intrinsic ultracontractivity and small perturbation for one dimensional generalized diffusion operators abstract we consider a one dimensional generalized diffusion operator g on an open interval which is not necessarily bounded we characterize sp i e 1 is a small perturbation of the operator g in terms of the classification of the boundary points from the view point of diffusion processes and give a sufficient condition for iu i e the associated heat kernel is intrinsically ultracontractive

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

intrinsic ultracontractivity and eigenfunction estimates for schrodinger operators let d be a domain in rn n 2 and let h h0 v where h0 is a divergence form uniformly elliptic operator with dirichlet boundary conditions and v is in the kato class intrinsic ultracontractivity is proved for the semigroup of h when d is holder domain of order 0 or a uniformly holder domain of order for 0 2 for every 2 there exists a uniformly holder domain of order for which the dirichlet laplacian is not intrinsically ultracontractive for a large class of domains it is shown that the heat kernels for h0 and h decay at the same rate at the boundary applications are given to the lifetime of conditioned brownian motion some of our results seem to be new even for smooth domains

mean field behavior for the survival probability and the percolation point to surface connectivity we consider the critical survival probability up to time t for oriented percolation and the contact process and the point to surface of the ball of radius t connectivity for critical percolation let t denote both quantities we prove in a unifled fashion that if t exhibits a power law and both the two point function and its certain restricted version exhibit the same mean fleld behavior then t t i1 for the time oriented models with d 4 and t t i2 for percolation with d 7

the survival probability for critical spread out oriented percolation above 4 1 dimensions i induction we consider critical spread out oriented percolation above 4 1 dimensions our main result is that the extinction probability at time n i e the probability for the origin to be connected to the hyperplane at time n but not to the hyperplane at time n 1 decays like 1 bn 2 as n to infty where b is a finite positive constant this in turn implies that the survival probability at time n i e the probability that the origin is connected to the hyperplane at time n decays like 1 bn as n to infty the latter has been shown in an earlier paper to have consequences for the geometry of large critical clusters and for the incipient infinite cluster the present paper is part i in a series of two papers in part ii we derive a lace expansion for the survival probability adapted so as to deal with point to plane connections this lace expansion leads to a nonlinear recursion relation for the survival probability in part i we use this recursion relation to deduce the asymptotics via induction

dirichlet conditions at infinity for parabolic and elliptic equations we investigate existence and uniqueness of bounded solutions of parabolic equations with unbounded coefficients in rn r under specific assumptions we establish existence of solutions satisfying prescribed conditions at infinity depending on the direction along which infinity is approached moreover the large time behavior of such solutions is addressed we consider also elliptic equations in rn with similar conditions at infinity

tree graph inequalities and critical behavior in percolation models various inequalities are derived and used for the study of the critical behavior in independent percolation models in particular we consider the critical exponent associated with the expected cluster sizex and the structure of then site connection probabilities n x1 xn it is shown that quite generally 1 the upper critical dimension above which attains the bethe lattice value 1 is characterized both in terms of the geometry of incipient clusters and a diagramatic convergence condition for homogeneousd dimensional lattices with x y o x y d 2 atp p c our criterion shows that 1 if 6 d 3 the connectivity functions n are generally bounded by tree diagrams which involve the two point function we conjecture that above the critical dimension the asymptotic behavior of n in the critical regime is actually given by such tree diagrams modified by a nonsingular vertex factor other results deal with the exponential decay of the cluster size distribution and the function 2 x y

hyperscaling inequalities for the contact process and oriented percolation the contact process and oriented percolation are expected to exhibit the same critical behavior in any dimension above their upper critical dimension dc they exhibit the same critical behavior as the branching process assuming existence of the critical exponents we prove a pair of hyperscaling inequalities which together with the results of refs 16 and 18 implies dc 4

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

intrinsic ultracontractivity of a schrodinger semigroup in rn abstract we give a possibly sharp sufficient condition on the electric potential q r n 0 in the schrodinger operator a q x on l 2 r n that guarantees that the schrodinger heat semigroup e a t t 0 on l 2 r n generated by a is intrinsically ultracontractive moreover if q x q x is radially symmetric we show that our condition on q is also necessary i e truly sharp it reads r 0 q r 1 2 d r for some r 0 0 our proofs make essential use of techniques based on a logarithmic sobolev inequality rosen s inequality proved via a new fenchel young inequality and a very precise asymptotic formula due to hartman and wintner

inference on counterfactual distributions counterfactual distributions are important ingredients for policy analysis and decomposition analysis in empirical economics in this article we develop modeling and inference tools for counterfactual distributions based on regression methods the counterfactual scenarios that we consider consist of ceteris paribus changes in either the distribution of covariates related to the outcome of interest or the conditional distribution of the outcome given covariates for either of these scenarios we derive joint functional central limit theorems and bootstrap validity results for regression based estimators of the status quo and counterfactual outcome distributions these results allow us to construct simultaneous confidence sets for function valued effects of the counterfactual changes including the effects on the entire distribution and quantile functions of the outcome as well as on related functionals these confidence sets can be used to test functional hypotheses such as no effect positive effect or stochastic dominance our theory applies to general counterfactual changes and covers the main regression methods including classical quantile duration and distribution regressions we illustrate the results with an empirical application to wage decompositions using data for the united states as a part of developing the main results we introduce distribution regression as a comprehensive and flexible tool for modeling and estimating the entire conditional distribution we show that distribution regression encompasses the cox duration regression and represents a useful alternative to quantile regression we establish functional central limit theorems and bootstrap validity results for the empirical distribution regression process and various related functionals publication abstract

positive harmonic functions on complete manifolds of negative curvature this paper studies the interaction between the geometry of complete riemannian manifolds of negative curvature and some aspects of function theory on these spaces the study of harmonic functions on the unit disc provides a classical and beautiful example of this interaction we recall some aspects of this below there is a well known representation of positive harmonic functions on the unit disc u due to herglotz 13 in terms of positive borel measures tt on the circle s1

partial identification of functionals of the joint distribution of potential outcomes in this paper we present a systematic study of partial identification of two general classes of functionals of the joint distribution of two potential outcomes when a bivariate sample from the joint distribution is not available to the econometrician assuming the identification of the conditional marginal distributions of potential outcomes and the distribution of the covariate vector we show that the identified sets for functionals in both classes are intervals and provide conditions under which the identified sets point identify the true value of the functionals in addition we establish sufficient and necessary conditions for the covariate information to be informative in the sense of shrinking the identified sets we focus on the application of our general results to evaluating distributional treatment effects of a binary treatment in two commonly used frameworks in the literature for evaluating average treatment effects the selection on observables framework and a latent threshold crossing model we characterize the role of the propensity score in the selection on observables framework and the role of endogenous selection in the latent threshold crossing model examples of policy parameters that our results apply include the correlation coefficient between the potential outcomes many inequality measures of the distribution of treatment effects and median of the distribution of the individual treatment effect

nonuniqueness of the positive dirichlet problem for parabolic equations in cylinders abstract we investigate uniqueness of nonnegative solutions to parabolic equations in cylinders of r n 1 with zero dirichlet boundary condition where bases of cylinders are unbounded domains we give necessary and or sufficient conditions for the uniqueness in terms of geometric properties of the bases

critical behavior and the limit distribution for long range oriented percolation i we consider oriented percolation on mathbb z d times mathbb z whose bond occupation probability is pd where p is the percolation parameter and d is a probability distribution on mathbb z d suppose that d x decays as x d for some 0 we prove that the two point function obeys an infrared bound which implies that various critical exponents take on their respective mean field values above the upper critical dimension d c 2 alpha wedge2 we also show that for every k the fourier transform of the normalized two point function at time n with a proper spatial scaling has a convergent subsequence to e c k alpha wedge2 for some c 0

self avoiding walk in 5 or more dimensions using an expansion based on the renormalization group philosophy we prove that for a t step weakly self avoiding random walk in five or more dimensions the variance of the endpoint is of order t and the scaling limit is gaussian as t

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

identifying treatment effects under data combination we consider the identification of counterfactual distributions and treatment effects when the outcome variables and conditioning covariates are observed in separate data sets under the standard selection on observables assumption the counterfactual distributions and treatment effect parameters are no longer point identified however applying the classical monotone rearrangement inequality we derive sharp bounds on the counterfactual distributions and policy parameters of interest

asymptotic behavior of the gyration radius for long range self avoiding walk and long range oriented percolation let 0 and l 2 1 1 and suppose that the 1 step distribution d for random walk on z d decays as d x jx lj d for large jxjsuch that

rigorous analysis of discontinuous phase transitions via mean field bounds we consider a variety of nearest neighbor spin models defined on the d dimensional hypercubic lattice z d our essential assumption is that these models satisfy the condition of reflection positivity we prove that whenever the associated mean field theory predicts a discontinuous transition the actual model also undergoes a discontinuous transition which occurs near the mean field transition temperature provided the dimension is sufficiently large or the first order transition in the mean field model is sufficiently strong as an application of our general theory we show that for d sufficiently large the 3 state potts ferromagnet on z d undergoes a first order phase transition as the temperature varies similar results are established for all q state potts models with q 3 the r component cubic models with r 4 and the o n nematic liquid crystal models with n 3

ultracontractivity and the heat kernel for schrodinger operators and dirichlet laplacians abstract abstract connections between integral kernels of positivity preserving semigroups and suitable lp contractivity properties are established then these questions are studied for the semigroups generated by v and h the dirichlet laplacian for an open connected region as an application under a suitable hypothesis sobolev estimates are proved valid up to of the form x c 0 x h k 2 where 0 is the unique positive l2 eigenfunction of h

heat kernel eigenfunctions and conditioned brownian motion in planar domains abstract some recent results concerning uniform convergence of the shape of the heat kernel to that of the first eigenfunction and the lifetime of h processes in bounded lipschitz domains in r n are extended to all planar domains of finite area

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

gaussian limit for critical oriented percolation in high dimensions in this paper we consider the spread out oriented bond percolation models inz d z withd 4 and the nearest neighbor oriented bond percolation model in sufficiently high dimensions let n n 1 2 be the random measures defined onr d by eta n a sum limits x in z d 1 a x sqrt n 1 0 0 to x n the mean of n denoted by bar eta n is the measure defined by bar eta n a e p eta n a we use the lace expansion method to show that the sequence of probability measures bar eta n r d 1 bar eta n converges weakly to a gaussian limit asn for everyp in the subcritical regime as well as the critical regime of these percolation models also we show that for these models the parallel correlation length xi p p c p 1 asp pc

mean field critical behavior for the contact process the contact process is a model of spread of an infectious disease combining with the result of ref 1 we prove that the critical exponents take on the mean field values for sufficiently high dimensional nearest neighbor models and for sufficiently spread out models with d 4 c as c and c 1 as c where and are the spread probability and the susceptibility of the infection respectively and c is the critical infection rate our results imply that the upper critical dimension for the contact process is at most 4

on the asymptotic decay of l2 solutions of one body schrodinger equations in unbounded domains the asymptotic decay of l 2 solutions of schrodinger equations v 0 in r x er n x r r is investigated where v x v 1 r v 2 x with v 1 for r and with some e 0 for large r under additional assumptions on the decay of v 1 pointwise upper bounds to and lower bounds to the spherical average of are given showing the same asymptotics for r for the case v const 0 for r investigated in 8 a simplified treatment is given

semismall perturbations semi intrinsic ultracontractivity and integral representations of nonnegative solutions for parabolic equations we consider nonnegative solutions of a parabolic equation in a cylinder d i where d is a noncompact domain of a riemannian manifold and i 0 t with 0 t or i 0 under the assumption ssp i e the constant function 1 is a semismall perturbation of the associated elliptic operator on d we establish an integral representation theorem of nonnegative solutions in the case i 0 t any nonnegative solution is represented uniquely by an integral on d 0 md 0 t where md is the martin boundary of d for the elliptic operator and in the case i 0 any nonnegative solution is represented uniquely by the sum of an integral on md 0 and a constant multiple of a particular solution we also show that ssp implies the condition siu i e the associated heat kernel is semi intrinsically ultracontractive

random walk on the incipient infinite cluster for oriented percolation in high dimensions we consider simple random walk on the incipient infinite cluster for the spread out model of oriented percolation on mathbb z d times mathbb z in dimensions d 6 we obtain bounds on exit times transition probabilities and the range of the random walk which establish that the spectral dimension of the incipient infinite cluster is frac 4 3 and thereby prove a version of the alexander orbach conjecture in this setting the proof divides into two parts one part establishes general estimates for simple random walk on an arbitrary infinite random graph given suitable bounds on volume and effective resistance for the random graph a second part then provides these bounds on volume and effective resistance for the incipient infinite cluster in dimensions d 6 by extending results about critical oriented percolation obtained previously via the lace expansion

self attracting self avoiding walk this article is concerned with self avoiding walks saw on mathbb z d that are subject to a self attraction the attraction which rewards instances of adjacent parallel edges introduces difficulties that are not present in ordinary saw ueltschi has shown how to overcome these difficulties for sufficiently regular infinite range step distributions and weak self attractions ueltschi in probab theory relat fields 124 2 189 203 2002 this article considers the case of bounded step distributions for weak self attractions we show that the connective constant exists and in d ge 5 carry out a lace expansion analysis to prove the mean field behaviour of the critical two point function hereby addressing a problem posed by den hollander random polymers vol 1974 springer verlag berlin 2009

martin boundaries of elliptic skew products semismall perturbations and fundamental solutions of parabolic equations abstract we consider positive solutions of elliptic partial differential equations on non compact domains of riemannian manifolds we explicitly determine martin boundaries and martin kernels for a class of elliptic equations in skew product form by exploiting and developing perturbation theory for elliptic equations and short long time estimates for fundamental solutions of parabolic equations

efficient estimation of average treatment effects using the estimated propensity score we are interested in estimating the average effect of a binary treatment on a scalar outcome if assignment to the treatment is exogenous or unconfounded that is independent of the potential outcomes given covariates biases associated with simple treatmentcontrol average comparisons can be removed by adjusting for differences in the covariates rosenbaum and rubin 1983 show that adjusting solely for differences between treated and control units in the propensity score removes all biases associated with differences in covariates although adjusting for differences in the propensity score removes all the bias this can come at the expense of efficiency as shown by hahn 1998 heckman ichimura and todd 1998 and robins mark and newey 1992 we show that weighting by the inverse of a nonparametric estimate of the propensity score rather than the true propensity score leads to an efficient estimate of the average treatment effect we provide intuition for this result by showing that this estimator can be interpreted as an empirical likelihood estimator that efficiently incorporates the information about the propensity score

pointwise eigenfunction estimates and intrinsic ultracontractivity type properties of feynman kac semigroups for a class of levy processes we introduce a class of l evy processes subject to specific regularity conditions and consider their feynman kac semigroups given under a kato class potential using new techniques first we analyze the rate of decay of eigenfunctions at infinity we prove bounds on subaveraging functions from which we derive two sided sharp pointwise estimates on the ground state and obtain upper bounds on all other eigenfunctions next by using these results we analyze intrinsic ultracontractivity and related properties of the semigroup refining them by the concept of ground state domination and asymptotic versions we establish the relationships of these properties derive sharp necessary and sufficient conditions for their validity in terms of the behavior of the l evy density and the potential at infinity define the concept of borderline potential for the asymptotic properties and give probabilistic and variational characterizations these results are amply illustrated by key examples

the critical contact process dies out by producing a finite box criterion for survival in a slab we show that the critical contact process dies out and that in two and more dimensions the critical parameter value coincides with the limit of slab thresholds using the techniques developed in this paper one may obtain the complete convergence and shape theorems for contact processes in all dimensions and for all infection rates our results apply also to the discrete time analogue of the contact process viz oriented percolation

an intrinsic metric approach to uniqueness of the positive cauchy neumann problem for parabolic equations abstract we study uniqueness of nonnegative solutions of the cauchy neumann problem for parabolic equations in unbounded domains and give a sufficient condition for the uniqueness of nonnegative solutions to hold we also give a parabolic harnack inequality with neumann boundary conditions

characterization of intrinsic ultracontractivity for one dimensional schrodinger operators abstract we consider a schrodinger operator l d 2 d x 2 v x on r where v is a real valued measurable function and give an explicit and simple characterization of intrinsic ultracontractivity iu of the schrodinger semigroup generated by l for a wide class of potentials by making use of it we also give new examples of potentials for which the semigroups satisfy iu or non iu

backbone scaling for critical lattice trees in high dimensions we prove that for critical spread out lattice trees in dimensions the unique paths from the origin to vertices of large tree distance converge to brownian motion this provides an important ingredient for proving weak convergence of the corresponding historical processes

a self avoiding walk with attractive interactions a self avoiding walk with small attractive interactions is described here the existence of the connective constant is established and the diffusive behavior is proved using the method of the lace expansion

random subgraphs of finite graphs ii the lace expansion and the triangle condition in a previous paper we defined a version of the percolation triangle condition that is suitable for the analysis of bond percolation on a finite connected transitive graph and showed that this triangle condition implies that the percolation phase transition has many features in common with the phase transition on the complete graph in this paper we use a new and simplified approach to the lace expansion to prove quite generally that for finite graphs that are tori the triangle condition for percolation is implied by a certain triangle condition for simple random walks on the graph the latter is readily verified for several graphs with vertex set 0 1 r 1 n including the hamming cube on an alphabet of r letters the n cube for r 2 the n dimensional torus with nearest neighbor bonds and n sufficiently large and the n dimensional torus with n 6 and sufficiently spread out long range bonds the conclusions of our previous paper thus apply to the percolation phase transition for each of the above examples

the central role of the propensity score in observational studies for causal effects abstract the results of observational studies are often disputed because of nonrandom treatment assignment for example patients at greater risk may be overrepresented in some treatment group this paper discusses the central role of propensity scores and balancing scores in the analysis of observational studies the propensity score is the estimated conditional probability of assignment to a particular treatment given a vector of observed covariates both large and small sample theory show that adjustment for the scalar propensity score is sufficient to remove bias due to all observed covariates applications include matched sampling on the univariate propensity score which is equal percent bias reducing under more general conditions than required for discriminant matching multivariate adjustment by subclassification on balancing scores where the same subclasses are used to estimate treatment effects for all outcome variables and in all subpopulations and visual representation of multivariate adjustment by a two dimensional plot author

self avoiding walks and trees in spread out lattices letgr be the graph obtained by joining all sites ofzd which are separated by a distance of at mostr let gr denote the connective constant for counting the self avoiding walks in this graph let gr denote the coprresponding constant for counting the trees embedded ingr then asr gr is asymptotic to the coordination numberkr ofgr while gr is asymptotic toekr however ifd is 1 or 2 then gr kr diverges to

a generalised inductive approach to the lace expansion the lace expansion is a powerful tool for analysing the critical behaviour of self avoiding walks and percolation it gives rise to a recursion relation which we abstract and study using an adaptation of the inductive method introduced by den hollander and the authors we give conditions under which the solution to the recursion relation behaves as a gaussian both in fourier space and in terms of a local central limit theorem these conditions are shown elsewhere to hold for sufficiently spread out models of networks of self avoiding walks in dimensions d 4 and for sufficiently spread out models of critical oriented percolation in dimensions d 1 5 providing a unified approach and an essential ingredient for a detailed analysis of the branching behaviour of these models

mean field driven first order phase transitions in systems with long range interactions we consider a class of spin systems on d with vector valued spins sx that interact via the pair potentials jx ysx sy the interactions are generally spread out in the sense that the jx y s exhibit either exponential or power law fall off under the technical condition of reflection positivity and for sufficiently spread out interactions we prove that the model exhibits a first order phase transition whenever the associated mean field theory signals such a transition as a consequence e g in dimensions d 3 we can finally provide examples of the 3 state potts model with spread out exponentially decaying interactions which undergoes a first order phase transition as the temperature varies similar transitions are established in dimensions d 1 2 for power law decaying interactions and in high dimensions for next nearest neighbor couplings in addition we also investigate the limit of infinitely spread out interactions specifically we show that once the mean field theory is in a unique state then in any sequence of translation invariant gibbs states various observables converge to their mean field values and the states themselves converge to a product measure

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

the lace expansion approach to ballistic behaviour for one dimensional weakly self avoiding walks we prove ballistic behaviour in dimension one fora model of weakly self avoiding walks where loops of length m are penalized by a factor e mp with p 0 1 and sufficiently large furthermore we prove that the fluctuations around the linear drift satisfy a centrallimit theorem the proof uses a variant of the lace expansion together with an inductive analysis of the arising recursion relation

construction of the incipient infinite cluster for spread out oriented percolation above 4 1 dimensions we construct the incipient infinite cluster measure iic for sufficiently spread out oriented percolation on d for d 1 4 1 we consider two different constructions for the first construction we define n e by taking the probability of the intersection of an event e with the event that the origin is connected to x n d summing this probability over x d and normalising the sum to get a probability measure we let n and prove existence of a limiting measure the iic for the second construction we condition the connected cluster of the origin in critical oriented percolation to survive to time n and let n under the assumption that the critical survival probability is asymptotic to a multiple of n 1 we prove existence of a limiting measure with in addition we study the asymptotic behaviour of the size of the level set of the cluster of the origin and the dimension of the cluster of the origin under our methods involve minor extensions of the lace expansion methods used in a previous paper to relate critical oriented percolation to super brownian motion for d 1 4 1

maximum and anti maximum principlesand eigenfunctions estimates via perturbation theory of positive solutions of elliptic equations in this paper we discuss some new results concerning perturbation theory for second order elliptic partial differential equations related to positivity properties of such equations we continue the study of some different notions of small perturbations and discuss their relations to comparisons of green s functions refined maximum and anti maximum principles ground state and the decay of eigenfunctions in particular we show that if v is a positive function which is a semismall perturbation of a subcritical schrodinger operator h defined on a domain omega subset mathbb r d and phi k k geq 0 are the dirichlet eigenfunctions of the equation hu lambda vu then for any k geq 0 the function phi k phi 0 is bounded and has a continuous extension up to the martin boundary of the pair omega h where phi 0 is the ground state of h with a principal eigenvalue lambda 0

a pointwise lower bound for positive solutions of a schrodinger equation in rn abstract strong positivity of the bounded inverse a 1of a schrodinger operator a q x inl2 r n is proved in the following form if a u f 0 inl2 r n withf 0 thenu c 1a e in r n here 1is the positive eigenfunction associated with the principal eigenvalue 1of a andcis a positive constant it is shown that this result is valid if and only if the potentialq x which is assumed to be strictly positive and locally bounded has a sufficiently fast growth as x this result is applied to linear and nonlinear elliptic boundary value problems in strongly ordered banach spaces whose positive cone is generated by the eigenfunction 1 in particular problems of existence and uniqueness are addressed

quantile and probability curves without crossing this paper proposes a method to address the longstanding problem of lack of monotonicity in estimation of conditional and structural quantile functions also known as the quantile crossing problem bassett and koenker 1982 the method consists in sorting or monotone rearranging the original estimated non monotone curve into a monotone rearranged curve we show that the rearranged curve is closer to the true quantile curve than the original curve in finite samples establish a functional delta method for rearrangement related operators and derive functional limit theory for the entire rearranged curve and its functionals we also establish validity of the bootstrap for estimating the limit law of the entire rearranged curve and its functionals our limit results are generic in that they apply to every estimator of a monotone function provided that the estimator satisfies a functional central limit theorem and the function satisfies some smoothness conditions consequently our results apply to estimation of other econometric functions with monotonicity restrictions such as demand production distribution and structural distribution functions we illustrate the results with an application to estimation of structural distribution and quantile functions using data on vietnam veteran status and earnings

on uniqueness and nonuniqueness of the positive cauchy problem for parabolic equations with unbounded coefficients where i xi a nonnegative solution u x t of the equation lu 0 in xt x 0 t with a nonnegative initial data u x 0 u0 x is called a solution of the positive cauchy problem such a solution can be thought of as the temperature field at the time t in an n dimensional body which occupies the domain x the coefficients aij x are the components of the matrix of the thermal conductivities while the function 1 x is the heat capacity of a unit hypervolume hence nonnegative solutions of the cauchy problem are natural objects to study in particular it is of interest to know whether the

ground state positivity negativity and compactness for a schrodinger operator in rn abstract we treat the schrodinger operator a q x on l 2 r n with the potential q r n q 0 bounded below and satisfying some reasonable hypotheses on the growth at infinity faster than x 2 as x we are concerned primarily with the compactness of the resolvent a i 1 of a as an operator on the banach space x x f l 2 r n f l r n f x ess sup r n f where denotes the ground state for a if is the ground state energy for a we show that the restricted operator a i 1 x x is not only bounded but also compact for in particular the spectra of a in l 2 r n and x coincide each eigenfunction belongs to x as another consequence we obtain a maximum and an anti maximum principles

critical two point functions and the lace expansion for spread out high dimensional percolation and related models we consider spread out models of self avoiding walk bond percolation lattice trees and bond lattice animals on mathbb z d having long finite range connections above their upper critical dimensions d 4 self avoiding walk d 6 percolation and d 8 trees and animals the two point functions for these models are respectively the generating function for self avoiding walks from the origin to x in mathbb z d the probability of a connection from 0 to x and the generating function for lattice trees or lattice animals containing 0 and x we use the lace expansion to prove that for sufficiently spread out models above the upper critical dimension the two point function of each model decays at the critical point as a multiple of x 2 d as x to infty we use a new unified method to prove convergence of the lace expansion the method is based on x space methods rather than the fourier transform our results also yield unified and simplified proofs of the bubble condition for self avoiding walk the triangle condition for percolation and the square condition for lattice trees and lattice animals for sufficiently spread out models above the upper critical dimension

structure of positive solutions to δ v u 0 in rn on etudie la structure de toutes les solutions positives d une equation de schrodinger stationnaire ou le potentiel v est suppose etre une fonction a valeur reelle appartenant a l p loc r n avec p n 2 pour n 2 et p 1 pour n 1 on considere une formule de representation explicite pour des solutions positives leurs asymptotes a l infini et le probleme de dirichlet avec une condition aux limites a l infini

on the role of the propensity score in efficient semiparametric estimation of average treatment effects in this paper the role of the propensity score in the efficient estimation of average treatment effects is examined under the assumption that the treatment is ignorable given some observed characteristics it is shown that the propensity score is ancillary for estimation of the average treatment effects the propensity score is not ancillary for estimation of average treatment effects on the treated it is suggested that the marginal value of the propensity score lies entirely in the dimension reduction efficient semiparametric estimators of average treatment effects and average treatment effects on the treated are shown to take the form of relevant sample averages of the data completed by the nonparametric imputation method it is shown that the projection on the propensity score is not necessary for efficient semiparametric estimation of average treatment effects on the treated even if the propensity score is known an application to the experimental data reveals that conditioning on the propensity score may even result in a loss of efficiency

critical behavior and the limit distribution for long range oriented percolation ii spatial correlation we prove that the fourier transform of the properly scaled normalized two point function for sufficiently spread out long range oriented percolation with index 0 converges to e c k alpha wedge2 for some c in 0 infty above the upper critical dimension dc equiv 2 alpha wedge 2 this answers the open question remained in the previous paper chen and sakai in probab theory relat fields 142 151 188 2008 moreover we show that the constant c exhibits crossover at 2 which is a result of interactions among occupied paths the proof is based on a new method of estimating fractional moments for the spatial variable of the lace expansion coefficients

an extension of maximum and anti maximum principles to a schrodinger equation in r2 abstract strong maximum and anti maximum principles are extended to weak l2 r 2 solutions u of the schrodinger equation u q x u u f x in l2 r 2 in the following form let 1 denote the positive eigenfunction associated with the principal eigenvalue 1 of the schrodinger operator a q x in l2 r 2 assume that q x q x f is a sufficiently smooth perturbation of a radially symmetric function f 0 and 0 f 1 c const a e in r 2 then there exists a positive number depending upon f such that for every 1 with 1 the inequality 1 u c 1 holds a e in r 2 where c is a positive constant depending upon f and it is shown that such an inequality is valid if and only if the potential q x which is assumed to be strictly positive and locally bounded has a superquadratic growth as x this result is applied to linear and nonlinear elliptic boundary value problems in strongly ordered banach spaces whose positive cone is generated by the eigenfunction 1 in particular problems of existence and uniqueness are addressed

mean field behavior for long and finite range ising model percolation and self avoiding walk we consider self avoiding walk percolation and the ising model with long and flnite range by means of the lace expansion we prove mean fleld behavior for these models if d 2 fi 2 for self avoiding walk and the ising model and d 3 fi 2 for percolation where d denotes the dimension and fi the power law decay exponent of the coupling function we provide a simplifled analysis of the lace expansion based on the trigonometric approach in borgs et al 13

partial distributional policy effects in this paper we propose a method to evaluate the effect of a counterfactual change in the unconditional distribution of a single covariate on the unconditional distribution of an outcome variable of interest both fixed and infinitesimal changes are considered we show that such effects are point identified under general conditions if the covariate affected by the counterfactual change is continuously distributed but are typically only partially identified if its distribution is discrete for the latter case we derive informa tive bounds making use of the available information we also discuss estimation and inference

integral representations of nonnegative solutions for parabolic equations and elliptic martin boundaries abstract we consider nonnegative solutions of a parabolic equation in a cylinder d 0 t where d is a noncompact domain of a riemannian manifold under the assumption iu i e the associated heat kernel is intrinsically ultracontractive we establish an integral representation theorem any nonnegative solution is represented uniquely by an integral on d 0 m d 0 t where m d is the martin boundary of d for the associated elliptic operator we apply it in a unified way to several concrete examples to explicitly represent nonnegative solutions we also show that iu implies the condition sp i e the constant function 1 is a small perturbation of the elliptic operator on d

the diffusion of self avoiding random walk in high dimensions we use the brydges spencer lace expansion to prove that the mean square displacement of at step strictly self avoiding random walk in thed dimensional square lattice is asymptotically of the formdt ast approaches infinity ifd is sufficiently large the diffusion constantd is greater than one

lace expansion for the ising model the lace expansion has been a powerful tool for investigating mean field behavior for various stochastic geometrical models such as self avoiding walk and percolation above their respective upper critical dimension in this paper we prove the lace expansion for the ising model that is valid for any spin spin coupling for the ferromagnetic case we also prove that the expansion coefficients obey certain diagrammatic bounds that are similar to the diagrammatic bounds on the lace expansion coefficients for self avoiding walk as a result we obtain gaussian asymptotics of the critical two point function for the nearest neighbor model with d gg 4 and for the spread out model with d 4 and l gg 1 without assuming reflection positivity

a new inductive approach to the lace expansion for self avoiding walks we introduce a new inductive approach to the lace expansion and apply it to prove gaussian behaviour for the weakly self avoiding walk on d where loops of length m are penalised by a factor e m p 0 4 p 0 2 d 4 in particular we derive results first obtained by brydges and spencer and revisited by other authors for the case d 4 p 0 in addition we prove a local central limit theorem with the exception of the case d 4 p 0

generalized approach to the non backtracking lace expansion the lace expansion is a powerful perturbative technique to analyze the critical behavior of random spatial processes such as the self avoiding walk percolation and lattice trees and animals the non backtracking lace expansion noble is a modification that allows us to improve its applicability in the nearest neighbor setting on the mathbb z d lattice for percolation lattice trees and lattice animals the noble gives rise to a recursive formula that we study in this paper at a general level we state assumptions that guarantee that the solution of this recursive formula satisfies the infrared bound in two related papers we show that these conditions are satisfied for percolation in d ge 11 for lattice trees in d ge 16 and for lattice animals in d ge 18

mean field critical behaviour for percolation in high dimensions the triangle condition for percolation states that sum limits x y tau 0 x tau 0 y cdot tau y 0 is finite at the critical point where x y is the probability that the sitesx andy are connected we use an expansion related to the lace expansion for a self avoiding walk to prove that the triangle condition is satisfied in two situations i for nearest neighbour independent bond percolation on thed dimensional hypercubic lattice ifd is sufficiently large and ii in more than six dimensions for a class of spread out models of independent bond percolation which are believed to be in the same universality class as the nearest neighbour model the class of models in ii includes the case where the bond occupation probability is constant for bonds of length less than some large number and is zero otherwise in the course of the proof an infrared bound is obtained the triangle condition is known to imply that various critical exponents take their mean field bethe lattice values gamma beta 1 delta delta t 2 t underset raise0 3em hbox smash scriptscriptstyle geqslant 2 and that the percolation density is continuous at the critical point we also prove thatv2 in i and ii wherev2 is the critical exponent for the correlation length

percolation critical exponents under the triangle condition for independent percolation models it is shown that if the diagrammatic triangle condition is satisfied then the critical exponents delta and hat beta exist and take their mean field values generalizing the criterion introduced in 1984 by aizenman and newman for the mean field value of gamma in nonoriented percolation the results apply to a broad class of nonoriented as well as oriented weakly homogeneous models in which the range of the connecting bonds need not be bounded for the nonoriented case the condition reduces to the finiteness at the critical point of nabla equiv sum x y tau 0 x tau x y tau y 0 with tau u nu the probability that the site u is connected to nu which was recently established by hara and slade for models with sufficiently spread out connections in d 6 dimensions our analysis proceeds through the derivation of complementary differential inequalities for the percolation order parameter m beta h whose value at h 0 yields the percolation density with beta parametrizing the bond or site occupation probabilities and with h h geq 0 a ghost field the conclusion is that under the triangle condition in the vicinity of the critical point beta c 0 m beta 0 approx beta beta c hat beta and m beta c h approx h 1 delta with hat beta 1 and delta 2

sharpness of the phase transition in percolation models the equality of two critical points the percolation threshold ph and the point p where the cluster size distribution ceases to decay exponentially is proven for all translation invariant independent percolation models on homogeneous d dimensional lattices d 1 the analysis is based on a pair of new nonlinear partial differential inequalities for an order parameter m h which for h q reduces to the percolation density p at the bond density p l e in the single parameter case these are 1 m hdm dh m2 mdm d and 2 dm d j mdm dh inequality 1 is intriguing in that its derivation provides yet another hint of a 3 structure in percolation models moreover through the elimination of one of its derivatives 1 yields a pair of ordinary differential inequalities which provide information on the critical exponents and one of these resembles an ising model inequality of fr hlich and sokal and yields the mean field bound 5 2 and the other implies the result of chayes and chayes that an inequality identical to 2 is known for ising models where it provides the basis for newman s universal relation 5 1 1 and for certain extrapolation principles which are now made applicable also to independent percolation these results apply to both finite and long range models with or without orientation and extend to periodic and weakly inhomogeneous systems

asymptotics of green functions and martin boundaries for elliptic operators with periodic coefficients abstract we give the asymptotics at infinity of a green function for an elliptic equation with periodic coefficients on r d basic ingredients in establishing the asymptotics are an integral representation of the green function and the saddle point method we also completely determine the martin compactification of r d with respect to an elliptic equation with periodic coefficients by using the exact asymptotics at infinity of the green function

high dimensional incipient infinite clusters revisited the incipient infinite cluster iic measure is the percolation measure at criticality conditioned on the cluster of the origin to be infinite using the lace expansion we construct the iic measure for high dimensional percolation models in three different ways extending previous work by the second named author and jarai we show that each construction yields the same measure indicating that the iic is a robust object furthermore our constructions apply to spread out versions of both finite range and long range percolation models we also get estimates on structural properties of the iic such as the volume of the intersection between the iic and euclidean balls