rescaled contact processes converge to super brownian motion in two or more dimensions we show that in dimensions two or more a sequence of long range contact processes suitably rescaled in space and time converges to a super brownian motion with drift as a consequence of this result we can improve the results of bramson durrett and swindle 1989 by replacing their order of magnitude estimates of how close the critical value is to 1 with sharp asymptotics

on the tail of the branching random walk local time consider a critical branching random walk on mathbb z d d ge 1 started with a single particle at the origin and let l x be the total number of particles that ever visit a vertex x we study the tail of l x under suitable conditions on the offspring distribution in particular our results hold if the offspring distribution has an exponential moment

a nonstandard construction of levy brownian motion a nonstandard construction of levy brownian motion on d is presented which extends r m anderson s nonstandard representation of brownian motion it involves a nonstandard construction of white noise and gives as a classical corollary a new white noise integral representation of levy brownian motion moreover a new invariance principle can be deduced in a similar way as donsker s invariance principles follows from anderson s construction

the survival probability and r point functions in high dimensions in this paper we investigate the survival probability n in high dimensional statistical physical models where n denotes the probability that the model survives up to time n we prove that if the r point functions scale to those of the canonical measure of super brownian motion and if certain self repellence and total population tail bound conditions are satised then n n 2 av where a is the asymptotic expected number of particles alive at timen andv is the vertex factor of the model our results apply to spread out lattice trees above 8 dimensions spread out oriented percolation above 4 1 dimensions and the spread out contact process above 4 1 dimensions in the case of oriented percolation this reproves a result by the rst author den hollander and slade which was proved using heavy lace expansion arguments at the cost of losing explicit error estimates we further derive several consequences of our result involving the scaling limit of the number of particles alive at time proportional to n our proofs are based on simple weak convergence arguments

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

rescaled lotka volterra models converge to super brownian motion we show that a sequence of stochastic spatial lotka volterra models suitably rescaled in space and time converges weakly to super brownian motion with drift the result includes both long range and nearest neighbor models the latter for dimensions three and above these theorems are special cases of a general convergence theorem for perturbations of the voter model

singularity of super brownian local time at a point catalyst in a one dimensional single point catalytic continuous super brownian motion studied by dawson and fleischmann the occupation density measure c at the catalyst s position c is shown to be a singular diffuse random measure the source of this qualitative new effect is the irregularity of the varying medium c describing the point catalyst the proof is based on a probabilistic characterization of the law of the palm canonical clusters x appearing in the levy khintchine representation of c in a historical process setting and the fact that these x have infinite left upper density with respect to lebesgue measure at the palm time point

super brownian motion with reflecting historical paths we consider super brownian motion whose historical paths reflect from each other unlike those of the usual historical super brownian motion we prove tightness for the family of distributions corresponding to a sequence of discrete approximations but we leave the problem of uniqueness of the limit open we prove a few results about path behavior for processes under any limit distribution in particular we show that for any 0 a typical increment of a reflecting historical path over a small time interval t is not greater than t 3 4

on the maximal displacement of subcritical branching random walks we study the maximal displacement of a one dimensional subcritical branching random walk initiated by a single particle at the origin for each n in mathbb n let m n be the rightmost position reached by the branching random walk up to generation n under the assumption that the offspring distribution has a finite third moment and the jump distribution has mean zero and a finite probability generating function we show that there exists rho 1 such that the function begin aligned g c n rho cn p m n ge cn quad hbox for each c 0 hbox and n in mathbb n end aligned satisfies the following properties there exist 0 underline delta le overline delta infty such that if c underline delta then begin aligned 0 liminf n rightarrow infty g c n le limsup n rightarrow infty g c n le 1 end aligned while if c overline delta then begin aligned lim n rightarrow infty g c n 0 end aligned moreover if the jump distribution has a finite right range r then overline delta r if furthermore the jump distribution is nearly right continuous then there exists kappa in 0 1 such that lim n rightarrow infty g c n kappa for all c underline delta we also show that the tail distribution of m sup n ge 0 m n namely the rightmost position ever reached by the branching random walk has a similar exponential decay without the cutoff at underline delta finally by duality these results imply that the maximal displacement of supercritical branching random walks conditional on extinction has a similar tail behavior

hyperfinite methods applied to the critical branching diffusion we apply the hyperfinite methods of re to the construction of a version of the critical branching diffusion studied by dawson et al several new sample path properties are derived from this construction

construction of immigration superprocesses with dependent spatial motion from one dimensional excursions a superprocess with dependent spatial motion and interactive immigration is constructed as the pathwise unique solution of a stochastic integral equation carried by a stochastic flow and driven by poisson processes of one dimensional excursions

large time behavior of interface solutions to the heat equation with fisher wright white noise the one dimensional heat equation driven by fisher wright white noise is studied from initial conditions with compact support solutions retain this compact support and die out in finite time there exist interface solutions which change from the value 1 to the value 0 in a finite region the motion of the interface location is shown to approach that of a brownian motion under rescaling solutions with a finite number of interfaces are approximated by a system of annihilating brownian motions

one dimensional stochastic partial differential equations and the branching measure diffusion an existence theorem in the spirit of keisler ke is proved for the simple one dimensional diffusion equation driven by white noise modulated by a non linear function of the solution this is used to obtain a density and a stochastic partial differential equation in one dimension for the critical branching diffusion studied by dawson et al

spatial epidemics and local times for critical branching random walks in dimensions 2 and 3 the behavior at criticality of spatial sir epidemic models in dimensions two and three is investigated in these models finite populations of size n are situated at the sites of the integer lattice and infectious contacts are limited to individuals at the same or at neighboring sites susceptible individuals once infected remain contagious for one unit of time and then recover after which they are immune to further infection it is shown that the measure valued processes associated with these epidemics suitably scaled converge in the large n limit either to a standard dawson watanabe process super brownian motion or to a dawson watanabe process with location dependent killing depending on the size of the the initially infected set a key element of the argument is a proof of adler s 1993 conjecture that the local time processes associated with branching random walks converge to the local time density process associated with the limiting super brownian motion

the structure of a random graph at the point of the phase transition consider the random graph models g n edges m and g n prob edge p with m m n 1 n 1 3 n 2 and p p n 1 an 1 3 n for 1 1 define an i component of a random graph as a component which has exactly i more edges than vertices call an i component with i 1 a complex component for both models we show that when a is constant the expected number of complex components is bounded almost surely a s each of these components if any exist has size of order n2 3 and the maximum value of i is bounded in probability we prove that a s the largest suspended tree in each complex component has size of order n2 3 and deletion of all suspended trees results in a smoothed graph of size of order n 3 with the maximum vertex degree 3 the total number of branching vertices i e of degree 3 is bounded in probability thus each complex component is almost surely topologically equivalent to a 3 regular multigraph of a uniformly bounded size lengths of the shortest cycle and of the shortest path between two branching vertices of a smoothed graph are each of order n 3 we find a relatively simple integral formula for the limit distribution of the numbers of complex components which implies in particular that all values of the complexity spectrum have positive limiting probabilities we also answer questions raised by erdos and renyi back in 1960 it is proven that there exists p a the limiting planarity probability with 0 p a 1 p oo 1 p oo 0 in particular g n m g n p resp is almost surely nonplanar iff m n 2 n 213 00 np i n 1 3 oo resp

using angular dispersion of gradient direction for detecting edge ribbons edges in a scene generally project to smooth continuous curves nearly everywhere in the image which results in low angular deviation of the intensity gradients in small neighborhoods straddling edges angular deviation is shown to be a measure of snr a theoretical analysis of angular deviation arising due to independent and identically distributed n 0 sigma sup 2 random noise is presented angular deviation thresholds for neighborhood sizes from 3 3 to 11 11 pixels are determined both from this analysis and numerical examples the proposed gradient angular dispersion detection algorithm detects edge elements by comparing the measured angular deviation with values computed for the minimum acceptable snr low values of deviation violate the no edge hypothesis the algorithm is shown to make good use of the limited dynamic range of the imaging system the sensitivity and selectivity of the strategy are both shown to be high

infinite canonical super brownian motion and scaling limits we construct a measure valued markov process which we call infinite canonical super brownian motion and which corresponds to the canonical measure of super brownian motion conditioned on non extinction infinite canonical super brownian motion is a natural candidate for the scaling limit of various random branching objects on mathbb z d when these objects are critical mean field and infinite we prove that icsbm is the scaling limit of the spread out oriented percolation incipient infinite cluster above 4 dimensions and of incipient infinite branching random walk in any dimension we conjecture that it also arises as the scaling limit in various other models above the upper critical dimension such as the incipient infinite lattice tree above 8 dimensions the incipient infinite cluster for unoriented percolation above 6 dimensions uniform spanning trees above 4 dimensions and invasion percolation above 6 dimensions

convergence of the viscosity method for isentropic gas dynamics a convergence theorem for the method of artificial viscosity applied to the isentropic equations of gas dynamics is established convergence of a subsequence in the strong topology is proved without uniform estimates on the derivatives using the theory of compensated compactness and an analysis of progressing entropy waves

super brownian motion as the unique strong solution to an spde a stochastic partial differential equation spde is derived for super brownian motion regarded as a distribution function valued process the strong uniqueness for the solution to this spde is obtained by an extended yamada watanabe argument similar results are also proved for the fleming viot process

a phase transition for measure valued sir epidemic processes we consider measure valued processes xt that solve the following martingale prob lem for a given initial measure x0 and for all smooth compactly supported test functions xt x0 1 z t 2 ds such processes arise as scaling limits of sir epidemic models we show that there exist critical values c d 0 for dimensions d 2 3 such that if c d then the solution survives forever with positive probability but if c d then the solution dies out in finite time with probability 1 for d 1 we prove that the solution dies out almost surely for all values of we also show that in dimensions d 2 3 the process dies out locally almost surely for any value of that is for any compact set k the process xt k 0 eventually

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

zero relaxation time limits to a hydrodynamic model of two carrier types for semiconductors in this paper we study the zero relaxation time limits to a one dimensional hydrodynamic model of two carrier types for semiconductors first we introduce the flux approximation coupled with the classical viscosity method to obtain the uniform l loc p p ge 1 bound of the approximation solutions rho i varepsilon delta and other estimates of u i varepsilon delta e varepsilon delta with the help of the high energy estimates jungel and peng comm partial differ equ 58 1007 1033 1999 then we apply the compensated compactness method coupled with the scaled variables technique marcati and natalini arch ration mech anal 129 129 145 1995 to prove the zero relaxation time limits with arbitrarily large initial data and arbitrary adiabatic exponents gamma i 1

existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in eulerian and lagrangian coordinates we prove the existence and compactness stability of entropy solutions for the hyperbolic systems of conservation laws corresponding to the isentropic gas dynamics where the pressure and density are related by a law for any 1 our results considerably extend and simplify the program initiated by diperna and provide a complete existence proof our methods are based on the compensated compactness and the kinetic formulation of systems of conservation laws 1996 john wiley sons inc

optimal local holder index for density states of superprocesses with 1 β branching mechanism for 0 0 if d 1 but locally unbounded otherwise moreover in the case of continuity we determine the optimal local holder index

a phase diagram for a stochastic reaction diffusion system in this paper a stochastic reaction diffusion system is considered which models the spread of a finite population reacting with a non renewable resource in the presence of individual based noise a two parameter phase diagram is established to describe the large time evolution distinguishing between certain death or possible life of the population

mutually catalytic branching in the plane finite measure states we study a pair of populations in r 2 which undergo diffusion and branching the system is interactive in that the branching rate of each type is proportional to the local density of the other type for a diffusion rate sufficiently large compared with the branching rate the model is constructed as the unique pair of finite measure valued processes which satisfy a martingale problem involving the collision local time of the solutions the processes are shown to have densities at fixed times which live on disjoint sets and explode as they approach the interface of the two populations in the long term limit global extinction of one type is shown the process constructed is a rescaled limit of the corresponding z 2 lattice model studied by d a dawson and e a perkins ann probab 26 1998 1088 1138 and resolves the large scale mass time space behavior of that model

precision edge contrast and orientation estimation the contrast and orientation estimation accuracy of several edge operators that have been proposed in the literature is examined both for the noiseless case and in the presence of additive gaussian noise the test image is an ideal step edge that has been sampled with a square aperture grid the effects of subpixel translations and rotations of the edge on the performance of the operators are studied it is shown that the effect of subpixel translations of an edge can generate more error than moderate noise levels methods with improved results are presented for sobel angle estimates and the nevatia babu operator and theoretical noise performance evaluations are also provided an edge operator based on two dimensional spatial moments is presented all methods are compared according to worst case and rms error in an ideal noiseless situation and rms error under various noise levels

a distribution function valued spde and its applications abstract in this paper we further study the stochastic partial differential equation first proposed by xiong 22 under localized conditions on its coefficients we prove a comparison theorem on its solutions and show that the solution is in fact distribution function valued we also establish pathwise uniqueness of the solution as applications we obtain the well posedness of martingale problems for two classes of measure valued diffusions interacting super brownian motions and interacting fleming viot processes properties of the two superprocesses such as the existence of density fields and the survival extinction behaviors are also studied

weak solutions to a hydrodynamic model for semiconductors and relaxation to the drift diffusion equation we investigate the relaxation problem for the hydrodynamic isentropic euler poisson system when the momentum relaxation time tends to zero very sharp estimates on the solutions independent of the relaxation time are obtained and used to establish compactness

robust and efficient image alignment based on relative gradient matching in this paper we present a robust image alignment algorithm based on matching of relative gradient maps this algorithm consists of two stages namely a learning based approximate pattern search and an iterative energy minimization procedure for matching relative image gradient the first stage finds some candidate poses of the pattern from the image through a fast nearest neighbor search of the best match of the relative gradient features computed from training database of feature vectors which are obtained from the synthesis of the geometrically transformed template image with the transformation parameters uniformly sampled from a given transformation parameter space subsequently the candidate poses are further verified and refined by matching the relative gradient images through an iterative energy minimization procedure this approach based on the matching of relative gradients is robust against nonuniform illumination variations experimental results on both simulated and real images are shown to demonstrate superior efficiency and robustness of the proposed algorithm over the conventional normalized correlation method

robust fft based scale invariant image registration with image gradients we present a robust fft based approach to scale invariant image registration our method relies on fft based correlation twice once in the log polar fourier domain to estimate the scaling and rotation and once in the spatial domain to recover the residual translation previous methods based on the same principles are not robust to equip our scheme with robustness and accuracy we introduce modifications which tailor the method to the nature of images first we derive efficient log polar fourier representations by replacing image functions with complex gray level edge maps we show that this representation both captures the structure of salient image features and circumvents problems related to the low pass nature of images interpolation errors border effects and aliasing second to recover the unknown parameters we introduce the normalized gradient correlation we show that using image gradients to perform correlation the errors induced by outliers are mapped to a uniform distribution for which our normalized gradient correlation features robust performance exhaustive experimentation with real images showed that unlike any other fourier based correlation techniques the proposed method was able to estimate translations arbitrary rotations and scale factors up to 6

critical points for spread out self avoiding walk percolation and the contact process above the upper critical dimensions we consider self avoiding walk and percolation in d oriented percolation in d and the contact process in d with pd being the coupling function whose range is proportional to l for percolation for example each bond is independently occupied with probability pd y x the above models are known to exhibit a phase transition when the parameter p varies around a model dependent critical point pc we investigate the value of pc when d 6 for percolation and d 4 for the other models and l 1 we prove in a unified way that pc 1 c d o l 2d where the universal term 1 is the mean field critical value and the model dependent term c d o l d is written explicitly in terms of the random walk transition probability d we also use this result to prove that pc 1 cl d o l d 1 where c is a model dependent constant our proof is based on the lace expansion for each of these models

holder continuity of the solutions for a class of nonlinear spde s arising from one dimensional superprocesses the holder continuity of the solution x t x to a nonlinear stochastic partial differential equation see 1 2 below arising from one dimensional superprocesses is obtained it is proved that the holder exponent in time variable is arbitrarily close to 1 4 improving the result of 1 10 in li et al to appear on probab theory relat fields the method is to use the malliavin calculus the holder continuity in spatial variable x of exponent 1 2 is also obtained by using this new approach this holder continuity result is sharp since the corresponding linear heat equation has the same holder continuity

gradient orientation pattern matching with the hamming distance abstract this paper presents a novel pattern matching technique that is robust to illumination changes and the occlusion problem the technique is based on the matching of gradient orientations in place of traditional image features such as intensities or gradients gradient orientations depend on the texture in an image they are known to be insensitive to changes of image intensities that are often caused by time varying illuminations or the auto gain control agc function of the camera moreover the proposed method employs a voting strategy in the process of matching gradient orientations the method works remarkably well even when a large part of the pattern is occluded with a foreign object consequently the proposed method is robust to both irregular lighting conditions and the occlusion problem

boundary preserving semianalytic numerical algorithms for stochastic differential equations construction of splitting step methods and properties of related nonnegativity and boundary preserving semianalytic numerical algorithms for solving stochastic differential equations sdes of ito type are discussed as the crucial assumption we oppose conditions such that one can decompose the original system of sdes into subsystems for which one knows either the exact solution or its conditional transition probability we present convergence proofs for a newly designed splitting step algorithm and simulation studies for numerous well known numerical examples ranging from stochastic dynamics occurring in asset pricing in mathematical finance cox ingersoll ross cir and constant elasticity of variance cev models to measure valued diffusion and super brownian motion stochastic pdes spdes as met in biology and physics

spatial epidemics critical behavior in one dimension in the simple mean field sis and sir epidemic models infection is transmitted from infectious to susceptible members of a finite population by independent p coin tosses spatial variants of these models are considered in which finite populations of size n are situated at the sites of a lattice and infectious contacts are limited to individuals at neighboring sites scaling laws for these models are given when the infection parameter p is such that the epidemics are critical it is shown that in all cases there is a critical threshold for the numbers initially infected below the threshold the epidemic evolves in essentially the same manner as its branching envelope but at the threshold evolves like a branching process with a size dependent drift the corresponding scaling limits are super brownian motions and dawson watanabe processes with killing respectively

weak invariance principles for local time we establish sufficient conditions to ensure that the suitably defined local times of a sequence of random walks converge weakly to brownian local time

on the filtration of historical brownian motion we show that the historical brownian motion may be recovered from ordinary super brownian motion when the dimension d of the underlying brownian motion is greater than 4 we outline a proof showing that this conclusion is false if d 3 the state of affairs in the critical dimension d 4 is left unresolved some extensions are given for 1 stable branching mechanisms where 0 1

a phase transition for a stochastic pde related to the contact process we consider the one dimensional heat equation with a semilinear term and with a nonlinear white noise term r durrett conjectured that this equation arises as a weak limit of the contact process with longrange interactions we show that our equation possesses a phase transition to be more precise we assume that the initial function is nonnegative with bounded total mass if a certain parameter in the equation is small enough then the solution dies out to 0 in finite time with probability 1 if this parameter is large enough then the solution has a positive probability of never dying out to 0 this result answers a question of durett

explicit stochastic integral representations for historical functionals it is known from previous work of the authors that any square integrable functional of a superprocess may be represented as a constant plus a stochastic integral against the associated orthogonal martingale measure here we give for a large class of such functionals an explicit description of the integrand that is analogous to clark s formula for the representation of certain brownian functionals as a consequence we develop a partial analogue of the wiener chaos expansion in the superprocess setting rather than work with superprocesses per se our results are stated and proved in the richer and more natural context of the associated historical process

kinetic formulation of the isentropic gas dynamics and p systems we consider the 2 x 2 hyperbolic system of isentropic gas dynamics in both eulerian or lagrangian variables also called the p system we show that they can be reformulated as a kinetic equation using an additional kinetic variable such a formulation was first obtained by the authors in the case of multidimensio nal scalar conservation laws a new phenomenon occurs here namely that the advection velocity is now a combination of the macroscopic and kinetic velocities various applications are given we recover the invariant regions deduce new l estimates using moments lemma and prove l w stability for 7 3

survival and coexistence in stochastic spatial lotka volterra models a spatially explicit stochastic lotka volterra model was introduced by neuhauser and pacala in neuhauser and pacala ann appl probab 9 1226 1259 1999 a low density limit theorem for this process was proved by the authors in cox and perkins ann probab 33 904 947 2005 showing that certain generalized rescaled lotka volterra models converge to super brownian motion with drift here we use this convergence result to extend what is known about the parameter regions for the lotka volterra process where i survival of one type holds and ii coexistence holds

joint continuity of the solutions to a class of nonlinear spdes for a one dimensional superprocess in random environment a nonlinear spde was derived by dawson et al ann inst henri poincare probab stat 36 2 167 180 2000 for its density process the time space joint continuity of the density process was left as an open problem in this paper we give an affirmative answer to this problem

super brownian motion with reflecting historical paths ii convergence of approximations we prove that the sequence of finite reflecting branching brownian motion forests defined by burdzy and le gall 1 converges in probability to the super brownian motion with reflecting historical paths this solves an open problem posed in 1 where only tightness was proved for the sequence of approximations several results on path behavior were proved in 1 for all subsequential limits they obviously hold for the unique limit found in the present paper

brownian excursions critical random graphs and the multiplicative coalescent let b t s 0 s be reflecting inhomogeneous brownian motion with drift t s at time s started with b t 0 0 consider the random graph script g sign n n 1 tn 4 3 whose largest components have size of order n 2 3 normalizing by n 2 3 the asymptotic joint distribution of component sizes is the same as the joint distribution of excursion lengths of b t corollary 2 the dynamics of merging of components as t increases are abstracted to define the multiplicative coalescent process the states of this process are vectors x of nonnegative real cluster sizes x i and clusters with sizes x i and x j merge at rate x i x j the multiplicative coalescent is shown to be a feller process on l 2 the random graph limit specifies the standard multiplicative coalescent which starts from infinitesimally small clusters at time the existence of such a process is not obvious

regularity and irregularity of bolds 1 beta stable super brownian motion this paper establishes the continuity of the density of 1 stable super brownian motion 0 1 for fixed times in d 1 and local unboundedness of the density in all higher dimensions where it exists we also prove local unboundedness of the density in time for a fixed spatial parameter in any dimension where the density exists and local unboundedness of the occupation density the local time in the spatial parameter for dimensions d 2 where the local time exists

the dimension of the boundary of super brownian motion we show that the hausdorff dimension of the boundary of d dimensional super brownian motion is 0 if d 1 4 2 sqrt 2 if d 2 and 9 sqrt 17 2 if d 3

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

improved upper bounds for the contact process critical value the best known upper bound for the critical value c of the basic one dimensional contact process is 2 most techniques for finding bounds on critical values have the property that they can be modified in order to obtain improved bounds this seemed not to be the case for the approach which yielded c 2 for the basic contact process in this paper we propose a technique for generating better bounds in this context to illustrate its use we carry out the full program in one case with the conclusion that c 1 942

nonstandard construction of the stochastic integral and applications to stochastic differential equations ii h j keisler has recently used a nonstandard theory of lto integration due to r m anderson to construct solutions of ito integral equations by solving an associated internal difference equation in this paper we use the same general approach to find solutions y t of semimartingale integral equations of the form y t c h t c f s y w dz s where z is a given semimartingale h is a right continuous process and f s w is continuous on the space of right continuous functions with left limits with the topology of uniform convergence on compacts in addition we generalize keisler s continuity theorem and give necessary and sufficient conditions for an internal martingale to be s continuous introduction this paper is a continuation of hoover and perkins 5 notation and definitions presented in 5 are used without further introduction and references to that paper are made by simply citing the number of the theorem or lemma due to the consecutive numbering of the two papers no ambiguity will arise finally the reader is referred to 0 of 5 for a brief description of the contents of this work 8 continuous local martingales we assume throughout this section that all processes are r or r valued since the main results theorems 8 1 8 5 and 8 6 then follow immediately for higher dimensions we will show theorem 8 1 that a continuous local martingale has a q1 local martingale lifting for any internal filtration t recall that in general theorem 5 6 asserted only that such a lifting exists for some internal filtration for this reason unless otherwise stated the internal filtration under consideration will be it i t e t which was used in the original definition of t note that if x t x q m is an s continuous process i e x w is s continuous a s then st x has continuous paths a s and x is a uniform lifting of st x in the sense of keisler 6 the following lifting theorem was first proved in a more specialized setting in panetta 9 the proof given here is different received by the editors january 22 1982 the results of this paper were presented at the second victoria symposium on nonstandard analysis victoria canada june 16 20 1980 a ms mos subject cllassifications 1970 primary 60h20 02h25 secondary 60g45

a new approach to the single point catalytic super brownian motion a new approach is provided to the super brownian motionx with a single point catalyst c as branching rate we start from a superprocessu with constant branching rate and spatial motion given by the 1 2 stable subordinator we prove that the occupation density measure c ofx at the catalystc is distributed as the total occupation time measure ofu furthermore we show thatxt is determined from c by an explicit representation formula heuristically a mass c ds of particles leaves the catalyst at times and then evolves according to ito s brownian excursion measure as a consequence of our representation formula the density fieldx ofx satisfies the heat equation outside ofc with a noisy boundary condition atc given by the singularly continuous random measure c in particular x isc outside the catalyst we also provide a new derivation of the singularity of the measure c

conceptual proofs of l log l criteria for mean behavior of branching processes the kesten stigum theorem is a fundamental criterion for the rate of growth of a supercritical branching process showing that an l log l condition is decisive in critical and subcritical cases results of kolmogorov and later authors give the rate of decay of the probability that the process survives at least n generations we give conceptual proofs of these theorems based on comparisons of galton watson measure to another measure on the space of trees this approach also explains yaglom s exponential limit law for conditioned critical branching processes via a simple characterization of the exponential distribution

occupation statistics of critical branching random walks in two or higher dimensions consider a critical nearest neighbor branching random walk on the d dimensional integer lattice initiated by a single particle at the origin let g n be the event that the branching random walk survives to generation n we obtain the following limit theorems conditional on the event g n for a variety of occupation statistics 1 let v n be the maximal number of particles at a single site at time n if the offspring distribution has finite th moment for some integer a 2 then in dimensions 3 and higher v n o p n 1 if the offspring distribution has an exponentially decaying tail then v n o p log n in dimensions 3 and higher and v n o p log n 2 in dimension 2 furthermore if the offspring distribution is nondegenerate then p vn log n g n 1 for some 0 2 let m n j be the number of multiplicity j sites in the nth generation that is sites occupied by exactly j particles in dimensions 3 and higher the random variables m n j n converge jointly to multiples of an exponential random variable 3 in dimension 2 the number of particles at a typical site i e at the location of a randomly chosen particle of the nth generation is of order o p log n and the number of occupied sites is o p n log n we also show that in dimension 2 there is particle clustering around a typical site

the bipolar hydrodynamic model for semiconductors and the drift diffusion equations we establish the existence of entropy solutions for a bipolar hydrodynamic model for semiconductors we show also that the limit as the relaxation time goes to zero of an appropriate scaled sequence of entropy solutions is a solution of the classical drift diffusion equations

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

hybrid image matching combining hausdorff distance with normalized gradient matching image matching has been a central problem in computer vision and image processing for decades most of the previous approaches to image matching can be categorized into the intensity based and edge based comparison hausdorff distance has been widely used for comparing point sets or edge maps since it does not require point correspondences in this paper we propose a new image similarity measure combining the hausdorff distance with a normalized gradient consistency score for image matching the normalized gradient consistency score is designed to compare the normalized image gradient fields between two images to alleviate the illumination variation problem in image matching by combining the edge based and intensity based information for image matching we are able to achieve robust image matching under different lighting conditions we show the superior robustness property of the proposed image matching technique through experiments on face recognition under different lighting conditions