h c1 maps and elliptic spdes with polynomial and exponential perturbations of nelson s euclidean free field abstract elliptic stochastic partial differential equations spde with polynomial and exponential perturbation terms defined in terms of nelson s euclidean free field on rd are studied using results by s kusuoka and a s ustunel and m zakai concerning transformation of measures on abstract wiener space spdes of this type arise in particular in euclidean quantum field theory with interactions of the polynomial or exponential type the probability laws of the solutions of such spdes are given by girsanov probability measures that are non linearly transformed measures of the probability law of nelson s free field defined on subspaces of schwartz space of tempered distributions

von neumann stability analysis of first order accurate discretization schemes for one dimensional 1d and two dimensional 2d fluid flow equations abstract in literature the von neumann stability analysis of simplified model equations such as the wave equation is typically used to determine stability conditions for the non linear partial differential fluid flow equations navier stokes and euler however practical experience suggests that such simplistic stability conditions are grossly inadequate for computations involving the system of coupled flow equations the goal of this paper is to determine stability conditions for the full system of fluid flow equations the euler equations are examined as any conditions derived for the euler equations will apply to the navier stokes ns equations in the limit of convection dominated flows a von neumann stability analysis is conducted for the one dimensional 1d and two dimensional 2d euler equations the system of equations is discretized on a staggered grid using finite difference discretization techniques the use of a staggered grid allows equivalence to finite volume discretization by combining the different discretization techniques ten solution schemes are formulated eight solution schemes are considered for the 1d euler equations and two schemes for the 2d euler equations for each scheme error amplification matrices are determined from the stability analysis stable and unstable regimes are identified and practical stability limits are predicted in terms of the maximum allowable cfl courant friedrichs lewy number as a function of mach number the predictions are verified for selected schemes using the riemann problem at incompressible and compressible mach numbers very good agreement is obtained between the analytically predicted and the experimentally observed cfl values the successfully tested stability limits are presented in graphical form which offer a viable alternative to complicated mathematical expressions often reported in published literature and should benefit everyday cfd computational fluid dynamics users the stability regions are used to discuss the effect of time integration explicit vs implicit density bias in continuity equation and momentum convection term linearization on stability a comparison of the predicted stability limits for 1d and 2d euler equations with commonly used stability conditions arising from the wave equation shows that the stability thresholds for the euler equations lie well below those predicted by the wave equation analysis in addition the 2d euler stability limits are more restrictive as compared to 1d euler limits since the present analysis accounts for the full system of fluid flow euler equations the derived stability conditions can be used by cfd practitioners to estimate a timestep or cfl number to guide the stability of their computations

covariant spdes and quantum field structures covariant stochastic partial differential equations spdes are studied in any dimension a special class of such equations is selected and it is proved that the solutions can be analytically continued to minkowski spacetime yielding tempered wightman distributions which are covariant obey the locality axiom and a weak form of the spectral axiom

central finite difference schemes for nonlinear dispersive waves abstract spatial finite difference and three level time schemes for the korteweg de vries kdv and related equations are studied the stability conditions are derived from a uniform framework based on the schur cohn theory of simple von neumann polynomials numerical results and comparisons with other methods are considered

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

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

corrigendum for the paper invariant tori for nearly integrable hamiltonian systems with degeneracy in the paper 1 the authors obtain a kam theorem for nearly integrable hamiltonian systems under the russmann s non degeneracy condition which is known to be sharpest one for small divisor conditions however the remark 1 3 is wrong because we have ignored the null set which may contain zeros of of high order such that 1 4 does not hold for all p the remark 1 3 might mislead the readers that the condition 1 5 of theorem b is equivalent to the russmann s non degeneracy condition actually the russmann s non degeneracy condition is equivalent to the condition 1 4 of theorem a as proved in 1 under the russmann s non degeneracy condition 1 4 as proved in the remark 3 1 the condition 1 5 holds if we replace n 1 by a sufficiently large number n depending on h and then the conclusion of theorem b remains valid if in the measure estimate n 1 is replaced by n

stochastic boundary value problems a white noise functional approach we give a program for solving stochastic boundary value problems involving functionals of multiparameter white noise as an example we solve the stochastic schrodinger equation ie391 1 wherev is a positive noisy potential we represent the potentialv by a white noise functional and interpret the product of the two distribution valued processesv andu as a wick productv u such an interpretation is in accordance with the usual interpretation of a white noise product in ordinary stochastic differential equations the solutionu will not be a generalized white noise functional but can be represented as anl 1 functional process

verification of the global markov property in some class of strongly coupled exponential interactions we verify the global markov property in some class of strongly coupled exponential interactions in two dimensional space time to obtain this result we apply the albeverio and ho egh krohn strategy the basic ingredients we use in order to employ this strategy are the fortuin kastelyn ginibre correlation inequalities

global flows with invariant gibbs measures for euler and navier stokes two dimensional fluids we construct a family of probability spaces p 0 associated with the euler equation for a two dimensional inviscid incompressible fluid which carries a pointwise flow t time evolution leavingp globally invariant t is obtained as the limit of galerkin approximations associated with euler equations p is also in invariant measure for a stochastic process associated with a navier stokes equation with viscosity stochastically perturbed by a white noise force

uniqueness and global markov property for euclidean fields the case of general polynomial interactions we give a general method for proving uniqueness and global markov property for euclidean quantum fields the method is based on uniform continuity of local specifications proved by using potential theoretical tools and exploitation of a suitable fkg order structure we apply this method to give a proof of uniqueness and global markov property for the gibbs states and to study extremality of gibbs states also in the case of non uniqueness in particular we prove extremality for 2 4 also in the case of non uniqueness and global markov property for weak coupling 2 4 which solves a long standing problem uniqueness and extremality holds also at any point of differentiability of the pressure with respect to the external magnetic field

weak poincare inequalities on domains defined by brownian rough paths we prove weak poincare inequalities on domains which are inverse images of open sets in wiener spaces under continuous functions of brownian rough paths the result is applicable to dirichlet forms on loop groups and connected open subsets of path spaces over compact riemannian manifolds

boosting the accuracy of finite difference schemes via optimal time step selection and non iterative defect correction abstract in this article we present a unified analysis of the simple technique for boosting the order of accuracy of finite difference schemes for time dependent partial differential equations pdes by optimally selecting the time step used to advance the numerical solution and adding defect correction terms in a non iterative manner the power of the technique which is applicable to time dependent semilinear scalar pdes where the leading order spatial derivative has a constant coefficient is its ability to increase the accuracy of formally low order finite difference schemes without major modification to the basic numerical algorithm through straightforward numerical analysis arguments we explain the origin of the boost in accuracy and estimate the computational cost of the resulting numerical method we demonstrate the utility of optimal time step ots selection combined with non iterative defect correction nidc on several different types of finite difference schemes for a wide array of classical linear and semilinear pdes in one and more space dimensions on both regular and irregular domains

a composite integration scheme for the numerical solution of systems of parabolic pdes in one space dimension abstract carroll j a composite integration scheme for the numerical solution of systems of parabolic pdes in one space dimension journal of computational and applied mathematics 46 1993 327 343 we apply the numerical method of lines to approximate the solution of a system of parabolic partial differential equations in one space dimension by approximating the spatial derivatives by discrete values of the solution at a set of mesh points in space we use these approximations to represent the partial differential equations at each mesh point giving a semidiscrete system of ordinary differential equations in the time direction the resulting ode system is integrated numerically using a second order l stable composite integration scheme the author this journal 1989 using variable stepsize integration an implementation of the composite scheme is applied to approximate the solution of some test problems selected from the literature using a uniform spatial mesh we present the results of a number of numerical experiments which allow an appraisal of the proposed approach with published works

generalized stochastic processes in algebras of generalized functions stochastic processes with paths in a generalized function algebra are defined and it is shown that there exists an embedding of generalized functional stochastic processes into such ones gaussian stochastic processes with paths in an algebra of generalized functions are characterized by their first and second moments and an application to stochastic differential equations is given

stationary solutions of stochastic parabolic and hyperbolic sine gordon equations the authors consider the damped sine gordon equation perturbed by thermal space time noise in the form it arises in the theory of the josephson junction and charge density waves they announce a rigorous proof that the coupling constant expansion of the solution of the initial value problem converges for small coupling taking the limit of the initial time t0 to infinity they obtain a stationary solution it is shown that the stationary solution is localized both in time and space even if the solution at finite time is not

solution of the monge ampere equation on wiener space for general log concave measures abstract in this work we prove that the unique 1 convex solution of the monge kantorovitch measure transportation problem between the wiener measure and a target measure which has an h log concave density in the sense of feyel and ustunel j funct anal 176 2000 400 428 w r t the wiener measure is also the strong solution of the monge ampere equation in the frame of infinite dimensional frechet spaces we further enhance the polar factorization results of the mappings which transform a spread measure to another one in terms of the measure transportation of monge kantorovitch and clarify the relation between this concept and the ito solutions of the monge ampere equation

equivalence of euclidean and wightman field theories a new inversion formula for the laplace transformation of tempered distributions with supports in the closed positive semiaxis is obtained the inverse laplace transform of a tempered distribution is defined by means of a limit of a special distribution constructed from this distribution the weak spectral condition on the euclidean green s functions implies that some of the limits needed for the inversion formula exist for any euclidean green s function with an even number of variables we then prove that the initial osterwalder schrader axioms 1 and the weak spectral condition are equivalent with the wightman axioms

difference schemes for the parabolic wave equation in ocean acoustics abstract we introduce and analyze a collection of difference schemes for the numerical solution of a model multidimensional equation of schrodinger type with applications to the three dimensional parabolic wave equation arising from the sound propagation in the ocean this collection of methods includes explicit and implicit schemes two level and three level schemes and real and complex schemes many of these are analogous to classical schemes for the heat equation and the wave equation but some schemes are unique to the schrodinger equation von neumann type stability results are given for all schemes numerical results arising from the application to an ocean acoustic problem are presented

variational calculation of laplace transforms via entropy on wiener space and applications abstract let w h be the classical wiener space where h is the cameron martin space which consists of the primitives of the elements of l 2 0 1 d t r d we denote by l a 2 h the equivalence classes w r t d t d whose lebesgue densities s u s w are almost surely adapted to the canonical brownian filtration if f is a wiener functional s t 1 e e f e f d is of finite relative entropy w r t we prove that j inf e f u 1 2 u h 2 u l a 2 h log e e f inf w f d h p w where p w is the set of probability measures on w b w and h is the relative entropy of w r t we call f a tamed functional if the inequality above can be replaced with equality we characterize the class of tamed functionals which is much larger than the set of essentially bounded wiener functionals we show that for a tamed functional the minimization problem of l h s has a solution u 0 if and only if u 0 i w u 0 is almost surely invertible and d u 0 d e f e e f and then u 0 is unique to do this we prove the theorem which says that the relative entropy of u 0 is equal to the energy of u 0 if and only if it has a a s left inverse we use these results to prove the strong existence of the solutions of stochastic differential equations with singular functional drifts and also to prove the non existence of strong solutions of some stochastic differential equations

a jost schroer theorem for string fields it is shown that the truncated wightman functions of three or more string localized fields vanish if they are solutions of a klein gordon equation in each variable as an application it is shown that a string field is a free field if its two point functions are those of a free field another application to perturbation theory is pointed out

persistence of lagrange invariant tori at tangent degeneracy abstract in the present paper without russmann s nondegenderate condition we show the persistence of kam invariant tori using this theorem we obtain the persistence of invariant tori for degenerate systems rising from restricted three body problems especially for lagrange equilibrium position l2 with certain degeneracy

some examples concerning the global markov property we present examples of interactions of classical lattice systems whose extremal gibbs states fail to have the global markov property one of the examples is translation invariant

stability analysis of six point finite difference schemes for the constant coefficient convective diffusion equation abstract a comprehensive and systematic study is presented to derive stability properties of various two level six point finite difference schemes in particular difference schemes of pade type for the approximation to the constant coefficient convective diffusion equation first the modified equivalent partial differential equation mepde for a general six point difference scheme is derive the mepde provides direct information on the order of accuracy of a difference scheme the von neumann and matrix methods are then employed to deduce the necessary and sufficient conditions for the numerical stability for the six point difference schemes an unified technique is developed to find the stability regions for the difference schemes some new second and third order six point difference schemes for the approximation of the constant coefficient convective diffusion equation are presented

measures induced on wiener space by monotone shifts in this work we study the absolute continuity of the image of the wiener measure under the transformations of the formt u the shiftu is a random variable with values in the cameron martin spaceh and is monotone in the sense that t h t h h 0 a s for allh inh

euclidean formulation of quantum field theory without positivity general properties of local quantum field theories qft without positivity are discussed in connection with their euclidean formulation modified euclidean axioms for local qft s without positivity are presented which allow us to recover by analytic continuation wightman functions satisfying the modified wightman axioms for indefinite metric qft s

an analysis of time discretization in the finite element solution of hyperbolic problems abstract the problem of the time discretization of hyperbolic equations when finite elements are used to represent the spatial dependence is critically examined a modified equation analysis reveals that the classical second order accurate time stepping algorithms i e the lax wendroff leap frog and crank nicolson methods properly combine with piecewise linear finite elements in advection problems only for small values of the time step on the contrary as the courant number increases the numerical phase error does not decrease uniformly at all wavelengths so that the optimal stability limit and the unit cfl property are not achieved these fundamental numerical properties can however be recovered while still remaining in the standard galerkin finite element setting by increasing the order of accuracy of the time discretization this is accomplished by exploiting the taylor series expansion in the time increment up to the third order before performing the galerkin spatial discretization using piecewise linear interpolations as a result it appears that the proper finite element equivalents of second order finite difference schemes are implicit methods of incremental type having third and fourth order global accuracy on uniform meshes taylor galerkin methods numerical results for several linear examples are presented to illustrate the properties of the taylor galerkin schemes in one and two dimensional calculations

phase diagrams and cluster expansions for low temperature models i the phase diagram low temperature phase diagrams of two dimensional open image in new window quantum field models are constructed let open image in new window lie in an r 1 dimensional space of perturbations of a polynomial withr degenerate minima perform a scaling open image in new window and assume 1 we constructk distinct states on left begin array 20 c r k end array right hypersurfaces of codimensionk 1 in the space of perturbations an expansion is used to exhibit exponential clustering of the schwinger functions of each of these states at the core of the construction is a general technique for finding the thermodynamically stable phases from a collection of competing minima we draw on ideas of pirogov and sinai 24 for this problem

wick products of the free bose field we consider wick products of the free bose field in the abstract context of a complex hilbert space h equipped with a self adjoint operator a satisfying a ei for some e 0 let k w be the free bose field over h and let h d a let h and k denote d a and d h respectively given their natural frechet topologies then for any 1 n h the wick product 1 n is constructed as a continuous sesquilinear form on k characterized by a generalization of the heisenberg commutation relations as an application we treat pointwise products of the free scalar field and its derivatives on r m m an arbitrary complete riemannian manifold for example if ck0 r m for large enough k then r m p p n corresponds to an operator with domain k if in addition is real valued n 2 and m is compact then this operator is essentially self adjoint

numerical analysis of two level finite difference schemes for unsteady diffusion convection problems for the linear diffusion convection model problem this paper develops the numerical analysis of two classes of three time level second order finite difference schemes weighed schemes similar to two step schemes for differential equations and lf schemes based on the classical conservative approximation of the wave equation as in our paper devoted to two level schemes fourier analysis is an essential tool which yields several key properties stability dissipation numerical diffusion and dispersion it also permits the analysis of parasitic solutions characteristic of three level schemes and their control in terms of the starting scheme the analysis is completed by giving results on the accuracy positivity and parabolicity of the numerical schemes the main conclusions are the expected superiority of these schemes over two level schemes when the diffusion phenomenon is not negligible cell reynolds number less than 1 and the very limited possibilities of obtaining satisfactory numerical results even with upstream differencing when convection is strongly predominating

some covariance inequalities in wiener space in this work we give an account of some covariance inequalities in abstract wiener space an fkg inequality is obtained with positivity and monotonicity being defined in terms of a given cone in the underlying cameron martin space the last part is dedicated to convex and log concave functionals including a proof of the gaussian conjecture for a particular class of log concave wiener functionals

second order explicit difference schemes for the space fractional advection diffusion equation in this paper two kinds of explicit second order difference schemes are developed to solve the space fractional advection diffusion equation the discretizations of fractional derivatives are based on the weighted and shifted grunwald difference operators developed in meerschaert and tadjeran j comput appl math 172 2004 65 77 tian et al arxiv 1201 5949 li and deng arxiv 1310 7671 the stability of the presented difference schemes are discussed by means of von neumann analysis the analysis shows that the presented numerical schemes are both conditionally stable the necessary conditions of stability is discussed finally the results of numerical experiments are given to illustrate the performance of the presented numerical methods

a stable explicit scheme for the ocean acoustic wave equation abstract a class of ocean acoustic wave propagation problems is represented by a parabolic equation of the schrodinger type using conventional explicit finite difference schemes e g the euler scheme to solve the parabolic wave equation is unstable thus important advantages of explicit schemes are completely missing this paper presents a conditionally stable explicit scheme by introducing an extra dissipative term this new explicit scheme is then applied to solve the ocean acoustic parabolic wave equation fully utilizing the advantages of explicit schemes the theoretical development the computational aspects and the advantages are discussed application of the scheme to a realistic ocean acoustic problem is included the solution obtained is compared with the unconditionally stable crank nicolson solution

high order finite volume schemes for the advection diffusion equation a high order finite volume method based on piecewise interpolant polynomials is proposed to discretize spatially the one dimensional and two dimensional advection diffusion equation evolution equations for the mean values of each control volume are integrated in time by a classical fourth order runge kutta since our work focuses on the behaviour of the spatial discretization the time step is chosen small enough to neglect the time integration error two dimensional interpolants are built by means of one dimensional interpolants it is shown that when the degree of the one dimensional interpolant q is odd the proper selection of a fixed stencil gives rise to centred schemes of order q 1 in order not to lose precision due to the change of stencil near boundaries the degree of the interpolants close to boundaries is raised to q 1 four test cases with small values of diffusion are integrated with high order methods it is shown that the spatial discretization of the advection diffusion equation with periodic boundary conditions leads to normal discretization matrices and asymptotic stability must be assured to bound the spatial discretization error once the asymptotic stability is assured by means of the spectra of the discretization matrix the spatial error is of the order of the truncation error however it is shown that the discretization of the advection diffusion equation with arbitrary boundary conditions gives rise to non normal matrices if asymptotic stability is assured the spatial order of steady solutions is of the order of the truncation error but for transient processes the order of the spatial error is determined by both the truncation error and the norm of the exponential matrix of the spatial discretization the use of the pseudospectra of the discretization matrix is proposed as a valuable tool to analyse the transient error of the numerical solution copyright 2001 john wiley sons ltd

multiscale kam theorem for hamiltonian systems abstract in this paper we study the persistence of invariant tori in nearly integrable multiscale hamiltonian systems with highorder degeneracy in the integrable part such hamiltonian systems arise naturally in planar and spatial lunar problems of celestial mechanics for which the persistence problem connects closely to the stability of the systems we introduce multiscale nondegenerate condition and multiscale diophantine condition comparable to the usual diophantine condition using quasilinear kam method we prove a multiscale kam theorem

fractional derivative quantum fields at positive temperature abstract this paper considers fractional generalization of finite temperature klein gordon kg field and vector potential in covariant gauge and static temporal gauge fractional derivative quantum field at positive temperature can be regarded as a collection of infinite number of fractional thermal oscillators generalized riemann zeta function regularization and heat kernel techniques are used to obtain the high temperature expansion of free energy associated with the fractional kg field we also show that quantization of the fractional derivative fields can be carried out by using the parisi wu stochastic quantization

sobolev functions on infinite dimensional domains abstract we study extensions of sobolev and bv functions on infinite dimensional domains along with some positive results we present a negative solution of the long standing problem of existence of sobolev extensions of functions in gaussian sobolev spaces from a convex domain to the whole space

tempered generalized functions and hermite expansions in this work we introduce an algebra of tempered generalized functions the tempered distributions are embedded in this algebra via their hermite expansions the fourier transform is naturally extended to this algebra in such a way that the usual relations involving multiplication convolution and differentiation are valid furthermore we give a generalized ito formula in this context and some applications to stochastic analysis

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

invariant tori for nearly integrable hamiltonian systems with degeneracy does not fall into a hyperplanethrough the origin the condition 1 3 is the sharpest one we have not seenits proof yet in this paper we will give a nondegeneracy condition by meansof the derivatives of h p which is equivalent to the condition 1 3 in analyticcase and under this nondegeneracy condition we obtain the russmann s results for analytic case furthermore the arguments of this paper are available to thenonanalytic case since our nondegeneracy conditions only involve the nite orderderivatives of h p main resultstheorem a suppose that h h p f p q is analytic in t

large deviation estimates in the stochastic quantization ofϕ 2 4 we study in the small noise limit the behaviour of field trajectories for the process constructed by the authors in connection with the stochastic quantization of 2 4 due to the presence of infinite renormalization the usual large deviation techniques do not apply immediately and a new strategy has to be developed we prove some estimates analogous to the freidlin ventzel inequalities from these it follows that the field trajectories suitably smeared in space over a scaler0 behave when the noise is small as the projection on the same scale of a field obeying a regularized stochastic equation with a large cut off however the estimates are not uniform in the cut off and an interesting feature of the problem is that the scale over which the field is smeared determines whether the noise is sufficiently small for the estimates to apply

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

strong solutions to the stochastic quantization equations we prove the existence and uniqueness of a strong solution of the stochastic quantization equation in dimension 2 for almost all initial data with respect to the invariant measure the method is based on a fixed point result in suitable besov spaces

numerical solution of heated subsidence mound problem in a porous medium abstract two different numerical models are constructed to solve a two dimensional subsidence mound problem heated along the moving wet dry interface one numerical model is based on cartesian coordinates while the other is based on polar coordinates in both approaches coordinate transformations are used that render the interface stationary the problem involves a system of three coupled equations an elliptic equation for a stream function a parabolic equation for the temperature and a non linear equation for the boundary location good agreement is found between the results of both methods graphic results are presented for the decay of a subsidence mound for different values of the various parameters in the model problem

path integral representation of fractional harmonic oscillator abstract fractional oscillator process can be obtained as the solution to the fractional langevin equation there exist two types of fractional oscillator processes based on the choice of fractional integro differential operators namely weyl and riemann liouville an operator identity for the fractional differential operators associated with the fractional oscillators is derived it allows the solution of fractional langevin equations to be obtained by simple inversion the relationship between these two fractional oscillator processes is studied the operator identity also plays an important role in the derivation of the path integral representation of the fractional oscillator processes relevant quantities such as two point and n point functions can be calculated from the generating functions

phase diagrams and cluster expansions for low temperaturep φ 2 models we give a cluster expansion for the schwinger functions of the stable phases found in part i the wightman axioms the mass gap and asymptoticity of perturbation theory follow

entropy invertibility and variational calculus of adapted shifts on wiener space in this work we study the necessary and sufficient conditions for a positive random variable whose expectation under the wiener measure is one to be represented as the radon nikodym derivative of the image of the wiener measure under an adapted perturbation of identity with the help of the associated innovation process we prove that the innovation conjecture holds if and only if the original process is almost surely invertible we also give variational characterizations of the invertibility of the perturbations of identity and the representability of a positive random variable whose total mass is equal to unity we prove in particular that an adapted perturbation of identity u iw u satisfying the girsanov theorem is invertible if and only if the kinetic energy of u is equal to the entropy of the measure induced with the action of u on the wiener measure in other words u is invertible iff 12 w u h2d wdu d logdu d d the relations with the monge kantorovitch measure transportation are also studied an application of these results to a variational problem related to large deviations is also given

monge kantorovitch measure transportation and monge ampere equation on wiener space let w h be an abstract wiener space assume two i i 1 2 probabilities on w w we give some conditions for the wasserstein distance between 1 and 2 with respect to the cameron martin space to be finite where the infimum is taken on the set of probability measures on w w whose first and second marginals are 1 and 2 in this case we prove the existence of a unique cyclically monotone map t iw with w h such that t maps 1 to 2 moreover if 2 then t is stochastically invertible i e there exists s w w such that s t iw 1 a s and t s iw 2 a s if in addition 1 then there exists a 1 convex function in the gaussian sobolev space such that these results imply that the quasi invariant transformations of the wiener space with finite wasserstein distance from can be written as the composition of a transport map t and a rotation i e a measure preserving map we give also 1 convex sub solutions and ito type solutions of the monge ampere equation on w

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

stable explicit schemes for equations of the schro dinger type most conventional explicit finite difference schemes e g euler s scheme for solving the parabolic equation of schrodinger type u t iu xx are unconditionally unstable this difficulty can be overcome by introducing a dissipative term to the conventional explicit schemes based on this approach we derive a class of new explicit finite difference schemes which are conditionally stable spans two time levels and are o k h 2 accurate we also determine the schemes from this class that have the least restrictive stability requirements it is interesting to note that the analog of the lax wendroff scheme is unstable

stability analysis of finite difference schemes for the advection diffusion equation we present a collection of stability results for finite difference approximations to the advection diffusion equation u t a u x b u xx the results are for centered difference schemes in space and include explicit and implicit schemes in time up to fourth order and schemes that use different space and time discretizations for the advective and diffusive terms the results are derived from a uniform framework based on the schur cohn theory of simple von neumann polynomials and are necessary and sufficient for the stability of the cauchy problem some of the results are believed to be new

generalized stochastic processes in algebras of generalized functions independence stationarity and spdes abstract stochastic processes are regarded in the framework of colombeau type algebras of generalized functions colombeau stochastic processes with independent values and stationary colombeau stochastic processes are studied with special attention paid to the property of translational invariance of generalized functions processes with stationary increments are characterized via stationarity of their gradient gaussian stationary solutions are analyzed for linear stochastic partial differential equations with generalized constant coefficients in the framework of colombeau stochastic processes

models of local relativistic quantum fields with indefinite metric in all dimensions a condition on a set of truncated wightman functions is formulated and shown to permit the construction of the hilbert space structure included in the morchio strocchi modified wightman axioms the truncated wightman functions which are obtained by analytic continuation of the truncated schwinger functions of euclidean scalar random fields and covariant vector quaternionic random fields constructed via convoluted generalized white noise are then shown to satisfy this condition as a consequence such random fields provide relativistic models for indefinite metric quantum field theory in dimension 4 vector case respectively in all dimensions scalar case

strong uniqueness for both dirichlet operators and stochastic dynamics to gibbs measures on a path space with exponential interactions abstract we prove l p uniqueness of dirichlet operators for gibbs measures on the path space c r r d associated with exponential type interactions in infinite volume by extending an spde approach presented in previous work by the last two named authors we also give an spde characterization of the corresponding dynamics in particular for the first time we prove existence and uniqueness of a strong solution for the spde though the self interaction potential is not assumed to be differentiable hence the drift is possibly discontinuous as examples to which our results apply we mention the stochastic quantization of p 1 exp 1 and trigonometric quantum fields in infinite volume in particular our results imply essential self adjointness of the generator of the stochastic dynamics for these models finally as an application of the strong uniqueness result for the spde we prove some functional inequalities for diffusion semigroups generated by the above dirichlet operators

a white noise approach to a class of non linear stochastic heat equations abstract cauchy problems for a class of non linear stochastic evolution equations are studied they are formulated as integral equations for generalized random fields by methods of white noise analysis s transformation characterization theorem etc these problems are reduced to fixed point problems in appropriatly constructed banach spaces this technique provides a systematic treatment of existence and uniqueness questions for a variety of equations the method is applied to non linear heat equations non linear volterra equations and non linear ordinary differential equations which may all be anticipating for each case examples are given such as the stochastic burgers equation and stochastic reaction diffusion equations

fourier methods with extended stability intervals for the korteweg de vries equation a full leap frog fourier method for integrating the korteweg de vries kdv equation u t uu x varepsilon u xxx 0 results in an o n 3 stability constraint on the time step where n is the number of fourier modes used this stability limit is much more restrictive than the accuracy limit for many applications in this paper we propose a method for which the stability limit is extended by treating the linear dispersive u xxx term implicitly thus timesteps can be taken up to an accuracy limit larger than the explicit stability limit the implicit method is implemented without solving linear systems by integrating in time in the fourier space and discretizing the nonlinear uu x term by leap frog a second method we propose uses basis functions which solve the linear part of the kdv equation and leap frog for time integration a linearized stability analysis of the proposed schemes proves that a version of the first scheme possesses a certain kind of unconditional stability and that

uniqueness of gibbs states for general p ϕ 2 weak coupling models by cluster expansion we consider quantum fields with weak coupling in two space time dimensions we prove that the set of their ultraregular gibbs states consists of only one point and this point is an extremal gibbs state

construction of relativistic quantum fields in the framework of white noise analysis we construct a class of euclidean invariant distributions h indexed by a function h holomorphic at zero these generalized functions can be considered as generalized densities w r t the white noise measure and their moments fulfill all osterwalder schrader axioms except for reflection positivity the case where f s h is 12s2 s r is a levy characteristic is considered in rev math phys 8 763 1996 under this assumption the moments of the euclidean invariant distributions h can be represented as moments of a generalized white noise measure ph here we enlarge this class by convolution with kernels g coming from euclidean invariant operators g the moments of the resulting euclidean invariant distributions hg also fulfill all osterwalder schrader axioms except for reflection positivity for no nontrivial case we succeeded in proving reflection positivity nevertheless an analytic extension to wightman functions can be performed these functions fulfill all wightman axioms except for the po

vanishing of one dimensional l 2 cohomologies of loop groups abstract let g be a simply connected compact lie group let l e g be the based loop group with the base point e which is the identity element let e be the pinned brownian motion measure on l e g and let l 2 1 t l e g e d p 1 t l e g e 1 p 2 be a closed 1 form on l e g using results in rough path analysis we prove that there exists a measurable function f on l e g such that d f moreover we prove that dim ker 0 for the hodge kodaira type operator acting on 1 forms on l e g

generalized solutions of the multidimensional stochastic burgers equation abstract we introduce a new concept of weak solution for the conservative stochastic burgers equation in any dimension the definition is based on weak solution concepts introduced by various authors in order to make sense of equations which do not have solutions in the sense of distributions in one dimension the solution reduces to the classical distributional solution of the 1 d stochastic burgers equation

invariant states on random and quantum fields φ bounds and white noise analysis a stochastic or quantum field theory admitting the araki relation and a bound can be represented in terms of generalized functionals of white noise this is especially so for p 2 models

on the uniform integrability of the radon nikodym densities for wiener measure abstract we give sufficient conditions for the uniform integrability of the radon nikodym derivatives of the images of wiener measure under the non linear transformations of the wiener space these results are then applied to the construction of the functional inverses of these transformations

the notion of convexity and concavity on wiener space abstract we define in the frame of an abstract wiener space the notions of convexity and of concavity for the equivalence classes of random variables as application we show that some important inequalities of the finite dimensional case have their natural counterparts in this setting

a two space dimensional semilinear heat equation perturbed by gaussian white noise a two space dimensional heat equation perturbed by a white noise in a bounded volume is considered the equation is perturbed by a non linearity of the type f au where means wick re ordering with respect to the free solution a are small parameters u denotes a solution f is the fourier transform of a complex measure with compact support

an estimate of the gap of spectrum of schrodinger operators which generate hyperbounded semigroups abstract logarithmic sobolev inequalities with potential functions on loop spaces were proved by l gross 1991 j funct anal 102 265 313 s aida 1996 in proceedings of the fifth gregynog symposium stochastic analysis and applications pp 1 19 world scientific singapore and f z gong and z m ma 1998 j funct anal 599 623 the generators of the dirichlet forms are schrodinger operators on loop spaces and they generate hyperbounded semigroups recently f z gong m rockner and l m wu 2000 poincare inequality for weighted first order sobolev spaces on loop spaces submitted proved the existence of the gap of spectrum at the lowest eigenvalue of the schrodinger operators by using m hino s exponential decay estimates of the semigroup 2000 osaka j math 37 603 624 and the author s strong ergodicity property of the semi group s aida 1998 j funct anal 158 152 185 in this paper we will give a lower bound on the gap by using a weak poincare inequality which was introduced by m rockner and f y wang 2000 weak poincare inequalities and l 2 convergence rates of markov semigroups preprint also we will give estimates on the distribution function of ground states using the weak poincare inequality

sobolev estimates for optimal transport maps on gaussian spaces abstract in this work we will take the standard gaussian measure as the reference measure and study the variation of optimal transport maps in sobolev spaces with respect to it as a by product an inequality which gives a precise link between the variation of entropy fisher information between source and target measures with the sobolev norm of the optimal transport map will be given as applications we will construct strong solutions to monge ampere equations in finite dimension as well as on the wiener space when the target measure satisfies the strong log concavity condition a result on the regularity on the optimal transport map on the wiener space will be obtained

positivity and positive definiteness in generalized function algebras the definitions of positivity and positive definiteness are extended to generalized function algebras in coherence with the corresponding notions for distributions versions of bochner s theorem for a positive definite colombeau generalized function are given