diffusion on self avoiding walk networks two properties of self avoiding walk networks constructed by connecting the nearest neighbour points of self avoiding walks on the square lattice are studied numerically the average end to end linear part of such networks is investigated and is found to be proportional to the length of the walk secondly random walks on this structure are studied and the associated spectral dimension of such networks is estimated to be 1 03 or 0 03 which implies that the effect of such connections affects the spectral dimension only weakly if at all

excessive measures and the existence of right semigroups and processes given a resolvent u on a lusin space e f the paper gives necessary and sufficient conditions in terms of the excessive measures that ensure the existence of a right process resp a right continuous semigroup on e f with resolvent u furthermore a notion of nonbranch points with respect to un is introduced also in terms of the excessive measures and various characterizations are given they show in particular the equivalence of this definition with those introduced and discussed by engelbert and wittmann

classical dirichlet forms on topological vector spaces the construction of the associated diffusion process given a minimal classical dirichlet form onl 2 e we construct the associated diffusion process heree is a locally convex topological vector space and is a not necessarily quasi invariant probability measure one the construction is carried out under certain assumptions one and which can be easily verified in many examples in particular we explicitly apply our results to time zero and space time quantum fields with or with out cut off

asymptotic properties of some confidence interval estimators for simulation output the classical confidence interval estimator commonly used in simulation is compared with four new estimators based on standardization of a time series presented in a previous paper these new interval estimators are shown to have asymptotic properties that strictly dominate the classical estimator when used with data from independent simulation replications or means of batched observations from a single simulation run two of the new estimators also can be used with a single unbatched replication of a simulation program a situation where the classical estimator is not defined

traces of harmonic functions and a new path space for the free quantum field the infinite dimensional ornstein uhlenbeck process associated with the free field of quantum field theory on rd is reconstructed as the traces of random harmonic functions these harmonic functions are the solutions of a generalised dirichlet problem with a random schwartz distribution as boundary data a detailed analysis of the associated trace space which is a subspace of a scaled sobolev space of negative fractional order leads to a new path space for the free field as well as to a convenient regularisation method because of the particular form of the trace space it is possible to give an explicit formula for the transition function of the process furthermore the dependence of the path space on the dimension d is discussed and a formulation of our result in terms of abstract wiener spaces is given

construction of convergent simplicial approximations of quantum fields on riemannian manifolds we construct simplicial approximations of random fields on riemannian manifolds of dimensiond we prove convergence of the fields to the continuum limit for arbitraryd in the gaussian case and ford 2 in the non gaussian case in particular we obtain convergence of the simplicial approximation to the continuum limit for quantum fields on riemannian manifolds with exponential interaction

infinite dimensional stochastic differential equations obtained by subordination and related dirichlet forms abstract new results related to the decomposition theorem of additive functionals associated to quasi regular dirichlet forms are presented a characterization of subordinate processes associated to quasi regular symmetric dirichlet forms in terms of the unique solutions of the corresponding martingale problems is obtained the subordinate of generalized ornstein uhlenbeck processes are exhibited explicitly in terms of generators dirichlet forms and unique pathwise solutions of stochastic differential equations sdes in the case where the state space is infinite dimensional as e g in euclidean quantum field theory the construction provides a characterization of the processes in terms of projections on the topological dual space and corresponding finite dimensional sdes

permuted standardized time series for steady state simulations we describe an extension procedure for constructing new standardized time series procedures from existing ones the approach is based on averaging over sample paths obtained by permuting path segments analytical and empirical results indicate that permuting improves standardized time series methods we compare permuting to an alternative extension procedure known as batching we demonstrate the permuting method by applying it to estimators based on the maximum and the area of a normalized path

classical dirichlet forms on topological vector spaces closability and a cameron martin formula abstract we prove a sufficient condition for the closability of classical dirichlet forms on l 2 e which is also necessary if all components of the dirichlet form are closable here e is a locally convex topological vector space and a not necessarily quasi invariant probability measure on e the same condition is shown to imply the existence of a closed extension whose domain can be described explicitly in the special case where is quasi invariant with respect to certain vectors in e our result generalises previous theorems on closability in addition we prove a cameron martin type formula for a large class of measures if e is finite dimensional our characterisation of closability is the analogue of the corresponding one dimensional result applications to quantum fields and the connection with the well studied case of abstract wiener spaces are discussed

monte carlo study of random walks on a 2d gasket fractal in an external field using monte carlo simulation random walks on a two dimensional sierpinski gasket in the presence of an external field were studied it was observed that the random walk motion of a particle displayed a crossover from anomalous diffusion to drift for a non zero bias field and the crossover time tcr was a decreasing function of the external bias field the associated dynamic exponents obtained in a computer simulation agree with the predictions of stinchcombe s scaling treatment 1985

note new confidence interval estimators using standardized time series we develop new asymptotically valid confidence interval estimators cie s for the underlying mean of a stationary simulation process the new estimators are weighted generalizations of schruben s standardized time series area cie we show that the weighted cie s have the same asymptotic expected length and variance of the length as the area cie but in the small sample environment the new cie s exhibit performance characteristics which are different from those of the area cie

central limit theorems for permuted regenerative estimators we prove strong laws of large numbers and central limit theorems for some permuted estimators from regenerative simulations these limit theorems provide the basis for constructing asymptotically valid confidence intervals for the permuted estimators

lp uniqueness of schrodinger operators and the capacitary positive improving property abstract we prove several lp uniqueness results for schrodinger operators l v by means of the feynman kac formula using the m p capacity theory for general markov semigroups we show that the associated feynman kac semigroup is positive improving in the sense of m p capacity improving the well known one in the sense of measure using that capacitary positive improving property and two new inequalities for generalized ornstein uhlenbeck generators we show the essential self adjointness of the ground state diffusion generator l l 2 associated with two dimensional euclidean quantum fields

optimal mean squared error batch sizes when an estimator of the variance of the sample mean is parameterized by batch size one approach for selecting batch size is to pursue the minimal mean squared error mse we show that the convergence rate of the variance of the sample mean and the bias of estimators of the variance of the sample mean asymptotically depend on the data process only through its marginal variance and the sum of the autocorrelations weighted by their absolute lags combining these results with variance results of goldsman and meketon we obtain explicit asymptotic approximations for mse optimal batch size optimal mse and robustness for four quadratic form estimators of the variance of the sample mean our empirical results indicate that the asymptotic approximations are reasonably accurate for sample sizes seen in practice although we do not discuss batch size estimation procedures the empirical results suggest that the explicit asymptotic batch size approximation which depends only on a summary measure which we re

resistance and related scaling properties of self avoiding random walks three scaling exponents for random and self avoiding chains are studied by direct renormalisation these describe the end to end resistance of a conducting chain the arc length of the shortest path traceable between the chain ends and the arc length of chain in no loop the scaling behaviours of these quantities are identical to first order in epsilon 4 d d is the space dimension for a long self avoiding chain the authors predict that each one one is proportional to the chain length

standardized time series l p norm variance estimators for simulations this paper studies a class of estimators for the variance parameter of a stationary stochastic process the estimators are based on lp norms of standardized time series and they generalize previously studied estimators due to schruben we show that the new estimators have some desirable properties they are asymptotically unbiased and have low asymptotic variance we also illustrate empirically the performance of the lp norm estimators on various stochastic processes

overlapping variance estimators for simulation to estimate the variance parameter i e the sum of covariances at all lags for a steady state simulation output process we formulate certain statistics that are computed from overlapping batches separately and then averaged over all such batches we form overlapping versions of the area and cramer von mises estimators using the method of standardized time series for these estimators we establish i their limiting distributions as the sample size increases while the ratio of the sample size to the batch size remains fixed and ii their mean square convergence to the variance parameter as both the batch size and the ratio of the sample size to the batch size increase compared with their counterparts computed from nonoverlapping batches the estimators computed from overlapping batches asymptotically achieve reduced variance while maintaining the same bias as the sample size increases moreover the new variance estimators usually achieve similar improvements compared with the conventional variance estimators based on nonoverlapping or overlapping batch means in follow up work we present several analytical and monte carlo examples and we formulate efficient procedures for computing the overlapping estimators with only order of sample size effort

random walk on self avoiding walk in external bias diffusion drift and trapping the effect of external bias on random walks rws on self avoiding walks on square lattices in the presence of bridges is studied by monte carlo simulation the qualitative as well as quantitative features of the latter problem are compared and contrasted with those of rws on percolation clusters in the presence of external bias

existence and uniqueness of solutions to stochastic functional differential equations in infinite dimensions abstract in this paper we present a general framework for solving stochastic functional differential equations in infinite dimensions in the sense of martingale solutions which can be applied to a large class of spde with finite delays e g d dimensional stochastic fractional navier stokes equations with delays d dimensional stochastic reaction diffusion equations with delays d dimensional stochastic porous media equations with delays moreover under local monotonicity conditions for the nonlinear terms we obtain the existence and uniqueness of strong solutions to spde with delays

dirichlet operators a priori estimates and the uniqueness problem abstract we study the equation du dt hu 0 where h is the operator associated with the dirichlet form of quasi invariant measure on hilbert space a priori estimates of the first and second order derivatives of the solutions of the equation are obtained then these estimates are applied to the problem of essential self adjointness of the operator h on domains which consist of smooth functions

strong uniqueness for sdes in hilbert spaces with nonregular drift we prove pathwise uniqueness for a class of stochastic dierential equations sde on a hilbert space with cylindrical wiener noise whose non linear drift parts are sums of the subdierential of a convex function and a bounded part this generalizes a classical result by one of the authors to innite dimensions our results also generalize

ergodicity ofl2 semigroups and extremality of gibbs states we extend the classical results of holley stroock on the characterization of extreme gibbs states for the ising model in terms of the irreducibility resp ergodicity of the corresponding glauber dynamics to the case of lattice systems with unbounded linear spin spaces we first develop a general framework to discuss questions of this type using classical dirichlet forms on infinite dimensional state spaces and their associated diffusions we then describe concrete applications to lattice models with polynomial interactions i e the discretep d models of euclidean quantum field theory in addition we prove the equivalence of extremality and shift ergodicity for tempered gibbs states of these models and also discuss this question in the general framework

large deviations w r t quasi every starting point for symmetric right processes on general state spaces in the work of donsker and varadhan fukushima and takeda and that of deuschel and stroock it has been shown that the lower bound for the large deviations of the empirical distribution of an ergodic symmetric markov process is given in terms of its dirichlet form we give a short proof generalizing this principle to general state spaces that include in particular infinite dimensional and non0metrizable examples our result holds w r t quasi every starting point of the markov process moreover we show the corresponding weak upper bound w r t quasi every starting point

dirichlet operators on loop spaces essential self adjointness and log sobolev inequality for each 0 1 and potential function v rd r we consider the dirichlet form e and the associated dirichlet operator h for the gibbs measure on the loop space e c 0 1 rd 0 1 the gibbs measure is related to the gibbs state of the quantum anharmonic oscillator with the potential v via the feynman kac formula we formulate dirichlet forms in the framework of rigged hilbert spaces which are related to the loop space e we then give an approximate criterion for the essential self adjointness of dirichlet operators associated with dirichlet forms given by probability measures on hilbert spaces under appropriate conditions on the potential we apply the approximate criterion to show that the dirichlet operator h is essentially self adjoint on the domain of smooth cylinder functions in addition if the potential satisfies a uniform convexity condition we prove that the dirichlet operator h has a gap at the lower end of spectrum we also show that the gibbs measure satisfie

uniformly integrable operators and large deviations for markov processes abstract in this paper we introduce and investigate the notion of uniformly integrable operators on l p e its relations to classical compactness and hypercontractivity are exhibited several consequences of this notion are established such as perron frobenius type theorems independence on p of the spectral radius in l p p continuity of spectral radius and especially the existence of spectral gap in the irreducible case we also present some infinitesimal criteria ensuring the uniform integrability of a positive semigroup or of its resolvent for a essentially irreducible markov process we show that the uniform integrability in l p of some type of resolvent associated with the transition semigroup implies the large deviation principle of level 3 with some rate function given by a modified donsker varadhan entropy functional we also prove that the uniform integrability condition becomes even necessary in the symmetric case finally we present several applications of our results to feynman kac semigroups to the thermodynamical limits of grand ensembles and to non symmetric markov processes given by girsanov s formula

uniqueness of generalized schrodinger operators and applications abstract we prove necessary and sufficient conditions on r d r 0 a e so that the operator s 2 1 w1 n has exactly one self adjoint extension on w1 p which generates a sub markovian semigroup on w1 n these conditions are shown to be always fulfilled if d 1 and are verified for a large class of functions if d 1 we also prove an infinite dimensional analogue of this result where r d is replaced by some topological vector space e and the lebesgue measure dx by a gaussian measure the role of is taken by some ornstein uhlenbeck operator with general linear drift and dom s bounded smooth cylinder functions both results are derived as special cases from a general perturbation result we also give the corresponding probabilistic version i e we prove uniqueness for the associated martingale problem we present in some detail applications to the case where is a ground state of a hamiltonian in quantum field theory settling a problem which has been open for some time furthermore we apply our results to prove uniqueness in distribution of the diffusion first constructed by g jona lasinio and p k mitter for the stochastic quantization of f 2 field theory

restricted markov uniqueness for the stochastic quantization of p φ 2 and its applications abstract in this paper we obtain restricted markov uniqueness of the generator and uniqueness of martingale probabilistically weak solutions for the stochastic quantization problem in both the finite and infinite volume case by clarifying the precise relation between the solutions to the stochastic quantization problem obtained by the dirichlet form approach and those obtained in 10 and in 24 we prove that the solution x z where x is obtained by the dirichlet form approach in 4 and z is the corresponding o u process satisfies the corresponding shifted equation see 1 4 below moreover we obtain that the infinite volume p 2 quantum field is an invariant measure for the x y z where y is the unique solution to the shifted equation

spectral dimension and the shortest path of saw s with multi neighbour interactions the spectral dimension and the shortest path of self avoiding walks saws with bridge length interaction range b 1 2 3 2 are studied numerically the spectral dimension is calculated by performing exact multi neighbour random walks on the monte carlo generated saw configurations it is found that the spectral dimension is not affected by the finite range of interactions finite bridge length and approach to ds 1 both in dimensions d 2 and 3 the shortest path length sn of the n step saw with local bridges is also investigated it is shown that sn n a n delta b c n the authors numerical simulation results indicate that the exponent delta is independent of the dimensionality and is about 3 16 in dimensions d 2 5

optimal tests for initialization bias in simulation output we present a family of tests for detecting initialization bias in the mean of a simulation output series using a hypothesis testing framework the null hypothesis is that the output mean does not change throughout the simulation run the alternative hypothesis specifies a general transient mean function the tests are asymptotically optimal based on cumulative sums of deviations about the sample mean a particular test in this family is applied to a variety of simulation models the test requires very modest computation and appears to be both robust and powerful

tightness of general c1 p capacities on banach space abstract we prove tightness of c 1 p capacities on a banach space e these capacities are defined in terms of measures on e which are not necessarily gaussian our methods are purely analytic and completely different from earlier approaches which were implemented under more restrictive assumptions

simulation output analysis using standardized time series the method of standardized time series sts was proposed by schruben as an approach for constructing asymptotic confidence intervals for the steady state mean from a single simulation run the sts method cancels out the variance constant while other methods attempt to consistently estimate the variance constant our goal in this paper is to generalize the sts method and to study some of its basic properties starting from a functional central limit theorem fclt for the sample mean of the simulated process a class of mappings of c 0 1 to is identified each of which leads to a sts confidence interval one of these mappings leads to the batch means method a lower bound is obtained for the expected length of the asymptotic as the run size becomes large sts confidence intervals this lower bound is not attained but can be approached arbitrarily closely by sts confidence intervals methods that consistently estimate the variance constant do realize this lower bound the variance of the length of a sts confidence interval is of larger order in the run length than is that for the regenerative method

a new topological approach to the l uniqueness of operators and the l1 uniqueness of fokker planck equations abstract the usual semigroups of kernels on a polish space e are in general not strongly continuous on l e with respect to the norm topology we introduce a new topology on l e such that they become c 0 semigroups for which we can establish a simplified hille yosida theorem the new topology will allow us to introduce the uniqueness of pre generator on l e which turns out to be equivalent to the l 1 uniqueness of the associated fokker planck equation among many others and it is intimately related with the liouville properties for l 1 harmonic functions the uniqueness of several second order elliptic differential operators in l are studied 1 one dimensional diffusion operators a x f b x f 2 schrodinger operators 1 2 v 3 multi dimensional diffusion generator 1 2

on closability of classical dirichlet forms we construct a measure on r2 such that the classical dirichlet form e f g f g d is closable but the partial dirichlet form ex f g xf xgd is not this proves the well known conjecture of m rockner

properties of standardized time series weighted area variance estimators we wish to estimate the variance of the sample mean from a continuous time stationary stochastic process this article expands on the results of a technical note goldsman and schruben 1990 by using the theory of standardized time series to investigate weighted generalizations of schruben s area variance estimator we find a simple expression for the bias of the weighted area variance estimator and we give weights which yield variance estimators with lower asymptotic bias than certain other popular estimators we use the weighted area variance estimators to derive asymptotically valid confidence interval estimators cies for the mean of a stationary stochastic process although the weighted area cies have the same asymptotic expected value and variance of the length as schruben s area cie we show that the new cies sometimes yield coverages which are closer to the nominal value

shortest path of saws with bridges series results recent monte carlo simulation results of yang and chakrabarti 1990 suggest that the shortest path sn of an n stepped self avoiding walk saw with finite range interactions of bridges has a finite size scaling behaviour sn n approximately a n delta b c n where the exponent delta is superuniversal delta approximately 0 19 for all dimensions d studied 2 or d or 5 the authors report the small n series enumeration results for sn for saws with nearest neighbour bridges on square up to n 18 triangular up to n 11 and simple cubic up to n 12 lattices the estimated values of delta for different lattices approximately 0 22 or 0 01 for d 2 and approximately 0 26 or 0 01 for d 3 have been compared with the above monte carlo estimate and the indication of a superuniversal behaviour for delta has been discussed

uniqueness of nelsons diffusions let 2 be a generalized schrodinger operator or generator of nelsons diffusion defined on c 0 d where is a continuous and strictly positive function on an open domain d d such that lloc2 d some results are given about the two questions below i whether does generate a unique semigroup in l1 d 2dx ii whether the semigroup determined by is strong feller

quantum interaction φ44 the construction of quantum field defined as a bilinear form we construct the solution t x of the quantum wave equation m2 3 0 as a bilinear form which can be expanded over wick polynomials of the free in field and where 3 t x is defined as the normal ordered product with respect to the free in field the constructed solution is correctly defined as a bilinear form on d d where d is a dense linear subspace in the fock space of the free in field on d d the diagonal of the wick symbol of this bilinear form satisfies the nonlinear classical wave equation

small world phenomena and the statistics of linear polymers a regular lattice in which the sites can have long range connections at a distance l with a probabilty p l l in addition to the short range nearest neighbour connections shows small world behaviour for 0 c in the most appropriate physical example of such a system namely the linear polymer network the exponent is related to the exponents of the corresponding n vector model in the n 0 limit and its value is less than c still the polymer networks do not show small world behaviour here we show that this is due to a small value constraint on the number q of long range connections per monomer in the network in the general q space we obtain a phase boundary separating regions with and without small world behaviour and show that the polymer network falls marginally in the regular lattice region

multivariate standardized time series for steady state simulation output analysis the theory of standardized time series initially proposed to estimate a single steady state mean from the output of a simulation is extended to the case where more than one steady state mean is to be estimated simultaneously under mild assumptions on the stochastic process representing the output of the simulation namely a functional central limit theorem we obtain asymptotically valid confidence regions for a multivariate steady state mean based on multivariate standardized time series we provide examples of multivariate standardized time series including the multivariate versions of the batch means method and schruben s standardized sum process large sample properties of confidence regions obtained from multivariate standardized time series are discussed we show that as in the univariate case the asymptotic expected volume of confidence regions produced by standardized time series procedures is larger than that obtained from a consistent estimation procedure we present and discuss experimental results that illustrate our theory

an investigation of finite sample behavior of confidence interval estimators we investigate the small sample behavior and convergence properties of confidence interval estimators cies for the mean of a stationary discrete process we consider cies arising from nonoverlapping batch means overlapping batch means and standardized time series all of which are commonly used in discrete event simulation the performance measures of interest are the coverage probability and the expected value and variance of the half length we use empirical and analytical methods to make detailed comparisons regarding the behavior of the cies for a variety of stochastic processes all the cies under study are asymptotically valid however they are usually invalid for small sample sizes we find that for small samples the bias of the variance parameter estimator figures significantly in cie coverage performance the less bias the better a secondary role is played by the marginal distribution of the stationary process we also point out that some cies require fewer observations before manifesting

stochastic analysis on product manifolds dirichlet operators on differential forms abstract we define a de rham complex over a product manifold infinite product of compact manifolds and dirichlet operators on differential forms associated with differentiable measures in particular gibbs measures which generalize the notions of bochner and de rham laplacians we give probabilistic representations for corresponding semigroups and study properties of the corresponding stochastic dynamics

nonlinear parabolic stochastic differential equations with additive colored noise on bf r d times bf r _ a regulated stochastic quantization we prove the existence of solutions to the nonlinear parabolic stochastic differential equation partial mathord left vphantom partial partial t right kern nulldelimiterspace partial t delta varphi v varphi eta c for polynomialsv of even degree with positive leading coefficient and c a gaussian colored noise process onr d r when c is colored enough that the gaussian solution to the linear problem has holder continuous covariance the nongaussian processes are almost surely realized by continuous functions uniqueness regularity properties asymptotic perturbation expansions and nonperturbative fluctuation bounds are obtained for the infinite volume processes these equations are a cutoff version of the parisi wu stochastic quantization procedure forp d models and the results of this paper rigorously establish the nonperturbative nature of regularization via modification of the noise process in the limit c gaussian white noise we find that the asymptotic expansion and the rigorous bounds agree for processes corresponding to the regulated stochastic quantization of super renormalizable and small coupling strictly renormalizable scalar field theories and disagree for nonrenormalizable models

confidence interval estimation using standardized time series observations of a stationary stochastic process can be transformed into a standardized time series this paper presents a lemma giving the asymptotic properties of this standardized series under quite general conditions in particular the conditions are satisfied by stationary discrete event simulations confidence intervals can be constructed using this lemma for illustration we develop two easily computed interval estimators for the process mean when independent replications of the series are available such as in computer simulation experiments these interval estimators may be combined with the classical confidence interval estimator these interval estimators also tend to compensate for simulation initialization bias if the sign of the bias is known in an empirical study using three elementary simulated processes the interval estimators presented here compare favorably with the classical interval estimator in a recent paper goldsman and schruben goldsman d l schruben 1982 asymptotic properties of some confidence interval estimators tech rep 544 school of o r i e cornell university ithaca ny show that the asymptotic properties of the confidence intervals presented in this paper strictly dominate those of classical confidence intervals

fluctuations and their glauber dynamics in lattice systems abstract we consider a class of unbounded spin systems containing in particular anharmonic classical crystals and construct the stochastic dynamics in the space of macroscopic fluctuations starting from a given microscopic stochastic time evolution

uniqueness of schrodinger operators restricted in a domain abstract let s 2 v be the schrodinger s operator defined on c 0 d where d is a open domain in r d by means of the asymptotic behavior of v near the boundary d we give the necessary and sufficient conditions to the essential markovian self adjointness of s for the nonnegative potential v and to the uniqueness of s in l 1 d for general v

stochastic quantization of field theory in finite and infinite volume abstract using the theory of dirichlet forms a distribution valued symmetric markov process is constructed so that it has the 4 2 measure as its invariant probability measure several properties of this process are proved and a finite to infinite volume limit theorem is established

entropy reversible diffusion processes and markov uniqueness abstract consider a symmetric bilinear form e defined on c c r d by formula in this paper we study the stochastic process associated with the smallest closed markovian extension of e c c and give a new proof of markov uniqueness i e the uniqueness of a closed markovian extension based on purely probabilistic arguments we also give another purely analytic one as a consequence we show that all invariant measures are reversible provided they are of finite energy the problem of uniqueness of such measures is also partially solved

binary jumps in continuum i equilibrium processes and their scaling limits let denote the space of all locally finite subsets configurations in rd a stochastic dynamics of binary jumps in continuum is a markov process on in which pairs of particles simultaneously hop over rd in this paper we study an equilibrium dynamics of binary jumps for which a poisson measure is a symmetrizing and hence invariant measure the existence and uniqueness of the corresponding stochastic dynamics are shown we next prove the main result of this paper a big class of dynamics of binary jumps converge in a diffusive scaling limit to a dynamics of interacting brownian particles we also study another scaling limit which leads us to a spatial birth and death process in continuum a remarkable property of the limiting dynamics is that its generator possesses a spectral gap a property which is hopeless to expect from the initial dynamics of binary jumps

poincare inequality for weighted first order sobolev spaces on loop spaces abstract let e be the loop space over a compact connected riemannian manifold with a torsion skew symmetric connection let l d be the ornstein uhlenbeck operator on a nonempty connected component d of the loop space e and let v d r be the restriction on d of the potential in the logarithmic sobolev inequality found by l gross on the loop group by s aida and by f z gong and z m ma on the loop space respectively we prove that the schrodinger operator l v l d v always has a spectral gap at the bottom 0 v of its spectrum and thus has its ground state transformed operator 1 l v 0 v where is the unique ground state of l v in particular our result proves l gross s conjecture about the existence of a spectral gap for the ground state transform of the schrodinger operator studied by him on the loop group in addition in all the above cases we identify the domain of the dirichlet forms associated with the ground state transforms as weighted first order sobolev spaces with weight given by 2 thus establishing a poincare inequality for them all these results are consequences from some new results in this paper on dirichlet forms characterizing certain classes with spectral gaps and from results by s aida and m hino

two inequalities for symmetric diffusion semigroups underγ3 0 abstract from the meyer bakry inequality we deduce two inequalities one is an extension of the gagliardo nirenberg inequality for multiplication of sobolev functions the other extends an inequality due to maz ya for composition of nonnegative sobolev functions

brownian motions on metric graphs brownian motions on a metric graph are defined their generators are characterized as laplace operators subject to wentzell boundary at every vertex conversely given a set of wentzell boundary conditions at the vertices of a metric graph a brownian motion is constructed pathwise on this graph so that its generator satisfies the given boundary conditions

non self adjoint jacobi matrices with a rank one imaginary part we develop direct and inverse spectral analysis for finite and semi infinite non self adjoint jacobi matrices with a rank one imaginary part it is shown that given a set of n not necessarily distinct nonreal numbers in the open upper lower half plane uniquely determines an n n jacobi matrix with a rank one imaginary part having those numbers as its eigenvalues counting algebraic multiplicity algorithms of reconstruction for such finite jacobi matrices are presented a new model complementing the well known livsic triangular model for bounded linear operators with a rank one imaginary part is obtained it turns out that the model operator is a non self adjoint jacobi matrix we show that any bounded prime non self adjoint linear operator with a rank one imaginary part acting on some finite dimensional respectively separable infinite dimensional hilbert space is unitarily equivalent to a finite respectively semi infinite non self adjoint jacobi matrix this obtained theorem strengthens a classical result of stone established for self adjoint operators with simple spectrum we establish the non self adjoint analogs of the hochstadt and gesztesy simon uniqueness theorems for finite jacobi matrices with nonreal eigenvalues as well as an extension and refinement of these theorems for finite non self adjoint tri diagonal matrices to the case of mixed eigenvalues real and nonreal a unique jacobi matrix unitarily equivalent to the operator of integration ff x 2i xlf t dt in the hilbert space l2 0 l is found as well as spectral properties of its perturbations and connections with the well known bernoulli numbers we also give the analytic characterization of the weyl functions of dissipative jacobi matrices with a rank one imaginary part

uniqueness and the global markov property for euclidean fields the case of general exponential interaction the uniqueness and the global markov property for the regular gibbs measure corresponding to the interaction u lambda varphi lambda int limits lambda d 2 x int d varrho alpha e alpha varphi 0 x for 0 d a probability measure with support in 2 sqrt pi 2 sqrt pi is proved

dirichlet forms and dirichlet operators for gibbs measures of quantum unbounded spin systems essential self adjointness and log sobolev inequality for each 0 1 we consider the dirichlet form mu gamma and the associated dirichlet operator h mu gamma for the gibbs measure of quantum unbounded spin systems interacting via superstable and regular potential the gibbs measure is related to the gibbs state of the system via a functional euclidean integral procedure the configuration space for the spin systems is given by omega e mathbb z v e left omega in c left left 0 1 right mathbb r d right omega left 0 right omega left 1 right right we formulate dirichlet forms in the framework of rigged hilbert spaces which are related to the space under appropriate conditions on the potential we show that the dirichlet operator h mu left gamma right is essentially self adjoint on the domain of smooth cylinder functions we give sufficient conditions on the potential so that the corresponding gibbs measure is uniformly log concave ulc this property gives the spectral gap of the dirichlet operator h mu left gamma right at the lower end of the spectrum furthermore we prove that under the conditions of ulc the unique gibbs measure satisfies the log sobolev inequality ls we use an approximate argument used in the study of the same subjects for loop spaces which in turn is a modification of the method originally developed by s albeverio yu g kondratiev and m rockner

green s and dirichlet spaces associated with fine markov processes this is the second paper in a series devoted to green s and dirichlet spaces in the first paper we have investigated green s space k and the dirichlet space h associated with a symmetric markov transition function pt x b now we assume that p is a transition function of a fine markov process x and we prove that a the space h can be built from functions which are right continuous along almost all paths b the positive cone k in k can be identified with a cone m of measures on the state space c the positive cone h in h can be interpreted as the cone of green s potentials of measures m to every measurable set b in the state space e there correspond a subspace k b of k and a subspace h b of h the orthogonal projections of k onto k and of h onto h b can be expressed in terms of the hitting probabilities of b by the markov process x as the main tool we use additive functionals of x corresponding to measures m

universality of symmetric one dimensional random flights the asymptotic behaviour of all the moments of a class of symmetric one dimensional random flights at the first few orders are calculated different symmetric random flights are considered and it is shown that they simply differ up to a scaling in the number of jumps and that important statistical quantities are universal

formes de dirichlet generales et densite des variables aleatoires reelles sur l espace de wiener abstract it is shown that the main properties of dirichlet spaces can be proved without the classical local compacity hypothesis to this purely measurable framework we extend the functional calculus and the energy image density property these results are then applied to the ornstein uhlenbeck semi group on wiener space where it is obtained that the real random variables which are solutions of lipschitzian stochastic differential equations always have probability densities after perhaps an initial deterministic phase

on the contraction property of energy forms on infinite dimensional space abstract we prove that certain energy forms on infinite dimensional space with polynomial domain have the contraction or dirichlet property some consequences are discussed

folded overlapping variance estimators for simulation abstract we propose and analyze a new class of estimators for the variance parameter of a steady state simulation output process the new estimators are computed by averaging individual estimators from folded standardized time series based on overlapping batches composed of consecutive observations the folding transformation on each batch can be applied more than once to produce an entire set of estimators we establish the limiting distributions of the proposed estimators as the sample size tends to infinity while the ratio of the sample size to the batch size remains constant we give analytical and monte carlo results showing that compared to their counterparts computed from nonoverlapping batches the new estimators have roughly the same bias but smaller variance in addition these estimators can be computed with order of sample size work

random walk on self avoiding walk a model for conductivity of linear polymers random walks on self avoiding walks saws are studied using monte carlo techniques on a square lattice with nearest neighbour hopping along the chain and between saw points which are nearest neighbours on the embedding lattice the average of the square of the end to end distance for random walks of t steps on saws of length n is fitted to the scaling forms rt2 varies as ndelta tk for t or approximately ntheta where theta approximately 2 nu s k nu s being the average end to end distance exponent for saws the observed value of the exponent delta is supported by the authors real space renormalisation group result for the conductivity of saw chains the exponent k has been related to the effective fractal dimension of the saw chain

real space renormalisation group approach to random walks on self avoiding walks the t dependence of the average of the end to end distance rt2 of t step random walks on self avoiding walks is studied by small cell real space renormalisation the results are in qualitative agreement with earlier monte carlo data

self avoiding walk model for proteins a real space renormalisation group treatment diffusion on the model for proteins introduced by helman coniglio and tsallis 1984 hct is discussed using real space renormalisation group ideas the authors perform cell to cell transformations in two dimensions for both the hct model and the self avoiding walk the latter system is quasi linear and so they expect trivial diffusive behaviour given by dw 2df 8 3 diffusion on the hct model however is complicated by the possibility of jumps across nearest neighbour hydrogen bond bridges the authors results are not consistent with the prediction that dw 2 for the hct model rather they support the assertion that dwhct is close to but slightly greater than dwsaw

on the uniqueness problem for dirichlet operators abstract strong uniqueness in l 2 and in l 1 for dirichlet operators in finite dimensional spaces is studied

scaling behavior of the contact process in networks with long range connections we present simulation results for the contact process on regular cubic networks that are composed of a one dimensional lattice and a set of long edges with unbounded length networks with different sets of long edges are considered that are characterized by different shortest path dimensions and random walk dimensions we provide numerical evidence that an absorbing phase transition occurs at some finite value of the infection rate and the corresponding dynamical critical exponents depend on the underlying network furthermore the time dependent quantities exhibit log periodic oscillations in agreement with the discrete scale invariance of the networks in case of spreading from an initial active seed the critical exponents are found to depend on the location of the initial seed and break the hyperscaling law of the directed percolation universality class due to the inhomogeneity of the networks however if the cluster spreading quantities are averaged over initial sites the hyperscaling law is restored

a real space renormalisation group study of the ising model on self avoiding walk chains here the authors apply a small cell real space renormalisation group technique to study the critical behaviour of the nearest neighbour interacting ising model on self avoiding walk saw chains in the limit of vanishing interactions between nearest neighbours which are not connected by the walk itself the treatment reproduces exactly the critical behaviour of the linear chain ising model when all the nearest neighbour interactions are allowed the treatment gives a vanishing tc with the small cells chosen for the ising model on saws on the square and triangular lattices the study of the critical behaviour of such models indicates some interesting departures from that of the ising model on linear chains

ergodicity for the stochastic dynamics of quasi invariant measures with applications to gibbs states the convex set maof quasi invariant measures on a locally convex spaceewith given shift radon nikodym derivatives i e cocycles a atk k k0 t ris analyzed the extreme points of maare characterized and proved to be non empty a specification of lattice type is constructed so that macoincides with the set of the corresponding gibbs states as a consequence via a well known method due to dynkin and follmer a unique representation of an arbitrary element in main terms of extreme ones is derived furthermore the corresponding classical dirichlet forms e d e and their associated semigroups t t t 0onl2 e are discussed under a mild positivity condition it is shown that mais extreme if and only if e d e is irreducible or equivalently t t t 0is ergodic this implies time ergodicity of associated diffusions applications to gibbs states of classical and quantum lattice models as well as those occuring in euclidean quantum field theory are presented in particular it is proved that the stochastic quantization of a guerra rosen simon gibbs state on d r2 ininfinite volumewith polynomial interaction is ergodic if the gibbs state is extreme i e is a pure phase which solves a long standing open problem

non symmetric dirichlet forms on semifinite von neumann algebras abstract the theory of non symmetric dirichlet forms is generalized to the non abelian setting also establishing the natural correspondences among dirichlet forms sub markovian semigroups and sub markovian resolvents within this context some results on the allowed functional calculus for closed derivations on hilbert algebras are obtained examples of non symmetric dirichlet forms given by derivations on hilbert algebras are studied

on the small time behavior of ornstein uhlenbeck processes with unbounded linear drifts in this paper we study the small time behavior of ornstein ulenbeck processes with unbounded linear drifts on hilbert spaces the large deviations estimates are obtained the general theory of dirichlet forms lyons zheng s decompositions and the convolution representation of the processes play an important role

specifications and martin boundaries for scr p phi _2 random fields it is shown thatp 2 gibbs states in the sense of guerra rosen and simon are given by a specification the construction of the specification is based on finding a proper version of the interaction density given by the polynomialp the existence of this version follows from the fact that all powers of the solution of a dirichlet problem for an open bounded setu with boundary data given by a distribution are integrable onu as a consequence the martin boundary theory for specifications can be applied top 2 random fields it follows that anyp 2 gibbs state can be represented in terms of extreme gibbs states in certain cases the extreme gibbs states are characterized in terms of harmonic functions it follows in particular that for any given boundary condition introduced so far the associated cutoffp 2 measure has a representation as an integral over harmonic functions

transportation information inequalities for markov processes in this paper one investigates the transportation information tci inequalities tc i for all probability measures on a metric space mathcal x d where is a given probability measure tc is the transportation cost from to with respect to the cost function c x y on mathcal x 2 i is the fisher donsker varadhan information of with respect to and 0 0 is a left continuous increasing function using large deviation techniques it is shown that tci is equivalent to some concentration inequality for the occupation measure of a reversible ergodic markov process related to i the tensorization property of tci and comparisons of tci with poincare and log sobolev inequalities are investigated several easy to check sufficient conditions are provided for special important cases of tci and several examples are worked out

cramer von mises variance estimators for simulations we study estimators for the variance parameter 2 of a stationary process the estimators are based on weighted cramer von mises statistics and certain weightings yield estimators that are first order unbiased for 2 we derive an expression for the asymptotic variance of the new estimators this expression is then used to obtain the first order unbiased estimator having the smallest variance among fixed degree polynomial weighting functions our work is based on asymptotic theory however we present exact and empirical examples to demonstrate the new estimators small sample robustness we use a single batch of observations to derive the estimators asymptotic properties and then we compare the new estimators among one another in real life applications one would use more than one batch we indicate how this generalization can be carried out

potential theory of infinite dimensional levy processes abstract we study the potential theory of a large class of infinite dimensional levy processes including brownian motion on abstract wiener spaces the key result is the construction of compact lyapunov functions i e excessive functions with compact level sets then many techniques from classical potential theory carry over to this infinite dimensional setting thus a number of potential theoretic properties and principles can be proved answering long standing open problems even for the brownian motion on abstract wiener space as e g formulated by r carmona in 1980 in particular we prove the analog of the known result that the cameron martin space is polar in the levy case and apply the technique of controlled convergence to solve the dirichlet problem with general not necessarily continuous boundary data

feynman kac semigroups ground state diffusions and large deviations we study the generalized schrodinger operator l v where l is the generator of a symmetric markov semigroup pt on l2 e m and the corresponding dirichlet form ev by means of the cramer functional v we give necessary and sufficient conditions for ev to be lower bounded and for the feynman kac semigroup pvt to be bounded some sufficient conditions for the essential self adjointness of l v are also given by means of large deviations we find a new condition which ensures the existence of ground state of l v and we construct the ground state process q t whose generator is given in the diffusion case by l l 1 where is the square field operator associated to l the self adjointness of l is discussed as applications we consider perturbation of the semigroups of second quantization on an abstract wiener space the time evolution of euclidean quantum fields and stochastic quantization

stochastic embedding of dynamical systems most physical systems are modeled by an ordinary or a partial differential equation like the n body problem in celestial mechanics in some cases for example when studying the long term behavior of the solar system or for complex systems there exist elements which can influence the dynamics of the system which are not well modeled or even known one way to take these problems into account consists of looking at the dynamics of the system on a larger class of objects that are eventually stochastic in this paper we develop a theory for the stochastic embedding of ordinary differential equations we apply this method to lagrangian systems in this particular case we extend many results of classical mechanics namely the least action principle the euler lagrange equations and noether s theorem we also obtain a hamiltonian formulation for our stochastic lagrangian systems many applications are discussed at the end of the paper

the essential self adjointness of generalized schrodinger operators a necessary and sufficient condition is given for the generalized schrodinger operator a 12 i 1n di di to be essentially self adjoint in l2 dx under general assumptions on and for arbitrary domains in rn in particular if is strictly positive and locally lipschitz continuous on rn then a is essentially self adjoint examples of non essential self adjointness and a complete discussion of the one dimensional case are also given these results have applications to the problem of the essential self adjointness of quantum hamiltonians and to the uniqueness problem of markov processes

critical dynamics of heisenberg spins on self avoiding walk chains the dynamical exponent z for the critical spin wave dynamics of nearest neighbour interacting heisenberg spins on a self avoiding walk saw chain is estimated here using a scaling picture and also applying a real space renormalisation group technique to some quasilinear fractal lattices the results indicate z 2dt where d is the fractal dimensionality of the saw chain lattice and t is the exponent for the length of the shortest nearest neighbour connecting path of the saw