solvability of linear local and nonlocal robin problems over c ω abstract let r n be a bounded lipschitz domain we first consider an elliptic boundary value problem with general robin boundary conditions the boundary conditions can be either local or nonlocal depending on the conditions imposed on the elliptic operator we prove that this boundary value problem is uniquely solvable and moreover we show that such weak solution is holder continuous on we also prove that a realization of the associated differential operator with generalized local or nonlocal robin boundary conditions generates an analytic c 0 semigroup of angle 2 over c we conclude by applying the elliptic regularity theory to solve the corresponding cauchy problem over c

some aspects of polya tree distributions for statistical modelling the definition and elementary properties of polya tree distributions are reviewed two theorems are presented showing that polya trees can be constructed to concentrate arbitrarily closely about any desired pdf and that polya tree priors can put positive mass in every relative entropy neighborhood of every positive density with finite entropy thereby satisfying a consistency condition such theorems are false for dirichlet processes models are constructed combining partially specified polya trees with other information such as monotonicity or unimodality it is shown how to compute bounds on posterior expectations over the class of all priors with the given specifications a numerical example is given a theorem of diaconis and freedman about dirichlet processes is generalized to polya trees allowing polya trees to be the models for errors in regression problems finally empirical bayes models using dirichlet processes are generalized to polya trees an example from berry and christensen is reanalyzed with a polya tree model

global regularity for a class of quasi linear local and nonlocal elliptic equations on extension domains abstract we consider the solvability of quasi linear elliptic equations with either local or nonlocal neumann robin or wentzell boundary conditions defined in the generalized sense on a bounded w 1 p extension domain whose boundary is an upper d set for an appropriate d 0 then we extend the fine regularity theory for weak solutions of the elliptic equations with the above boundary conditions known for bounded lipschitz domains to bounded w 1 p extension domains whose boundaries are upper d sets by showing that such weak solutions are globally holder continuous consequently we generalize substantially the class of bounded domains where weak solutions of boundary value problems of type neumann robin or wentzell may be uniformly continuous up to the boundary

critical behaviour of self avoiding walk in five or more dimensions we use the lace expansion to prove that in five or more dimensions the standard self avoiding walk on the hypercubic integer lattice behaves in many respects like the simple random walk in particular it is shown that the leading asymptotic behaviour of the number of step self avoiding walks is purely exponential that the mean square displacement is asymptotically linear in the number of steps and that the scaling limit is gaussian in the sense of convergence in distribution to brownian motion some related facts are also proved these results are optimal according to the widely believed conjecture that the self avoiding walk behaves unlike the simple random walk in dimensions two three and four

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

quasi linear boundary value problems with generalized nonlocal boundary conditions abstract we investigate a quasi linear boundary value problem of the form div u p 2 u 0 involving a general boundary map and mixed neumann boundary conditions on a bounded lipschitz domain we show existence uniqueness and holder continuity of the weak solution of this mixed boundary value problem and obtain maximum principles for this class of mixed equations as a consequence we obtain uniform continuity up to the boundary to solutions associated with a class of electrical models described by maxwell s equations with nonlocal boundary conditions an extension to boundary value problems with generalized nonlocal robin boundary conditions is also achieved

a weighted occupation time for a class of measured valued branching processes a weighted occupation time is defined for measure valued processes and a representation for it is obtained for a class of measure valued branching random motions on rd considered as a process in its own right the first and second order asymptotics are found as time t specifically the finiteness of the total weighted occupation time is determined as a function of the dimension d and when infinite a central limit type renormalization is considered yielding gaussian or asymmetric stable generalized random fields in the limit in one gaussian case the results are contrasted in high versus low dimensions

on the upper critical dimension of lattice trees and lattice animals we give a rigorous proof of mean field critical behavior for the susceptibility 1 2 and the correlation length v 1 4 for models of lattice trees and lattice animals in two cases i for the usual model with trees or animals constructed from nearest neighbor bonds in sufficiently high dimensions and ii for a class of spread out or long range models in which trees and animals are constructed from bonds of various lengths above eight dimensions this provides further evidence that for these models the upper critical dimension is equal to eight the proof involves obtaining an infrared bound and showing that a certain square diagram is finite at the critical point and uses an expansion related to the lace expansion for the self avoiding walk

existence of solutions to quasi linear elliptic problems with generalized robin boundary conditions abstract we characterize the l q solvability of a class of quasi linear elliptic equations involving the p laplace operator with generalized nonlinear robin type boundary conditions on bad domains some uniqueness results are also given

quasilinear parabolic equations with nonlinear wentzell robin type boundary conditions let rn be a bounded domain with lipschitz boundary a c with a 0 on let be the restriction to of the n 1 dimensional hausdorff measure and let b r 0 be measurable in the first variable and assume that for a e x b x is a proper convex lower semicontinuous functional we prove in the first part that for every p 1 the operator ap div a u p 2 u with nonlinear wentzell robin type boundary conditions apu b u p 2 u n u 0on generates a nonlinear submarkovian c0 semigroup on suitable l2 space here n x denotes the unit outer normal at x and for a e x the maximal monotone graph x denotes the subdifferential b x of the functional b x we also assume that b l and satisfies b x b0 0 a e on for some constant b0 as a consequence we obtain that there exist consistence nonexpansive nonlinear semigroups on suitable lq spaces for all q 1 in the second part we show some domination results

a quasi linear neumann problem of ambrosetti prodi type on extension domains abstract let r n be a bounded domain with the extension property whose boundary is an upper d set with respect to a measure for n 2 and d n p n we investigate the solvability of the ambrosetti prodi problem for the p laplace operator p with neumann boundary conditions using a priori estimates regularity theory a sub supersolution method and the leray schauder degree theory we obtain a necessary condition for the non existence of solutions in the weak sense the existence of at least one minimal solution and the existence of at least two distinct solutions moreover we establish global holder continuity for weak solutions of the neumann problem of ambrosetti prodi type on a large class of bad domain extending the solvability of this type of elliptic problem for the first time to a wide class of non smooth domains

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

fine regularity for elliptic and parabolic anisotropic robin problems with variable exponents abstract we investigate a class of quasi linear elliptic and parabolic anisotropic problems with variable exponents over a general class of bounded non smooth domains which may include non lipschitz domains such as domains with fractal boundary and rough domains we obtain solvability and global regularity results for both the elliptic and parabolic robin problem some a priori estimates as well as fine properties for the corresponding nonlinear semigroups are established as a consequence we generalize the global regularity theory for the robin problem over non smooth domains by extending it for the first time to the variable exponent case and furthermore to the anisotropic variable exponent case

the brownian cactus ii upcrossings and local times of super brownian motion we study properties of the random metric space called the brownian map for every r 0 we consider the connected components of the complement of the open ball of radius r centered at the root and we let mathbf n r varepsilon be the number of those connected components that intersect the complement of the ball of radius r varepsilon we then prove that varepsilon 3 mathbf n r varepsilon converges as varepsilon rightarrow 0 to a constant times the density at r of the profile of distances from the root in terms of the brownian cactus this gives asymptotics for the number of points at height r that have descendants at height r varepsilon our proofs are based on a similar approximation result for local times of super brownian motion by upcrossing numbers our arguments make a heavy use of the brownian snake and its special markov property

blow up and global existence of solutions to an inhomogeneous parabolic system abstract we study the critical phenomenon of a weakly coupled inhomogeneous parabolic system which is largely unexplored even though the homogeneous problem has been widely studied the conditions on the powers of the nonlinear terms are new compared with the well known homogeneous case the method appears to be different from the standard ones in the literature an application to the lame emden system is also given

on the range of lattice models in high dimensions we investigate the scaling limit of the range the set of visited vertices for a general class of critical lattice models starting from a single initial particle at the origin conditions are given on the random sets and an associated ancestral relation under which conditional on longterm survival the rescaled ranges converge weakly to the range of super brownian motion as random sets these hypotheses also give precise asymptotics for the limiting behaviour of the probability of exiting a large ball that is for the extrinsic one arm probability we show that these conditions are satisfied by the voter model in dimensions d ge 2 sufficiently spread out critical oriented percolation and critical contact processes in dimensions d 4 and sufficiently spread out critical lattice trees in dimensions d 8 the latter result proves conjecture 1 6 of van der hofstad et al ann probab 45 278 376 2017 and also has important consequences for the behaviour of random walks on lattice trees in high dimensions

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

quasi linear variable exponent boundary value problems with wentzell robin and wentzell boundary conditions let p c0 1 be such that 1 p p 1 p p let rn rn be a bounded w1 p w1 p extension domain and let be an upper d ahlfors measure supported on with d n p n d n p n we investigate the solvability of a class of quasi linear boundary value problems involving the p p laplace and p p laplace beltrami operators and either classical wentzell robin boundary conditions p ud u p 2 u dhn 1 u p 2ud 0on or general fully wentzell type boundary conditions of the form p ud p ud u p 2 u dhn 1 u p 2ud 0on where l d l d is such that infx x 0 for some constant 0 0 0 0 and u dhn 1 denotes the generalized p p normal derivative on in the interpretative sense we prove that the realization of the p p laplace operator with both of the above boundary conditions generate nonlinear ultracontractive submarkovian c0c0 semigroups on l2 dx l2 d l2 dx l2 d and hence their associated first order cauchy problems are both well posed on lq dx lq d lq dx lq d for all measurable function q with 1 q q 1 q q in addition we investigate the associated quasi linear elliptic problem with general wentzell boundary conditions and obtain existence uniqueness and global regularity of weak solutions to this equation

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

super brownian motion path properties and hitting probabilities sample path properties of super brownian motion including a one sided modulus of continuity and exact hausdorff measure function of the range and closed support are obtained analytic estimates for the probability of hitting balls lead to upper bounds on the hausdorff measure of the set of k multiple points and a sufficient condition for a set to be polar

path processes and historical superprocesses a superprocessx over a markov process can be obtained by a passage to the limit from a branching particle system for which describes the motion of individual particles the historical process hat xi for is the process whose state at timet is the path of over time interval 0 t the superprocess hat x over hat xi the historical superprocess over reflects not only the particle distribution at any fixed time but also the structure of family trees the principal property of a historical process hat xi is that hat xi s is a function of hat xi t for alls t every process with this property is calleda path process we develop a theory of superprocesses over path processes whose core is the integration with respect to measure functionals by applying this theory to historical superprocesses we construct the first hitting distributions and prove a special markov property for superprocesses

robin boundary value problems on arbitrary domains we develop a theory of generalised solutions for elliptic boundary value problems subject to robin boundary conditions on arbitrary domains which resembles in many ways that of the dirichlet problem in particular we establish lp lq estimates which turn out to be the best possible in that framework we also discuss consequences to the spectrum of robin boundary value problems finally we apply the theory to parabolic equations

local lipschitz continuity of the inverse of the fractional p laplacian holder type continuity and continuous dependence of solutions to associated parabolic equations on bounded domains let p 1 s 0 1 and rn an arbitrary bounded open set in the first part we consider the inverse s p p s 1 of the fractional p laplace operator p s with the dirichlet boundary condition we show that in the singular case p 1 2 the operator s p is locally lipschitz continuous on l and that global lipschitz continuity cannot be achieved we use this result to show that in the case n sp if 2nn 2s p 2 0 q npn sp 2 and are small constants then the nonlinear problem psu u qu u npn sp 2u hin u 0 on rn has at least one weak solution in the second part of the paper we prove that the operator p s generates a nonlinear submarkovian semigroup ss p t t 0 on l2 if p 2 and sp 0 such that for every t 0 and u v lq q 2 we have ss p t u ss p t v l c s t s u v lq s where s s and s are explicit constants depending only on n s p q

distance statistics in quadrangulations with a boundary or with a self avoiding loop we consider quadrangulations with a boundary and derive explicit expressions for the generating functions of these maps with either a marked vertex at a prescribed distance from the boundary or two boundary vertices at a prescribed mutual distance in the map for large maps this yields explicit formulae for the bulk boundary and boundary boundary correlators in the various encountered scaling regimes a small boundary a dense boundary and a critical boundary regime the critical boundary regime is characterized by a one parameter family of scaling functions interpolating between the brownian map and the brownian continuum random tree we discuss the cases of both generic and self avoiding boundaries which are shown to share the same universal scaling limit we finally address the question of the bulk loop distance statistics in the context of planar quadrangulations equipped with a self avoiding loop here again a new family of scaling functions describing critical loops is discovered

nonlocal robin laplacians and some remarks on a paper by filonov on eigenvalue inequalities abstract the aim of this paper is twofold first we characterize an essentially optimal class of boundary operators which give rise to self adjoint laplacians in l 2 d n x with nonlocal and local robin type boundary conditions on bounded lipschitz domains r n n n n 2 second we extend friedlander s inequalities between neumann and dirichlet laplacian eigenvalues to those between nonlocal robin and dirichlet laplacian eigenvalues associated with bounded lipschitz domains following an approach introduced by filonov for this type of problems

a probabilistic approach to one class of nonlinear differential equations we establish connections between positive solutions of one class of nonlinear partial differential equations and hitting probabilities and additive functionals of superdiffusion processes as an application we improve results on superprocesses by using the recent progress in the theory of removable singularities for differential equations

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

random planar lattices and integrated superbrownian excursion in this paper a surprising connection is described between a specific brand of random lattices namely planar quadrangulations and aldous integrated superbrownian excursion ise as a consequence the radius r n of a random quadrangulation with n faces is shown to converge up to scaling to the width r r l of the support of the one dimensional ise or precisely more generally the distribution of distances to a random vertex in a random quadrangulation is described in its scaled limit by the random measure ise shifted to set the minimum of its support in zero the first combinatorial ingredient is an encoding of quadrangulations by trees embedded in the positive half line reminiscent of cori and vauquelin s well labelled trees the second step relates these trees to embedded discrete trees in the sense of aldous via the conjugation of tree principle an analogue for trees of vervaat s construction of the brownian excursion from the bridge from probability theory we need a new result of independent interest the weak convergence of the encoding of a random embedded plane tree by two contour walks to the brownian snake description of ise our results suggest the existence of a continuum random map describing in term of ise the scaled limit of the dynamical triangulations considered in two dimensional pure quantum gravity

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

quasilinear parabolic and elliptic equations with nonlinear boundary conditions abstract this paper is concerned with a class of quasilinear parabolic and elliptic equations in a bounded domain with both dirichlet and nonlinear neumann boundary conditions the equation under consideration may be degenerate or singular depending on the property of the diffusion coefficient the consideration of the class of equations is motivated by some heat transfer problems where the heat capacity and thermal conductivity are both temperature dependent the aim of the paper is to show the existence and uniqueness of a global time dependent solution of the parabolic problem existence of maximal and minimal steady state solutions of the elliptic problem including conditions for the uniqueness of a solution and the asymptotic behavior of the time dependent solution in relation to the steady state solutions applications are given to some heat transfer problems and an extended logistic reaction diffusion equation

the structure of unicellular maps and a connection between maps of positive genus and planar labelled trees a unicellular map is a map which has only one face we give a bijection between a dominant subset of rooted unicellular maps of given genus and a set of rooted plane trees with distinguished vertices the bijection applies as well to the case of labelled unicellular maps which are related to all rooted maps by marcus and schaeffer s bijection this gives an immediate derivation of the asymptotic number of unicellular maps of given genus and a simple bijective proof of a formula of lehman and walsh on the number of triangulations with one vertex from the labelled case we deduce an expression of the asymptotic number of maps of genus g with n edges involving the ise random measure and an explicit characterization of the limiting profile and radius of random bipartite quadrangulations of genus g in terms of the ise

a class of quasi linear parabolic and elliptic equations with nonlocal robin boundary conditions abstract let p 1 n r n a bounded w 1 p extension domain and let be an upper d ahlfors measure on with d n p n we show in the first part that for every p 2 n n 2 n 1 n a realization of the p laplace operator with nonlinear generalized nonlocal robin boundary conditions generates a nonlinear strongly continuous submarkovian semigroup on l 2 and hence the associated first order cauchy problem is well posed on l q for every q 1 in the second part we investigate existence uniqueness and regularity of weak solutions to the associated quasi linear elliptic equation more precisely global a priori estimates of weak solutions are obtained

the range of tree indexed random walk in low dimensions we study the range rn of a random walk on the d dimensional lattice z d indexed by a random tree with n vertices under the assumption that the random walk is centered and has fi nite fourth moments we prove in dimension d 3 that n d 4 rn converges in distribution to the lebesgue measure of the support of the integrated super brownian excursion ise an auxiliary result shows that the suitably rescaled local times of the tree indexed ran dom walk converge in distribution to the density process of ise we obtain similar results for the range of critical branching random walk in z d d 3 as an intermediate estimate we get exact asymptotics for the probability that a critical branching random walk starting with a single particle at the origin hits a distant point the results of the present article complement those derived in higher dimensions in our earlier work 1 introduction in the present paper we continue our study of asymptotics for the num ber of distinct sites of the lattice visited by a tree indexed random walk we consider discrete plane trees which are rooted ordered trees that can be viewed as describing the genealogy of a population starting with one ancestor or root which is denoted by the symbol given such a treet and a probability measure on z d we can consider the random walk with jump distribu tion indexed by the treet this means that we assign a random spatial location zt u 2 z d to every vertex u oft in the following way first the spatial location zt of the root is the origin of z d then we assign independently to every edge e of the tree t a random variable ye distributed according to and we let the spatial location zt u of the vertex u be the sum of the quantities ye over all edges e belonging to the simple path from to u in the tree the number of distinct spatial locations is called the range of the tree indexed random walk zt in our previous work 13 we stated the following result let be a probability distribution on z d which is symmetric with finite support and is not supported on a strict subgroup of z d and for every integer n 1 lett n be a random tree uniformly distributed over all plane trees with n vertices conditionally given t n let zt n be a random walk with jump distribution indexed byt n and let rn stand for the range of zt n then

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

combinatorics of bicubic maps with hard particles we present a purely combinatorial solution of the problem of enumerating planar bicubic maps with hard particles this is done by the use of a bijection with a particular class of blossom trees with particles obtained by an appropriate cutting of the maps although these trees have no simple local characterization we prove that their enumeration may be performed upon introducing a larger class of admissible trees with possibly doubly occupied edges and summing them with appropriate signed weights the proof relies on an extension of the cutting procedure allowing for the presence on the maps of special non sectile edges the admissible trees are characterized by simple local rules allowing eventually for an exact enumeration of planar bicubic maps with hard particles we also discuss generalizations for maps with particles subject to more general exclusion rules and show how to re derive the enumeration of quartic maps with ising spins in the present framework of admissible trees we finally comment on a possible interpretation in terms of branching processes

the number and size of branched polymers in high dimensions we consider two models of branched polymers lattice trees on thed dimensional hypercubic lattice i the nearest neighbor model in sufficiently high dimensions and ii a spread out or long range model ford 8 in which trees are constructed from bonds of length less than or equal to a large parameterl we prove that for either model the critical exponent for the number of branched polymers exists and equals 5 2 and that the critical exponentv for the radius of gyration exists and equals 1 4 this improves our earlier results for the corresponding generating functions the proof uses the lace expansion together with an analysis involving fractional derivatives which has been applied previously to the self avoiding walk in a similar context

the heat equation with nonlinear generalized robin boundary conditions abstract let r n be a bounded domain and let be an admissible measure on we show in the first part that if has the h 1 extension property then a realization of the laplace operator with generalized nonlinear robin boundary conditions formally given by u d x u d 0 on generates a strongly continuous nonlinear submarkovian semigroup s b s b t t 0 on l 2 we also obtain that this semigroup is ultracontractive in the sense that for every u v l p p 2 and every t 0 one has s b t u s b t v c 1 e c 2 t t n 2 p u v p for some constants c 1 c 2 0 in the second part we prove that if is a bounded lipschitz domain one can also define a realization of the laplacian with nonlinear robin boundary conditions on c and this operator generates a strongly continuous and contractive nonlinear semigroup

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