clustering of linearly interacting diffusions and universality of their long time limit distribution let k d d 1 be a compact convex set and a countable abelian group we study a stochastic process x in k equipped with the product topology where each coordinate solves a sde of the form dx i t j a j i x j t x i t dt x i t db i t here a is the kernel of a continuous time random walk on and is a continuous root of a diffusion matrix w on k if x t converges in distribution to a limit x and the symmetrized random walk with kernel a s i a i a i is recurrent then each component x i is concentrated on x k x 0 and the coordinates agree i e the system clusters both these statements fail if a s is transient under the assumption that the class of harmonic functions of the diffusion matrix w is preserved under linear transformations of k we show that the system clusters for all spatially ergodic initial conditions and we determine the limit distribution of the components this distribution turns out to be universal in all recurrent kernels a s on abelian groups

phase ordering and persistence in a class of stochastic processes interpolating between the ising and voter models we study the dynamics of a class of two dimensional stochastic processes depending on two parameters which may be interpreted as two different temperatures respectively associated to interfacial and to bulk noise special lines in the plane of parameters correspond to the ising model voter model and majority vote model the dynamics of this class of models may be described formally in terms of reaction diffusion processes for a set of coalescing annihilating and branching random walkers we use the freedom allowed by the space of parameters to measure by numerical simulations the persistence probability of a generic model in the low temperature phase where the system coarsens this probability is found to decay at large times as a power law with a seemingly constant exponent we also discuss the connection between persistence and the nature of the interfaces between domains

analytic representation of bayes labeling and bayes clustering operators for random labeled point processes clustering algorithms typically group points based on some similarity criterion but without reference to an underlying random process to make clustering algorithms rigorously predictive in fact there exists a probabilistic theory of clustering in the context of random labeled point sets in which clustering error is defined in terms of the process in the present paper given an underlying point process we develop a general analytic procedure for finding an optimal clustering operator the bayes clusterer that corresponds to the bayes classifier in classification theory we provide detailed solutions under gaussian models owing to computational complexity we also develop approximations of the bayes clusterer

interacting fisher wright diffusions in a catalytic medium we study the longtime behaviour of interacting systems in a randomly fluctuating space time medium and focus on models from population genetics there are two prototypes of spatial models in population genetics spatial branching processes and interacting fisher wright diffusions quite a bit is known on spatial branching processes where the local branching rate is proportional to a random environment catalytic medium

renormalization of the voter model in equilibrium we consider the d dimensional voter model for d 3 our interest is the large scale limit of the equilibrium state of the voter model where we prove the d 3 results of 1 for d 4 which turn out to be of a different nature than for d 3 for this purpose we use the historical process we establish some surprising facts about the green s function of random walks in dimension d 4 which lead to the different features in d 3 versus d 4 secondly we prove an analogous result for the voter model on the hierarchical group

multiple time scale analysis of interacting diffusions we consider the questions how can the long term behavior of large systems of interacting components be described in terms of infinite systems on what time scale does the infinite system give a qualitatively correct description and what happens at large resp critical time scales lety n t be a solution y i n t i n n of the system of stochastic differential equations w i t are i i d brownian motions dy i n t left frac 1 2n 1 sum limits j n n y j n t y i n t right dt sqrt 2g y i n t dw i t in the mckean vlasov limit n we obtain the infinite independent system dy i infty t e y i infty t y i infty t dt sqrt 2g y i infty t dw i t i in z this infinite system has a one parameter set of invariant measures v theta mathop otimes limits x in z gamma theta with the equilibrium measure of dx t theta x t dt sqrt 2g x t dw t letq s be the transition kernel of the diffusion with generator u g x left frac partial partial x right 2 with u g x int g y gamma x dy then one main result is that asn mathcal l y n s 2n 1 rightarrow int q s theta d theta v theta theta e y 0 this provides a new example of a phenomenon also exhibited by the voter model and branching random walk in particular we are also able to modify our model by adding the termcn 1 a y i n t dt to obtain the first example in which the analog ofq s converges to an honest equilibrium instead of absorption in traps as in all models previously studied in the literature finally we discuss a hierarchical model with two levels from the point of view discussed above but now in two fast time scales

a phase transition for the coupled branching process part i the ergodic theory in the range of finite second moments we consider a particular markov process tu on s s n the random variable tu x is interpreted as the number of particles atx at timet the initial distribution of this process is a translation invariant measure withf x d the evolution is as follows at rateb x a particle is born atx but moves instantaneously toy chosen with probabilityq x y all particles at a site die at ratepd withp 0 1 d and individual particles die independently from each other at rate 1 p d every particle moves independently of everything else according to a continuous time random walk

coalescing random walks and voter model consensus times on the torus in mathbb z d voter models on z d and on the torus of site n in z d are considered the second ones get trapped with probability 1 as t tends to infinity at all 0 s or all 1 s the asymptotic growth of these trapping times are examined as n tends to infinity this is done with limit thewems for coalescing random walk systems on the torus of side n in z d

survival of contact processes on the hierarchical group we consider contact processes on the hierarchical group where sites infect other sites at a rate depending on their hierarchical distance and sites become healthy with a constant recovery rate if the infection rates decay too fast as a function of the hierarchical distance then we show that the critical recovery rate is zero on the other hand we derive sufficient conditions on the speed of decay of the infection rates for the process to exhibit a nontrivial phase transition between extinction and survival for our sufficient conditions we use a coupling argument that compares contact processes on the hierarchical group with freedom two with contact processes on a renormalized lattice an interesting novelty in this renormalization argument is the use of a result due to rogers and pitman on markov functionals

a conformal boundary for space times based on light like geodesics the 3 dimensional case a new causal boundary which we will term the l boundary inspired by the geometry of the space of light rays and invariant by conformal diffeomorphisms for space times of any dimension m 3 proposed by one of the authors r j low the space of null geodesics and a new causal boundary lecture notes in physics 692 springer 2006 pp 35 50 is analyzed in detail for space times of dimension 3 under some natural assumptions it is shown that the completed space time becomes a smooth manifold with boundary and its relation with geroch kronheimer penrose causal boundary is discussed a number of examples illustrating the properties of this new causal boundary as well as a discussion on the obtained results will be provided

variable selection for model based clustering we consider the problem of variable or feature selection for model based clustering the problem of comparing two nested subsets of variables is recast as a model comparison problem and addressed using approximate bayes factors a greedy search algorithm is proposed for finding a local optimum in model space the resulting method selects variables or features the number of clusters and the clustering model simultaneously we applied the method to several simulated and real examples and found that removing irrelevant variables often improved performance compared with methods based on all of the variables our variable selection method consistently yielded more accurate estimates of the number of groups and lower classification error rates as well as more parsimonious clustering models and easier visualization of results

different clustering regimes in systems of hierarchically interacting diffusions we study a system of interacting diffusions dx t a x t x t dt g x t dw t indexed by the hierarchical group as a genealogical two genotype model where x t denotes the frequency of say type a with hierarchically determined degrees of relationship between colonies in the case of short interaction range it is known that the system clusters that is locally one genotype dies out we focus on the description of the different regimes of cluster growth which is shown to depend on the interaction kernel a via its recurrent potential kernel one of these regimes will be further investigated by mean field methods for general interaction range we shall also relate the behavior of large finite systems indexed by finite subsets n of to that of the infinite system on the way we will establish relations between hitting times of random walks and their potentials

finite and infinite systems of interacting diffusions we study the problem of relating the long time behavior of finite and infinite systems of locally interacting components we consider in detail a class of lincarly interacting diffusionsx t x i t i d in the regime where there is a one parameter family of nontrivial invariant measures for these systems there are naturally defined corresponding finite systems x n t left x i n t i in lambda n right with lambda n n n d cap mathbb z d our main result gives a comparison between the laws ofx t n andx n t n for timest n asn the comparison involves certain mixtures of the invariant measures for the infinite system

on the long term behavior of some finite particle systems we consider the problem of comparing large finite and infinite systems with locally interacting components and present a general comparison scheme for the case when the infinite system is nonergodic we show that this scheme holds for some specific models one of these is critical branching random walk onz d let t denote this system and let denote a finite version of t defined on the torus n n d z d ford 3 we prove that for stationary shift ergodic initial measures with density that ift n andt n 2n 1 d s 0 asn then v 0 is the set of extremal invariant measures for the infinite system t andq s is the transition function of feller s branching diffusion we prove several extensions and refinements of this result the other systems we consider are the voter model and the contact process

efficient computation of isometry invariant distances between surfaces we present an efficient computational framework for isometry invariant comparison of smooth surfaces we formulate the gromov hausdorff distance as a multidimensional scaling like continuous optimization problem in order to construct an efficient optimization scheme we develop a numerical tool for interpolating geodesic distances on a sampled surface from precomputed geodesic distances between the samples for isometry invariant comparison of surfaces in the case of partially missing data we present the partial embedding distance which is computed using a similar scheme the main idea is finding a minimum distortion mapping from one surface to another while considering only relevant geodesic distances we discuss numerical implementation issues and present experimental results that demonstrate its accuracy and efficiency

solving variational problems and partial differential equations mapping into general target manifolds a framework for solving variational problems and partial differential equations that define maps onto a given generic manifold is introduced in this paper we discuss the framework for arbitrary target manifolds while the domain manifold problem was addressed in j comput phys 174 2 2001 759 the key idea is to implicitly represent the target manifold as the level set of a higher dimensional function and then implement the equations in the cartesian coordinate system where this embedding function is defined in the case of variational problems we restrict the search of the minimizing map to the class of maps whose target is the level set of interest in the case of partial differential equations we re write all the equation s geometric characteristics with respect to the embedding function we then obtain a set of equations that while defined on the whole euclidean space are intrinsic to the implicitly defined target manifold and map into it this permits the use of classical numerical techniques in cartesian grids regardless of the geometry of the target manifold the extension to open surfaces and submanifolds is addressed in this paper as well in the latter case the submanifold is defined as the intersection of two higher dimensional hypersurfaces and all the computations are restricted to this intersection examples of the applications of the framework here described include harmonic maps in liquid crystals where the target manifold is a hypersphere probability maps where the target manifold is a hyperplane chroma enhancement texture mapping and general geometric mapping between high dimensional manifolds

renormalization of hierarchically interacting isotropic diffusions we study a renormalization transformation arising in an infinite system of interacting diffusions the components of the system are labeled by the n dimensional hierarchical lattice n 2 and take values in the closure of a compact convex set bar d subset mathbb r d d geqslant 1 each component starts at some d and is subject to two motions 1 an isotropic diffusion according to a local diffusion rate g bar d to 0 infty chosen from an appropriate class 2 a linear drift toward an average of the surrounding components weighted according to their hierarchical distance in the local mean field limit n block averages of diffusions within a hierarchical distance k on an appropriate time scale are expected to perform a diffusion with local diffusion rate f k g where f k g f c k circ circ f c 1 g is the kth iterate of renormalization transformations fc c 0 applied to g here the ck measure the strength of the interaction at hierarchical distance k we identify fc and study its orbit f k g k 0 we show that there exists a fixed shape g such that limk kf k g g for all g where the k are normalizing constants in terms of the infinite system this property means that there is complete universal behavior on large space time scales our results extend earlier work for d 1 and bar d 0 1 resp 0 the renormalization transformation fc is defined in terms of the ergodic measure of a d dimensional diffusion in d 1 this diffusion allows a yamada watanabe type coupling its ergodic measure is reversible and the renormalization transformation fc is given by an explicit formula all this breaks down in d 2 which complicates the analysis considerably and forces us to new methods part of our results depend on a certain martingale problem being well posed

diffusive clustering in an infinite system of hierarchically interacting diffusions we study a countable system of interacting diffusions on the interval 0 1 indexed by a hierarchical group a particular choice of the interaction guaranties we are in the diffusive clustering regime this means clusters of components with values either close to 0 or close to 1 grow on various different scales however single components oscillate infinitely often between values close to 0 and close to 1 in such a way that they spend fraction one of their time together and close to the boundary the processes in the whole class considered and starting with a shift ergodic initial law have the same qualitative properties universality

comparison of interacting diffusions and an application to their ergodic theory a general comparison argument for expectations of certain multitime functionals of infinite systems of linearly interacting diffusions differing in the diffusion coefficient is derived as an application we prove clustering occurs in the case when the symmetrized interaction kernel is recurrent and the components take values in an interval bounded on one side the technique also gives an alternative proof of clustering in the case of compact intervals

characterization stability and convergence of hierarchical clustering methods we study hierarchical clustering schemes under an axiomatic view we show that within this framework one can prove a theorem analogous to one of kleinberg 2002 in which one obtains an existence and uniqueness theorem instead of a non existence result we explore further properties of this unique scheme stability and convergence are established we represent dendrograms as ultrametric spaces and use tools from metric geometry namely the gromov hausdorff distance to quantify the degree to which perturbations in the input metric space affect the result of hierarchical methods

recurrence and ergodicity of interacting particle systems an important technical result proposition 2 3 and its proof present in the original submission was erroneously omitted when this paper was published in 2000 the missing text which should have appeared on page 243 directly before section 3 is included in this erratum together with a short account of its context simultaneously with the publication of this erratum the electronic version of the paper in vol 116 no 2 2000 will be completed by insertion of the missing text readers of probability theory and related fields who have access to the electronic version of this erratum will also have access via a url to the full intact paper

cluster formation in a stepping stone model with continuous hierarchically structured sites a stepping stone model with site space a continuous hierarchical group is constructed via duality with a system of delayed coalescing stable levy processes this model can be understood as a continuum limit of discrete state space two allele genetics models with hierarchically structured resampling and migration the existence of a process rescaling limit on suitably related large space and time scales is established and interpreted in terms of the dynamics of cluster formation this paper was inspired by recent work of klenke

the geometry of the space of null geodesics the topology and geometry of the space of null geodesics n of a space time m are used to study the causal structure of the space time itself in particular the question of whether the topology of n is hausdorff or admits a compatible manifold structure carries information on the global structure of m and the transversality properties of the intersections of skies of points tell whether the points are conjugate points on a null geodesic

stochastic dynamics on hypergraphs and the spatial majority rule model this article starts by introducing a new theoretical framework to model spatial systems which is obtained from the framework of interacting particle systems by replacing the traditional graphical structure that defines the network of interactions with a structure of hypergraph this new perspective is more appropriate to define stochastic spatial processes in which large blocks of vertices may flip simultaneously which is then applied to define a spatial version of the galam s majority rule model in our spatial model each vertex of the lattice has one of two possible competing opinions say opinion 0 and opinion 1 as in the popular voter model hyperedges are updated at rate one which results in all the vertices in the hyperedge changing simultaneously their opinion to the majority opinion of the hyperedge in the case of a tie in hyperedges with even size a bias is introduced in favor of type 1 which is motivated by the principle of social inertia our analytical results along with simulations and heuristic arguments suggest that in any spatial dimensions and when the set of hyperedges consists of the collection of all n n blocks of the lattice opinion 1 wins when n is even while the system clusters when n is odd which contrasts with results about the voter model in high dimensions for which opinions coexist this is fully proved in one dimension while the rest of our analysis focuses on the cases when n 2 and n 3 in two dimensions

on null and causal pseudoconvex space times the condition of pseudoconvexity has important implications for geodesic structures of space time low showed that this condition guarantees the existence of a smooth structure of the space of geodesics recently the place of this property within the causality ladder is in the attention there is a conjecture which states that every strongly causal space time is causally simple if and only if it is null pseudoconvex in this paper we study the difference of causal and null pseudoconvexity in space times and introduce the limit geodesic segment as an equivalence condition to the pseudoconvexity then we show that in the case of open subspaces of n minkowski space time this is causal geodesic connectedness finally we prove a refined form of the conjecture that says every strongly causal space time is causally simple if and only if it is maximal null pseudoconvex

diffusive clustering in the two dimensional voter model on donne des resultats de simulation et une enquete concernant les resultats connus sur le modele du votant dans trois cas distincts trois ou plusieurs dimensions une dimension et deux dimensions

ergodic theorems for coupled random walks and other systems with locally interacting components in 5 the second author introduced a variety of new infinite systems with locally interacting components on the basis of computations for the finite analogues of these systems he made conjectures ragarding their limiting behavior as t this paper is devoted to the construction of these processes and to the proofs of these conjectures we restrict ourselves primarily to spatially homogeneous situations interesting problems remain unsolved in inhomogeneous cases two features distinguish these processes from most other infinite particle systems which have been studied one is that the state spaces of these systems are noncompact the other that even though the invariant measures are not generally of product form one can nevertheless compute explicitly the first and second moments of the number of particles per site in equilibrium the second moment computations are of inherent interest of course and they play an important role in the proofs of the ergodic theorems as well

network comparison embeddings and interiors this paper presents methods to compare networks where relationships between pairs of nodes in a given network are defined we define such network distance by searching for the optimal method to embed one network into another network prove that such distance is a valid metric in the space of networks modulo permutation isomorphisms and examine its relationship with other network metrics the network distance defined can be approximated via multidimensional scaling however the lack of structure in networks results in suboptimal approximations to alleviate such a problem we consider methods to define the interiors of networks we show that comparing interiors induced from a pair of networks yields the same result as the actual network distance between the original networks practical implications are explored by showing the ability to discriminate networks generated by different models

metrics in the space of high order networks this paper presents methods to compare high order networks defined as weighted complete hypergraphs collecting relationship functions between elements of tuples they can be considered as generalizations of conventional networks where only relationship functions between pairs are defined important properties between relationships of tuples of different lengths are established particularly when relationships encode dissimilarities or proximities between nodes two families of distances are then introduced in the space of high order networks the distances measure differences between networks we prove that they are valid metrics in the spaces of high order dissimilarity and proximity networks modulo permutation isomorphisms practical implications are explored by comparing the coauthorship networks of two popular signal processing researchers the metrics succeed in identifying their respective collaboration patterns

consolidation rates for two interacting systems in the plane this paper is a sequel of a paper of cox and griffeath diffusive clustering in the two dimensional voter model we continue our study of the voter model and coalescing random walks on the two dimensional integer lattice some exact asymptotics concerning the rate of clustering in the former process and the coalescence rate of the latter are derived we use these results to prove a limit law announced in that earlier paper concerning the size of the largest square centered at the origin which is of solid color at a large time t

persistent brain network homology from the perspective of dendrogram the brain network is usually constructed by estimating the connectivity matrix and thresholding it at an arbitrary level the problem with this standard method is that we do not have any generally accepted criteria for determining a proper threshold thus we propose a novel multiscale framework that models all brain networks generated over every possible threshold our approach is based on persistent homology and its various representations such as the rips filtration barcodes and dendrograms this new persistent homological framework enables us to quantify various persistent topological features at different scales in a coherent manner the barcode is used to quantify and visualize the evolutionary changes of topological features such as the betti numbers over different scales by incorporating additional geometric information to the barcode we obtain a single linkage dendrogram that shows the overall evolution of the network the difference between the two networks is then measured by the gromov hausdorff distance over the dendrograms as an illustration we modeled and differentiated the fdg pet based functional brain networks of 24 attention deficit hyperactivity disorder children 26 autism spectrum disorder children and 11 pediatric control subjects

fixation in a cyclic lotka volterra model we study a cyclic lotka volterra model of n interacting species populating a d dimensional lattice in the realm of a kirkwood approximation a critical number of species nc d above which the system fixates is determined analytically we find nc d 5 14 23 in dimensionsdd 1 2 3 in remarkably good agreement with simulation results in two dimensions a cyclic variant of the lotka volterra model of interacting populations originally introduced by vito volterra for the description of the struggle for existence among species 1 2 has then appeared in a number of apparently unrelated fields ranging from plasma physics 3 to integrable systems 4 5 recently the cyclic lotka volterra model also known as the cyclic voter model has attracted considerable interest as it was realized that introduction of the spatial structure drastically enriches the dynamics 6 11 namely if species live on a one dimensional 1d lattice a homogeneous initial state evolves into a coarsening mosaic of interacting species this heterogeneous spatial structure spontaneously develops when the number of species is sufficiently small n nc fixation occurs i e the system approaches a frozen state little is known in higher dimensions not even the existence of nc has yet been established theoretically or numerically in simulations on two dimensional 2d lattices with n 6 10 species no sign of fixation has been found and instead a reactive steady state has been observed 6 11 in this letter we investigate the cyclic lotka volterra model in the framework of a kirkwood like approximation this approach predicts a finite nc in all spatial dimensions in the following we shall use the language of the voter model 12 consider the cyclic voter model with n possible opinions each site of a d dimensional cubic lattice is occupied by a voter which has an opinion labelled by with d 1 n voters can change their opinions in a cyclic manner 1 modulo n according to the opinions of their neighbourhood specifically the following sequential dynamics is implemented i we randomly choose a site of opinion say and one of its 2d nearest neighbours of opinion ii if d 1 then the chosen site changes its opinion from to d 1 and iii otherwise opinion does not change we set the timescale so that in unit time each site of the lattice is chosen once on average when nd 2 the cyclic voter model is

ergodic theorems for infinite systems of locally interacting diffusions let x t x i t i in mathbb z d be the solution of the system of stochastic differential equations dx i t bigg sum j in mathbb z d a i j x j t x i t bigg dt sqrt 2g x i t dw i t quad i in mathbb z d here g lbrack 0 1 rbrack rightarrow mathbb r satisfies g 0 on 0 1 g 0 g 1 0 g is lipschitz a i j is an irreducible random walk kernel on mathbb z d and w i t i in mathbb z d is a family of standard independent brownian motions on mathbb r x t is a markov process on x lbrack 0 1 rbrack mathbb z d this class of processes was studied by notohara and shiga the special case g v v 1 v has been studied extensively by shiga we show that the long term behavior of x t depends only on hat a i j a i j a j i 2 and is universal for the entire class of g considered if hat a i j is transient then there exists a family nu theta theta in lbrack 0 1 rbrack of extremal translation invariant equilibria each nu theta is mixing and has density theta int x 0 d nu theta if hat a i j is recurrent then the set of extremal translation invariant equilibria consists of the point masses delta 0 delta 1 the process starting in a translation invariant shift ergodic measure mu on x with int x 0 d mu theta converges weakly as t rightarrow infty to nu theta if hat a i j is transient and to 1 theta delta 0 theta delta 1 if hat a i j is recurrent our results in the recurrent case remove a mild assumption on g imposed by notohara and shiga for the case hat a i j transient we use methods developed for infinite particle systems by liggett and spitzer for the case hat a i j recurrent we use a duality comparison argument

hierarchical models of interacting diffusions multiple time scale phenomena phase transition and pattern of cluster formation the purpose of this paper is to explore the connection between multiple space time scale behaviour for block averages and phase transitions respectively formation of clusters in infinite systems with locally interacting components the essential object is the associated markov chain which describes the joint distribution of the block averages at different time scales a fixed point and stability property of a particular dynamical system under a renormalisation procedure is used to explain this pattern of cluster formation and the fact that the longtime behaviour is universal in entire classes of evolutions

multiple scale analysis of clusters in spatial branching models in this paper we will investigate the long time behaviour of critical branching brownian motion and finite variance super brownian motion the so called dawson watanabe process on r d these processes are known to be persistent if d 3 that is there exist non trivial equilibrium measures if d 2 they cluster i e the processes converge to the 0 configuration while the surviving mass piles up in so called clusters we study the spatial profile of the clusters in the critical dimension d 2 via multiple space scale analysis we will also investigate the long time behaviour of these models restricted to finite boxes in d 2 on the way we develop coupling and comparison methods for spatial branching models

hierarchical clustering given confidence intervals of metric distances this paper considers metric the exact dissimilarities between pairs of points are not unknown but known to belong to some interval the goal is to study methods for the determination of hierarchical clusters i e a family of nested partitions indexed by a resolution parameter induced from the given distance intervals of the dissimilarities our construction of hierarchical clustering methods is based on defining admissible methods to be those methods that satisfy the axioms of value nodes in a metric space with two nodes are clustered together at the convex combination of the upper and lower bounds determined by a parameter and transformation when both distance bounds are reduced the output may become more clustered but not less two admissible methods are constructed and are shown to provide universal bounds in the space of admissible methods practical implications are explored by clustering moving points via snapshots and by clustering coauthorship networks representing collaboration between researchers from different communities the proposed clustering methods succeed in identifying underlying hierarchical clustering structures via the maximum and minimum distances in all snapshots as well as in differentiating collaboration patterns in journal publications between different research communities based on bounds of network distances

continuum sites stepping stone models coalescing exchangeable partitions and random trees analogues of stepping stone models are considered where the sitespace is continuous the migration process is a general markov process and the type space is infinite such processes were defined in previous work of the second author by specifying a feller transition semigroup in terms of expectations of suitable functionals for systems of coalescing markov processes an alternative representation is obtained here in terms of a limit of interacting particle systems it is shown that under a mild condition on the migration process the continuum sites stepping stone process has continuous sample paths the case when the migration process is brownian motion on the circle is examined in detail using a duality relation between coalescing and annihilating brownian motion this duality relation is also used to show that a tree like random compact metric space that is naturally associated to an in nite family of coalescing brownian motions on the circle has hausdorff and packing dimension both almost surely equal to 1 2 and moreover this space is capacity equivalent to the middle 1 2 cantor set and hence also to the brownian zero set

on the attracting orbit of a nonlinear transformation arising from renormalization of hierarchically interacting diffusions abstract this paper analyzes the n fold composition of a non linear integral operator acting on a class of functions on 0 attracting orbits and attracting fixed points are identified various results of convergence to these orbits and to these fixed points are derived the proofs are based on order preserving properties and comparison techniques a key role is played by the eigenfunctions of the operator which are used as comparison objects the results imply that the space time scaling limit of an infinite system of interacting diffusions has universal behavior independent of model parameters the paper can be read independently of part i

rescaled voter models converge to super brownian motion we show that a sequence of voter models suitably rescaled in space and time converges weakly to super brownian motion the result includes both nearest neighbor and longer range voter models and complements a limit theorem of mueller and tribe in one dimension

a soluble kinetic model for spinodal decomposition we compare the two dimensional voter model with approximate theories for spinodal decomposition the cluster size distribution and the short time dynamics of the voter model are studied by means of a monte carlo simulation the time dependent structure factor and the long time scaling of the voter dynamics are known analytically