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94
Heat kernel estimates for stablelike processes on dsets, Stochastic Process
 Appl
"... The notion of dset arises in the theory of function spaces and in fractal geometry. Geometrically selfsimilar sets are typical examples of dsets. In this paper stablelike processes on dsets are investigated, which include reflected stable processes in Euclidean domains as a special case. More ..."
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Cited by 108 (42 self)
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The notion of dset arises in the theory of function spaces and in fractal geometry. Geometrically selfsimilar sets are typical examples of dsets. In this paper stablelike processes on dsets are investigated, which include reflected stable processes in Euclidean domains as a special case. More precisely, we establish parabolic Harnack principle and derive sharp twosided heat kernel estimate for such stablelike processes. Results on the exact Hausdorff dimensions for the range of stablelike processes are also obtained.
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 86 (13 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
Nonlocal Dirichlet forms and symmetric jump processes
 Transactions of the American Mathematical Society
, 1999
"... We consider the symmetric nonlocal Dirichlet form (E, F) given by E(f, f) = (f(y) − f(x)) 2 J(x, y)dxdy Rd Rd with F the closure of the set of C 1 functions on R d with compact support with respect to E1, where E1(f, f): = E(f, f) + ∫ R d f(x) 2 dx, and where the jump kernel J satisfies κ1y − x ..."
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Cited by 55 (23 self)
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We consider the symmetric nonlocal Dirichlet form (E, F) given by E(f, f) = (f(y) − f(x)) 2 J(x, y)dxdy Rd Rd with F the closure of the set of C 1 functions on R d with compact support with respect to E1, where E1(f, f): = E(f, f) + ∫ R d f(x) 2 dx, and where the jump kernel J satisfies κ1y − x  −d−α ≤ J(x, y) ≤ κ2y − x  −d−β for 0 < α < β < 2, x − y  < 1. This assumption allows the corresponding jump process to have jump intensities whose size depends on the position of the process and the direction of the jump. We prove upper and lower estimates on the heat kernel. We construct a strong Markov process corresponding to (E, F). We prove a parabolic Harnack inequality for nonnegative functions that solve the heat equation with respect to E. Finally we construct an example where the corresponding harmonic functions need not be continuous.
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 51 (4 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.
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 51 (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 51 (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
Uniformly elliptic operators on Riemannian manifolds
 J. Diff. Geom
, 1992
"... Given a Riemannian manifold (M, g), we study the solutions of heat equations associated with second order differential operators in divergence form that are uniformly elliptic with respect to g. Typical examples of such operators are the Laplace operators of Riemannian structures which are quasiiso ..."
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Cited by 42 (4 self)
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Given a Riemannian manifold (M, g), we study the solutions of heat equations associated with second order differential operators in divergence form that are uniformly elliptic with respect to g. Typical examples of such operators are the Laplace operators of Riemannian structures which are quasiisometric to g. We first prove some Poincare and Sobolev inequalities on geodesic balls. Then we use Moser's iteration to obtain Harnack inequalities. Gaussian estimates, uniqueness theorems, and other applications are also discussed. These results involve local or global lower bound hypotheses on the Ricci curvature of g. Some of them are new even when applied to the Laplace operator of (M, g). 1.
Risk communication
 Proceedings of the national conference on risk communication, Conservation Foundation,Washington, DC
, 1987
"... We consider Schrodinger semigroups e. IH, H =A+V on Iw ” with VcIxl ’ as 1x1rco, O<c<[(l/2)(n2)] * with H>O. We determine the exact power law divergence of I~e‘Hi~p,p and of some IIe‘Hlly,p as maps from Lp to Lq. The results are expressed most naturally in terms of the power a fo ..."
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Cited by 37 (1 self)
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We consider Schrodinger semigroups e. IH, H =A+V on Iw ” with VcIxl ’ as 1x1rco, O<c<[(l/2)(n2)] * with H>O. We determine the exact power law divergence of I~e‘Hi~p,p and of some IIe‘Hlly,p as maps from Lp to Lq. The results are expressed most naturally in terms of the power a for which there exists a positive resonance 9 such that Hq = 0, q(x) 1.x‘.:Ta 1991 Academic Press, Inc. 1.
Gaussian Upper Bounds For The Heat Kernel On Arbitrary Manifolds
, 1997
"... In this paper, we develop a universal way of obtaining Gaussian upper bounds of the heat kernel on Riemannian manifolds. By the word "Gaussian" we mean those estimates which contain a Gaussian exponential factor similar to one which enters the explicit formula for the heat kernel of the co ..."
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Cited by 36 (2 self)
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In this paper, we develop a universal way of obtaining Gaussian upper bounds of the heat kernel on Riemannian manifolds. By the word "Gaussian" we mean those estimates which contain a Gaussian exponential factor similar to one which enters the explicit formula for the heat kernel of the conventional Laplace operator in R...
Characterization of subGaussian heat kernel . . .
, 2000
"... SubGaussian estimates for random walks are typical of fractal graphs. We characterize them in the strongly recurrent case, in terms of resistance estimates only, without assuming elliptic Harnack inequalities. ..."
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Cited by 32 (4 self)
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SubGaussian estimates for random walks are typical of fractal graphs. We characterize them in the strongly recurrent case, in terms of resistance estimates only, without assuming elliptic Harnack inequalities.