Results 1  10
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54
SemiSelfsimilar Processes
, 1999
"... A notion of semiselfsimilarity of R d valued stochastic processes is introduced as a natural extension of the selfsimilarity. Several topics on semiselfsimilar processes are studied: 1. The existence of the exponent of semiselfsimilar processes. 2. Characterization for semiselfsimilar processes ..."
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Cited by 58 (3 self)
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A notion of semiselfsimilarity of R d valued stochastic processes is introduced as a natural extension of the selfsimilarity. Several topics on semiselfsimilar processes are studied: 1. The existence of the exponent of semiselfsimilar processes. 2. Characterization for semiselfsimilar processes as scaling limits. 3. Relationship between semiselfsimilar processes with independent increments and semiselfdecomposable distributions. 4. Construction of semiselfsimilar processes with stationary increments. Semistable processes where all joint distributions are multivariate semistable are also discussed in connection with semiselfsimilar processes.
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
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 32 (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 30 (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.
Transition density estimates for Brownian motion on scale irregular Sierpinski gaskets
 Commun. Math. Phys
, 1997
"... We construct Brownian motion on a class of fractals which are spatially homogeneous but which do not have any exact selfsimilarity. We obtain transition density estimates for this process which are up to constants best possible. 1 Introduction There is now a fairly extensive literature on the heat ..."
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Cited by 21 (7 self)
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We construct Brownian motion on a class of fractals which are spatially homogeneous but which do not have any exact selfsimilarity. We obtain transition density estimates for this process which are up to constants best possible. 1 Introduction There is now a fairly extensive literature on the heat equation on fractal spaces, and on the spectral properties of such spaces. Most of these papers treat sets F which have exact selfsimilarity, so that there exist 11 contractions / i : F ! F such that / i (F ) " / j (F ) is (in some sense) small when i 6= j, and F = [ i / i (F ): (1.1) In the simplest cases, such as the nested fractals of Lindstrøm [18], F ae R d , the / i are linear, and / i (F ) " / j (F ) is finite when i 6= j. For very regular fractals such as nested fractals, or Sierpinski carpets, it is possible to construct a diffusion X t with a semigroup P t which is symmetric with respect to ¯, the Hausdorff measure on F , and to obtain estimates on the density p t (x; y) of P ...
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.
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.
Growth of selfsimilar graphs
, 2001
"... Abstract. Locally finite selfsimilar graphs with bounded geometry and without bounded geometry as well as nonlocally finite selfsimilar graphs are characterized by the structure of their cell graphs. Geometric properties concerning the volume growth and distances in cell graphs are discussed. The ..."
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Cited by 9 (4 self)
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Abstract. Locally finite selfsimilar graphs with bounded geometry and without bounded geometry as well as nonlocally finite selfsimilar graphs are characterized by the structure of their cell graphs. Geometric properties concerning the volume growth and distances in cell graphs are discussed. The length scaling factor ν and the volume scaling factor µ can be defined similarly to the corresponding parameters of continuous selfsimilar sets. There are different notions of growth dimensions of graphs. For a rather general class of selfsimilar graphs it is proved that all these dimensions coincide and that they can be calculated in the same way as the Hausdorff dimension of continuous selfsimilar fractals: log µ dim X = log ν. 1.
functions on selfsimilar graphs and bounds for the spectrum
 of the Laplacian, Ann. Inst. Fourier (Grenoble
"... Selfsimilar graphs can be seen as discrete versions of fractals (more precisely: compact, complete metric spaces defined as the fixed set of an iterated system of contractions, see Hutchinson in [15]). The simple random walk is a crucial tool in order to study diffusion on fractals, see Barlow and ..."
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Cited by 9 (1 self)
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Selfsimilar graphs can be seen as discrete versions of fractals (more precisely: compact, complete metric spaces defined as the fixed set of an iterated system of contractions, see Hutchinson in [15]). The simple random walk is a crucial tool in order to study diffusion on fractals, see Barlow and
Random Walks on Sierpiński Graphs: Hyperbolicity and Stochastic Homogenization
, 2002
"... We introduce two new techniques to the analysis on fractals. One is based on the presentation of the fractal as the boundary of a countable Gromov hyperbolic graph, whereas the other one consists in taking all possible “backward” extensions of the above hyperbolic graph and considering them as the c ..."
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Cited by 9 (0 self)
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We introduce two new techniques to the analysis on fractals. One is based on the presentation of the fractal as the boundary of a countable Gromov hyperbolic graph, whereas the other one consists in taking all possible “backward” extensions of the above hyperbolic graph and considering them as the classes of a discrete equivalence relation on an appropriate compact space. Illustrating these techniques on the example of the Sierpiński gasket (the associated hyperbolic graph is called the Sierpiński graph), we show that the Sierpiński gasket can be identified with the Martin and the Poisson boundaries for fairly general classes of Markov chains on the Sierpiński graph.