the bolthausen sznitman coalescent and the genealogy of continuous state branching processes we use bochner s subordination to give a representation of the genealogical structure associated with general continuous state branching processes we then apply this representation to connections between a branching process introduced by neveu and the coalescent process recently investigated by bolthausen sznitman and others

some exact solutions of the ideal mhd equations through symmetry reduction method abstract we use the symmetry reduction method based on lie group theory to obtain some exact solutions the so called invariant solutions of the ideal magnetohydrodynamic equations in 3 1 dimensions in particular these equations are invariant under a galilean similitude lie algebra for which the classification by conjugacy classes of r dimensional subalgebras 1 r 4 was already known we restrict our study to the three dimensional galilean similitude subalgebras that give us systems composed of ordinary differential equations here some examples of these solutions are presented with a brief physical interpretation

the scaling limit of the incipient infinite cluster in high dimensional percolation ii integrated super brownian excursion for independent nearest neighbor bond percolation on zd with d 6 we prove that the incipient infinite cluster s two point function and three point function converge to those of integrated super brownian excursion ise in the scaling limit the proof is based on an extension of the new expansion for percolation derived in a previous paper and involves treating the magnetic field as a complex variable a special case of our result for the two point function implies that the probability that the cluster of the origin consists of n sites at the critical point is given by a multiple of n 3 2 plus an error term of order n 3 2 e with e 0 this is a strong version of the statement that the critical exponent is given by 2

markov snakes and superprocesses we suggest the name markov snakes for a class of path valued markov processes introduced recently by j f le gall in connection with the theory of branching measure valued processes le gall applied this class to investigate path properties of superdiffusions and to approach probabilistically partial differential equations involving a nonlinear operator v v2 we establish an isomorphism theorem which allows to translate results on continuous superprocesses into the language of markov snakes and vice versa by using this theorem we get limit theorems for discrete markov snakes

nonclassical symmetry reductions of the three dimensional incompressible navier stokes equations the nonclassical reduction method as pioneered by bluman and cole j meth mech 18 1025 42 is used to examine symmetries of the full three dimensional unsteady incompressible navier stokes equations of fluid mechanics the procedure when applied to a system of partial differential equations yields reduced sets of equations with one fewer independent variables we find eight possibilities for reducing the navier stokes equations in the three spatial and one temporal dimensions to sets of partial differential equations in three independent variables some of these reductions are derivable using the lie group method of classical symmetries but the remainder are genuinely nonclassical further investigations of one of our eight forms shows how it is possible to derive novel exact solutions of the navier stokes equations by the nonclassical method

complex geometry and the asymptotics of harish chandra modules for real reductive lie groups i let g be a connected semisimple real matrix group it is now apparent that the representation theory of g is intimately connected with the complex geometry of the flag variety 9 by studying appropriate orbit structures on 9 we are naturally led to representation theory in the category of harish chandra modules x or the representation theory of category 9 the jacquet functor j f 9 has proved a useful tool in converting yfw problems into 9 problems which are often more tractable in this paper we advance the philosophy that the complex geometry of 9 associated to yf and 9 interacts in a natural way with the functor j leading to deep new information on the structure of jacquet modules this in turn gives new insight into the structure of certain nilpotent cohomology groups associated to harish chandra modules our techniques are based upon many of the ideas present in the proof of the kazhdan lusztig conjectures and bernstein s proof of the jantzen conjecture

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

a class of path valued markov processes and its applications to superprocesses let s be a continuous markov process satisfying certain regularity assumptions we introduce a path valued strong markov process associated with s which is closely related to the so called superprocess with spatial motion s in particular a subseth of the state space of s intersects the range of the superprocess if and only if the set of paths that hith is not polar for the path valued process the latter property can be investigated using the tools of the potential theory of symmetric markov processes a set is not polar if and only if it supports a measure of finite energy the same approach can be applied to study sets that are polar for the graph of the superprocess in the special case when s is a diffusion process we recover certain results recently obtained by dynkin

the brownian snake and solutions of δu u2 in a domain we investigate the connections between the path valued process called the brownian snake and nonnegative solutions of the partial differential equation u u2 in a domain of d in particular we prove two conjectures recently formulated by dynkin the first one gives a complete characterization of the boundary polar sets which correspond to boundary removable singularities for the equation u u2 the second one establishes a one to one correspondence between nonnegative solutions that are bounded above by a harmonic function and finite measures on the boundary that do not charge polar sets this correspondence can be made explicit by a probabilistic formula involving a special class of additive functionals of the brownian snake our proofs combine probabilistic and analytic arguments an important role is played by a new version of the special markov property which is of independent interest

subtree prune and regraft a reversible real tree valued markov process we use dirichlet form methods to construct and analyze a reversible markov process the stationary distribution of which is the brownian continuum random tree this process is inspired by the subtree prune and regraft spr markov chains that appear in phylogenetic analysis a key technical ingredient in this work is the use of a novel gromov hausdorff type distance to metrize the space whose elements are compact real trees equipped with a probability measure also the investigation of the dirichlet form hinges on a new path decomposition of the brownian excursion

snakes and spiders brownian motion on ℝ trees we consider diffusion processes on a class of trees the processes are defined in a manner similar to that of le gall s brownian snake each point in the tree has a real valued height or generation and the height of the diffusion process evolves as a brownian motion when the height process decreases the diffusion retreats back along a lineage whereas when the height process increases the diffusion chooses among branching lineages according to relative weights given by a possibly infinite measure on the family of lineages the class of trees we consider can have branch points with countably infinite branching and lineages along which the branch points have points of accumulation

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

invariance principles for random bipartite planar maps it is conjectured in the physics literature that properly rescaled random planar maps when conditioned to have a large number of faces should converge to a limiting surface whose law does not depend up to scaling factors on details of the class of maps that are sampled previous works on the topic starting with chassaing schaeffer have shown that the radius of a random quadrangulation with n faces converges in distribution once rescaled by n 1 4 to the diameter of the brownian snake up to a scaling constant using a bijection due to bouttier di francesco guitter between bipartite planar maps and a family of labeled trees we show the corresponding invariance principle for a class of random maps that follow a boltzmann distribution the radius of such maps conditioned to have n faces or n vertices and under a criticality assumption converges in distribution once rescaled by n 1 4 to a scaled version of the diameter of the brownian snake convergence results for the so called profile of maps are also provided as well as convergence of rescaled bipartite maps to the brownian map as introduced by marckert mokkadem the proofs of these results rely on a new invariance principle for two type spatial galton watson trees

branching particle systems and superprocesses we start from a model of a branching particle system with immigration and with death rate and branching mechanism depending on time and location then we consider a limit case when the mass of particles and their life times are small and their density is high this way we construct a measure valued process x t which we call a superprocess replacing the underlying markov process xi t by the corresponding historical process xi leq t we construct a measure valued process m t in functional spaces which we call a historical superprocess the moment functions for superprocesses are evaluated linear positive additive functionals are studied they are used to construct a continuous analog of a random tree obtained by stopping every particle at a time depending on its path say at the first exit time from a domain a related special markov property for superprocesses is proved which is useful for applications to certain nonlinear partial differential equations the concluding section is devoted to a survey of the literature and the terminology on markov processes used in the paper is explained in the appendix

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

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

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

dynamical pruning of rooted trees with applications to 1 d ballistic annihilation we introduce generalized dynamical pruning on rooted binary trees with edge lengths that encompasses a number of discrete and continuous pruning operations including the tree erasure and horton pruning the pruning removes parts of a tree t starting from the leaves according to a pruning function defined on descendant subtrees within t we prove the invariance of critical binary galton watson tree with exponential edge lengths with respect to the generalized dynamical pruning for an arbitrary admissible pruning function these results facilitate analysis of the continuum 1 d ballistic annihilation model a a rightarrow varnothing for a constant particle density and initial velocity that alternates between the values of pm 1 we show that the model s shock wave is isometric to the level set tree of the potential function and the model evolution is equivalent to the generalized dynamical pruning of the shock wave tree

reduction and exact solutions of the navier stokes equations the authors construct a complete set of g 1 3 inequivalent ansatze of codimension 1 for the navier stokes ns field which reduce the ns equations to systems of ordinary differential equations ode having solved these odes they thereby obtain solutions of the ns equations formulae of group multiplication of solutions are given several non lie ansatze are discussed

some properties of the range of super brownian motion we consider a super brownian motion x its canonical measures can be studied through the path valued process called the brownian snake we obtain the limiting behavior of the volume of the neighborhood for the range of the brownian snake and as a consequence we derive the analogous result for the range of super brownian motion and for the support of the integrated super brownian excursion then we prove the support of xt is capacity equivalent to 0 1 2 in d d 3 and the range of x as well as the support of the integrated super brownian excursion are capacity equivalent to 0 1 4 in d d 5

last exit decompositions and regularity at the boundary of transition probabilities the purpose of this paper is to give a probabilistic approach to studying the regularity at the boundary of the transition probabilities of certain hypoelliptic diffusions with boundary conditions the main tools are last exit decompositions of brownian motion the malliavin calculus the theory of excursions and the calculus of variations on brownian excursions

vortices and invariant surfaces generated by symmetries for the 3d navier stokes equations abstract we show that certain infinitesimal operators of the lie point symmetries of the incompressible 3d navier stokes equations give rise to vortex solutions with different characteristics this approach allows an algebraic classification of vortices and throws light on the alignment mechanism between the vorticity and the vortex stretching vector s where s is the strain matrix the symmetry algebra associated with the navier stokes equations turns out to be infinite dimensional new vortical structures generalizing in some cases well known configurations such as for example the burgers and lundgren solutions are obtained and discussed in relation to the value of the dynamic angle arctan s s a systematic treatment of the boundary conditions invariant under the symmetry group of the equations under study is also performed and the corresponding invariant surfaces are recognized

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

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

the uniform random tree in a brownian excursion to any brownian excursione with duration e and anyt 1 t p 0 e we associate a branching tree withp branches denoted byt p e t 1 t p which is closely related to the structure of the minima ofe our main theorem states that ife is chosen according to the ito measure and t 1 t p according to lebesgue measure on 0 e p the treet p e t 1 t p is distributed according to the uniform measure on the set of trees withp branches the proof of this result yields additional information about the subexcursions ofe corresponding to the different branches of the tree thus generalizing a well known representation theorem of bismut if we replace the ito measure by the law of the normalized excursion a simple conditioning argument leads to another remarkable result originally proved by aldous with a very different method

dynamically stretched vortices as solutions of the 3d navier stokes equations abstract a well known limitation with stretched vortex solutions of the 3d navier stokes and euler equations such as those of burgers type is that they possess uni directional vorticity which is stretched by a strain field that is decoupled from them it is shown here that these drawbacks can be partially circumvented by considering a class of velocity fields of the type u u1 x y t u2 x y t x y t z w x y t where u1 u2 and w are functions of x y and t but not z it turns out that the equations for the third component of vorticity 3 and w decouple more specifically solutions of burgers type can be constructed by introducing a strain field into u such that u 2 x 2 y z y x w the strain rate t is solely a function of time and is related to the pressure via a riccati equation 2 p zz t 0 a constraint on pzz t is that it must be spatially uniform the decoupling of 3 and w allows the equation for 3 to be mapped to the usual general 2d problem through the use of lundgren s transformation while that for w can be mapped to the equation of a 2d passive scalar when 3 stretches then w compresses and vice versa various solutions for w are discussed and some 2 periodic dependent solutions for w are presented which take the form of a convergent power series in a similarity variable hence the vorticity r 1 w w r 3 has nonzero components in the azimuthal and radial as well as the axial directions for the euler problem the equation for w can sustain a vortex sheet type of solution where jumps in w occur when passes through multiples of 2

remarks on approximate fixed points abstract a mapping f from a topological space x to a metric space m d is said to be r continuous r 0 if for any x x there exists a neighborhood u x of x such that diam f u x r in 1961 victor klee proved that an r continuous self mapping of a compact convex subset of a banach space always has a invariant point for any r recently h x phu extended klee s result to mappings that are almost r continuous in this note we extend phu s result to compact hyperconvex spaces and to geodesically bounded complete r trees we also extend klee s stability result to certain geodesic spaces

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

the contour of splitting trees is a levy process splitting trees are those random trees where individuals give birth at a constant rate during a lifetime with general distribution to i i d copies of themselves the width process of a splitting tree is then a binary homogeneous crump mode jagers cmj process and is not markovian unless the lifetime distribution is exponential or a dirac mass at here we allow the birth rate to be infinite that is pairs of birth times and life spans of newboms form a poisson point process along the lifetime of their mother with possibly infinite intensity measure a splitting tree is a random so called chronological tree each element of a chronological tree is a so called existence point of some individual vertex in a discrete tree where is a nonnegative real number called chronological level time we introduce a total order on existence points called linear order and a mapping from the tree into the real line which preserves this order the inverse of is called the exploration process and the projection of this inverse on chronological levels the contour process for splitting trees truncated up to level r we prove that a thus defined contour process is a levy process reflected below and killed upon hitting 0 this allows one to derive properties of i splitting trees conceptual proof of le gall le jan s theorem in the finite variation case exceptional points coalescent point process and age distribution ii cmj processes one dimensional marginals conditionings limit theorems and asymptotic numbers of individuals with infinite versus finite descendances

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

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

a representation of exchangeable hierarchies by sampling from random real trees a hierarchy on a set s also called a total partition of s is a collection mathcal h of subsets of s such that s in mathcal h each singleton subset of s belongs to mathcal h and if a b in mathcal h then a cap b equals either a or b or varnothing every exchangeable random hierarchy of positive integers has the same distribution as a random hierarchy mathcal h associated as follows with a random real tree mathcal t equipped with root element 0 and a random probability distribution p on the borel subsets of mathcal t given mathcal t p let t 1 t 2 ldots be independent and identically distributed according to p and let mathcal h comprise all singleton subsets of mathbb n and every subset of the form j t j in f x as x ranges over mathcal t where f x is the fringe subtree of mathcal t rooted at x there is also the alternative characterization every exchangeable random hierarchy of positive integers has the same distribution as a random hierarchy mathcal h derived as follows from a random hierarchy mathscr h on 0 1 and a family u j of i i d uniform 0 1 random variables independent of mathscr h let mathcal h comprise all sets of the form j u j in b as b ranges over the members of mathscr h

weak transversality and partially invariant solutions new exact solutions are obtained for several nonlinear physical equations namely the navier stokes and euler systems an isentropic compressible fluid system and a vector nonlinear schrodinger equation the solution method makes use of the symmetry group of the system in situations when the standard lie method of symmetry reduction is not applicable

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

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

the scaling limit of lattice trees in high dimensions we prove that above eight dimensions the scaling limit of sufficiently spread out lattice trees is the variant of super brownian motion known as integrated super brownian excursion ise as conjectured by aldous the same is true for nearest neighbour lattice trees in sufficiently high dimensions the proof uses the lace expansion

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

the genealogy of continuous state branching processes with immigration recent works by j f le gall and y le jan 15 have extended the genealogical structure of galton watson processes to continuous state branching processes cb we are here interested in processes with immigration cbi

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

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

the depth first processes of galton watson trees converge to the same brownian excursion in this paper we show a strong relation between the depth first processes associated to galton watson trees with finite variance conditioned by the total progeny the depth first walk the depth first queue process the height process a consequence is that these processes suitably normalized converge to the same brownian excursion this provides an alternative proof of aldous one of the convergence of the depth first walk to the brownian excursion which does not use the existence of a limit tree the methods that we introduce allow one to compute some functionals of trees or discrete excursions for example we compute the limit law of the process of the height of nodes with a given out degree and the process of the height of nodes root of a given subtree

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

the real tree in this note it is shown that there exists a natural metric on the set formula of bounded subsets of r containing their infimum which endowstwith the structure of an r tree so that for everyt t the number of connected components oft t coincides with the cardinality p r of the set of subsets of r in addition the set of ends oftis explicitly determined and various further features oftare discussed too

a continuum tree valued markov process we present a construction of a levy continuum random tree crt associated with a super critical continuous state branching process using the so called exploration process and a girsanov theorem we also extend the pruning procedure to this super critical case let be a critical branching mechanism we set let or be the set of values of for which is a conservative branching mechanism the pruning procedure allows to construct a decreasing levy crt valued markov process mathcal t theta theta in theta such that mathcal t theta has branching mechanism it is sub critical if 0 and super critical if 0 we then consider the explosion time a of the crt the smallest negative time for which the continuous state branching process cb associated with mathcal t theta has finite total mass i e the length of the excursion of the exploration process that codes the crt is finite we describe the law of a as well as the distribution of the crt just after this explosion time the crt just after explosion can be seen as a crt conditioned not to be extinct which is pruned with an independent intensity related to a we also study the evolution of the crt valued process after the explosion time this extends results from aldous and pitman on galton watson trees for the particular case of the quadratic branching mechanism we show that after explosion the total mass of the cb behaves like the inverse of a stable subordinator with index 1 2 this result is related to the size of the tagged fragment for the fragmentation of aldous s crt

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

percolation on random triangulations and stable looptrees we study site percolation on angel and schramm s uniform infinite planar triangulation we compute several critical and near critical exponents and describe the scaling limit of the boundary of large percolation clusters in all regimes subcritical critical and supercritical we prove in particular that the scaling limit of the boundary of large critical percolation clusters is the random stable looptree of index 3 2 which was introduced in curien and kortchemski random stable looptrees arxiv 1304 1044 2014 we also give a conjecture linking looptrees of any index alpha in 1 2 with scaling limits of cluster boundaries in random triangulations decorated with o n models

continuum tree asymptotics of discrete fragmentations and applications to phylogenetic models given any regularly varying dislocation measure we identify a natural self similar fragmentation tree as scaling limit of discrete fragmentation trees with unit edge lengths as an application we obtain continuum random tree limits of aldous s beta splitting models and ford s alpha models for phylogenetic trees this confirms in a strong way that the whole trees grow at the same speed as the mean height of a randomly chosen leaf

a group analysis of the 2d navier stokes fourier equations abstract we study a 2 1 dimensional model of an incompressible thermoconducting fluid named navier stokes fourier system we apply a group theoretical analysis in correspondence of the generators of the symmetry group allowed by this model exact solutions are found some of them show possible interesting physical interpretations in our first exploration this feature is illustrated by dealing with simple special cases

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

the scaling limit of the incipient infinite cluster in high dimensional percolation i critical exponents this is the first of two papers on the critical behavior of bond percolation models in high dimensions in this paper we obtain strong joint control of the critical exponents and for the nearest neighbor model in very high dimensions d 6 and for sufficiently spread out models in all dimensions d 6 the exponent describes the low frequency behavior of the fourier transform of the critical two point connectivity function while describes the behavior of the magnetization at the critical point our main result is an asymptotic relation showing that in a joint sense 0 and 2 the proof uses a major extension of our earlier expansion method for percolation this result provides evidence that the scaling limit of the incipient infinite cluster is the random probability measure on d known as integrated super brownian excursion ise in dimensions above 6 in the sequel to this paper we extend our methods to prove that the scaling limits of the incipient infinite cluster s two point and three point functions are those of ise for the nearest neighbor model in dimensions d 6

sub exponential tail bounds for conditioned stable bienayme galton watson trees we establish uniform sub exponential tail bounds for the width height and maximal outdegree of critical bienayme galton watson trees conditioned on having a large fixed size whose offspring distribution belongs to the domain of attraction of a stable law this extends results obtained for the height and width by addario berry devroye and janson in the finite variance case

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

embeddings of harish chandra modules homology and the composition series problem the case of real rank one let g be a connected semisimple matrix group of real rank one fix a minimal parabolic subgroup p man and form the normalized principal series representations 1i u in the case of regular infinitesimal character we explicitly determine in terms of langlands classification all irreducible submodules and quotients of 1i u as a corollary all embeddings of an irreducible harish chandra module into principal series are computed the number of such embeddings is always less than or equal to three these computations are equivalent to the determination of zero n homology

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