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43
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 46 (9 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
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 27 (3 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
Random walk on the incipient infinite cluster on trees
 Illinois J. Math
"... Abstract. Let G be the incipient infinite cluster (IIC) for percolation on a homogeneous tree of degree n0 + 1. We obtain estimates for the transition density of the the continuous time simple random walk Y on G; the process satisfies anomalous diffusion and has spectral dimension 4 ..."
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Cited by 26 (7 self)
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Abstract. Let G be the incipient infinite cluster (IIC) for percolation on a homogeneous tree of degree n0 + 1. We obtain estimates for the transition density of the the continuous time simple random walk Y on G; the process satisfies anomalous diffusion and has spectral dimension 4
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.
A priori Hölder estimate, parabolic Harnack principle and heat kernel estimates for diffusions with jumps
, 2008
"... In this paper, we consider the following type of nonlocal (pseudodifferential) operators L on R d: Lu(x) = 1 2 d� i,j=1 ∂xi ..."
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Cited by 16 (13 self)
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In this paper, we consider the following type of nonlocal (pseudodifferential) operators L on R d: Lu(x) = 1 2 d� i,j=1 ∂xi
The Art of Random Walks
 Lecture Notes in Mathematics 1885
, 2006
"... 1.1 Basic definitions and preliminaries................ 8 ..."
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Cited by 15 (5 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 13 (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.
ASYMPTOTICS OF THE TRANSITION PROBABILITIES OF THE SIMPLE RANDOM WALK ON SELFSIMILAR GRAPHS
, 2002
"... It is shown explicitly how selfsimilar graphs can be obtained as ‘blowup ’ constructions of finite cell graphs Ĉ. This yields a larger family of graphs than the graphs obtained by discretising continuous selfsimilar fractals. For a class of symmetrically selfsimilar graphs we study the simple ra ..."
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Cited by 12 (3 self)
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It is shown explicitly how selfsimilar graphs can be obtained as ‘blowup ’ constructions of finite cell graphs Ĉ. This yields a larger family of graphs than the graphs obtained by discretising continuous selfsimilar fractals. For a class of symmetrically selfsimilar graphs we study the simple random walk on a cell graph Ĉ, starting in a vertex v of the boundary of Ĉ. It is proved that the expected number of returns to v before hitting another vertex in the boundary coincides with the resistance scaling factor. Using techniques from complex rational iteration and singularity analysis for Green functions we compute the asymptotic behaviour of the nstep transition probabilities of the simple random walk on the whole graph. The results of Grabner and Woess for the Sierpiński graph are generalised to the class of symmetrically selfsimilar graphs and at the same time the error term of the asymptotic expression is improved. Finally we present a criterion for the occurrence of oscillating phenomena of the nstep transition probabilities.
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.