time reversal for infinite dimensional diffusions in this paper we analize the reversibility of the diffusion property for the solution of certain infinite dimensional systems of stochastic differential equations necessary and sufficient conditions ensuring this reversibility are given the proofs use the techniques of the stochastic calculus of variations

white noise analysis for levy processes we construct a white noise theory for levy processes the starting point of this theory is a chaos expansion for square integrable random variables we use this approach to malliavin calculus to prove the following white noise generalization of the clark haussmann ocone formula for levy processes f e f m 1 0te dt m f ft y t m dt here e f is the generalized expectation the operators dt m f m 1 are generalized malliavin derivatives is the wick product and for all m 1y t m is the white noise of power jump processes yt m in particular y t 1 is the white noise of the levy process the formula holds for all f g l2 where g is a space of stochastic distributions and is a white noise probability measure finally we give an application of this formula to partial observation minimal variance hedging problems in financial markets driven by levy processes

stochastic galerkin framework with locally reduced bases for nonlinear two phase transport in heterogeneous formations abstract the generalized polynomial chaos method with multiwavelet basis functions is applied to the buckley leverett equation we consider a spatially homogeneous domain modeled as a random field the problem is projected onto stochastic basis functions which yields an extended system of partial differential equations analysis and numerical methods leading to reduced computational cost are presented for the extended system of equations the accurate representation of the evolution of a discontinuous stochastic solution over time requires a large number of stochastic basis functions adaptivity of the stochastic basis to reduce computational cost is challenging in the stochastic galerkin setting since the change of basis affects the system matrix itself to achieve adaptivity without adding overhead by rewriting the entire system of equations for every grid cell we devise a basis reduction method that distinguishes between locally significant and insignificant modes without changing the actual system matrices results are presented for problems in one and two spatial dimensions with varying number of stochastic dimensions we show how to obtain stochastic velocity fields from realistic permeability fields and demonstrate the performance of the stochastic galerkin method with local basis reduction the system of conservation laws is discretized with a finite volume method and we demonstrate numerical convergence to the reference solution obtained through monte carlo sampling

forward backward and symmetric stochastic integration we define three types of non causal stochastic integrals forward backward and symmetric our approach consists in approximating the integrator two optics are considered the first one is based on traditional usual stochastic calculus and the second one on wiener distributions

the malliavin calculus a functional analytic approach in his groundbreaking articles s 91 malliavin introduced a technique for obtaining elliptic regularity results using function space calculus in his formulation the function space calculus on which his theory rests is the martingale calculus coming from the ornstein uhlenbeck process on wiener space cf 121 for a detailed discussion of malliavin s theory from this point of view although this formulation has a great deal to recommend it and is particularly pleasing to afficionados of the so called brownian sheet most analysts including those who are reasonably facile with the machinery of probability theory are unlikely to find the technical difficulties inherent in this approach outweighed by the eventual rewards shigekawa 111 discovered that there is an alternative formulation of malliavin s theory in shigekawa s formulation no mention need be made of the ornstein uhlenbeck process instead what he relies on is a sobolev type extension of the frechet dervative with wiener measure playing the role played by lebesgue measure in the finite dimensional context a third formulation of malliavin s theory was recently provided by bismut l bismut s idea is to exploit the quasi invariance of wiener measure as it is manifested in a beautiful relation due to haussmann 4 the present article has two aims in the first place section 1 2 and 3 are devoted to yet another formulation of malliavin s calculus although the end results are precisely the same as those obtained in 121 no reference is made to the ornstein uhlenbeck process on which the original theory depended instead the machinery is entirely that of elementary functional analysis and is devoid of any sophisticated martingale calculus it has been my intent that these three sections should be readily accessible to any analyst who has had some acquintance with wiener measure

transformation of the wiener measure under non invertible shifts in this paper we consider the transformation of measure induced by a not necessarily invertible perturbation of the identity the radon nikodym density for the image of the wiener measure and the associated girsanov type density are derived an application of these results yields an extension of the degree theorem

sheaf theoretic investigation of cip method abstract the cip method is a numerical computational method in the computational fluid dynamics discovered by takewaki nishiguchi and yabe h takewaki a nishiguchi t yabe the cubic interpolated pseudo particle cip method for solving hyperbolic type equations j comput phys 61 1985 261 268 which shows nice properties as a numerical computational method but it is known that its mathematical essentials have not been revealed well in this article we investigate the cip method from an algebraic viewpoint then it is shown that the cip method is described well in terms of the sheaf theory

a convex log concave correlation inequality for gaussian measure and an application to abstract wiener spaces this paper deals with a generalization of a result due to brascamp and lieb which states that in the space of probabilities with log concave density with respect to a gaussian measure on open image in new window this gaussian measure is the one which has strongest moments we show that this theorem remains true if we replace x by a general convex function then we deduce a correlation inequality for convex functions quite better than the one already known finally we prove results concerning stochastic analysis on abstract wiener spaces through the notion of approximate limit

onsager machlup functionals for non trace class spde s an onsager machlup functional limit is derived for a class of spde s whose principal part is not trace class both nondegenerate and degenerate limits are obtained and are illustrated by examples the proof uses fkg type inequalities

the laplacian energy of threshold graphs and majorization we study the laplacian energy of threshold graphs inspired by the recent results of vinagre del vecchio justo and trevisan 22 in particular we compute the degree sequences of threshold graphs that maximize or minimize the laplacian energy for a fixed number of vertices and edges the analysis involves combinatorial methods using ferrers diagrams and ideas from majorization theory some new inequalities for threshold degree sequences are obtained in this process in the review process a referee pointed out that recently and independently helmberg and trevisan 14 obtained very similar results and we discuss this connection

stochastic calculus with anticipating integrands we study the stochastic integral defined by skorohod in 24 of a possibly anticipating integrand as a function of its upper limit and establish an extended ito formula we also introduce an extension of stratonovich s integral and establish the associated chain rule in all the results the adaptedness of the integrand is replaced by a certain smoothness requirement

linear preservers of circulant majorization on rn abstract let m n m be the set of all n m real matrices an n n real matrix a is called circulant doubly stochastic if it is a convex combination of circulant permutation matrices for x y r n x is said to be circulant majorized by y written as x c y if there exists a circulant doubly stochastic matrix d such that x d y in this paper the concept of circulant majorization is investigated and then the linear preservers and strong linear preservers of this concept are characterized on r n

integration par parties dans l espace de wiener et approximation du temps local let be a real valued stochastic process having a continuous local timel u t u 0 t t andxe t e x t t 0 the regularization ofx by means of the convolution with the approximation of unity e the main theorem in this paper theorem 3 5 is a generalization of various results about the approximation for fixedu of the local timel u by means of a convenient normalization of the numbernxe u of crossings of the processxe with the levelu especially this theorem extends to a class of not necessarily markovian continuous martingales a result of this type for one dimensional diffusions due to azais a2 the methods of proof combine some estimations of the moments of the number of crossings with a level of a regular stochastic processes with stochastic analysis techniques based upon integration by parts in the wiener space

four applications of majorization to convexity in the calculus of variations abstract the resemblance between the horn thompson theorem and a recent theorem by dacorogna marcellini tanteri indicates that schur convexity and the majorization relation are relevant for applications in the calculus of variations and its related notions of convexity such as rank one convexity or quasiconvexity in theorem 6 6 we give simple necessary and sufficient conditions for an isotropic objective function to be rank one convex on the set of matrices with positive determinant majorization is used in order to give a very short proof of a theorem of thompson and freede r c thompson l j freede eigenvalues of sums of hermitian matrices iii j res nat bur standards b 75b 1971 115 120 ball j m ball constitutive inequalities and existence theorems in nonlinear elastostatics in r j knops ed nonlinear analysis and mechanics heriot watt symposium vol 1 res notes math 17 pitman 1977 pp 187 241 or le dret h le dret sur les fonctions de matrices convexes et isotropes cr acad sci paris serie i 310 1990 617 620 concerning the convexity of a class of isotropic functions which appear in nonlinear elasticity next we prove theorem 7 3 a lower semicontinuity result for functionals with the form w d x d x with w f h ln v f here f r f u f v f r f is the usual polar decomposition of f gl n r and ln v f is hencky s logarithmic strain we close this paper with a compact proof of dacorogna marcellini tanteri theorem based only on classical results about majorization the mentioned resemblance of this theorem with the horn thompson theorem is thus explained

existence of a smooth density for the filter in nonlinear filtering with infinite dimensional noise abstract in this paper we prove by means of malliavin calculus that under a local hormander condition the unnormalized filter associated with a nonlinear filtering problem in which the system process depends on an infinite dimensional noise admits a smooth density with respect to the lebesgue measure

absolutely continuous flows generated by sobolev class vector fields in finite and infinite dimensions abstract we prove the existence of the global flow ut generated by a vector field a from a sobolev class w1 1 on a finite or infinite dimensional space x with a measure provided is sufficiently smooth and that a a and a where a is the divergence with respect to are exponentially integrable in addition the measure is shown to be quasi invariant under ut in the case x r n and p dx where p w1 1loc r n is a locally uniformly positive probability density a sufficient condition is exp c a exp c a p p l1 for all c in the infinite dimensional case we get analogous results for measures differentiable along sufficiently many directions examples of measures which fit our framework important for applications are symmetric invariant measures of infinite dimensional diffusions and gibbs measures typically in both cases such measures are essentially non gaussian our result in infinite dimensions significantly extends previously studied cases where was a gaussian measure finally we study flows generated by vector fields whose values are not necessarily in the cameron martin space

generalized stochastic integrals and the malliavin calculus the paper first reviews the skorohod generalized stochastic integral with respect to the wiener process over some general parameter space t and it s relation to the malliavin calculus as the adjoint of the malliavin derivative some new results are derived and it is shown that every sufficiently smooth process ut t t can be decomposed into the sum of a malliavin derivative of a wiener functional and a process whose generalized integral over t vanishes using the results on the generalized integral the bismut approach to the malliavin calculus is generalized by allowing non adapted variations of the wiener process yielding sufficient conditions for the existence of a density which is considerably weaker than the previously known conditions

linear stochastic differential equations and wick products we establish the existence and uniqueness of the solution to a multidimensional linear skorohod stochastic differential equation with deterministic diffusion matrix using the notions of wick product andstransform if the diffusion matrix is constant and has real eigenvalues the solution is a stochastic process with moments of all orders provided that the initial condition is differentiable up to a suitable order the case of a diffusion matrix in the first wiener chaos is discussed in the last section

linear operators preserving directional majorization abstract a characterization of linear operators preserving directional majorization is given the result extends a theorem of t ando linear algebra appl 118 1989 163 and the proof also gives the characterization of linear operators preserving multivariate majorization recently obtained by beasley and lee

geometric analysis of conditional independence on wiener space in this paper we study the conditional independence of sigma fields on the wiener space using the tools of the stochastic calculus of variations particular emphasis is given to the relation between the splitting of the random tangent spaces associated to the sigma fields and the conditional independence

valuation of boundary linked assets by stochastic boundary value problems solved with a wavelet collocation algorithm this article studies the valuation of boundary linked assets and their derivatives in continuous time markets valuing boundary linked assets requires the solution of a stochastic differential equation with boundary conditions which often is not markovian we propose a wavelet collocation algorithm for solving a milstein approximation to the stochastic boundary problem its convergence properties are studied furthermore we value boundary linked derivatives using malliavin calculus and monte carlo methods we apply these ideas to value european call options of boundary linked assets

transformation of wiener measure under anticipative flows lett f be a transformation from the wiener space to itself with the range off assumed to be in the cameron martin space the absolute continuity and the density function associated witht is considered t is assumed to be embedded in or defined through a parameterizationt t f t andf t is assumed to be differentiable int the paper deals first with the case where the range of thet derivative off t is also in the cameron martin space and new representations for the radon nikodym derivative and the carleman fredholm determinant are derived the case where thet derivative off t is not in the cameron martin space is considered next and results on the absolute continuity and the density function under conditions which are considerably weaker than previously known conditions are presented

permutohedra and minimal matrices the notions of minimality uniqueness and additivity originated in discrete tomography they have applications to kronecker products of characters of the symmetric group and arise as the optimal solutions of quadratic transportation problems here we introduce the notion of real minimality and give geometric characterizations of all these notions for a matrix a by considering the intersection of the permutohedron determined by a with the transportation polytope in which a lies we also study the computational complexity of deciding if the properties of being additive real minimal unique and minimal hold for a given matrix and show how to efficiently construct some matrix with any of these properties

absolute continuity of the law of an infinite dimensional wiener functional with respect to the wiener probability in this paper we study conditions ensuring that the law of ac 0 1 valued functional defined on an abstract wiener space is absolutely continuous with respect to the wiener measure onc 0 1 these conditions extend those established byp malliavin 12 13 for finite dimensional wiener functionals and those of 15 for hilbert valued functionals

ito s lemma without non anticipatory conditions a stochastic integral with respect to brownian motion which extends ito s integral to anticipatory integrands is constructed and investigated this stochastic integral is different from the skorokhod integral the ito lemma is proved for this integral

representation of the distributions on wiener space and stochastic calculus of variations abstract in this work we construct projections and dual projections with respect to the past of the wiener process for the cameron martin space valued distributions on the wiener space with their help we then prove on the one hand an extension of clark s formula to the representation of the scalar distributions with the extended ito integral and on the other hand we give a decomposition of the vector valued distributions by exactly calculating the kernel of the divergence operator

finite dimensional approximation of riemannian path space geometry abstract in this paper we construct a finite dimensional approximation for the geometry on the path space over a compact riemannian manifold this approximation allows to construct the horizontal lift of the ornstein uhlenbeck process on the path space through the markovian connection we also prove a representation formula for the heat semigroup on adapted vector fields as well as a commutation formula for its derivative

positivity preserving schemes in multidimensions the performance of the muscl hancock upwind scheme is examined on problems of two dimensional advection it is found that when the advection direction is skewed relative to the mesh most discrete total variation tvd limiters give rise to large spurious oscillations near discontinuities the cause of these oscillations is traced to reconstructions that are not bounded by neighboring cell averages it is proved that if the reconstruction in each cell is bounded by the cell averages of first order neighbors then the muscl hancock scheme is positivity preserving a simple limiter that achieves such bounded reconstructions is presented next along with a variant that is uniformly second order accurate numerical experiments show that the new schemes are accurate and efficient and compare favorably with other schemes

application of malliavin calculus to a class of stochastic differential equations in this article we deal with stochastic differential equations driven by an infinite dimensional brownian motion under some non degeneracy conditions the existence and smoothness of the density for the law of the solution is proved

occupation densities for stochastic integral processes in the second wiener chaos the two point distributions of skorohod integral processes in the second wiener chaos are mainly described by a hilbert schmidt operatort giving the mutual interaction of infinitely many gaussian components and by simple multiplication operators so are the fourier transforms of their occupation measures this enables us to use the well known fourier analytic criterion discovered and elaborated by berman to derive integral conditions for the existence of their occupation densities in terms of associated hilbert schmidt operators ift is a trace class operator we get a necessary and sufficient criterion if it is not still a sufficient one in a case in which the interaction is particularly simple we verify the appropriate integral condition and show that the results are essentially beyond the reach of enlargement of filtrations techniques of semimartingale theory

malliavin derivatives and derivatives of functionals of the wiener process with respect to a scale parameter let f c 0w be a functional of the wiener process with variance parameter c 2 0 and let f cw be an extension of f c 0w to f cw c in 0 c 0 relations are derived between the malliavin derivatives between the derivatives with respect to the scale parameter partial f rho cw partial rho p 1 and noncoherent derivatives such as de f cw sqrt varepsilon c tilde w mid w d varepsilon varepsilon 0 where tilde w is another wiener process independent of w and between the generator of the nontime homogeneous ornstein uhlenbeck process

a change of variables formula for stratonovich integrals and existence of solutions for two point stochastic boundary value problems existence results for a class of two point non adaptive stochastic boundary value problems are derived by combining a new change of variables formula with optimal control techniques the class of equations treated is significantly wider than that for which previously known existence results apply

a conditional approach to the anticipating girsanov transformation we study the law of a stochastic differential equation d xi t d omega t k 1 xi omega dt where the drift anticipates the future behavior of the brownian path for example the endpoint we first investigate anticipation of the endpoint using a conditional girsanov transformation and methods of malliavin calculus a combination with results of buckdahn 2 leads to new versions of the anticipating girsanov transformation of ramer and kusuoka and in particular to explicit formulas for the carleman fredholm determinant

the ito formula for anticipative processes with nonmonotonous time scale via the malliavin calculus this work is devoted to derive ito type formulae for anticipative stochastic processes with nonmonotonous time using the malliavin calculus techniques and the fundamental theorem of the differential calculus the same method is applied also to give an ito ventcell type formula in the anticipative case

adaptive spacetime discontinuous galerkin method for hyperbolic advection diffusion with a non negativity constraint summary applications where the diffusive and advective time scales are of similar order give rise to advection diffusion phenomena that are inconsistent with the predictions of parabolic fickian diffusion models non fickian diffusion relations can capture these phenomena and remedy the paradox of infinite propagation speeds in fickian models in this work we implement a modified frame invariant form of cattaneo s hyperbolic diffusion relation within a spacetime discontinuous galerkin advection diffusion model an h adaptive spacetime meshing procedure supports an asynchronous patch by patch solution procedure with linear computational complexity in the number of spacetime elements this localized solver enables the selective application of optimization algorithms in only those patches that require inequality constraints to ensure a non negative concentration solution in contrast to some previous methods we do not modify the numerical fluxes to enforce non negative concentrations thus the element wise conservation properties that are intrinsic to discontinuous galerkin models are defined with respect to physically meaningful riemann fluxes on the element boundaries we present numerical examples that demonstrate the effectiveness of the proposed model and we explore the distinct features of hyperbolic advection diffusion response in subcritical and supercritical flows copyright 2015 john wiley sons ltd

majorization doubly stochastic matrices and comparison of eigenvalues abstract 1 basic properties of majorization 2 isotone maps and algebraic operations 3 double sub and superstochasticity 4 doubly stochastic matrices 5 doubly stochastic matrices with minimum permanent 6 comparison of eigenvalues 7 doubly stochastic maps

generalized brownian functionals and the solution to a stochastic partial differential equation abstract a class of generalized wiener functionals related to those of hida and watanabe is introduced and the malliavin calculs associated with these functionals is developed these notions are applied to the derivation of a solution to the stochastic partial differential equation y t ly y where l is a second order partial differential operator in the n space variables denotes a white noise in the n 1 space time variables y denotes a skorohod type stochastic integral and is a non random function of the n 1 independent variables

markov field property of stochastic differential equations the purpose of this paper is to prove a characterization of the conditional independence of two independent random variables given a particular functional of them in terms of a factorization property as an application we discuss the markov field property for solutions of stochastic differential equations with a boundary condition involving the values of the process at times t 0 and t 1

on the structure of independence on wiener space motives par des resultats de travaux anterieurs sur la caracterisation de l independance de variables aleatoires dans l espace de wiener les auteurs etudient ici la relation entre l independance de certaines classes de sous algebres et l orthogonalite des h derivees faibles des variables aleatoires qui engendrent ces sous algebres ou c h est l espace de wiener classique

stochastic calculus of variations for stochastic partial differential equations abstract this paper develops the stochastic calculus of variations for hilbert space valued solutions to stochastic evolution equations whose operators satisfy a coercivity condition an application is made to the solutions of a class of stochastic pde s which includes the zakai equation of nonlinear filtering in particular a lie algebraic criterion is presented that implies that all finite dimensional projections of the solution define random variables which admit a density this criterion generalizes hypoellipticity type conditions for existence and regularity of densities for finite dimensional stochastic differential equations

applications of the degree theorem to absolute continuity on wiener space let h p be an abstract wiener space and define a shift on byt f wheref is anh valued random variable we study the absolute continuity of the measurespot 1and f p ot 1 with respect top using the techniques of the degree theory of wiener maps where f det2 1 f exp f 1 2 f 2

anticipative girsanov transformations the transformations of measures induced by left w int limits 0 bullet k s ds right with ks possibly anticipating the wiener process ws is discussed and a girsanovtype theorem under rather weak assumptions on ks is derived

on linear preservers of g tridiagonal majorization on rn abstract for vectors x y r n it is said that x is g tridiagonal majorized by y written as x g t y if there exists a tridiagonal g doubly stochastic matrix d such that x d y in this paper we continue the previous work armandnejad and gashool 2012 3 and give a complete characterization of linear operators preserving g t on r n

a quantum nonadapted ito formula and stochastic analysis in fock scale abstract a generalized definition of quantum stochastic qs integrals and differentials is given in the free of adaptiveness and dimensionality form in terms of malliavin derivative on a projective fock space and their uniform continuity with respect to the inductive limite convergence is proved a new form of qs calculus based on an inductive algebraic structure in an indefinite space is developed and a nonadaptive generalization of the qs ito formula for its representation in fock space is derived 1 the problem of solution of general qs evolution equations in a hilbert space is solved in terms of the constructed operator representation of chronological products defined in the indefinite space and the isometry and homomorphism property respectively for operators and maps of these solutions corresponding to the peseudounitary and homomorphism property of the qs integrable generators is proved

boundary value processes estimation and identification recent results obtained for boundary value processes and the associated smoothing and identification problems are presented in this paper both lumped and distributed parameter models are considered some open problems are discussed and the fundamental mathematical difficulties that arise in studying nonlinear extensions of the proposed models are mentioned

on linear preservers of right matrix majorization abstract an n m matrix a is said to be matrix majorized or more precisely matrix majorized from the right by an n m matrix b and write a b if there exists a row stochastic matrix r such that a br we characterize the linear operators that preserve the matrix majorization

a two sided stochastic integral and its calculus let x be a forward diffusion and y a backward diffusion both defined on 0 1 x t and y t being respectively adapted to the past of a wiener process w and to its future increments we construct a two sided stochastic integral of the form mathop smallint limits 0 t phi u x u y u dw u which generalizes the backward and forward ito integrals simultaneously our construction is quite intuitive and leads to a generalized stochastic calculus it is also shown that for each fixed t our integral coincides with that defined by skorohod in 18

principal majorization ideals and optimization for a given vector b rn let the principal majorization ideal m b be the set of vectors with nonincreasing coordinates that are majorized by b m b is a polytope and we study the 1 skeleton and lattice properties of this set certain optimization problems involving m b are studied and a related class of matrices which contains the positive semidefinite matrices is investigated

girsanov theorem for anticipative shifts on poisson space we study the absolute continuity of the image measure of the canonical poisson probability measure under nonlinear shifts the radon nykodim density function is expressed using a carleman fredholm determinant and a divergence operator results are obtained for non necessarily invertible transformations under almost sure differentiability hypothesis

on the relation between the stratonovich and ogawa integrals l auteur montre que l existence de l integrale d ogawa implique l existence de l integrale de stratonovich et que les deux sont egales sous condition d une definition convenable de l integrale stochastique de stratonovich non adaptee

calcul stochastique non adapte pour des processus a deux parametres formules de changement de variables de type stratonovitch et de type skorohod nous etablissons une formule de changement de variables de type stratonovitch pour des processus non adaptes indexes par 0 1 2 qui est analogue a celle existant en calcul deterministe pour cela nous definissons une integrale de stratonovitch simple resp double nous en deduisons une formule de changement de variables de type skorohod ne contenant pas d integrale curviligne nous utilisons des regularisations du processus de wiener obtenues par convolution avec une approximation de l unite

majorization classes of integral matrices abstract the class a r s of 0 1 matrices with given row and column sum vectors r and s is well studied here we introduce and investigate the more general class a b s of integral matrices with given column sum vector s and with rows that satisfy majorization constraints each row is majorized by a given vector a row in b a characterization of nonemptyness of this class was recently given we present algorithms for constructing a matrix in a b s and study several properties of such classes for instance we show connectedness using certain transformations that generalize interchanges for 0 1 matrices

linear skorohod stochastic differential equations let andb be bounded processes on the wiener space p c 0 1 which are possibly anticipating the brownian motionw t t and let be a bounded random variable we deduce the existence and uniqueness of a solutionx for the linear equation with skorohod integral x t eta int limits 0 t sigma s x s dw s int limits 0 t b s x s ds t in 0 1 1 under rather weak assumptions on and no additional requirement onb and the description of the solutionx requires to study the family t t t 0 1 of transformationt t of into itself associated to 1 by the equation t t omega int limits 0 t wedge sigma s t s omega ds omega in omega t in 0 1

row and column majorization on mn m abstract let m n m be the set of all n m matrices with entries in r for a b m n m it is said that a is row majorized respectively column majorized by b if every row respectively column of a is majorized by the corresponding row respectively column of b i e for every i 1 i n there exists a doubly stochastic matrix d i such that a i b i d i where a i and b i are the i th rows of a and b respectively in this paper the relations row and column majorization on m n m are studied and also all linear operators t m n m m n m preserving or strongly preserving row or column majorization will be characterized

some relations among classes of σ fields on wiener space in this work we study sigma fields and their tangent spaces on the wiener space which are invariant in some sense with respect to the basic operators of the malliavin calculus

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

majorization permutahedra and 0 1 matrices a majorization permutahedron m v is a polytope associated with a majorization x v in rn defined by m v x rn x v several properties of these polytopes are investigated and a connection to discrete convexity is established these results are used to obtain a generalization of the gale ryser theorem for 0 1 matrices with given line sums

quantum and non causal stochastic calculus the quantum stochastic calculus initiated by hudson and parthasarathy and the non causal stochastic calculus originating with the papers of hitsuda and skorohod are two potent extensions of the ito calculus currently enjoying intensive development the former provides a quantum probabilistic extension of schrodinger s equation enabling the construction of a markov process for a quantum dynamical semigroup the latter allows the treatment of stochastic differential equations which involve terms which anticipate the future in this paper the close relationship between these theories is displayed and a noncausal quantum stochastic calculus already in demand from physics is described

covariance inequalities induced by biorthogonal operators abstract the logarithmic derivative of a weakly second order probability measure on a general vector space x is an example of an operator from x into l 2 which preserves the pairing of x and x under the inner product of l 2 this biorthogonality property is exploited to obtain by means of comparison a lower bound on s covariance operator the logarithmic derivative is then explicitly evaluated for probability measures induced by the solutions to stochastic ordinary and partial differential equations