on a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with nonglobally monotone coefficients we develop a perturbation theory for stochastic differential equations sdes by which we mean both stochastic ordinary differential equations sodes and stochastic partial differential equations spdes in particular we estimate the l p distance between the solution process of an sde and an arbitrary ito process which we view as a perturbation of the solution process of the sde by the l q distances of the differences of the local characteristics for suitable p q 0 as one application of the developed perturbation theory we establish strong convergence rates for numerical approximations of a class of sodes with nonglobally monotone coefficients as another application of the developed perturbation theory we prove strong convergence rates for spatial spectral galerkin approximations of solutions of semilinear spdes with nonglobally monotone nonlinearities including cahn hilliard cook type equations and stochastic burgers equations further applications of the developed perturbation theory include regularity analyses of solutions of sdes with respect to their initial values as well as small noise analyses for ordinary and partial differential equations

a stochastic nonlinear schrodinger equation with multiplicative noise we study a conservative stochastic nonlinear schrodinger equation with a multiplicative noise we show the global existence and uniqueness of square integrable solutions for subcritical nonlinearities the critical exponent being the same in dimension 1 or 2 as the critical exponent of the deterministic equation

galerkin approximations for the stochastic burgers equation existence and uniqueness for semilinear stochastic evolution equations with additive noise by means of finite dimensional galerkin approximations is established and the convergence rate of the galerkin approximations to the solution of the stochastic evolution equation is estimated these abstract results are applied to several examples of stochastic partial differential equations spdes of evolutionary type including a stochastic heat equation a stochastic reaction diffusion equation and a stochastic burgers equation the estimated convergence rates are illustrated by numerical simulations the main novelty in this article is the estimation of the difference of the finite dimensional galerkin approximations and of the solution of the infinite dimensional spde uniformly in space i e in the l infty topology instead of the usual hilbert space estimates in the l 2 topology that were shown before

ergodicity of the 2d navier stokes equations with degenerate stochastic forcing the stochastic 2d navier stokes equations on the torus driven by degenerate noise are studied we characterize the smallest closed invariant subspace for this model and show that the dynamics restricted to that subspace is ergodic in particular our results yield a purely geometric characterization of a class of noises for which the equation is ergodic in l 2 0 t 2 unlike previous works this class is independent of the viscosity and the strength of the noise the two main tools of our analysis are the asymptotic strong feller property introduced in this work and an approximate integration by parts formula the first when combined with a weak type of irreducibility is shown to ensure that the dynamics is ergodic the second is used to show that the first holds un

geometrical implications of certain infinite dimensional decompositions we investigate the connections between the global structure of a banach space i e the existence of certain finite and infinite dimensional de compositions and the geometrical properties of the closed convex bounded sub sets of such a space i e the existence of extremal and other topologically distin guished points the global structures of various supposedly pathological examples of banach spaces constructed by r c james turn out to be more universal than expected for instance james tree type resp james matrix type decompositions characterize banach spaces with the point of continuity property resp the radon nikodym property moreover the convex point of continuity property is stable under the formation of james infinitely branching tree type sums of infinite dimensional factors we also give several counterex amples to various questions relating some topological and geometrical concepts in banach spaces

time discretization of stochastic 2 d navier stokes equations with a penalty projection method a time discretization of the stochastic incompressible navier stokes problem by penalty method is analyzed some error estimates are derived combined and eventually arrive at a speed of convergence in probability of order 1 4 of the main algorithm for the pair of variables velocity and pressure also using the law of total probability we obtain the strong convergence of the scheme for both variables

probability and pathwise order of convergence of a semidiscrete scheme for the stochastic manakov equation it is well accepted by physicists that the manakov pmd equation is a good model to describe the evolution of nonlinear electric fields in optical fibers with randomly varying birefringence in the regime of the diffusion approximation theory an effective asymptotic dynamics has recently been obtained to describe this evolution this equation is called the stochastic manakov equation in this article we propose a semidiscrete version of a crank nicolson scheme for this limit equation and we analyze the approximation error allowing sufficient regularity of the initial data we prove that the numerical scheme has probability order 1 2 and almost sure order 1 2 epsilon for any epsilon in 0 1 2

absolute continuity and numerical approximation of stochastic cahn hilliard equation with unbounded noise diffusion abstract in this article we consider the absolute continuity and numerical approximation of the solution of the stochastic cahn hilliard equation with unbounded noise diffusion we first obtain the holder continuity and malliavin differentiability of the solution of the stochastic cahn hilliard equation by using the strong convergence of the spectral gakerkin approximation then we prove the existence and strict positivity of the density function of the law of the exact solution for the stochastic cahn hilliard equation with sublinear growth diffusion coefficient which fills a gap for the existed result when the diffusion coefficient satisfies a growth condition of order 1 3 1 to approximate the density function of the exact solution we propose a full discretization based on the spatial spectral galerkin approximation and the temporal drift implicit euler scheme furthermore a general framework for deriving the strong convergence rate of the full discretization is developed based on the variation approach and the factorization method consequently we obtain the sharp mean square convergence rates in both time and space via sobolev interpolation inequalities and semigroup theories to the best of our knowledge this is the first result on the convergence rate of full discretizations for the considered equation

error inequalities for quintic and biquintic discrete hermite interpolation in this paper we shall develop a class of discrete hermite interpolates in one and two independent variables further we offer explicit error bounds in norm for the quintic and biquintic discrete hermite interpolates some numerical examples are included to illustrate the results obtained

an accelerated exponential time integrator for semi linear stochastic strongly damped wave equation with additive noise abstract this paper is concerned with the strong approximation of a semi linear stochastic wave equation with strong damping driven by additive noise based on a spatial discretization performed by a spectral galerkin method we introduce a kind of accelerated exponential time integrator involving linear functionals of the noise under appropriate assumptions we provide error bounds for the proposed full discrete scheme it is shown that the scheme achieves higher strong order in time direction than the order of temporal regularity of the underlying problem which allows for higher convergence rate than usual time stepping schemes particularly for the space time white noise case in two or three spatial dimensions the scheme still exhibits a good convergence performance another striking finding is that even for the velocity with low regularity the scheme always promises first order strong convergence in time numerical examples are finally reported to confirm our theoretical findings

error estimates for discrete spline interpolation in this paper we shall develop a class of discrete spline interpolates in one and two independent variables further explicit error bounds in norm are derived for the quintic and biquintic discrete spline interpolates we also present some numerical examples to illustrate the results obtained

higher order pathwise numerical approximations of spdes with additive noise in this article the pathwise numerical approximation of semilinear parabolic stochastic partial differential equations spdes driven by additive noise is considered a new numerical scheme for the time and space discretization of such spdes is proposed and this scheme is shown to converge for such spdes faster than standard numerical schemes such as the linear implicit euler scheme or the linear implicit crank nicholson scheme the suggested scheme takes advantage of the smoothing effect of the dominant linear operator and of two linear functionals of the noise process of the spde the abstract setting under which the scheme is analyzed includes spdes driven by fractional brownian motions too

on the near differentiability property of banach spaces abstract let be a scalar measure of bounded variation on a compact metrizable abelian group g suppose that has the property that for any measure whose fourier stieltjes transform vanishes at the measure has radon nikod m derivative with respect to the haar measure on g then l pigno and s saeki showed that itself has radon nikod m derivative such property is not shared by vector measures in general we say that a banach space x has the near differentiability property if every x valued measure of bounded variation shares the above property we prove that banach spaces with the radon nikod m property have the near differentiability property while banach spaces with the near differentiability property enjoy the near radon nikod m property we also show that the banach spaces l 1 0 1 and l 1 h 0 1 have the near differentiability property lastly we show that banach spaces with the near differentiability property have type ii radon nikod m property whenever is a riesz subset of type 0 of g

c 1 hierarchical bases abstract in this note we derive estimates for the condition numbers of stiffness matrices relative to certain c1 conforming hierarchical bases for fourth order elliptic boundary value problems

full discretization of semilinear stochastic wave equations driven by multiplicative noise a fully discrete approximation of the semilinear stochastic wave equation driven by multiplicative noise is presented a standard linear finite element approximation is used in space and a stochastic trigonometric method is used for the temporal approximation this explicit time integrator allows for mean square error bounds independent of the space discretization and thus does not suffer from a step size restriction as in the often used stormer verlet leapfrog scheme furthermore it satisfies an almost trace formula i e a linear drift of the expected value of the energy of the problem numerical experiments are presented and confirm the theoretical results

the analytic complete continuity property abstract we introduce the notion of the analytic complete continuity property of banach spaces we give different characterizations of this property we show that this property is different from known related properties such as the complete continuity property and the analytic radon nikod m property

on the effect of a noise on the solutions of the focusing supercritical nonlinear schrodinger equation we investigate the influence of a random perturbation of white noise type on the finite time blow up of solutions of a focusing supercritical nonlinear schrodinger equation we prove that contrary to the deterministic case any initial data gives birth to a solution which develops singularities moreover the singularities appear immediately we use a stochastic generalization of the variance identity and a control argument

approximating stochastic evolution equations with additive white and rough noises in this paper we analyze galerkin approximations for stochastic evolution equations driven by an additive gaussian noise which is temporally white and spatially fractional with hurst index less than or equal to 1 2 first we regularize the noise by the wong zakai approximation and obtain its optimal order of convergence then we apply the galerkin method to discretize the stochastic evolution equations with regularized noises optimal error estimates are obtained for the galerkin approximations in particular our error estimates remove an infinitesimal factor which appears in the error estimates of various numerical methods for stochastic evolution equations in existing literatures

convergence of finite element solutions of stochastic partial integro differential equations driven by white noise numerical approximation of a stochastic partial integro differential equation driven by a space time white noise is studied by truncating a series representation of the noise with finite element method for spatial discretization and convolution quadrature for time discretization sharp order convergence of the numerical solutions is proved up to a logarithmic factor numerical examples are provided to support the theoretical analysis

finite difference solutions of a non linear schrodinger equation abstract this paper describes a finite difference method to approximate a schrodinger equation with a power non linearity a special case of this equation is currently used to model the propagation of a laser beam in a plasma the main feature of the method we present is that it satisfies a discrete analogue of some important conservation laws of the equations we present numerical results which show in particular the propagation and the formation of solitions in the one dimensional case

finite element approximation of the cahn hilliard cook equation we study the nonlinear stochastic cahn hilliard equation perturbed by additive colored noise we show almost sure existence and regularity of solutions we introduce spatial approximation by a standard finite element method and prove error estimates of optimal order on sets of probability arbitrarily close to 1 we also prove strong convergence without known rate

a trigonometric method for the linear stochastic wave equation a fully discrete approximation of the linear stochastic wave equation driven by additive noise is presented a standard finite element method is used for the spatial discretization and a stochastic trigonometric scheme for the temporal approximation this explicit time integrator allows for error bounds independent of the space discretization and thus does not have a step size restriction as in the often used stormer verlet leap frog scheme moreover it enjoys a trace formula as does the exact solution of our problem these favorable properties are demonstrated with numerical experiments

weak approximation of stochastic partial differential equations the nonlinear case we study the error of the euler scheme applied to a stochastic partial dierential equation we prove that as it is often the case the weak order of convergence is twice the strong order a key ingredient in our proof is malliavin calculus which enables us to get rid of the irregular terms of the error we apply our method to the case a semilinear stochastic heat equation driven by a space time white noise

applications of the theory of semi embeddings to banach space theory let x and y be banach spaces and t x y an injective bounded linear operator t is called a semi embedding if t maps the closed unit ball of x to a closed subset of y this concept was introduced by lotz peck and porta proc edinburgh math soc 22 1979 233 240 it is proved that if x semi embeds in y and x is separable then x has the radon nikodym property provided y does it is shown that if l1 semi embeds in y then y fails the schur property and contains a subspace isomorphic to l1 as a consequence of the proof it is shown that if x is a subspace of l1 either l1 embeds in x or l1 embeds in l1x the simpler result that l1 does not semi embed in c0 is treated separately this result is used to deduce the classic result of menchoff that there exists a singular probability measure on the circle with fourier coefficients vanishing at infinity some generalizations of the notion of semi embedding are given and several complements and open questions are discussed

decompositions of banach lattices into direct sums we consider the problem of decomposing a banach lattice z as a direct sum z x y where x and y are complemented subspaces satisfying a condition of incomparability e g every operator from y to x is strictly singular we treat both the atomic and nonatomic cases in particular we answer a question of wojtaszczyk by showing that l1 ffll2 has unique structure as a nonatomic banach lattice one of the most important problems in the theory of banach lattices which is still open is whether any complemented subspace of a banach lattice must be linearly isomorphic to a banach lattice the main difficulty seems to lie in the fact that most of the criteria for a banach space to be isomorphic to a lattice do not really distinguish between lattices and their complemented subspaces we do not actually treat this question in the present paper but rather consider the situation z x d3 y where z is a banach lattice and x and y two complemented subspaces which are assumed to satisfy different conditions that make them distinct in some or another sense this line of research was initiated by p wojtaszczyk 128 and also by i s edelstein and p wojtaszczyk 3 who proved that if z has a normalized unconditional basis zn n l i e it is a separable atomic lattice so that every linear operator from y into x is compact then zn n l splits into two disjoint parts which are respectively equivalent to bases of x and y in particular both x and y have unconditional bases the proof of this result is based on a fundamental theorem from 28 and 3 which is mentioned below as theorem a we give here a different proof which does not make use of theorem a but instead is based on a simple change of signs result from 2 which is described below as theorem b we also consider the case when the compactness assumption above is replaced by the total incomparability of x and y for which we prove a similar result provided x and y have unconditional bases unfortunately the most interesting case when every operator from y into x is assumed to be strictly singular which was raised as an open problem in 28 remains unsolved we conclude the section devoted to the atomic case with a simple theorem on block bases of a space with unconditional basis zn 1 whose span is complemented such a block basis splits into two disjoint parts the first equivalent to a subsequence of zn n received by the editors september 5 1986 1980 mathematics subject classification 1985 revision primary 46b30 46b15 secondary 46a40 this research was supported by grant no 84 00210 from the united states israel binational science foundation bsf jerusalem israel the research of the first author was also supported in part by nsf grant dms 8500938 the research of the second author was also supported in part by nsf grant dms 8601401 r 1987 american mathematical society ooo2 9947 87 1 00 25 per page

a random map implementation of implicit filters implicit particle filters for data assimilation generate high probability samples by representing each particle location as a separate function of a common reference variable this representation requires that a certain underdetermined equation be solved for each particle and at each time an observation becomes available we present a new implementation of implicit filters in which we find the solution of the equation via a random map as examples we assimilate data for a stochastically driven lorenz system with sparse observations and for a stochastic kuramoto sivashinsky equation with observations that are sparse in both space and time

some characterizations of the analytic radon nikodým property in banach spaces the analytic radon nikod m property for a complex banach space x is characterized in several ways analogous to the common characterizations of the radon nikod m property for example the banach space x has the analytic radon nikod m property if and only if any operator from l1 to x that factors through l1h01 is representable the characterizations are used to study complemented subspaces of l1h01 and to define the analytic radon nikod m property for real banach spaces

strong convergence rate of finite difference approximations for stochastic cubic schrodinger equations abstract in this paper we derive a strong convergence rate of spatial finite difference approximations for both focusing and defocusing stochastic cubic schrodinger equations driven by a multiplicative q wiener process beyond the uniform boundedness of moments for high order derivatives of the exact solution the key requirement of our approach is the exponential integrability of both the exact and numerical solutions by constructing and analyzing a lyapunov functional and its discrete correspondence we derive the uniform boundedness of moments for high order derivatives of the exact solution and the first order derivative of the numerical solution which immediately yields the well posedness of both the continuous and discrete problems the latter exponential integrability is obtained through a variant of a criterion given by cox hutzenthaler and jentzen arxiv 1309 5595 as a by product of this exponential integrability we prove that the exact and numerical solutions depend continuously on the initial data and obtain a large deviation type result on the dependence of the noise with first order strong convergence rate

finite element approximation of the linear stochastic wave equation with additive noise semidiscrete finite element approximation of the linear stochastic wave equation lswe with additive noise is studied in a semigroup framework optimal error estimates for the deterministic problem are obtained under minimal regularity assumptions these are used to prove strong convergence estimates for the stochastic problem the theory presented here applies to multidimensional domains and spatially correlated noise numerical examples illustrate the theory

the three space problem for l 1 we construct a subspace x of l1 such that x is an 1 sum of spaces isomorphic to 11 but such that l1 x does not contain a copy of l1 we also construct two banach spaces e1 e2 that do not contain a copy of li but such that e1 x e2 contains a copy of l1 moreover the projections this content downloaded from 207 46 13 150 on thu 28 jul 2016 05 31 04 utc all use subject to http about jstor org terms the three space problem for l 29 of li on each factor are one to one and the images of the unit ball of l1 are closed these examples settle questions of j lindenstrauss p pelczynski j bourgain and h p rosenthal equipe d analyse tour 46 u a au c n r s n0 754 universitt paris vi 4 place jussieu 75230 paris cedex 05 department of mathematics ohio state university 231 w 18th avenue columbus ohio 43210 this content downloaded from 207 46 13 150 on thu 28 jul 2016 05 31 04 utc all use subject to http about jstor org terms

an exponential wagner platen type scheme for spdes the strong numerical approximation of semilinear stochastic partial differential equations spdes driven by infinite dimensional wiener processes is investigated there are a number of results in the literature that show that euler type approximation methods converge under suitable assumptions to the exact solutions of such spdes with strong order 1 2 or at least with strong order 1 2 varepsilon where varepsilon 0 is arbitrarily small recent results extend these results and show that milstein type approximation methods converge under suitable assumptions to the exact solutions of such spdes with strong order 1 varepsilon it has also been shown that splitting up approximation methods converge under suitable assumptions with strong order 1 to the exact solutions of such spdes in this article an exponential wagner platen type numerical approximation method for such spdes is proposed and shown to converge under suitable assumptions with strong order 3 2

shape preserving piecewise rational interpolation with higher order continuity abstract a united form of the classical hermite interpolation and shape preserving interpolation is presented in this paper the presented interpolation method provides higher order continuous shape preserving interpolation splines the given interpolants are explicit piecewise rational expressions without solving a linear or nonlinear system of consistency equations by setting parameter values the interpolation curve can be generated by choosing the classical piecewise hermite interpolation polynomials or the presented piecewise rational expressions for monotonicity preserving and convexity preserving interpolation the appropriate values of a parameter are given on each subinterval numerical examples indicate that the given method produces visually pleasing curves

stability and convergence of the two parameter cubic spline collocation method for delay differential equations in this paper we propose the cubic spline collocation method with two parameters for solving delay differential equations ddes some results of the local truncation error and the convergence of the spline collocation method are given we also obtain some results of the linear stability and the nonlinear stability of the method for ddes in particular we design an algorithm to obtain the ranges of the two parameters a b which are necessary for the p stability of the collocation method some illustrative examples successfully verify our theoretical results

higher order strong approximations of semilinear stochastic wave equation with additive space time white noise novel fully discrete schemes are developed to numerically approximate a semilinear stochastic wave equation driven by additive space time white noise a spectral galerkin method is proposed for the

pathwise holder convergence of the implicit linear euler scheme for semi linear spdes with multiplicative noise in this article we prove pathwise holder convergence with optimal rates of the implicit euler scheme for the abstract stochastic cauchy problem begin aligned left begin aligned du t au t dt f t u t dt g t u t dw h t quad t in 0 t u 0 x 0 end aligned right end aligned here a is the generator of an analytic c 0 semigroup on a umd banach space x w h is a cylindrical brownian motion in a hilbert space h and the functions f 0 t times x rightarrow x theta f and g 0 t times x rightarrow fancyscript l h x theta g satisfy appropriate local lipschitz conditions the results are applied to a class of second order parabolic spdes driven by multiplicative space time white noise

on the stochastic korteweg de vries equation abstract we consider a stochastic korteweg de vries equation forced by a random term of white noise type this can be a model of water waves on a fluid submitted to a random pressure we prove existence and uniqueness of solutions inh1 r in the case of additive noise and existence of martingales solutions inl2 r in the case of multiplicative noise copyright 1998 academic press nous etudions une equation de korteweg de vries stochastique comportant une force aleatoire de type bruit blanc cette equation peut decrire les ondes de surface sur un fluide soumis a un champ de pression aleatoire nous montrons l existence et l unicite de solutions dansh1 r lorsque le bruit est additif puis dans le cas du bruit multiplicatif nous etablissons l existence de solutions martingales dansl2 r

methods for the numerical solution of the nonlinear schroedinger equation optimal l2 rates of convergence are established for several fully discrete schemes for the numerical solution of the nonlinear schroedinger equation both finite differences and finite elements are considered for the discretization in space while the integration in time is treated either by the leap frog technique or by a modified crank nicolson procedure which generalizes the one suggested by delfour fortin and payne and possesses two useful conserved quantities

strong convergence rate of splitting schemes for stochastic nonlinear schrodinger equations abstract in this paper we show that solutions of stochastic nonlinear schrodinger nls equations can be approximated by solutions of coupled splitting systems based on these systems we propose a new kind of fully discrete splitting schemes which possess algebraic strong convergence rates for stochastic nls equations key ingredients of our approach are using the exponential integrability and stability of the corresponding splitting systems and numerical approximations in particular under very mild conditions we derive the optimal strong convergence rate o n 2 1 2 of the spectral splitting crank nicolson scheme where n and denote the dimension of the approximate space and the time step size respectively

hierarchical conforming finite element methods for the biharmonic equation the paper deals with hierarchical bases in spaces of conforming c 1 elements in connection with the approximate solution of the biharmonic equation delta 2 u f quad text in omega qquad u frac partial u partial n 0 quad text on partial omega on a plane polygonal domain omega two different composite finite elements are studied piecewise quadratic powell sabin elements and piecewise cubic elements of clough tocher type the main result are estimates for the condition numbers of the corresponding discretization matrices that show that a conjugate gradient method applied to the hierarchical discretization the so called hierarchical multilevel method will yield suboptimal convergence rates in comparison with standard multigrid schemes

a semi discrete scheme for the stochastic nonlinear schrodinger equation we study the convergence of a semi discretized version of a numerical scheme for a stochastic nonlinear schrodinger equation the nonlinear term is a power law and the noise is multiplicative with a stratonovich product our scheme is implicit in the deterministic part of the equation as is usual for conservative equations we also use an implicit discretization of the noise which is better suited to stratonovich products we consider a subcritical nonlinearity so that the energy can be used to obtain an a priori estimate however in the semi discrete case no ito formula is available and we have to use a discrete form of this tool also in the course of the proof we need to introduce a cut off of the diffusion coefficient which allows to treat the nonlinearity then we prove convergence by a compactness argument due to the presence of noise and to the implicit discretization of the noise this is rather complicated and technical we finally obtain convergence of the discrete solutions in various topologies

galerkin finite element methods for stochastic parabolic partial differential equations we study the finite element method for stochastic parabolic partial differential equations driven by nuclear or space time white noise in the multidimensional case the discretization with respect to space is done by piecewise linear finite elements and in time we apply the backward euler method the noise is approximated by using the generalized l2 projection operator optimal strong convergence error estimates in the l2 and dot h 1 norms with respect to the spatial variable are obtained the proof is based on appropriate nonsmooth data error estimates for the corresponding deterministic parabolic problem the computational analysis and numerical example are given

numerical simulation of focusing stochastic nonlinear schrodinger equations abstract in this paper we numerically investigate nonlinear schrodinger equations with a stochastic contribution which is of white noise type and acts either as a potential multiplicative noise or as a forcing term additive noise in the subcritical case we recover similar results as in the case of the korteweg de vries equation in the critical or supercritical case we observe that depending on its smoothness the noise may have different effects spatially smooth noises amplify blow up phenomena whereas delta correlated multiplicative noises prevent blow up formation note that in this latter case very few results are known both from a theoretical and a numerical point of view

semigroup splitting and cubature approximations for the stochastic navier stokes equations approximation of the marginal distribution of the solution of the stochastic navier stokes equations on the two dimensional torus by high order numerical methods is considered the corresponding rates of convergence are obtained for a splitting scheme and the method of cubature on wiener space applied to a spectral galerkin discretization of degree n while the estimates exhibit a strong n dependence convergence is obtained for appropriately chosen time step sizes results of numerical simulations are provided and confirm the applicability of the methods

on the structure of non weakly compact operators on banach lattices in this paper we study extensions of theorems of hagler and stegall on lp spaces 11 and of rosenthal 24 25 and pelczynski 21 on c k spaces to more general banach spaces with some lattice structure which do not contain complemented copies of i r we show that i fx is a separable banach space such that co does not embed into x and t is a bounded linear operator from x into some banach space y such that t y is not separable then i if x has local unconditional structure h la gordon and lewis 7

existence of strong solutions for ito s stochastic equations via approximations given strong uniqueness for an ito s stochastic equation with discontinuous coefficients we prove that its solution can be constructed on any probability space by using for example euler s polygonal approximations stochastic equations in d and in domains in d are considered

stochastic symplectic and multi symplectic methods for nonlinear schrdinger equation with white noise dispersion we indicate that the nonlinear schrdinger equation with white noise dispersion possesses stochastic symplectic and multi symplectic structures based on these structures we propose the stochastic symplectic and multi symplectic methods which preserve the continuous and discrete charge conservation laws respectively moreover we show that the proposed methods are convergent with temporal order one in probability numerical experiments are presented to verify our theoretical results

numerical approximation of stochastic evolution equations convergence in scale of hilbert spaces the research of hakima bessaih was supported by the nsf united states grant dms 1418838 the research of erika hausenblas was supported by the austrian science fund fwf austria p25968 the research of tsiry randrianasolo was supported by the austrian science fund fwf austria p25968 and the german research council germany as part of the collaborative research center sfb 1283

well posedness and optimal regularity of stochastic evolution equations with multiplicative noises abstract in this paper we establish the well posedness and optimal trajectory regularity for the solution of stochastic evolution equations with generalized lipschitz type coefficients driven by general multiplicative noises to ensure the well posedness of the problem the linear operator of the equations is only need to be a generator of a c 0 semigroup and the proposed noises are quite general which include space time white noise and rougher noises when the linear operator generates an analytic c 0 semigroup we derive the optimal trajectory regularity of the solution through a generalized criterion of factorization method

radon nikodým properties associated with subsets of countable discrete abelian groups with any subset of a countable discrete abelian we associate with it three banach space properties these properties are radon nikodym type properties the relationship between these properties is investigated the results are applied to give new characterizations of riesz subsets and rosenthal subsets of countable discrete abelian groups

gδ embeddings in hilbert space abstract it is shown that a separable banach space x has the point of weak to norm continuity property resp the radon nikodym property if and only if there exists a compact g embedding resp an h embedding from x into l2 this solves several questions of j bourgain and h p rosenthal j funct anal 52 1983 it is also shown that every non relatively compact sequence in a banach space with property pc has a difference subsequence which is a boundedly complete basic sequence this solves a question of pelczynski and extends some results of w b johnson and h p rosenthal studia math 43 1972 77 92 various related questions asked in the above bourgain rosenthal reference and by g a edgar and r f wheeler pac j math 115 1984 and n ghoussoub and h p rosenthal math ann 264 1983 321 332 are also settled

exponential integrators for stochastic maxwell s equations driven by ito noise abstract this article presents explicit exponential integrators for stochastic maxwell s equations driven by both multiplicative and additive noises by utilizing the regularity estimate of the mild solution we first prove that the strong order of the numerical approximation is 1 2 for general multiplicative noise combining a proper decomposition with the stochastic fubini s theorem the strong order of the proposed scheme is shown to be 1 for additive noise moreover for linear stochastic maxwell s equation with additive noise the proposed time integrator is shown to preserve exactly the symplectic structure the evolution of the energy as well as the evolution of the divergence in the sense of expectation several numerical experiments are presented in order to verify our theoretical findings

a degree by degree recursive construction of hermite spline interpolants based on the classical hermite spline interpolant h 2 n 1 which is the piecewise interpolation polynomial of class c n 1 and degree 2n 1 a piecewise interpolation polynomial h 2 n of degree 2n is given the formulas for computing h 2 n by h 2 n 1 and computing h 2 n 1 by h 2 n are shown thus a simple recursive method for the construction of the piecewise interpolation polynomial set h j is presented the piecewise interpolation polynomial h 2 n satisfies the same interpolation conditions as the interpolant h 2 n 1 and is an optimal approximation of the interpolant h 2 n 1 some interesting properties are also proved

regularity analysis for stochastic partial differential equations with nonlinear multiplicative trace class noise abstract in this article spatial and temporal regularity of the solution process of a stochastic partial differential equation spde of evolutionary type with nonlinear multiplicative trace class noise is analyzed

numerical analysis of semilinear stochastic evolution equations in banach spaces the solution of stochastic evolution equations generally relies on numerical computation here usually the main idea is to discretize the spde spatially obtaining a system of sdes that can be solved by e g the euler scheme in this paper we investigate the discretization error of semilinear stochastic evolution equations in lp spaces resp banach spaces the space discretization may be done by galerkin approximation for the time discretization we consider the implicit euler the explicit euler scheme and the crank nicholson scheme in the last section we give some examples i e we consider an spdes driven by nuclear wiener noise approximated by wavelets and delay equation approximated by finite differences

on the splitting up method and stochastic partial differential equations we consider two stochastic partial differential equations du varepsilon t l ru varepsilon t f r t dv varepsilon t r m k u varepsilon t g k t circ dy t k qquad hspace 5pt varepsilon 0 1 driven by the same multidimensional martingale y yk and by different increasing processes vr0 vr1 r 1 2 d1 where lr and mk are second and first order partial differential operators and stands for the stratonovich differential we estimate the moments of the supremum in t of the sobolev norms of u1 t u0 t in terms of the supremum of the differences break vr0t vr1t hence we obtain moment estimates for the error of a multistage splitting up method for stochastic pdes in particular for the equation of the unnormalized conditional density in nonlinear filtering

weak approximation of the stochastic wave equation we investigate the accuracy of approximation of e f u t where u t email protected 0 is the solution of the stochastic wave equation driven by the space time white noise and f is an r valued function defined on the hilbert space l 2 r the approximation is done by the leap frog scheme we show that under certain conditions on f the approximation by the leap frog scheme is of order two

on fully discrete galerkin methods of second order temporal accuracy for the nonlinear schrodinger equation we approximate the solutions of an initial and boundary value problem for nonlinear schrodinger equations with emphasis on the cubic nonlinearity by two fully discrete finite element schemes based on the standard galerkin method in space and two implicit crank nicolson type second order accurate temporal discretizations for both schemes we study the existence and uniqueness of their solutions and provel 2 error bounds of optimal order of accuracy for one of the schemes we also analyze one step of newton s method for solving the nonlinear systems that arise at every time step we then implement this scheme using an iterative modification of newton s method that at each time stept n requires solving a number of sparse complex linear systems with a matrix that does not change withn the effect of this inner iteration is studied theoretically and numerically

grade two fluids on non smooth domain driven by multiplicative noise existence uniqueness and regularity the author presented a lecture on the subject of this paper at the workshop on numerics for stochastic partial differential equations and their applications held at the johann radon institute for computational and applied mathematics ricam linz austria during the period 04 16 december 2016

a recursive construction of hermite spline interpolants and applications let f k be the hermite spline interpolant of class c k and degree 2k 1 to a real function f which is defined by its values and derivatives up to order k at some knots of an interval a b we present a quite simple recursive method for the construction of f k we show that if at the step k the values of the kth derivative of f are known then f k can be obtained as a sum of f k 1 and of a particular spline g k 1 of class c k 1 and degree 2k 1 beyond the simplicity of the evaluation of g k 1 we prove that it has other interesting properties we also give some applications of this method in numerical approximation

conjugate operators that preserve the nonconvergence of bounded martingales we show that the conjugate t of an operator t x y with x and y banach spaces satisfies the following dichotomy either t preserves the nonconvergence of bounded martingales in y or there exists a compact operator k x y such that the kernel n t k fails the radon nikod m property

from rough path estimates to multilevel monte carlo new classes of stochastic differential equations can now be studied using rough path theory see e g t j lyons m caruana and t levy differential equations driven by rough paths springer berlin 2007 or p k friz and m hairer a course on rough paths universitext springer cham 2014 in this paper we investigate from a numerical analysis point of view stochastic differential equations driven by gaussian noise in the aforementioned sense our focus lies on numerical implementations and more specifically on the savings possible via multilevel methods our analysis relies on a subtle combination of pathwise estimates gaussian concentration and multilevel ideas numerical examples are given which both illustrate and confirm our findings

overcoming the order barrier in the numerical approximation of stochastic partial differential equations with additive space time noise we consider the numerical approximation of parabolic stochastic partial differential equations driven by additive space time white noise we introduce a new numerical scheme for the time discretization of the finite dimensional galerkin stochastic differential equations which we call the exponential euler scheme and show that it converges in the strong sense faster than the classical numerical schemes such as the linear implicit euler scheme or the crank nicholson scheme for this equation with the general noise in particular we prove that our scheme applied to a semilinear stochastic heat equation converges with an overall computational order 1 3 which exceeds the barrier order 1 6 for numerical schemes using only basic increments of the noise process reported previously by contrast our scheme takes advantage of the smoothening effect of the laplace operator and of a linear functional of the noise and therefore overcomes this order barrier

martingale and stationary solutions for stochastic navier stokes equations we prove the existence of martingale solutions and of stationary solutions of stochastic navier stokes equations under very general hypotheses on the diffusion term the stationary martingale solutions yield the existence of invariant measures when the transition semigroup is well defined the results are obtained by a new method of compactness

statistics of soliton bearing systems with additive noise we present a consistent method to calculate the probability distribution of soliton parameters in systems with additive noise even though the noise is weak we are interested in probabilities of large fluctuations generally non gaussian which are beyond perturbation theory our method is a development of the instanton formalism method of optimal fluctuation based on a saddle point approximation in the path integral we first