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39
SubGaussian estimates of heat kernels on infinite graphs
 Duke Math. J
, 2000
"... We prove that a two sided subGaussian estimate of the heat kernel on an infinite weighted graph takes place if and only if the volume growth of the graph is uniformly polynomial and the Green kernel admits a uniform polynomial decay. ..."
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Cited by 30 (10 self)
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We prove that a two sided subGaussian estimate of the heat kernel on an infinite weighted graph takes place if and only if the volume growth of the graph is uniformly polynomial and the Green kernel admits a uniform polynomial decay.
Harnack inequalities and subGaussian estimates for random walks
 Math. Annalen
, 2002
"... We show that a fiparabolic Harnack inequality for random walks on graphs is equivalent, on one hand, to so called fiGaussian estimates for the transition probability and, on the other hand, to the conjunction of the elliptic Harnack inequality, the doubling volume property, and the fact that the m ..."
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Cited by 29 (6 self)
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We show that a fiparabolic Harnack inequality for random walks on graphs is equivalent, on one hand, to so called fiGaussian estimates for the transition probability and, on the other hand, to the conjunction of the elliptic Harnack inequality, the doubling volume property, and the fact that the mean exit time in any ball of radius R is of the order R . The latter condition can be replaced by a certain estimate of a resistance of annuli.
Manifolds and Graphs With Slow Heat Kernel Decay
 Invent. Math
, 1999
"... We give upper estimates on the long time behaviour of the heat kernel on a noncompact Riemannian manifold and infinite graphs, which only depend on a lower bound of the volume growth. We also show that these estimates are optimal. ..."
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Cited by 24 (2 self)
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We give upper estimates on the long time behaviour of the heat kernel on a noncompact Riemannian manifold and infinite graphs, which only depend on a lower bound of the volume growth. We also show that these estimates are optimal.
Which Values of the Volume Growth and Escape Time Exponent Are Possible for a Graph?
, 2001
"... Let \Gamma = (G; E) be an infinite weighted graph which is Ahlfors ffregular, so that there exists a constant c such that c , where V (x; r) is the volume of the ball centre x and radius r. Define the escape time T (x; r) to be the mean exit time of a simple random walk on \Gamma starting at ..."
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Cited by 24 (3 self)
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Let \Gamma = (G; E) be an infinite weighted graph which is Ahlfors ffregular, so that there exists a constant c such that c , where V (x; r) is the volume of the ball centre x and radius r. Define the escape time T (x; r) to be the mean exit time of a simple random walk on \Gamma starting at x from the ball centre x and radius r. We say \Gamma has escape time exponent fi ? 0 if there exists a constant c such that c T (x; r) cr for r 1. Well known estimates for random walks on graphs imply that ff 1 and 2 fi 1 + ff.
Random walks on graphical Sierpinski carpets
"... We consider random walks on a class of graphs derived from Sierpinski carpets. We obtain upper and lower bounds (which are nonGaussian) on the transition probabilities which are, up to constants, the best possible. We also extend some classical Sobolev and Poincare inequalities to this setting. ..."
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Cited by 19 (3 self)
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We consider random walks on a class of graphs derived from Sierpinski carpets. We obtain upper and lower bounds (which are nonGaussian) on the transition probabilities which are, up to constants, the best possible. We also extend some classical Sobolev and Poincare inequalities to this setting.
Brownian Motion in a Brownian Crack
 Ann. Appl. Probab
, 1998
"... . Let D be the Wiener sausage of width " around twosided Brownian motion. The components of 2dimensional reflected Brownian motion in D converge to 1dimensional Brownian motion and iterated Brownian motion, resp., as " goes to 0. 1. Introduction. Our paper is concerned with a model for a diffus ..."
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Cited by 16 (6 self)
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. Let D be the Wiener sausage of width " around twosided Brownian motion. The components of 2dimensional reflected Brownian motion in D converge to 1dimensional Brownian motion and iterated Brownian motion, resp., as " goes to 0. 1. Introduction. Our paper is concerned with a model for a diffusion in a crack. This should not be confused with the "crack diffusion model" introduced by Chudnovsky and Kunin (1987) which proposes that cracks have the shape of a diffusion path. The standard Brownian motion is the simplest of models proposed by Chudnovsky and Kunin. An obvious candidate for a "diffusion in a Brownian crack" is the "iterated Brownian motion" or IBM (we will define IBM later in the introduction). The term IBM has been coined in Burdzy (1993) but the idea is older than that. See Burdzy (1993, 1994) and Khoshnevisan and Lewis (1997) for the review of literature and results on IBM. The last paper is the only article known to us which considers the problem of diffusion in a cr...
The Art of Random Walks
 Lecture Notes in Mathematics 1885
, 2006
"... 1.1 Basic definitions and preliminaries................ 8 ..."
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Cited by 13 (4 self)
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1.1 Basic definitions and preliminaries................ 8
Heat kernels on metricmeasure spaces and an application to semilinear elliptic equations
 Trans. Amer. Math. Soc
, 2003
"... Abstract. We consider a metric measure space (M, d,µ) andaheat kernel pt(x, y) on M satisfying certain upper and lower estimates, which depend on two parameters α and β. We show that under additional mild assumptions, these parameters are determined by the intrinsic properties of the space (M, d, µ) ..."
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Cited by 12 (4 self)
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Abstract. We consider a metric measure space (M, d,µ) andaheat kernel pt(x, y) on M satisfying certain upper and lower estimates, which depend on two parameters α and β. We show that under additional mild assumptions, these parameters are determined by the intrinsic properties of the space (M, d, µ). Namely, α is the Hausdorff dimension of this space, whereas β, called the walk dimension, is determined via the properties of the family of Besov spaces W σ,2 on M. Moreover, the parameters α and β are related by the inequalities 2 ≤ β ≤ α +1. We prove also the embedding theorems for the space W β/2,2, and use them to obtain the existence results for weak solutions to semilinear elliptic equations on M of the form −Lu + f(x, u) =g(x), where L is the generator of the semigroup associated with pt. The framework in this paper is applicable for a large class of fractal domains, including the generalized Sierpiński carpet in Rn. 1.
Construction of diffusion processes on fractals, dsets, and general metric measure spaces
, 2003
"... We give a su#cient condition to construct nontrivial symmetric di#usion processes on a locally compact metric measure space (M, #, ). These processes are associated with local regular Dirichlet forms which are obtained as continuous parts of #limits for approximating nonlocal Dirichlet forms. Fo ..."
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Cited by 10 (3 self)
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We give a su#cient condition to construct nontrivial symmetric di#usion processes on a locally compact metric measure space (M, #, ). These processes are associated with local regular Dirichlet forms which are obtained as continuous parts of #limits for approximating nonlocal Dirichlet forms. For various fractals, we can use existing estimates to verify our assumptions. This shows that our general method of constructing di#usions can be applied to these fractals.
The AlexanderOrbach conjecture holds in high dimensions
 Invent. Math
"... Abstract. We examine the incipient infinite cluster (IIC) of critical percolation in regimes where meanfield behavior have been established, namely when the dimension d is large enough or when d> 6 and the lattice is sufficiently spread out. We find that random walk on the IIC exhibits anomalous di ..."
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Cited by 10 (2 self)
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Abstract. We examine the incipient infinite cluster (IIC) of critical percolation in regimes where meanfield behavior have been established, namely when the dimension d is large enough or when d> 6 and the lattice is sufficiently spread out. We find that random walk on the IIC exhibits anomalous diffusion with the spectral dimension ds = 4 3, that is, pt(x,x) = t−2/3+o(1). This establishes a conjecture of Alexander and Orbach [4]. En route we calculate the onearm exponent with respect to the intrinsic distance. 1.