interior regularity criteria for suitable weak solutions of the magnetohydrodynamic equations we present new interior regularity criteria for suitable weak solutions of the magnetohydrodynamic equations in dimension three a suitable weak solution is regular near an interior point z if the scaled lx tp q norm of the velocity with 1 3 p 2 q 2 1 q is sufficiently small near z and if the scaled lx tl m norm of the magnetic field with 1 3 l 2 m 2 1 m is bounded near z similar results are also obtained for the vorticity and for the gradient of the vorticity furthermore with the aid of the regularity criteria we exhibit some regularity conditions involving pressure for weak solutions of the magnetohydrodynamic equations

remarks on the singular set of suitable weak solutions for the three dimensional navier stokes equations abstract in this study s denotes the possible interior singular set of suitable weak solutions for the three dimensional navier stokes equations we improve the known upper box counting dimension of this set from 360 277 1 300 given by 24 to 975 758 1 286 we also show that s r log e r 0 0 27 113 which extends the previous corresponding results concerning the improvement of the classical caffarelli kohn nirenberg theorem by a logarithmic factor given by choe and lewis 3 j funct anal 175 348 369 2000 and by choe and yang 4 comm math phys 336 171 198 2015 the proof is inspired by a new e regularity criterion which was proved by guevara and phuc 7 calc var 56 68 2017

the structure of the allelic partition of the total population for galton watson processes with neutral mutations we consider a sub critical galton watson process with neutral mutations infinite alleles model and decompose the entire population into clusters of individuals carrying the same allele we specify the law of this allelic partition in terms of the distribution of the number of clone children and the number of mutant children of a typical individual the approach combines an extension of harris representation of galton watson processes and a version of the ballot theorem some limit theorems related to the distribution of the allelic partition are also given

the dual tree of a recursive triangulation of the disk in the recursive lamination of the disk one tries to add chords one after another at random a chord is kept and inserted if it does not intersect any of the previously inserted ones curien and le gall ann probab vol 39 pp 2224 2270 2011 have proved that the set of chords converges to a limit triangulation of the disk encoded by a continuous process m based on a new approach resembling ideas from the so called contraction method in function spaces we prove that when properly rescaled the planar dual of the discrete lamination converges almost surely in the gromov hausdorff sense to a limit real tree t which is encoded by m this confirms a conjecture of curien and le gall 1 introduction and main results in 18 curien and le gall introduce the model of random recursive triangulations of the disk the construction goes as follows atn 1 two points are sampled independently with uniform distribution on the circle they are connected by a chord a straight line which splits the disk into two fragments later on at each step two independent points are sampled uniformly at random on the circle and are connected by a chord if the latter does not intersect any of the previously inserted chords in other words the two points are connected by a chord if they both fall in the same fragment this gives rise to a sequence of laminations of the disk for us a lamination will be a collection of chords which may only intersect at their end points at timen the lamination ln consists of the union of the chords inserted up to timen as an increasing closed subset of the disk ln converges and it is proved in 18 that

a tree approach to p variation and to integration we consider a real valued path it is possible to associate a tree to this path and we explore the relations between the tree the properties of p variation of the path and integration with respect to the path in particular the fractal dimension of the tree is estimated from the variations of the path and young integrals with respect to the path as well as integrals from the rough paths theory are written as integrals on the tree examples include some stochastic paths such as martingales levy processes and fractional brownian motions for which an estimator of the hurst parameter is given

the minkowski dimension of interior singular points in the incompressible navier stokes equations abstract we study the possible interior singular points of suitable weak solutions to the three dimensional incompressible navier stokes equations we present an improved parabolic upper minkowski dimension of the possible singular set which is bounded by 95 63 the result also continue to hold for the three dimensional incompressible magnetohydrodynamic equations without any difficulty

hausdorff measure of boundary singular points in the magnetohydrodynamic equations abstract we study boundary singular points of suitable weak solutions to the three dimensional incompressible magnetohydrodynamic equations by using the generalised hausdorff measure we estimate the size of boundary singular points and present the improved range of powers of logarithmic factors

some dimension results for super brownian motion the dawson watanabe super brownian motion has been intensively studied in the last few years in particular there has been much work concerning the hausdorff dimension of certain remarkable sets related to super brownian motion we contribute to this study in the following way let yt t 0 be a super brownian motion on d d 2 andh be a borel subset of d we determine the hausdorff dimension of t 0 suppyt h o improving and generalizing a result of krone we also obtain a new proof of a result of tribe which gives whend 4 the hausdorff dimension of cup t in rm b suppy t as a function of the dimension ofb

regenerative real trees in this work we give a description of all finite measures on the space of rooted compact r trees which satisfy a certain regenerative property we show that any infinite measure which satisfies the regenerative property is the law of a levy tree that is the law of a tree valued random variable that describes the genealogy of a population evolving according to a continuous state branching process on the other hand we prove that a probability measure with the regenerative property must be the law of the genealogical tree associated with a continuous time discrete state branching process

two alternative philosophies for estimation of the parameters of time series two alternative philosophical frameworks for the two problems of estimating the time invariant or time variant autocorrelation function and its fourier transform for stationary and cyclostationary time series are compared one is based on the stochastic process model and the other is based on the nonstochastic time series model it is then explained that results on estimator bias and variance for these two problems couched within the stochastic process framework have analogs within the nonstochastic framework the bias and variance results for cyclostationary time series that are available within these two frameworks are then summarized

path properties of superprocesses with a general branching mechanism we first consider a super brownian motion x with a general branching mechanism using the brownian snake representation with subordination we get the hausdorff dimension of supp x t the topological support of x t and more generally the hausdorff dimension of u t b supp x t we also provide estimations on the hitting probability of small balls for those random measures we then deduce that the support is totally disconnected in high dimension eventually considering a super stable process with a general branching mechanism we prove that in low dimension this random measure is absolutely continuous with respect to the lebesgue measure

fragmentation associated with levy processes using snake we consider the height process of a levy process with no negative jumps and its associated continuous tree representation using levy snake tools developed by le gall le jan and duquesne le gall with an underlying poisson process we construct a fragmentation process which in the stable case corresponds to the self similar fragmentation described by miermont for the general fragmentation process we compute a family of dislocation measures as well as the law of the size of a tagged fragment we also give a special markov property for the snake which is of its own interest

on caccioppoli s inequalities of stokes equations and navier stokes equations near boundary abstract we study caccioppoli s inequalities of the non stationary stokes equations and navier stokes equations our analysis is local near boundary and we prove that in contrast to the interior case the caccioppoli s inequalities of the stokes equations and the navier stokes equations in general fail near boundary

statistically self similar fractals letx be a complete separable bounded metric space and a borel probability measure on the space con x n of alln tuples of contractions ofx with the topology of pointwise convergence then there exists a unique self similar probability measurep on the spacek x of all non empty compact subsets ofx here a measurep onk x is called self similar if for every borel setb k x p b int p n left k o k n 1 left bigcup limits i 0 n 1 s i k i in b right right d mu s o s n 1 if for a e s 0 sn 1 eachs i has an inverse which satisfies a lipschitz condition then there is an 0 such that forp a e k k x the hausdorff dimensionh dim k is equal to ifx d is compact and has non empty interior and if a e s 0 sn 1 consists of similarities which satisfy a certain disjointness condition w r t x then is determined by the equation int sum limits i 0 n 1 lip s i alpha d mu s o s n 1 1 where lip s i denotes the smallest lipschitz constant fors i under fairly general assumptions the dimensional hausdorff measure ofp a e k k x equals o if andx are chosen in a rather special way thenp a e k k x is the graph of a homeomorphism of 0 1 or a curve or the graph of a continuous function

exact packing measure of the range of psi super brownian motions we consider super processes whose spatial motion is the d dimensional brownian motion and whose branching mechanism psi is critical or subcritical such processes are called psi super brownian motions if d 2 varvec gamma varvec gamma 1 where varvec gamma in 1 2 is the lower index of psi at infty then the total range of the psi super brownian motion has an exact packing measure whose gauge function is g r log log 1 r varphi 1 1 r log log 1 r 2 where varphi psi prime circ psi 1 more precisely we show that the occupation measure of the psi super brownian motion is the g packing measure restricted to its total range up to a deterministic multiplicative constant only depending on d and psi this generalizes the main result of duquesne ann probab 37 6 2431 2458 2009 that treats the quadratic branching case for a wide class of psi the constant 2 varvec gamma varvec gamma 1 is shown to be equal to the packing dimension of the total range

beta coalescents and continuous stable random trees coalescents with multiple collisions also known as a coalescents were introduced by pitman and sagitov in 1999 these processes describe the evolution of particles that undergo stochastic coagulation in such a way that several blocks can merge at the same time to form a single block in the case that the measure a is the beta 2 a a distribution they are also known to describe the genealogies of large populations where a single individual can produce a large number of offspring here we use a recent result of birkner et al to prove that beta coalescents can be embedded in continuous stable random trees about which much is known due to the recent progress of duquesne and le gall our proof is based on a construction of the donnelly kurtz lookdown process using continuous random trees which is of independent interest this produces a number of results concerning the small time behavior of beta coalescents most notably we recover an almost sure limit theorem of the present authors for the number of blocks at small times and give the multifractal spectrum corresponding to the emergence of blocks with atypical size also we are able to find exact asymptotics for sampling formulae corresponding to the site frequency spectrum and the allele frequency spectrum associated with mutations in the context of population genetics

the exploration process of inhomogeneous continuum random trees and an extension of jeulin s local time identity we study the inhomogeneous continuum random trees icrt that arise as weak limits of birthday trees we give a description of the exploration process a function defined on 0 1 that encodes the structure of an icrt and also of its width process determining the size of layers in order of height these processes turn out to be transformations of bridges with exchangeable increments which have already appeared in other icrt related topics such as stochastic additive coalescence the results rely on two different constructions of birthday trees from processes with exchangeable increments on weak convergence arguments and on general theory on continuum random trees

on the singular set in the navier stokes equations we consider suitably weak solutions u p to the incompressible navier stokes equations and under various assumptions on u obtain estimates for the size of its singular set one of our results improves a well known theorem of caffarelli kohn and nirenberg

rayleigh processes real trees and root growth with re grafting the real trees form a class of metric spaces that extends the class of trees with edge lengths by allowing behavior such as infinite total edge length and vertices with infinite branching degree aldous s brownian continuum random tree the random tree like object naturally associated with a standard brownian excursion may be thought of as a random compact real tree the continuum random tree is a scaling limit as n of both a critical galton watson tree conditioned to have total population size n as well as a uniform random rooted combinatorial tree with n vertices the aldous broder algorithm is a markov chain on the space of rooted combinatorial trees with n vertices that has the uniform tree as its stationary distribution we construct and study a markov process on the space of all rooted compact real trees that has the continuum random tree as its stationary distribution and arises as the scaling limit as n of the aldous broder chain a key technical ingredient in this work is the use of a pointed gromov hausdorff distance to metrize the space of rooted compact real trees berkeley statistics technical report no 654 february 2004 revised october 2004 to appear in probability theory and related fields

exit times for an increasing levy tree valued process we give an explicit construction of the increasing tree valued process introduced by abraham and delmas using a random point process of trees and a grafting procedure this random point process will be used in companion papers to study record processes on levy trees we use the poissonian structure of the jumps of the increasing tree valued process to describe its behavior at the first time the tree grows higher than a given height using a spinal decomposition of the tree similar to the classical bismut and williams decompositions we also give the joint distribution of this exit time and the ascension time which corresponds to the first infinite jump of the tree valued process

branching processes in levy processes laplace functionals of snakes and superprocesses we use the exploration process introduced in a previous work to develop a new construction of superprocesses with a general branching mechanism this construction depends on a path valued process called the levy snake which is of independent interest our method of proof involves a calculation of the laplace functional of the occupation field of the levy snake this calculation relies on an evaluation of the corresponding moment functionals which requires precise information about the underlying genealogical structure

a regularity criterion at one scale without pressure for suitable weak solutions to the navier stokes equations abstract in this paper we continue our work in 15 to derive e regularity criteria at one scale without pressure for suitable weak solutions to the navier stokes equations we establish an e regularity criterion below of suitable weak solutions for any 0 q 1 u 5 2 d x d t e as an application we extend the previous corresponding results concerning the improvement of the classical caffarelli kohn nirenberg theorem by a logarithmic factor

the cramer rao lower bound for directions of arrival of gaussian cyclostationary signals the cramer rao lower bound on the variance of estimates of directions of arrival signal strengths and carrier frequencies and phases of cyclostationary signals arriving at an array is considered numerical evaluations of the bound for the estimates of directions of arrival show that it can be orders of magnitude lower than the corresponding bound for stationary signals despite the great number of unknown parameters in the former case comparisons with the mean squared error of direction estimates obtained by the cyclic least squares algorithm show that the bound is not overly optimistic

cycloergodic properties of discrete parameter nonstationary stochastic processes it is shown that a large class of nonstationary discrete parameter stochastic processes possess novel ergodic properties which are referred to as em cycloergodic properties specifically it is shown that periodic components of time varying probabilistic parameters can be consistently estimated from time averages on one sample path the cycloergodic theory developed herein extends and generalizes existing ergodic theory for asymptotically mean stationary and n stationary cyclostationary processes and is presented in both wide sense and strict sense contexts

hausdorff measure of the singular set in the incompressible magnetohydrodynamic equations we derive a local energy inequality for weak solutions of the three dimensional magnetohydrodynamic equations combining biot savart law interpolation inequalities and the local energy inequality we prove a partial regularity theorem for suitable weak solutions furthermore we obtain an improved estimate for the logarithmic hausdorff dimension of the singular set of suitable weak solutions

the hausdorff measure of the support of two dimensional super brownian motion we show that two dimensional super brownian motion is a multiple of the h hausdorff measure on its closed support where h r r 2 log 1 r log log log 1 r this complements known results in dimensions greater than 2

spinal partitions and invariance under re rooting of continuum random trees we develop some theory of spinal decompositions of discrete and continuous fragmentation trees specifically we consider a coarse and a fine spinal integer partition derived from spinal tree decompositions we prove that for a two parameter poisson dirichlet family of continuous fragmentation trees including the stable trees of duquesne and le gall the fine partition is obtained from the coarse one by shattering each of its parts independently according to the same law as a second application of spinal decompositions we prove that among the continuous fragmentation trees stable trees are the only ones whose distribution is invariant under uniform re rooting

the multiplicative coalescent inhomogeneous continuum random trees and new universality classes for critical random graphs one major open conjecture in the area of critical random graphs formulated by statistical physicists and supported by a large amount of numerical evidence over the last decade braunstein et al in phys rev lett 91 16 168701 2003 wu et al in phys rev lett 96 14 148702 2006 braunstein et al int j bifurc chaos 17 07 2215 2255 2007 chen et al in phys rev lett 96 6 068702 2006 is as follows for a wide array of random graph models with degree exponent tau in 3 4 distances between typical points both within maximal components in the critical regime as well as on the minimal spanning tree on the giant component in the supercritical regime scale like n tau 3 tau 1 in this paper we study the metric space structure of maximal components of the multiplicative coalescent in the regime where the sizes converge to excursions of levy processes without replacement aldous and limic electron in j probab 3 3 59 1998 yielding a completely new class of limiting random metric spaces a by product of the analysis yields the continuum scaling limit of one fundamental class of random graph models with degree exponent tau in 3 4 where edges are rescaled by n tau 3 tau 1 yielding the first rigorous proof of the above conjecture the limits in this case are compact tree like random fractals with a dense collection of hubs infinite degree vertices a finite number of which are identified with leaves to form shortcuts in a special case we show that the minkowski dimension of the limiting spaces equal tau 2 tau 3 a s in stark contrast to the erd s renyi scaling limit whose minkowski dimension is 2 a s it is generally believed that dynamic versions of a number of fundamental random graph models as one moves from the barely subcritical to the critical regime can be approximated by the multiplicative coalescent in work in progress the general theory developed in this paper is used to prove analogous limit results for other random graph models with degree exponent tau in 3 4 our proof makes crucial use of inhomogeneous continuum random trees icrt which have previously arisen in the study of the entrance boundary of the additive coalescent we show that tilted versions of the same objects using the associated mass measure describe connectivity properties of the multiplicative coalescent since convergence of height processes of corresponding approximating mathbf p trees is not known we use general methodology in athreya et al 2014 and develop novel techniques relying on first showing convergence in the gromov weak topology and then extending this to gromov hausdorff prokhorov convergence by proving a global lower mass bound

biased random walk on critical galton watson trees conditioned to survive we consider the biased random walk on a critical galton watson tree conditioned to survive and confirm that this model with trapping belongs to the same universality class as certain one dimensional trapping models with slowly varying tails indeed in each of these two settings we establish closely related functional limit theorems involving an extremal process and also demonstrate extremal aging occurs

geodesics in large planar maps and in the brownian map we study geodesics in the random metric space called the brownian map which appears as the scaling limit of large planar maps in particular we completely describe geodesics starting from the distinguished point called the root and we characterize the set s of all points that are connected to the root by more than one geodesic the set s is dense in the brownian map and homeomorphic to a non compact real tree furthermore for every x in s the number of distinct geodesics from x to the root is equal to the number of connected components of s x in particular points of the brownian map can be connected to the root by at most three distinct geodesics our results have applications to the behavior of geodesics in large planar maps

on the regularity of weak solutions to the magnetohydrodynamic equations abstract in this paper we study the regularity of weak solution to the incompressible magnetohydrodynamic equations we obtain some sufficient conditions for regularity of weak solutions to the magnetohydrodynamic equations which is similar to that of incompressible navier stokes equations moreover our results demonstrate that the velocity field of the fluid plays a more dominant role than the magnetic field does on the regularity of solution to the magneto hydrodynamic equations

central limit theorems for the spectra of classes of random fractals we discuss the spectral asymptotics of some open subsets of the real line with random fractal boundary and of a random fractal the continuum random tree in the case of open subsets with random fractal boundary we establish the existence of the second order term in the asymptotics almost surely and then determine when there will be a central limit theorem which captures the fluctuations around this limit we will show examples from a class of random fractals generated from dirichlet distributions as this is a relatively simple setting in which there are sets where there will and will not be a central limit theorem the brownian continuum random tree can also be viewed as a random fractal generated by a dirichlet distribution the first order term in the spectral asymptotics is known almost surely and here we show that there is a central limit theorem describing the fluctuations about this though the positivity of the variance arising in the central limit theorem is left open in both cases these fractals can be described through a general crump mode jagers branching process and we exploit this connection to establish our central limit theorems for the higher order terms in the spectral asymptotics our main tool is a central limit theorem for such general branching processes which we prove under conditions which are weaker than those previously known

a new proof of the caffarelli kohn nirenberg theorem here we give a self contained new proof of the partial regularity theorems for solutions of incompressible navier stokes equations in three spatial dimensions these results were originally due to scheffer and caffarelli kohn and nirenberg our proof is much more direct and simpler 1998 john wiley sons inc

partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations in this paper we study the local behavior of the solutions to the three dimensional magnetohydrodynamic equations we are interested in both the uniform gradient estimates for smooth solutions and regularity of weak solutions it is shown that in some neighborhood of x0 t0 the gradients of the velocity field u and the magnetic field b are locally uniformly bounded in l norm as long as that either the scaled local l2 norm of the gradient or the scaled local total energy of the velocity field is small and the scaled local total energy of the magnetic field is uniformly bounded these estimates indicate that the velocity field plays a more dominant role than that of the magnetic field in the regularity theory as an immediately corollary we can derive an estimates of hausdorff dimension on the possible singular set of a suitable weak solution as in the case of pure fluid various partial regularity results are obtained as consequences of our blow up estimates

on a functional contraction method methods for proving functional limit laws are developed for sequences of stochastic processes which allow a recursive distributional decomposition either in time or space our approach is an extension of the so called contraction method to the space c 0 1 of continuous functions endowed with uniform topology and the space d 0 1 of cadlag functions with the skorokhod topology the contraction method originated from the probabilistic analysis of algorithms and random trees where characteristics satisfy natural distributional recurrences it is based on stochastic fixed point equations where probability metrics can be used to obtain contraction properties and allow the application of banach s fixed point theorem we develop the use of the zolotarev metrics on the spaces c 0 1 and d 0 1 in this context applications are given in particular a short proof of donsker s functional limit theorem is derived and recurrences arising in the probabilistic analysis of algorithms are discussed

nonparametric time series analysis for periodically correlated processes correlation functions of continuous time periodically correlated processes can be represented by a fourier series with coefficient functions it is shown that the usual estimator for stationary covariances formed from a single sample path of the process can be simply modified to provide a consistent in quadratic mean estimator for any of the coefficient functions resulting from the aforementioned representation it is shown that if the process is gaussian and b sub k tau is a fourier integral with respect to a density function g sub k lambda a two dimensional periodogram formed from a single sample function can be smoothed along a line of constant difference frequency to provide a consistent estimator for g sub k lambda this natural extension of the well known procedure for stationary processes provides a method for nonparametric spectral analysis of periodically correlated processes

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

smaller population size at the mrca time for stationary branching processes we consider an elementary model of random size varying population governed by a stationary continuous state branching process we compute the distributions of various variables related to the most recent common ancestor mrca the time to the mrca the size of the current population and the size of the population just before the mrca in particular we observe a natural mild bottleneck effect as the size of the population just before the mrca is stochastically smaller than the size of the current population we also compute the number of individuals involved in the last coalescent event of the genealogical tree that is the number of individuals at the time of the mrca who have descendants in the current population by studying more precisely the genealogical structure of the population we get asymptotics for the number of ancestors just before the current time we give explicit computations in the case of the quadratic branching mechanism in this case the size of the population at the mrca is in mean 2 3 of the size of the current population we also provide in this case the fluctuations for the renormalized number of ancestors

a space time property of a class of measure valued branching diffusions si d on montre que la diffusion ramifiee d dimensionnelle d indice etudiee par dawson et d autres distribue sa masse sur un support aleatoire de facon uniforme relativement a la mesure de hausdorff

branching processes in levy processes the exploration process the main idea of the present work is to associate with a general continuous branching process an exploration process that contains the desirable information about the genealogical structure the exploration process appears as a simple local time functional of a levy process with no negative jumps whose laplace exponent coincides with the branching mechanism function this new relation between spectrally positive levy processes and continuous branching processes provides a unified perspective on both theories in particular we derive the adequate formulation of the classical ray knight theorem for such levy processes as a consequence of this theorem we show that the path continuity of the exploration process is equivalent to the almost sure extinction of the branching process

ray knight representation of flows of branching processes with competition by pruning of levy trees we introduce flows of branching processes with competition which describe the evolution of general continuous state branching populations in which interactions between individuals give rise to a negative density dependence term this generalizes the logistic branching processes studied by lambert ann appl probab 15 2 1506 1535 2005 following the approach developed by dawson and li ann probab 40 2 813 857 2012 we first construct such processes as the solutions of certain flow of stochastic differential equations we then propose a novel genealogical description for branching processes with competition based on interactive pruning of levy trees and establish a ray knight representation result for these processes in terms of the local times of suitably pruned forests

self similar fragmentations derived from the stable tree i the basic object we consider is a certain model of continuum random tree called the stable tree we construct a fragmentation process f t t 0 out of this tree by removing the vertices located under height t thanks to a self similarity property of the stable tree we show that the fragmentation process is also self similar the semigroup and other features of the fragmentation are given explicitly asymptotic results are given as well as a couple of related results on continuous state branching processes

the scaling limit of the minimum spanning tree of the complete graph consider the minimum spanning tree mst of the complete graph with nn vertices when edges are assigned independent random weights endow this tree with the graph distance renormalized by n1 3n1 3 and with the uniform measure on its vertices we show that the resulting space converges in distribution as n n to a random compact measured metric space in the gromov hausdorff prokhorov topology we additionally show that the limit is a random binary rr tree and has minkowski dimension 33 almost surely in particular its law is mutually singular with that of the brownian continuum random tree or any rescaled version thereof our approach relies on a coupling between the mst problem and the erd s renyi random graph we exploit the explicit description of the scaling limit of the erd s renyi random graph in the so called critical window established in probab theory related fields 152 2012 367 406 and provide a similar description of the scaling limit for a critical minimum spanning forest contained within the mst in order to accomplish this we introduce the notion of rr graphs which generalise rr trees and are of independent interest

the topological structure of scaling limits of large planar maps we discuss scaling limits of large bipartite planar maps if p 2 is a fixed integer we consider for every integer n 2 a random planar map mn which is uniformly distributed over the set of all rooted 2p angulations with n faces then at least along a suitable subsequence the metric space consisting of the set of vertices of mn equipped with the graph distance rescaled by the factor n 1 4 converges in distribution as n towards a limiting random compact metric space in the sense of the gromov hausdorff distance we prove that the topology of the limiting space is uniquely determined independently of p and of the subsequence and that this space can be obtained as the quotient of the continuum random tree for an equivalence relation which is defined from brownian labels attached to the vertices we also verify that the hausdorff dimension of the limit is almost surely equal to 4

probabilistic and fractal aspects of levy trees we investigate the random continuous trees called levy trees which are obtained as scaling limits of discrete galton watson trees we give a mathematically precise definition of these random trees as random variables taking values in the set of equivalence classes of compact rooted trees which is equipped with the gromov hausdorff distance to construct levy trees we make use of the coding by the height process which was studied in detail in previous work we then investigate various probabilistic properties of levy trees in particular we establish a branching property analogous to the well known property for galton watson trees conditionally given the tree below level a the subtrees originating from that level are distributed as the atoms of a poisson point measure whose intensity involves a local time measure supported on the vertices at distance a from the root we study regularity properties of local times in the space variable and prove that the support of local time is the full level set except for certain exceptional values of a corresponding to local extinctions we also compute several fractal dimensions of levy trees including hausdorff and packing dimensions in terms of lower and upper indices for the branching mechanism function which characterizes the distribution of the tree we finally discuss some applications to super brownian motion with a general branching mechanism

measures for branching exit markov systems and their applications to differential equations semilinear equations lu u where l is an elliptic differential operator and is a positive function can be investigated by using l superdiffusions in a special case u u2 a powerful probabilistic tool the brownian snake introduced by le gall was successfully applied by him and his school to get deep results on solutions of this equation some of these results but not all of them were extended by dynkin and kuznetsov to general equations by applying superprocesses an important role in the theory of the brownian snake and its applications is played by measures open image in new windowx on the space of continuous paths our goal is to introduce analogous measures related to superprocesses and to general branching exit markov systems they are defined on the space of measures and we call them open image in new window measures using open image in new window measures allows to combine some advantages of brownian snakes and of superprocesses as tools for a study of semilinear pdes

self similar fragmentations derived from the stable tree ii splitting at nodes we study a natural fragmentation process of the so called stable tree introduced by duquesne and le gall which consists in removing the nodes of the tree according to a certain procedure that makes the fragmentation self similar with positive index explicit formulas for the semigroup are given and we provide asymptotic results we also give an alternative construction of this fragmentation using paths of levy processes hence echoing the two alternative constructions of the standard additive coalescent by fragmenting the brownian continuum random tree or using brownian paths respectively due to aldous pitman and bertoin

on the distribution of betweenness centrality in random trees abstract betweenness centrality is a quantity that is frequently used to measure how central a vertex v is it is defined as the sum over pairs of vertices other than v of the proportions of shortest paths that pass through v in this paper we study the distribution of the betweenness centrality in random trees and related subcritical graph families specifically we prove that the betweenness centrality of the root vertex in a simply generated tree is usually of linear order but has a mean of order n 3 2 we also show that a randomly chosen vertex typically also has linear order betweenness centrality and that the maximum betweenness centrality in a simply generated tree is of order n 2 we obtain limiting distributions for the betweenness centrality of the root vertex and of a randomly chosen vertex as well as for the maximum betweenness centrality and we also show that the centroid has positive probability in the limit to be the vertex of maximum betweenness centrality some similar results also hold for subcritical graph classes which will be briefly discussed finally we study random recursive trees and other families of increasing trees where the situation is quite different here the root betweenness centrality is of quadratic order as is the maximum betweenness centrality the betweenness centrality of a random vertex on the other hand is again of linear order again we also have limiting distributions upon suitable normalisation

the standard additive coalescent regard an element of the set x 1 x 2 x 1 x 2 0 i x i 1 as a fragmentation of unit mass into clusters of masses x i the additive coalescent of evans and pitman is the valued markov process in which pairs of clusters of masses x i x j merge into a cluster of mass x i x j at rate x i x j they showed that a version x t t of this process arises as a n weak limit of the process started at time 1 2 log n with n clusters of mass 1 n we show this standard additive coalescent may be constructed from the continuum random tree of aldous by poisson splitting along the skeleton of the tree we describe the distribution of x t on at a fixed time t we show that the size of the cluster containing a given atom as a process in t has a simple representation in terms of the stable subordinator of index 1 2 as t we establish a gaussian limit for centered and normalized cluster sizes and study the size of the largest cluster

scaling limits of random planar maps with large faces we discuss asymptotics for large random planar maps under the assumption that the distribution of the degree of a typical face is in the domain of attraction of a stable distribution with index alpha in 1 2 when the number n of vertices of the map tends to infinity the asymptotic behavior of distances from a distinguished vertex is described by a random process called the continuous distance process which can be constructed from a centered stable process with no negative jumps and index alpha in particular the profile of distances in the map rescaled by the factor n 1 2 alpha converges to a random measure defined in terms of the distance process with the same rescaling of distances the vertex set viewed as a metric space converges in distribution as n to infty at least along suitable subsequences towards a limiting random compact metric space whose hausdorff dimension is equal to 2 alpha

sub gaussian tail bounds for the width and height of conditioned galton watson trees we study the height and width of a galton watson tree with offspring distribution satisfying e 1 0 var conditioned on having exactly n nodes under this conditioning we derive sub gaussian tail bounds for both the width largest number of nodes in any level and height greatest level containing a node the bounds are optimal up to constant factors in the exponent under the same conditioning we also derive essentially optimal upper tail bounds for the number of nodes at level k for 1 k n

scaling limit of the recurrent biased random walk on a galton watson tree we show that the trace of the null recurrent biased random walk on a galton watson tree properly renormalized converges to the brownian forest our result extends to the setting of the random walk in random environment on a galton watson tree

the brownian map is the scaling limit of uniform random plane quadrangulations we prove that uniform random quadrangulations of the sphere with n faces endowed with the usual graph distance and renormalized by n 1 4 converge as n in distribution for the gromov hausdorff topology to a limiting metric space we validate a conjecture by le gall by showing that the limit is up to a scale constant the so called brownian map which was introduced by marckert mokkadem and le gall as the most natural candidate for the scaling limit of many models of random plane maps the proof relies strongly on the concept of geodesic stars in the map which are configurations made of several geodesics that only share a common endpoint and do not meet elsewhere

recursive self similarity for random trees random triangulations and brownian excursion recursive self similarity for a random object is the property of being decomposable into independent rescaled copies of the original object certain random combinatorial objects trees and triangulations possess approximate versions of recursive self similarity and then their continuous limits possess exact recursive self similarity in particular since the limit continuum random tree can be identified with brownian excursion we get a nonobvious recursive self similarity property for brownian excursion

the packing measure of the range of super brownian motion we prove that the total range of super brownian motion with quadratic branching mechanism has an exact packing measure with respect to the gauge function g r r 4 log log1 r 3 in super critical dimensions d geq 5 more precisely we prove that the total occupation measure of super brownian motion is equal to the g packing measure restricted to its range up to a deterministic multiplicative constant that only depends on space dimension d

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

tree based models for random distribution of mass a mathematical model for distribution of mass ind dimensional space based upon randomly embedding random trees into space is introduced and studied the model is a variant of thesuper brownian motion process studied by mathematicians we present calculations relating to i the distribution of position of a typical mass element ii moments of the center of mass iii large deviation behavior and iv a recursive self similarity property

a limit theorem for the contour process of condidtioned galton watson trees in this work we study asymptotics of the genealogy of galton watson processes conditioned on the total progeny we consider a fixed aperiodic and critical offspring distribution such that the rescaled galton watson processes converges to a continuous state branching process csbp with a stable branching mechanism of index 1 2 we code the genealogy by two different processes the contour process and the height process that le gall and le jan recently introduced we show that the rescaled height process of the corresponding galton watson family tree with one ancestor and conditioned on the total progeny converges in a functional sense to a new process the normalized excursion of the continuous height process associated with the stable csbp we deduce from this convergence an analogous limit theorem for the contour process in the brownian case 2 the limiting process is the normalized brownian excursion that codes the continuum random tree the result is due to aldous who used a different method

random stable laminations of the disk we study large random dissections of polygons we consider random dissections of a regular polygon with n sides which are chosen according to boltzmann weights in the domain of attraction of a stable law of index 2 1 2 as n goes to infinity we prove that these random dissections converge in distribution towards a random compact set called the random stable lamination if 2 we recover aldous brownian triangulation however if 2 1 2 large faces remain in the limit and a dierent random compact set appears we show that the random stable lamination can be coded by the continuous time height function associated to the normalized excursion of a strictly stable spectrally positive levy process of index using this coding we establish that the hausdor dimension of the stable random lamination is almost surely 2 1

growth of levy trees we construct random locally compact real trees called levy trees that are the genealogical trees associated with continuous state branching processes more precisely we define a growing family of discrete galton watson trees with i i d exponential branch lengths that is consistent under bernoulli percolation on leaves we define the levy tree as the limit of this growing family with respect to the gromov hausdorff topology on metric spaces this elementary approach notably includes supercritical trees and does not make use of the height process introduced by le gall and le jan to code the genealogy of sub critical continuous state branching processes we construct the mass measure of levy trees and we give a decomposition along the ancestral subtree of a poisson sampling directed by the mass measure

statistical tests for large tree structured data we develop a general statistical framework for the analysis and inference of large tree structured data with a focus on developing asymptotic goodness of fit tests we first propose a consistent statistical model for binary trees from which we develop a class of invariant tests using the model for binary trees we then construct tests for general trees by using the distributional properties of the continuum random tree which arises as the invariant limit for a broad class of models for tree structured data based on conditioned galton watson processes the test statistics for the goodness of fit tests are simple to compute and are asymptotically distributed as 2 and f random variables we illustrate our methods on an important application of detecting tumour heterogeneity in brain cancer we use a novel approach with tree based representations of magnetic resonance images and employ the developed tests to ascertain tumor heterogeneity between two groups of patients

scaling limits of random polya trees polya trees are rooted trees considered up to symmetry we establish the convergence of large uniform random polya trees with arbitrary degree restrictions to aldous continuum random tree with respect to the gromov hausdorff metric our proof is short and elementary and it is based on a novel decomposition it shows that the global shape of a random polya tree is essentially dictated by a large galton watson tree that it contains we also derive sub gaussian tail bounds for both the height and the width which are optimal up to constant factors in the exponent

limits of the boundary of random planar maps we discuss asymptotics for the boundary of critical boltzmann planar maps under the assumption that the distribution of the degree of a typical face is in the domain of attraction of a stable distribution with parameter alpha in 1 2 first in the dense phase corresponding to alpha in 1 3 2 we prove that the scaling limit of the boundary is the random stable looptree with parameter 1 alpha 1 2 second we show the existence of a phase transition through local limits of the boundary in the dense phase the boundary is tree like while in the dilute phase corresponding to alpha in 3 2 2 it has a component homeomorphic to the half plane as an application we identify the limits of loops conditioned to be large in the rigid o n loop model on quadrangulations proving thereby a conjecture of curien and kortchemski

consistent estimation of the cyclic autocorrelation the cyclic autocorrelation is often used to describe nonstationary random processes the authors investigate the conditions under which the cyclic autocorrelation can be estimated consistently in mean square for discrete time gaussian processes they extend and generalize results of hurd 1989 and refine results of boyles and gardner 1983 they derive necessary and sufficient conditions for consistency in mean square of an estimator which are in the form of a single sum of autocorrelation coefficients in the form of a double sum of autocorrelation coefficients in the bifrequency domain and in terms of the average spectrum they also discuss the rate of convergence for this estimator

scaling limits of random planar maps with a unique large face we study random bipartite planar maps defined by assigning nonnegative weights to each face of a map we prove that for certain choices of weights a unique large face having degree proportional to the total number of edges in the maps appears when the maps are large it is furthermore shown that as the number of edges n of the planar maps goes to infinity the profile of distances to a marked vertex rescaled by n 1 2 is described by a brownian excursion the planar maps with the graph metric resealed by n 1 2 are then shown to converge in distribution toward aldous brownian tree in the gromov hausdorff topology in the proofs we rely on the bouttier di francesco guitter bijection between maps and labeled trees and recent results on simply generated trees where a unique vertex of a high degree appears when the trees are large

subordination of trees and the brownian map we discuss subordination of random compact mathbb r trees we focus on the case of the brownian tree where the subordination function is given by the past maximum process of brownian motion indexed by the tree in that particular case the subordinate tree is identified as a stable levy tree with index 3 2 as a more precise alternative formulation we show that the maximum process of the brownian snake is a time change of the height process coding the levy tree we then apply our results to properties of the brownian map in particular we recover in a more precise form a recent result of miller and sheffield identifying the metric net associated with the brownian map

the stable trees are nested we show that we can construct simultaneously all the stable trees as a nested family more precisely if 1 alpha alpha prime le 2 we prove that hidden inside any alpha stable tree we can find a version of an alpha prime stable tree rescaled by an independent mittag leffler type distribution this tree can be explicitly constructed by a pruning procedure of the underlying stable tree or by a modification of the fragmentation associated with it our proofs are based on a recursive construction due to marchal which is proved to converge almost surely towards a stable tree

the continuum random tree iii let mathscr r k k geq 1 be random trees with k leaves satisfying a consistency condition removing a random leaf from mathscr r k gives mathscr r k 1 then under an extra condition this family determines a random continuum tree mathscr l which it is convenient to represent as a random subset of l 1 this leads to an abstract notion of convergence in distribution as n rightarrow infty of rescaled random trees mathscr j n on n vertices to a limit continuum random tree mathscr l the notion is based upon the assumption that for fixed k the subtrees of mathscr j n determined by k randomly chosen vertices converge to mathscr r k as our main example under mild conditions on the offspring distribution the family tree of a galton watson branching process conditioned on total population size equal to n can be rescaled to converge to a limit continuum random tree which can be constructed from brownian excursion

scaling limit for the ant in a simple high dimensional labyrinth we prove that after suitable rescaling the simple random walk on the trace of a large critical branching random walk in mathbb z d converges to the brownian motion on the integrated super brownian excursion when d 14