gibbs measures on brownian currents motivated by applications to quantum field theory we consider gibbs measures for which the reference measure is wiener measure and the interaction is given by a double stochastic integral and a pinning external potential in order properly to characterize these measures through dlr equations we are led to lift wiener measure and other objects to a space of configurations where the basic observables are not only the position of the particle at all times but also the work done by test vector fields we prove existence and basic properties of such gibbs measures in the small coupling regime by means of cluster expansion

the inverse mean curvature flow and the riemannian penrose inequality let m be an asymptotically flat 3 manifold of nonnegative scalar curvature the riemannian penrose inequality states that the area of an outermost minimal surface n in m is bounded by the adm mass m according to the formula n 16 m 2 we develop a theory of weak solutions of the inverse mean curvature flow and employ it to prove this inequality for each connected component of n using geroch s monotonicity formula for the adm mass our method also proves positivity of bartnik s gravitational capacity by computing a positive lower bound for the mass purely in terms of local geometry

the diffusive phase of a model of self interacting walks we consider simple random walk onzd perturbed by a factor exp t pjt wheret is the length of the walk and j t sum nolimits 0 leqslant i 0 ford 2 the diffusion constant is equal to 1 but ford 2 it is renormalized ford 1 andp 3 2 we prove diffusion for all real positive or negative ford 2 the scaling limit is brownian motion but ford 2 it is the edwards model with the wrong sign of the coupling when 0 which governs the limiting behaviour the latter arises since for p frac 4 d 2 t p j t is the discrete self intersection local time this establishes existence of a diffusive phase for this model existence of a collapsed phase for a very closely related model has been proven in work of bolthausen and schmock

the weyl problem in warped product spaces in this paper we discuss the weyl problem in warped product spaces we apply the method of continuity and prove the openness of the weyl problem a counterexample is constructed to show that the isometric embedding of the sphere with canonical metric is not unique up to an isometry if the ambient warped product space is not a space form then we study the rigidity of the standard sphere if we fixed its geometric center in the ambient space finally we discuss a shi tam type of inequality for the schwarzschild manifold as an application of our findings

markov processes as a tool in field theory abstract in a recent paper brydges frohlich and spencer have successfully applied markov chains to classical spin systems using probabilistic methods in a more systematic way we improve some of their results the methods can be extended to fields with continuous parameter such as p fields

the range of stable random walks limit theorems are proved for the range of d dimensional random walks in the domain of attraction of a stable process of index p3 the range rn is the number of distinct sites of zd visited by the random walk before time n our results depend on the value of the ratio 3 d the most interesting results are obtained for 2 3 1 which can only occur if d 1 we prove the convergence in distribution of rn e rn toward some constant times the lebesgue measure of the range of the limiting stable process some of our results require regularity assumptions on the characteristic function of x

uniqueness of isometric immersions with the same mean curvature abstract motivated by the quasi local mass problem in general relativity we study the rigidity of isometric immersions with the same mean curvature into a warped product space as a corollary of our main result two star shaped hypersurfaces in a spatial schwarzschild or ads schwarzschild manifold with nonzero mass differ only by a rotation if they are isometric and have the same mean curvature we also prove similar results if the mean curvature condition is replaced by an 2 curvature condition

a renormalization result for the intersection local time of lattice random walks ind 3 dimensions in this paper we derive some general conditions on stable walks inz d under which the central limit theorem holds for their normalized intersection local time in particular we prove that the process given by the normalized intersection local time of the simple random walk inz d withd 3 is weakly convergent to the standard brownian motion

large deviations for the one dimensional edwards model in this article we prove a large deviation principle for the empirical drift of a one dimensional brownian motion with self repellence called the edwards model our results extend earlier work in which a law of large numbers and a central limit theorem were derived in the edwards model a path of length t receives a penalty e ht w hereht is the selfintersection local time of the path and 0 is a parameter called the strength of self repellence we identify the rate function in the large deviation principle for the endpoint of the path as 2 3 i 1 3 with i given in terms of the principal eigenvalues of a one parameter family of sturm liouville operators we show that there exist numbers 0 b b such that 1 i is linearly decreasing on 0 b 2 i is real analytic and strictly convex on b 3 i is continuously differentiable at b and 4 i has a unique zero at b the latter fact identifies b as the asymptotic drift of the endpoint the critical drift b is associated with a crossover in the optimal strategy of the path for b b the path assumes local drift b during the full time t while for 0 b b it assumes local drift b during time b b 2b t and local drift b during the remaining time b b 2b t thus in the second regime the path makes an overshoot of size b b 2 t so as to reduce its intersection local time

the gibbs ensemble of a vortex filament we introduce a statistical ensemble for a single vortex filament of a three dimensional incompressible fluid the core of the vortex is modeled by a quite generic stochastic process we prove the existence of the partition function for both positive and a limited range of negative temperatures

asymptotic receiver structures for joint maximum likelihood time delay estimation and channel identification using gaussian signals the problem of jointly estimating the relative time delay and the impulse response linking two received discrete time gaussian signals is addressed using two different methods possible structures for the joint maximum likelihood ml estimator are proposed when the observation interval is long compared to both the delay to estimate and the correlation time of the various random processes involved these structures generalize the cross correlation method with prefiltering that implements the ml estimation of pure time delays

gaussian and non gaussian random fields associated with markov processes to every markov process with a symmetric transition density there correspond two random fields over the state space a gaussian field the free field and the occupation field t which describes amount of time the particle spends at each state a relation between these two random fields is established which is useful both for the field theory and theory of markov processes

a new discrete edwards model and a new polymer measure in two dimensions the new discrete edwards models in this paper are defined in terms of the so called restricted intersection local times of the lattice random walk in two dimensions we study the asymptotic behaviours of these new discrete edwards models in the superrenormalizable cases in particular by approximating these models we can construct new polymer measures in two dimensions which are different from the original polymer measures obtained by approximating the original discrete edwards models the new discrete edwards models can be thought of as zero component lattice 4 fields with different cutoffs in the free and interacting parts

maximum likelihood estimation of time varying delay part i this paper presents for the first time the exact theoretical solution to the problem of maximum likelihood ml estimation of time varying delay d t between a random signal s t received at one point in the presence of uncorrelated noise and the time delayed scaled version s t d t of that signal received at another point in the presence of uncorrelated noise the signal is modeled as a sample function of a nonstationary gaussian random process and the observation interval is arbitrary the analysis of this paper represents a generalization of that of knapp and carter 1 who derived the ml estimator for the case that the delay is constant d t d 0 the signal process is stationary and the received processes are observed over the infinite interval we show that the ml estimator of d t can be implemented in any of four canonical forms which in general are time varying systems we also show that our results reduce to a generalized cross correlator for the special case treated in 1

the intersection local time of the westwater process in this paper self intersection properties of the westwater process are investigated as a result we obtain that the westwater process has an intersection local time bar alpha x 0 s t 1 which is holder continuous with respect tox s t r3 0 1 2 1 2 1 and the hausdorff dimension of the double time set is 1 2 as for brownian motion inr3

proprietes d intersection des marches aleatoires i convergence vers le temps local d intersection we study intersection properties of multi dimensional random walks letx andy be two independent random walks with values in d d 3 satisfying suitable moment assumptions and letin denote the number of common points to the paths ofx andy up to timen the sequence in suitably normalized is shown to converge in distribution towards the intersection local time of two independent brownian motions results are applied to the proof of a central limit theorem for the range of a two dimensional recurrent random walk thus answering a question raised by n c jain and w e pruitt

a local time approach to the self intersections of brownian paths in space we study the brownian functional alpha x b int limits b int delta x w t w s dsdt wherewt is a brownian path in two or three dimensions forb off the diagonal we identify x b with a local time and establish the holder continuity of x b in bothx andb

harmonic analysis of additive levy processes let x1 xn denote n independent d dimensional levy processes and consider the n parameter random field mathfrak x t x 1 t 1 cdots x n t n first we demonstrate that for all nonrandom borel sets f subseteq bf r d the minkowski sum mathfrak x bf r n oplus f of the range mathfrak x bf r n of mathfrak x with f can have positive d dimensional lebesgue measure if and only if a certain capacity of f is positive this improves our earlier joint effort with yuquan zhong by removing a certain condition of symmetry in khoshnevisan et al ann probab 31 2 1097 1141 2003 moreover we show that under mild regularity conditions our necessary and sufficient condition can be recast in terms of one potential densities this rests on developing results in classical non probabilistic harmonic analysis that might be of independent interest as was shown in khoshnevisan et al ann probab 31 2 1097 1141 2003 the potential theory of the type studied here has a large number of consequences in the theory of levy processes presently we highlight a few new consequences

the probability of intersection of independent random walks in four dimensions lets1 ands2 be independent simple random walks of lengthn inz4 starting at 0 andx0 respectively if x0 2 n it is shown that the probability that the paths intersect is of order logn 1 ifx0 0 it is shown that the probability of no intersection of the paths decays no faster than logn 1 and no slower than logn 1 2 it is conjectured that logn 1 2 is the actual decay rate

random walks and intersection local time with each random walk on z 2 we associate a functional related to the number of steps which the walk spends in sites occupied at least k times we show that if our random walk is in the domain of attraction of a stable process of order greater than 2 2k 2 2k 1 then our functional converges to the intersection local time of the limiting process

a simple method for finite range decomposition of quadratic forms and gaussian fields we present a simple method to decompose the green forms corresponding to a large class of interesting symmetric dirichlet forms into integrals over symmetric positive semi definite and finite range properly supported forms that are smoother than the original green form this result gives rise to multiscale decompositions of the associated gaussian free fields into sums of independent smoother gaussian fields with spatially localized correlations our method makes use of the finite propagation speed of the wave equation and chebyshev polynomials it improves several existing results and also gives simpler proofs

scalar curvature deformation and a gluing construction for the einstein constraint equations on a compact manifold the scalar curvature map at generic metrics is a local surjection f m we show that this result may be localized to compact subdomains in an arbitrary riemannian manifold the method is extended to establish the existence of asymptotically flat scalar flat metrics on n n 3 which are spherically symmetric hence schwarzschild at infinity i e outside a compact set such metrics provide cauchy data for the einstein vacuum equations which evolve into nontrivial vacuum spacetimes which are identically schwarzschild near spatial infinity

isometric embeddings into the minkowski space and new quasi local mass the definition of quasi local mass for a bounded space like region in space time is essential in several major unsettled problems in general relativity the quasi local mass is expected to be a type of flux integral on the boundary two surface sigma partial omega and should be independent of whichever space like region sigma bounds an important idea which is related to the hamiltonian formulation of general relativity is to consider a reference surface in a flat ambient space with the same first fundamental form and derive the quasi local mass from the difference of the extrinsic geometries this approach has been taken by brown york 4 5 and liu yau 16 17 see also related works 3 6 9 12 14 15 28 32 to define such notions using the isometric embedding theorem into the euclidean three space however there exist surfaces in the minkowski space whose quasilocal mass is strictly positive 19 it appears that the momentum information needs to be accounted for to reconcile the difference in order to fully capture this information we use isometric embeddings into the minkowski space as references in this article we first prove an existence and uniqueness theorem for such isometric embeddings we then solve the boundary value problem for jang s 13 equation as a procedure to recognize such a surface in the minkowski space in doing so we discover a new expression of quasi local mass for a large class of admissible surfaces see theorem a and remark 1 1 the new mass is positive when the ambient space time satisfies the dominant energy condition and vanishes on surfaces in the minkowski space it also has the nice asymptotic behavior at spatial infinity and null infinity some of these results were announced in 29

factorization properties of optimum space time processors in nonstationary environments a structural analysis of space time log likelihood processors llp that applies to arbitrary signal transmission models consisting of n sources m sensors a time varying linear channel and nonstationary gaussian source signal and sensor noise processes is presented the approach is based on representing the time varying linear channel as a bounded linear operator l with closed range by exploiting the properties of such operators and the specific structure of the array covariance function it is shown that the classical m dimensional integral equations defining the llp can be transformed into equivalent n dimensional integral equations as a result it is always possible to factor the llp into a cascade of three specialized time varying subprocessors namely a space time whitening filter an m input n output unitary beamformer ub and an n input quadratic postprocessor qpp both the ub and the qpp are given an interpretation and their most important features are indicated to illustrate the theory several examples of its application to signal models involving time varying delays are given

on edwards model for polymer chains ii the self consistent potential we obtain an existence and uniqueness theorem for the self consistent potential in edwards model for polymer chains and confirm the asymptotic analysis proposed by him on the basis of wkb arguments

the random walk representation of classical spin systems and correlation inequalities ii the skeleton inequalities on utilise la representation marche aleatoire pour prouver les premieres d une nouvelle famille d inegalites de correlation pour des modeles reticules ferromagnetiques 4 ces inegalites etablissent que les sommes partielles finies du developpement en perturbation resomme du propagateur pour la fonction a 4 points forment un ensemble alterne de limites superieures et inferieures rigoureuses pour la fonction a 4 points exactes

the first passage sets of the 2d gaussian free field convergence and isomorphisms in a previous article we introduced the first passage set fps of constant level a of the two dimensional continuum gaussian free field gff on finitely connected domains informally it is the set of points in the domain that can be connected to the boundary by a path along which the gff is greater than or equal to a this description can be taken as a definition of the fps for the metric graph gff and in the current article we prove that the metric graph fps converges towards the continuum fps in the hausdorff distance we also draw numerous consequences in particular we obtain a relatively simple proof of the fact that certain natural interfaces of the metric graph gff converge to hbox sle 4 level lines these results improve our understanding of the continuum gff by strengthening its relationship with the critical brownian loop soup indeed a new construction of the fps using clusters of brownian loops and excursions helps to strengthen the known gff isomorphism theorems and allows us to use brownian loop soup techniques to prove technical results on the geometry of the gff we also obtain a new representation of brownian loop soup clusters and as a consequence we prove that the clusters of a critical brownian loop soup admit a non trivial minkowski content in the gauge r mapsto log r 1 2 r 2

construction of an edwards probability measure on c r r in this article we prove that the measures q t associated to the one dimensional edwards model on the interval 0 t converge to a limit measure when t goes to infinity in the following sense for all s 0 and for all events a s depending on the canonical process only up to time s t a s a s moreover we prove that if p is wiener measure there exists a martingale d s s such that a s e 1 s d s and we give an explicit expression for this martingale

on the construction of the three dimensional polymer measure the three dimensional polymer measure was first constructed by westwater in 1980 with a very complicated proof we give an alternative construction for small coupling parameter which is based on the approach by brydges frohlich sokal in quantum field theory and bovier felder frohlich using skeleton inequalities the main new features are the proof of convergence which had been open in the brydges frohlich sokal construction and the construction of the measure on the space of paths with fixed time length

the random walk representation of classical spin systems and correlation inequalities ferromagnetic lattice spin systems can be expressed as gases of random walks interacting via a soft core repulsion by using a mixed spinrandom walk representation we present a unified approach to many recently established correlation inequalities as an application of these inequalities we obtain a simple proof of the mass gap for the 4 2 quantum field model we also establish new upper bounds on critical temperatures

polymers in a weak random potential in dimension four rigorous renormalization group analysis correlation functions of the edwards model of polymers at weak coupling are defined and studied at the critical point in dimension four by a rigorous renormalization group method which validates at any order perturbative renormalization group results on their behaviour at large distances remainders are controlled by a new argument which enlarges the use of methods of constructive field theory to models of statistical physics

construction and borel summability of infrared φ 4 4 by a phase space expansion we construct the thermodynamic limit of the critical massless 4 model in 4 dimensions with an ultraviolet cutoff by means of a partly renormalized phase space expansion this expansion requires in a natural way the introduction of effective or running constants and the infrared asymptotic freedom of the model i e the decay of the running coupling constant plays a crucial role we prove also that the correlation functions of the model are the borel sums of their perturbation expansion

multiple intersection local time of planar brownian motion as a particular hida distribution abstract subspaces of the space of hida distributions are quantified in which the multiple intersection local time of planar brownian motion renormalized by taking off the term of order zero in its chaos decomposition lives this is done by means of a new formula expressing the expectation of higher products of multiple wiener ito integrals by certain functions of the scalar products of their kernels

geometric analysis of φ4 fields and ising models parts i and ii we provide here the details of the proof announced in 1 that ind 4 dimensions the even 4 euclidean field theory with a lattice cut off is inevitably free in the continuum limit in the single phase regime the analysis is nonperturbative and is based on a representation of the field variables or spins in ising systems as source sink creation operators in a system of random currents which may be viewed as the mediators of correlations in this dual representation the onset of long range order is attributed to percolation in an ensemble of sourceless currents and the physical interaction in the 4 field and other aspects of the critical behavior in ising models are directly related to the intersection properties of long current clusters an insight into the criticality of the dimensiond 4 is derived from an analogy foreseen by k symanzik with the intersection properties of paths of brownian motion other results include the proof that in certain respect the critical behavior in ising models is in exact agreement with the mean field approximation in high dimensionsd 4 but not in the low dimensiond 2 for which we establish the universality of hyperscaling

curvature estimates for immersed hypersurfaces in riemannian manifolds we establish mean curvature estimate for immersed hypersurface with nonnegative extrinsic scalar curvature in riemannian manifold n n 1 bar g through regularity study of a degenerate fully nonlinear curvature equation in general riemannian manifold the estimate has a direct consequence for the weyl isometric embedding problem of mathbb s 2 g in 3 dimensional warped product space n 3 bar g we also discuss isometric embedding problem in spaces with horizon in general relativity like the anti de sitter schwarzschild manifolds and the reissner nordstrom manifolds

self avoiding fractional brownian motion the edwards model in this work we extend varadhan s construction of the edwards polymer model to the case of fractional brownian motions in d for any dimension d 2 with arbitrary hurst parameters h 1 d

random fields associated with multiple points of the brownian motion abstract let xt be the brownian motion in r d the random set t1 tn z xtl xtn z in r d n is empty a s except in the following cases a n 1 d 1 2 b d 2 n 2 3 c d 3 n 2 in each of these cases a family of random measures m concentrated on is constructed takes values in a certain class of measures on r d measures m characterize the time space location of self intersections for brownian paths if n d 1 then m dt dz dz nz dt where n2 is the local time at z in the case n 2 the set can be identified with the set of brownian loops the measure m explodes on the diagonal t1 t2 and to study small loops a random distribution which regularizes m is constructed

sketches with curvature the curve indicator random field and markov processes while complaints about typical edge operators are common proposals articulating a notion of the perfect edge map are comparatively rare hindering the improvement of contour enhancement techniques to address this situation we suggest that one objective of visual contour computation is the estimation of a clean sketch from a corrupted rendition the latter modeling noisy and low contrast edge or line operator responses to an image our formal model of this clean sketch is the curve indicator random field cirf whose role is to provide a basis for defining edge likelihood models by eliminating the parameter along each curve to create an image of curves for curves modeled with stationary markov processes this ideal edge prior is non gaussian and its moment generating functional has a form closely related to the feynman kac formula this sketch model leads to a nonlinear minimum mean squared error contour enhancement filter that requires the solution of two elliptic partial differential equations the framework is also independent of the order of the contour model allowing us to introduce a markov process model for contour curvature we analyze the distribution of such curves and show that its mode is the euler spiral a curve minimizing changes in curvature example computations using the contour enhancement filter with the curvature based contour model are provided highlighting how the filter is curvature selective even when curvature is absent in the input

asymptotic completeness and multiparticle structure in field theories the ideas developed in part i ref 1 are applied to the recently constructed massive gross neveu model we define in this case an irreducible kernel satisfying a regularized bethe salpeter equation which is convenient to derive asymptotic completeness in the 2 particle region as in part i the method allows direct graphical definition of general irreducible kernels and is well suited to the analysis of asymptotic completeness and related results in more general energy regions

self avoiding walk in five or more dimensions i the critical behaviour we use the lace expansion to study the standard self avoiding walk in thed dimensional hypercubic lattice ford 5 we prove that the numbercn ofn step self avoiding walks satisfiescn a n where is the connective constant i e 1 and that the mean square displacement is asymptotically linear in the number of steps i e v 1 2 a bound is obtained forcn x the number ofn step self avoiding walks ending atx the correlation length is shown to diverge asymptotically like z 1 2 the critical two point function is shown to decay at least as fast as x 2 and its fourier transform is shown to be asymptotic to a multiple ofk 2 ask 0 i e 0 we also prove that the scaling limit is gaussian in the sense of convergence in distribution to brownian motion the infinite self avoiding walk is constructed in this paper we prove these results assuming convergence of the lace expansion the convergence of the lace expansion is proved in a companion paper

proof of the riemannian penrose inequality using the positive mass theorem we prove the riemannian penrose conjecture an important case of a con jecture 41 made by roger penrose in 1973 by defining a new flow of metrics this flow of metrics stays inside the class of asymptotically flat riemannian 3 manifolds with nonnegative scalar curvature which contain minimal spheres in particular if we consider a riemannian 3 manifold as a totally geodesic submanifold of a space time in the context of general relativity then outermost minimal spheres with total area a correspond to apparent horizons of black holes contributing a massa 16 scalar cur vature corresponds to local energy density at each point and the rate at which the metric becomes flat at infinity corresponds to total mass also called the adm mass the riemannian penrose conjecture then states that the total mass of an asymptotically flat 3 manifold with nonnegative scalar curvature is greater than or equal to the mass contributed by the black holes the flow of metrics we define continuously evolves the original 3 metric to a schwarzschild 3 metric which represents a spherically symmetric black hole in vacuum we define the flow such that the area of the minimal spheres which flow outward and hence the mass contributed by the black holes in each of the metrics in the flow is constant and then use the positive mass theorem to show that the total mass of the metrics is nonincreasing then since the total mass equals the mass of the black hole in a schwarzschild metric the riemannian penrose conjecture follows we also refer the reader to the beautiful work of huisken and ilmanen 30 who used inverse mean curvature flows of surfaces to prove that the total mass is at least the mass contributed by the largest black hole in sections 1 and 2 we motivate the problem discuss important quantities like total mass and horizons of black holes and state the positive mass theorem and the penrose conjecture for riemannian 3 manifolds in section 3 we give the proof of the riemannian penrose