Martin T.; Bass, Richard F.
 Brownian Motion and Harmonic Analysis on Sierpinski Carpets We consider a class of fractal subsets of $\R^d$ formed in a manner analogous to the construction of the Sierpinski carpet. We prove a uniform Harnack inequality for positive harmonic functions; study the heat equation, and obtain upper and lower bounds on the heat kernel which are, up to constants, the best possible; construct a locally isotropic diffusion $X$ and determine its basic properties; and extend some classical Sobolev and Poincar\'e inequalities to this setting. Keywords:Sierpinski carpet, fractal, Hausdorff dimension, spectral dimension, Brownian motion, heat equation, harmonic functions, potentials, reflecting Brownian motion, coupling, Harnack inequality, transition densities, fundamental solutionsCategories:60J60, 60B05, 60J35
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