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85
Brownian motion and harmonic analysis on Sierpinski carpets
 MR MR1701339 (2000i:60083
, 1999
"... Abstract. 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 con ..."
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Cited by 66 (12 self)
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Abstract. 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 Poincaré inequalities to this setting. 1
Random walk on supercritical percolation clusters
 ANN. PROBAB
, 2003
"... We obtain Gaussian upper and lower bounds on the transition density qt(x, y) of the continuous time simple random walk on a supercritical percolation cluster C ∞ in the Euclidean lattice. The bounds, analogous to Aronsen’s bounds for uniformly elliptic divergence form diffusions, hold with constants ..."
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Cited by 44 (3 self)
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We obtain Gaussian upper and lower bounds on the transition density qt(x, y) of the continuous time simple random walk on a supercritical percolation cluster C ∞ in the Euclidean lattice. The bounds, analogous to Aronsen’s bounds for uniformly elliptic divergence form diffusions, hold with constants ci depending only on p (the percolation probability) and d. The irregular nature of the medium means that the bound for qt(x, ·) only holds for t ≥ Sx(ω), where the constant Sx(ω) depends on the percolation configuration ω.
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 44 (5 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.
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 43 (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.
On the relation between elliptic and parabolic Harnack inequalities
, 2001
"... We show that, if a certain Sobolev inequality holds, then a scaleinvariant elliptic Harnack inequality suces to imply its a priori stronger parabolic counterpart. Neither the relative Sobolev inequality nor the elliptic Harnack inequality alone suces to imply the parabolic Harnack inequality in que ..."
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Cited by 42 (5 self)
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We show that, if a certain Sobolev inequality holds, then a scaleinvariant elliptic Harnack inequality suces to imply its a priori stronger parabolic counterpart. Neither the relative Sobolev inequality nor the elliptic Harnack inequality alone suces to imply the parabolic Harnack inequality in question; both are necessary conditions. As an application, we show the equivalence between parabolic Harnack inequality for on M , (i.e., for @ t + ) and elliptic Harnack inequality for @ 2 t + on R M . 1
Transition density estimates for diffusion processes on post critically finite selfsimilar fractals
 Proc. London Math. Soc. (3) 78:2 (1999), 431–458. MR 99m:60118 Zbl 1027.60087
"... The recent development of analysis on fractal spaces is physically motivated by the study of diffusion in disordered media. The natural questions that arise concern the existence and uniqueness of a suitable Laplace operator, and the behaviour of the associated heat semigroup, on a space which is fr ..."
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Cited by 30 (1 self)
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The recent development of analysis on fractal spaces is physically motivated by the study of diffusion in disordered media. The natural questions that arise concern the existence and uniqueness of a suitable Laplace operator, and the behaviour of the associated heat semigroup, on a space which is fractal. The
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 29 (4 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 28 (5 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.
Twosided estimates on the density of Brownian motion with singular drift
 Ill. J. Math
, 2006
"... Abstract. Let µ = (µ 1,..., µ d) be such that each µ i is a signed measure on R d belonging to the Kato class Kd,1. The existence and uniqueness of a continuous Markov process X on R d, called a Brownian motion with drift µ, was recently established by Bass and Chen. In this paper we study the poten ..."
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Cited by 21 (20 self)
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Abstract. Let µ = (µ 1,..., µ d) be such that each µ i is a signed measure on R d belonging to the Kato class Kd,1. The existence and uniqueness of a continuous Markov process X on R d, called a Brownian motion with drift µ, was recently established by Bass and Chen. In this paper we study the potential theory of X. We show that X has a continuous density q µ and that there exist positive constants ci, i = 1, · · · , 9, such that and c1e −c2t − t d 2 e − c3 x−y2 2t ≤ q µ (t, x, y) ≤ c4e c5t − t d 2 e − c6 x−y2 2t ∇xq µ (t, x, y)  ≤ c7e c8t − t d+1 2 e − c9 x−y2 2t for all (t, x, y) ∈ (0, ∞) × R d × R d. We further show that, for any bounded C 1,1 domain D, the density q µ,D of X D, the process obtained by killing X upon exiting from D, has the following estimates: for any T> 0, there exist positive constants Ci, i = 1, · · · , 5, such that and C1(1 ∧ ρ(x) √ t)(1 ∧ ρ(y) √ t)t − d 2 e − C 2 x−y2 t ≤ q µ,D (t, x, y) ≤ C3(1 ∧ ρ(x) √)(1 ∧ t ρ(y)