random walks on graphs electric networks and fractals this paper is devoted to the problem of the nearest neighbour random walk on infinite graphs we investigate the rw x n started from a fixed vertex x0 x v of the graph g v e and the expected value of the first exit time t n from the n ball b n in g it will be shown that if g is sufficiently regular then d underset raise0 3em hbox smash scriptscriptstyle geqslant d omega d r underset raise0 3em hbox smash scriptscriptstyle geqslant 2 and d r d 2 d omega where d r is the rw dimension d is the fractal dimension and d is the exponent of the growth of the resistance of b n though the method of the paper was developed for the case when x0 x is a fixed vertex of the graph we hope that the result can be generalized

espaces de lipschitz et inegalites de poincare resume in the setting of infinite graphs and non compact riemannian manifolds we show that suitable families of poincare inequalities yield global embeddings of sobolev spaces into lipschitz spaces as well as trudinger type inequalities this applies for example to cocompact coverings and to manifolds that are roughly isometric to a manifold with nonnegative ricci curvature in the process we give several reformulations of the sobolev inequalities and in particular show their equivalence with some l p faber krahn inequalities we also give an interpretation of some of our results in terms of distances on graphs associated with the l p norm of the gradient

sub gaussian estimates of heat kernels on infinite graphs we prove that a two sided sub gaussian 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

eigenvalues of graphs and digraphs we show that given a digraph x there is a vertex transitive digraph y such that the minimal polynomial of x divides that of y two consequences of this are that there exist vertex transitive digraphs with nondiagonalizable adjacency matrices and that any algebraic integer is the eigenvalue of some vertex transitive digraph

divergence form operators on fractal like domains we consider elliptic operators l in divergence form on certain domains in rd with fractal volume growth the domains we look at are pre sierpinski carpets which are derived from higher dimensional sierpinski carpets we prove a harnack inequality for non negative l harmonic functions on these domains and establish upper and lower bounds for the corresponding heat equation

random walks on discrete cylinders with large bases and random interlacements following the recent work of sznitman probab theory related fields 145 2009 143 174 we investigate the microscopic picture induced by a random walk trajectory on a cylinder of the form g n x z where g n is a large finite connected weighted graph and relate it to the model of random interlacements on infinite transient weighted graphs under suitable assumptions the set of points not visited by the random walk until a time of order g n 2 in a neighborhood of a point with z component of order g n converges in distribution to the law of the vacant set of a random interlacement on a certain limit model describing the structure of the graph in the neighborhood of the point the level of the random interlacement depends on the local time of a brownian motion the result also describes the limit behavior of the joint distribution of the local pictures in the neighborhood of several distant points with possibly different limit models as examples of g n we treat the d dimensional box of side length n the sierpinski graph of depth n and the d ary tree of depth n where d 2

spectral dimension of trees with a unique infinite spine using generating functions techniques we develop a relation between the hausdorff and spectral dimension of trees with a unique infinite spine furthermore it is shown that if the outgrowths along the spine are independent and identically distributed then both the hausdorff and spectral dimension can easily be determined from the probability generating function of the random variable describing the size of the outgrowths at a given vertex provided that the probability of the height of the outgrowths exceeding n falls off as the inverse of n we apply this new method to both critical non generic trees and the attachment and grafting model which is a special case of the vertex splitting model resulting in a simplified proof for the values of the hausdorff and spectral dimension for the former and novel results for the latter

estimates of transition densities for brownian motion on nested fractals we obtain upper and lower bounds for the transition densities of brownian motion on nested fractals compared with the estimate on the sierpinski gasket the results require the introduction of a new exponent dj related to the shortest path metric and chemical exponent on nested fractals further holder order of the resolvent densities sample paths and local times are obtained the results are obtained using the theory of multi type branching processes

resolvent positive operators resolvent positive operators on an ordered banach space with generating and normal positive cone are by definition linear possibly unbounded operators whose resolvent exists and is positive on a right half line even though these operators are defined by a simple purely algebraic condition analogues of the basic results of the theory of q semigroups can be proved for them in fact if a is resolvent positive and has a dense domain then the cauchy problem associated with a has a unique solution for every initial value in the domain of a and the solution is positive if the initial value is positive also the converse is true if we assume that a has a non empty resolvent set and d a n e is dense in moreover every positive resolvent is a laplace stieltjes transform of a so called integrated semigroup and conversely every such increasing non degenerate integrated semigroup defines a unique resolvent positive operator

obtaining upper bounds of heat kernels from lower bounds we show that a near diagonal lower bound of the heat kernel of a dirichlet form on a metric measure space with a regular measure implies an on diagonal upper bound if in addition the dirichlet form is local and regular then we obtain a full off diagonal upper bound of the heat kernel provided the dirichlet heat kernel on any ball satisfies a near diagonal lower estimate this reveals a new phenomenon in the relationship between the lower and upper bounds of the heat kernel c 2007 wiley periodicals inc

harnack inequalities and sub gaussian estimates for random walks we show that the beta parabolic harnack inequality for random walks on graphs is equivalent on one hand to the sub gaussian estimate 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 beta the latter condition can be replaced by a certain estimate of the resistance of annuli

analytic fragmentation semigroups and continuous coagulation fragmentation equations with unbounded rates abstract in this paper we show that continuous fragmentation operators are sectorial for a large range of physically relevant fragmentation rates and use this fact to prove classical solvability of the combined coagulation fragmentation equation with unbounded coagulation kernels

on the resistance of the sierpinski carpet let f text n be the nth stage in the construction of the sierpinski carpet let r text n be the electrical resistance of f text n when the left and right sides are each short circuited and a voltage is applied between them we prove that there exists a constant rho such that textstyle frac 1 4 rho n leq r n leq 4 rho n the motivation for this result came from the problem of establishing a the existence and b the value of the spectral dimension of the sierpinski carpet in this and a subsequent paper we settle a and give bounds for b

on a class of continuous coagulation fragmentation equations abstract a model for the dynamics of a system of particles undergoing simultaneously coalescence and breakup is considered each particle being assumed to be fully identified by its size existence of solutions to the corresponding evolution integral partial differential equation is shown for product type coagulation kernels with a weak fragmentation the failure of density conservation or gelation is also investigated in some particular cases

energy inequalities for cutoff functions and some applications let x d m be a metric measure space with a local regular dirichlet form we establish necessary and sufficient conditions for upper heat kernel bounds with subdiffusive space time exponent to hold this characterization is stable under rough isometries that is it is preserved under bounded perturbations of the dirichlet form further we give a criterion for stochastic completeness in terms of a sobolev inequality for cutoff functions as an example we show that this criterion applies to an anomalous diffusion on a geodesically incomplete fractal space where the well established criterion in terms of volume growth fails

local nash inequality and inhomogeneity of heat kernels the local nash inequality is introduced as a natural extension of the classical nash inequality yielding a space homogeneous upper heat kernel estimate the local nash inequality contains local information of the heat kernel and is a necessary condition for the space inhomogeneous heat kernel estimate involving the volume of balls like the one obtained by li and yau for a complete riemannian manifold with non negative ricci curvature under the volume doubling property the local nash inequality combined with the exit time estimate is shown to be equivalent to a sub gaussian off diagonal upper estimate of the heat kernel allowing space inhomogeneity

random walks on supercritical percolation clusters 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 c1 in the euclidean lattice the bounds analogous to aronsen s bounds for uniformly elliptic divergence form diusions 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 where the constant sx depends on the percolation congura tion

on structures of quasi thin table algebras and applications to association schemes abstract in this paper we develop a structure theory for a class of table algebras i e the so called quasi thin table algebras they are direct generalizations of quasi thin association schemes although this class of table algebras is rather close to finite groups its structural properties are quite different from those of finite groups the thin radical and thin residue give rise to two group structures for every table algebra they play the central role in our discussions applications to association schemes are also discussed and some new results about quasi thin association schemes are obtained in particular we will give necessary and sufficient conditions under which a commutative quasi thin association scheme has multiplicities 1 or 2 we will need these results in the study of the characterization and classification of double quasi thin association schemes in 22

weak solutions to the continuous coagulation equation with multiple fragmentation the existence of weak solutions to the continuous coagulation equation with multiple fragmentation is shown for a class of unbounded coagulation and fragmentation kernels the fragmentation kernel having possibly a singularity at the origin this result extends previous ones where either boundedness of the coagulation kernel or no singularity at the origin for the fragmentation kernel was assumed

heat kernel fluctuations for a resistance form with non uniform volume growth in this article we consider the problem of estimating the heat kernel on measure metric spaces equipped with a resistance form such spaces admit a corresponding resistance metric that reflects the conductivity properties of the set in this situation it has been proved that when there is uniform polynomial volume growth with respect to the resistance metric the behaviour of the on diagonal part of the heat kernel is completely determined by this rate of volume growth however recent results have shown that for certain random fractal sets there are global and local point wise fluctuations in the volume as r 0 and so these uniform results do not apply motivated by these examples we present global and local on diagonal heat kernel estimates when the volume growth is not uniform and demonstrate that when the volume fluctuations are non trivial there will be non trivial fluctuations of the same order up to exponents in the short time heat kernel asymptotics we also provide bounds for the off diagonal part of the heat kernel these results apply to deterministic and random self similar fractals and metric space dendrites the topological analogues of graph trees

time fractional poisson equations representations and estimates abstract in this paper we study existence and uniqueness of strong as well as weak solutions for general time fractional poisson equations we show that there is an integral representation of the solutions of time fractional poisson equations with zero initial values in terms of semigroup for the infinitesimal spatial generator l and the corresponding subordinator associated with the time fractional derivative this integral representation has an integral kernel q t x y which we call the fundamental solution for the time fractional poisson equation if the semigroup for l has an integral kernel we further show that q t x y can be expressed as a time fractional derivative of the fundamental solution for the homogeneous time fractional equation under the assumption that the associated subordinator admits a conjugate subordinator moreover when the laplace exponent of the associated subordinator satisfies the weak scaling property and its distribution is self decomposable we establish two sided estimates for the fundamental solution q t x y through explicit estimates of transition density functions of subordinators

the einstein relation for random walks on graphs this paper investigates the einstein relation the connection between the volume growth the resistance growth and the expected time a random walk needs to leave a ball on a weighted graph the einstein relation is proved under different set of conditions in the simplest case it is shown under the volume doubling and time comparison principles this and the other set of conditions provide the basic framework for the study of sub diffusive behavior of the random walks on weighted graphs

manifolds and graphs with slow heat kernel decay we give upper estimates on the long time behavior of the heat kernel on noncompact riemannian manifolds and infinite graphs which only depend on a lower bound of the volume growth we also show that these estimates are sharp

random walk on the incipient infinite cluster for oriented percolation in high dimensions we consider simple random walk on the incipient infinite cluster for the spread out model of oriented percolation on mathbb z d times mathbb z in dimensions d 6 we obtain bounds on exit times transition probabilities and the range of the random walk which establish that the spectral dimension of the incipient infinite cluster is frac 4 3 and thereby prove a version of the alexander orbach conjecture in this setting the proof divides into two parts one part establishes general estimates for simple random walk on an arbitrary infinite random graph given suitable bounds on volume and effective resistance for the random graph a second part then provides these bounds on volume and effective resistance for the incipient infinite cluster in dimensions d 6 by extending results about critical oriented percolation obtained previously via the lace expansion

power dissipation in fractal feynman sierpinski ac circuits this paper studies the concept of power dissipation in infinite graphs and fractals associated with passive linear networks consisting of non dissipative elements in particular we analyze the so called feynman sierpinski ladder a fractal ac circuit motivated by feynman s infinite ladder that exhibits power dissipation and wave propagation for some frequencies power dissipation in this circuit is obtained as a limit of quadratic forms and the corresponding power dissipation measure associated with harmonic potentials is constructed the latter measure is proved to be continuous and singular with respect to an appropriate hausdorff measure defined on the fractal dust of nodes of the network

spectral dimension and random walks on the two dimensional uniform spanning tree we study the simple random walk on the uniform spanning tree on mathbb z 2 we obtain estimates for the transition probabilities of the random walk the distance of the walk from its starting point after n steps and exit times of both euclidean balls and balls in the intrinsic graph metric in particular we prove that the spectral dimension of the uniform spanning tree on mathbb z 2 is 16 13 almost surely

volume growth and heat kernel estimates for the continuum random tree in this article we prove global and local point wise volume and heat kernel bounds for the continuum random tree we demonstrate that there are almost surely logarithmic global fluctuations and log logarithmic local fluctuations in the volume of balls of radius r about the leading order polynomial term as r 0 we also show that the on diagonal part of the heat kernel exhibits corresponding global and local fluctuations as t 0 almost surely finally we prove that this quenched almost sure behaviour contrasts with the local annealed averaged over all realisations of the tree volume and heat kernel behaviour which is smooth

estimates of heat kernels for non local regular dirichlet forms in this paper we present new heat kernel upper bounds for a certain class of non local regular dirichlet forms on metric measure spaces including fractal spaces we use a new purely analytic method where one of the main tools is the parabolic maximum principle we deduce off diagonal upper bound of the heat kernel from the on diagonal one under the volume regularity hypothesis restriction of the jump kernel and the survival hypothesis as an application we obtain two sided estimates of heat kernels for non local regular dirichlet forms with finite effective resistance including settings with the walk dimension greater than 2

stability of parabolic harnack inequalities let g e be a graph with weights a xy for which a parabolic harnack inequality holds with space time scaling exponent 2 suppose xy is another set of weights that are comparable to a xy we prove that this parabolic harnack inequality also holds for g e with the weights a xy we also give stable necessary and sufficient conditions for this parabolic harnack inequality to hold

quenched invariance principles for walks on clusters of percolation or among random conductances in this work we principally study random walk on the supercritical infinite cluster for bond percolation on d we prove a quenched functional central limit theorem for the walk when d 4 we also prove a similar result for random walk among i i d random conductances along nearest neighbor edges of d when d 1

weak uncertainty principle for fractals graphs and metric measure spaces we develop a new approach to formulate and prove the weak uncertainty inequality which was recently introduced by okoudjou and strichartz we assume either an appropriate measure growth condition with respect to the effective resistance metric or in the absence of such a metric we assume the poincare inequality and reverse volume doubling property we also consider the weak uncertainty inequality in the context of nash type inequalities our results can be applied to a wide variety of metric measure spaces including graphs fractals and manifolds

on the existence of moments of solutions to fragmentation equations abstract we consider the multiple fragmentation equations with polynomially bounded fragmentation rates both in the discrete and continuous cases the theory of semigroups of operators on frechet spaces is used to produce a simple proof that if moments of all non negative orders of solutions are initially finite then they remain finite for all future times moreover a class of fragmentation processes is identified in which the existence of the first moment of the initial distribution suffices for the existence of all other moments for positive times

random walk on the range of random walk we study the random walk x on the range of a simple random walk on d in dimensions d 4 when d 5 we establish quenched and annealed scaling limits for the process x which show that the intersections of the original simple random walk path are essentially unimportant for d 4 our results are less precise but we are able to show that any scaling limit for x will require logarithmic corrections to the polynomial scaling factors seen in higher dimensions furthermore we demonstrate that when d 4 similar logarithmic corrections are necessary in describing the asymptotic behavior of the return probability of x to the origin

heat kernel estimates and parabolic harnack inequalities for symmetric dirichlet forms abstract in this paper we consider the following symmetric dirichlet forms on a metric measure space m d e f g e c f g m m f x f y g x g y j d x d y where e c is a strongly local symmetric bilinear form and j d x d y is a symmetric radon measure on m m under general volume doubling condition on m d and some mild assumptions on scaling functions we establish stability results for upper bounds of heat kernel resp two sided heat kernel estimates in terms of the jumping kernels the cut off sobolev inequalities and the faber krahn inequalities resp the poincare inequalities we also obtain characterizations of parabolic harnack inequalities our results apply to symmetric diffusions with jumps even when the underlying spaces have walk dimensions larger than 2

harmonic analysis for resistance forms abstract in this paper we define the green functions for a resistance form by using effective resistance and harmonic functions then the green functions and harmonic functions are shown to be uniformly lipschitz continuous with respect to the resistance metric making use of this fact we construct the green operator and the measure valued laplacian the domain of the laplacian is shown to be a subset of uniformly lipschitz continuous functions while the domain of the resistance form in general consists of uniformly 1 2 holder continuous functions

minorations pour les chaines de markov unidimensionnelles we prove a space time lower estimate for reversible random walks on graphs having linear growth this estimate complements a similar upper bound that follows from a work of hebisch and saloff coste

brownian motion on a random recursive sierpinski gasket we introduce a random recursive fractal based on the sierpinski gasket and construct a diffusion upon the fractal via a dirichlet form this form and its symmetrizing measure are determined by the electrical resistance of the fractal the effective resistance provides a metric with which to discuss the properties of the fractal and the diffusion the main result is to obtain uniform upper and lower bounds for the transition density of the brownian motion on the fractal in terms of this metric the bounds are not tight as there are logarithmic corrections due to the randomness in the environment and the behavior of the shortest paths in the effective resistance metric is not well understood the results are deduced from the study of a suitable general branching process

volume and time doubling of graphs and random walks the strongly recurrent case this paper proves upper and lower off diagonal sub gaussian transition probability estimates for strongly recurrent random walks under sufficient and necessary conditions besides the known conditions volume doubling and the elliptic harnack inequality a new property is introduced time doubling 2001 john wiley sons inc

lower estimates of heat kernels for non local dirichlet forms on metric measure spaces abstract we are concerned with heat kernel estimates for a non local dirichlet form on an ahlfors regular metric measure space we use an analytic approach to obtain the full lower stable like estimate of the heat kernel from the near diagonal lower estimate combining with other known results we obtain certain equivalent conditions for two sided stable like estimates of the heat kernel the results can be simplified in certain cases for example when the dirichlet form admits an effective resistance

transition density estimates for diffusion processes on post critically finite self similar fractals 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 classes of fractals for which these questions were rst answered were classes of exactly self similar fractals with strong spatial symmetry such as nested or af ne nested fractals see for example 2 8 the existence of a laplacian and estimates on the heat kernel were obtained by considering the associated diffusion process and using the symmetry of the space the uniqueness of the laplacian for nested and af ne nested fractals has recently been solved through consideration of their dirichlet forms 23 in 15 the framework of post critically nite which we abbreviate to p c f self similar sets was introduced in order to capture the notion of exactly selfsimilar nitely rami ed fractals as used in the physics literature finitely rami ed fractals have the property that the intersection of any connected subset of the fractal with the rest of the set should occur only at a nite number of points this makes these structures much easier to analyse than in nitely rami ed sets such as the sierpinski carpet 5 the p c f self similar sets do not have spatial symmetry in general and have provided a mathematical test bed for analysis on fractals in this paper we will obtain uniform short time estimates on the heat kernel associated with a natural laplacian on the fractal a laplacian can be constructed on a p c f self similar fractal as a limit of discrete laplacians on graph approximations to the fractal based on the rami cation points we construct these operators via their dirichlet forms which can be set on any l space with a full measure n in 23 criteria are given which give a partial answer to the existence and uniqueness of laplace type operators on p c f self similar sets in a series of papers 16 17 18 20 25 7 some of the interesting spectral properties of p c f sets have been elucidated it has been shown that there can exist localized eigenfunctions if there is a high degree of symmetry in the set and corresponding laplace operator this corresponds to oscillation in the leading order term for the asymptotics of the eigenvalue counting function such behaviour occurs for nested fractals 7 and p c f sets with strong harmonic

characterization of sub gaussian heat kernel estimates on strongly recurrent graphs sub gaussian 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

off diagonal upper estimates for the heat kernel of the dirichlet forms on metric spaces we give equivalent characterizations for off diagonal upper bounds of the heat kernel of a regular dirichlet form on the metric measure space in two settings for the upper bounds with the polynomial tail typical for jump processes and for the upper bounds with the exponential tail for diffusions our proofs are purely analytic and do not use the associated hunt process

metric based upscaling we consider divergence form elliptic operators in dimension n ge 2 with l coefficients although solutions of these operators are only holder continuous we show that they are differentiable c1 with respect to harmonic coordinates it follows that numerical homogenization can be extended to situations where the medium has no ergodicity at small scales and is characterized by a continuum of scales this new numerical homogenization method is based on the transfer of a new metric in addition to traditional averaged homogenized quantities from subgrid scales into computational scales error bounds can be given and this method can also be used as a compression tool for differential operators 2006 wiley periodicals inc

on the length of chains in a metric space abstract we obtain an upper bound on the minimal number of points in an e chain joining two points in a metric space this generalizes a bound due to hambly and kumagai 1999 for the case of resistance metric on certain self similar fractals as an application we deduce a condition on e chains introduced by grigor yan and telcs 2012 this allows us to obtain sharp bounds on the heat kernel for spaces satisfying the parabolic harnack inequality without assuming further conditions on the metric a snowflake transform on the euclidean space shows that our bound is sharp

a new proof of moser s parabolic harnack inequality using the old ideas of nash in 1958 nash published his fundamental work on the local holder continuity of solutions of second order parabolic equations with coefficients which need not be smooth 8 the primary purpose of that work was to study the properties of the fundamental solution corresponding to the parabolic operator and from these properties to derive regularity for a general solution though the work is often cited in the literature about weak solutions of elliptic and parabolic equations one feels that nash s ideas were never fully understood and maybe still are not and that because of this the more understandable and seemingly more fruitful ideas of de giorgi 4 and moser 6 7 were subsequently adopted

subsequential scaling limits of simple random walk on the two dimensional uniform spanning tree the first main result of this paper is that the law of the rescaled two dimensional uniform spanning tree is tight in a space whose elements are measured rooted real trees continuously embedded into euclidean space various properties of the intrinsic metrics measures and embeddings of the subsequential limits in this space are obtained with it being proved in particular that the hausdorff dimension of any limit in its intrinsic metric is almost surely equal to 8 5 in addition the tightness result is applied to deduce that the annealed law of the simple random walk on the two dimensional uniform spanning tree is tight under a suitable rescaling for the limiting processes which are diffusions on random real trees embedded into euclidean space detailed transition density estimates are derived

explicit formulas for heat kernels on diamond fractals this paper provides explicit pointwise formulas for the heat kernel on compact metric measure spaces that belong to a mathbb n times mathbb n parameter family of fractals which are regarded as projective limits of metric measure graphs and do not satisfy the volume doubling property the formulas are applied to obtain uniform continuity estimates of the heat kernel and to derive an expression of the fundamental solution of the free schrodinger equation the results also open up the possibility to approach infinite dimensional spaces based on this model

heat kernels on metric measure spaces and an application to semilinear elliptic equations we consider a metric measure space m d p and a heat kernel p t x v on m satisfying certain upper and lower estimates which depend on two parameters a and we show that under additional mild assumptions these parameters are determined by the intrinsic properties of the space m d namely a is the hausdorff dimension of this space whereas called the walk dimension is determined via the properties of the family of besov spaces w 2 on m moreover the parameters a and are related by the inequalities 2 a 1 we prove also the embedding theorems for the space w 2 2 and use them to obtain the existence results for weak solutions to semilinear elliptic equations on m of the form u f x u g x where is the generator of the semigroup associated with p t the framework in this paper is applicable for a large class of fractal domains including the generalized sierpinski carpet in r n

diffusion on the scaling limit of the critical percolation cluster in the diamond hierarchical lattice we construct critical percolation clusters on the diamond hierarchical lattice and show that the scaling limit is a graph directed random recursive fractal a dirichlet form can be constructed on the limit set and we consider the properties of the associated laplace operator and diffusion process in particular we contrast and compare the behaviour of the high frequency asymptotics of the spectrum and the short time behaviour of the on diagonal heat kernel for the percolation clusters and for the underlying lattice in this setting a number of features of the lattice are inherited by the critical cluster

the continuous coagulation equation with multiple fragmentation abstract we present a proof of the existence of solutions to the continuous coagulation equation with multiple fragmentation whenever the kernels satisfy certain growth conditions the proof relies on weak l 1 compactness methods applied to suitably chosen approximating equations the question of uniqueness is also considered

harmonic calculus on limits of networks and its application to dendrites abstract the main purpose of this paper is to give a general framework to analysis on fractals including finitely ramified self similar sets random fractals canter sets dendrites and the recurrent case of infinitely ramified self similar sets we study a limit of a sequence of electrical networks satisfying certain compatibility which is a generalization of the decimation invariance of random walks playing an important role in constructing diffusion processes on self similar sets as a limit we propose notions of finite resistance forms and effective resistance metrics also it is shown that a finite resistance form induces an effective resistance metric and vice versa a finite resistance form becomes a regular dirichlet form if the domain is dense in l 2 space in such a case the effective resistance metric is thought of as the intrinsic metric for the dirichlet form in the latter half of the paper we apply the general framework to dendrites which are topological spaces containing no loops a shortest path metric where all arcs are geodesics on a dendrite is shown to be an effective resistance metric and the corresponding finite resistance form turns out to be a local regular dirichlet form furthermore we establish a simple relation between the hausdorff dimensions and the spectral dimensions of dendrites with shortest path metrics

dirichlet forms on fractals poincare constant and resistance we study dirichlet forms associated with random walks on fractal like finite grahs we consider related poincare constants and resistance and study their asymptotic behaviour we construct a markov semi group on fractals as a subsequence of random walks and study its properties finally we construct self similar diffusion processes on fractals which have a certain recurrence property and plenty of symmetries

global strict solutions to continuous coagulation fragmentation equations with strong fragmentation in this paper we give an elementary proof of the unique global in time solvability of the coagulation multiple fragmentation equation with polynomially bounded fragmentation and particle production rates and a bounded coagulation rate the proof relies on a new result concerning domain invariance for the fragmentation semigroup which is based on a simple monotonicity argument

parabolic harnack inequality and heat kernel estimates for random walks with long range jumps we investigate the relationships between the parabolic harnack inequality heat kernel estimates some geometric conditions and some analytic conditions for random walks with long range jumps unlike the case of diffusion processes the parabolic harnack inequality does not in general imply the corresponding heat kernel estimates

transition densities for brownian motion on the sierpinski carpet upper and lower bounds are obtained for the transition densitiesp t x y of brownian motion on the sierpinski carpet these are of the same form as those which hold for the sierpinski gasket in addition the joint continuity ofp t x y is proved the existence of the spectral dimension is established and the einstein relation connecting the spectral dimension the hausdorff dimension and the resistance exponent is shown to hold

the character values of commutative quasi thin schemes the main result of this article shows that all character values of commutative quasi thin schemes are cyclotomic algebraic integers in particular all of the eigenvalues of the adjacency matrices corresponding to relations in these association schemes are cyclotomic algebraic integers

the behaviour of aging functions in one dimensional bouchaud s trap model let i be a collection of i i d positive random variables with distribution in the domain of attraction of an stable law with 1 the symmetric bouchaud s trap model on is a markov chain x t whose transition rates are given by w xy 2 x 1 if x y are neighbours in we study the behaviour of two correlation functions x t w t x t w and open image in new window it is well known that for any of these correlation functions a time scale t f t w such that aging occurs can be found we study these correlation functions on time scales different from f t w and we describe more precisely the behaviour of a singular diffusion obtained as the scaling limit of bouchaud s trap model

decoupling inequalities and interlacement percolation on g z we study the percolative properties of random interlacements on g where g is a weighted graph satisfying certain sub gaussian estimates attached to the parameters 1 and 2 1 describing the respective polynomial growths of the volume on g and of the time needed by the walk on g to move to a distance we develop decoupling inequalities which are a key tool in showing that the critical level u for the percolation of the vacant set of random interlacements is always finite in our set up and that it is positive when 1 2 we also obtain several stretched exponential controls both in the percolative and non percolative phases of the model even in the case where g d d 2 several of these results are new