nonlinear fractional dynamics on a lattice with long range interactions abstract a unified approach has been developed to study nonlinear dynamics of a 1d lattice of particles with long range power law interaction a classical case is treated in the framework of the generalization of the well known frenkel kontorova chain model for the non nearest interactions quantum dynamics is considered following davydov s approach for molecular excitons in the continuum limit the problem is reduced to dynamical equations with fractional derivatives resulting from the fractional power of the long range interaction fractional generalizations of the sine gordon nonlinear schrodinger and hilbert schrodinger equations have been found there exists a critical value of the power s of the long range potential below the critical value s 3 s 1 2 we obtain equations with fractional derivatives while for s 3 we have the well known nonlinear dynamical equations with space derivatives of integer order long range interaction impact on the quantum lattice propagator has been studied we have shown that the quantum exciton propagator exhibits transition from the well known gaussian like behavior to a power law decay due to the long range interaction a link between 1d quantum lattice dynamics in the imaginary time domain and a random walk model has been discussed

random rotations of the wiener path let w h be an abstract wiener space and letr w be a strongly measurable random variable with values in the set of isometries onh suppose that rh is smooth in the sobolev sense and that it is a quasi nilpotent operator onh for everyh h it is shown that r w h is again a gaussian 0 h 2 random variable consequently if e i i w is a complete orthonormal basis ofh then tilde w sum nolimits i delta r w e i e i defines a measure preserving transformation a rotation onw it is also shown that if for some strongly measurable operator valued onh random variabler r w k h is 0 h 2 gaussian for allk h h thenr is an isometry and rh is quasi nilpotent for allh h the relation between the stochastic calculi for these wiener pathsw and tilde w as well as the conditions of the inverbibility of the map w to tilde w are discussed and the problem of the absolute continuity of the image of the wiener measure under euclidean motion on the wiener space i e w to tilde w composed with a shift is studied

map of discrete system into continuous continuous limits of discrete systems with long range interactions are considered the map of discrete models into continuous medium models is defined a wide class of long range interactions that give the fractional equations in the continuous limit is discussed the one dimensional systems of coupled oscillators for this type of long range interactions are considered the discrete equations of motion are mapped into the continuum equation with the riesz fractional derivative

fractional variational calculus and the transversality conditions this paper presents the euler lagrange equations and the transversality conditions for fractional variational problems the fractional derivatives are defined in the sense of riemann liouville and caputo the connection between the transversality conditions and the natural boundary conditions necessary to solve a fractional differential equation is examined it is demonstrated that fractional boundary conditions may be necessary even when the problem is defined in terms of the caputo derivative furthermore both fractional derivatives the riemann liouville and the caputo arise in the formulations even when the fractional variational problem is defined in terms of one fractional derivative only examples are presented to demonstrate the applications of the formulations

bayes factors based on test statistics summary traditionally the use of bayes factors has required the specification of proper prior distributions on model parameters that are implicit to both null and alternative hypotheses i describe an approach to defining bayes factors based on modelling test statistics because the distributions of test statistics do not depend on unknown model parameters this approach eliminates much of the subjectivity that is normally associated with the definition of bayes factors for standard test statistics including the 2 f t and z statistics the values of bayes factors that result from this approach have simple closed form expressions

fractional variations for dynamical systems hamilton and lagrange approaches fractional generalization of an exterior derivative for calculus of variations is defined the hamilton and lagrange approaches are considered fractional hamilton and euler lagrange equations are derived fractional equations are obtained by fractional variation of lagrangian and hamiltonian that have only integer derivatives

continuous limit of discrete systems with long range interaction discrete systems with long range interactions are considered continuous medium models as continuous limit of discrete chain system are defined long range interactions of chain elements that give the fractional equations for the medium model are discussed the chain equations of motion with long range interaction are mapped into the continuum equation with the riesz fractional derivative we formulate the consistent definition of continuous limit for the systems with long range interactions in this paper we consider a wide class of long range interactions that give fractional medium equations in the continuous limit the power law interaction is a special case of this class

the vanishing of l2 harmonic one forms on based path spaces abstract we prove the triviality of the first l 2 cohomology class of based path spaces of riemannian manifolds furnished with brownian motion measure and the consequent vanishing of l 2 harmonic one forms we give explicit formulae for closed and co closed one forms expressed as differentials of functions and co differentials of l 2 two forms respectively these are considered as extended clark ocone formulae a feature of the proof is the use of the temporal structure of path spaces to relate a rough exterior derivative operator on one forms to the exterior differentiation operator used to construct the de rham complex and the self adjoint laplacian on l 2 one forms this laplacian is shown to have a spectral gap

an implicit midpoint difference scheme for the fractional ginzburg landau equation this paper proposes and analyzes an efficient difference scheme for the nonlinear complex ginzburg landau equation involving fractional laplacian the scheme is based on the implicit midpoint rule for the temporal discretization and a weighted and shifted grunwald difference operator for the spatial fractional laplacian by virtue of a careful analysis of the difference operator some useful inequalities with respect to suitable fractional sobolev norms are established then the numerical solution is shown to be bounded and convergent in the l h 2 norm with the optimal order o 2 h 2 with time step and mesh size h the a priori bound as well as the convergence order holds unconditionally in the sense that no restriction on the time step in terms of the mesh size h needs to be assumed numerical tests are performed to validate the theoretical results and effectiveness of the scheme

de rham hodge kodaira operator on loop groups we consider a based loop group le g over a compact lie groupg endowed with its pinned wiener measure the law of the brownian bridge ong and we shall calculate the ricci curvature for differentialn forms over le g a type of bochner weitzenbock formula for general differentialn forms or shigekawa identity will be established

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

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

anl2estimate for riemannian anticipative stochastic integrals abstract we define a metric and a markovian connection on the path space of a riemannian manifold which are different from those introduced in cm and prove a corresponding weitzenbock formula an l 2 inequality for the divergence is obtained as a consequence

the generalized fractional trace variational identity and fractional integrable couplings of kaup newell hierarchy starting from fractional isospectral problems and general bilinear forms the generalized fractional trace identity is presented then a new explicit lie algebra is introduced for which the fractional integrable couplings of kaup newell hierarchy are derived furthermore we obtained the fractional hamiltonian structures of the fractional integrable couplings of kaup newell hierarchy the methods derived by us can be generalized to other fractional integrable couplings

high accuracy power series solutions with arbitrarily large radius of convergence for the fractional nonlinear schrodinger type equations fractional nonlinear differential equations present an interplay between two common and important effective descriptions used to simplify high dimensional or more complicated theories nonlinearity and fractional derivatives these effective descriptions thus appear commonly in physical and mathematical modeling we present a new series method providing systematic controlled accuracy for solutions of fractional nonlinear differential equations including the fractional nonlinear schrodinger equation and the fractional nonlinear diffusion equation the method relies on spatially iterative use of power series expansions our approach permits an arbitrarily large radius of convergence and thus solves the typical divergence problem endemic to power series approaches in the specific case of the fractional nonlinear schrodinger equation we find fractional generalizations of cnoidal waves of jacobi elliptic functions as well as a fractional bright soliton for the fractional nonlinear diffusion equation we find the combination of fractional and nonlinear effects results in a more strongly localized solution which nevertheless still exhibits power law tails albeit at a much lower density

a general form of the generalized taylor s formula with some applications in this article a new general form of fractional power series is introduced some results of classical power series are circulated and proved to this new form a new general form of the generalized taylor s formula is also obtained applications including fractional power series solutions for fdes are analyzed results reveal that the new expansion is effective for formulating the solutions in this article a new general form of fractional power series is introduced in the sense of the caputo fractional derivative using this approach some results of the classical power series are circulated and proved to this fractional power series whilst a new general form of the generalized taylor s formula is also obtained some applications including fractional power series solutions for higher order linear fractional differential equations subject to given nonhomogeneous initial conditions are provided and analyzed to guarantee and to confirm the performance of the proposed results the results reveal that the new fractional expansion is very effective straightforward and powerful for formulating the exact solutions in the form of a rapidly convergent series with easily computable components

logarithmic sobolev inequalities on loop groups let g be a compact lie group denote by m the brownian bridge measure on the loop group y g c 0 1 g g 0 g 1 e the finite energy subgroup of y determines in a natural way a gradient operation for functions on y the following logarithmic sobolev inequality is proven f2 log f dm gradf y 2 v y f y 2 dm f 2log f wherein f denotes the l2 m norm and v is a potential which is quadratic in the associated lie algebra valued brownian motion the inequality is derived by a method of inheritance from the known inequality for the g valued brownian motion

analysis on wiener spaces ii differential forms abstract the author gives the notions of p forms and p forms and shows some properties of them in particular a vanishing theorem of de rham cohomology of poincare type also he shows the hypoellipticity of certain elliptic operators on wiener space and a theorem of poinare inequality type

reconstruction for the signature of a rough path recently it was proved that the group of rough paths modulo tree like equivalence is isomorphic to the corresponding signature group through the signature map s a generalized notion of taking iterated path integrals however the proof of this uniqueness result does not contain any information on how to see the trajectory of a tree reduced rough path from its signature and a constructive understanding on the uniqueness result in particular on the inverse of s has become an interesting and important question the aim of this paper is to reconstruct a rough path from its signature in an explicit and universal way

transition to chaos in discrete nonlinear schrodinger equation with long range interaction discrete nonlinear schrodinger dnls equation describes a chain of oscillators with nearest neighbor interactions and a specific nonlinear term we consider its modification with long range interaction through a potential proportional to 1 l1 with fractional 2 and l as a distance between oscillators this model is called dnls it exhibits competition between the nonlinearity and a level of correlation between interacting far distanced oscillators that is defined by the value of we consider transition to chaos in this system as a function of and nonlinearity it is shown that decreasing of with respect to nonlinearity stabilize the system connection of the model to the fractional generalization of the nls called fnls in the long wave approximation is also discussed and some of the results obtained for dnls can be correspondingly extended to the fnls

the fractional quadratic form identity and hamiltonian structure of an integrable coupling of the fractional ablowitz kaup newell segur hierarchy starting from a general isospectral problem of fractional order we propose a fractional quadratic form identity from which the hamiltonian structure of an integrable coupling of the fractional ablowitz kaup newell segur hierarchy is derived the method can be generalized to other fractional integrable couplings

weak levi civita connection for the damped metric on the riemannian path space and vanishing of ricci tensor in adapted differential geometry abstract we shall establish in the context of adapted differential geometry on the path space p m o m a weitzenbock formula which generalizes that in a b cruzeiro and p malliavin j funct anal 177 2000 219 253 without hypothesis on the ricci tensor the renormalized ricci tensor will be vanished the connection introduced in a b cruzeiro and s fang 1997 j funct anal 143 400 414 will play a central role

markovian connection curvature and weitzenbock formula on riemannian path spaces abstract we shall consider on a riemannian path space p m o m the cruzeiro malliavin s markovian connection the laplace operator will be defined as the divergence of the gradient we shall compute explicitly the associated curvature tensor a weitzenbock formula will be established to this end we shall introduce an inner product between the tangent processes and simple vector fields

lpcontraction semigroups for vector valued functions abstract we discuss semigroups acting on vector valued functions we consider a comparison theorem between two semigroups a semigroup acting on scalar valued functions and a semigroup acting on vector valued functions the criterion is given by the abstract kato theorem by using this we give a sufficient condition for the criterion in the setting of square field operator we also consider the essential self adjointness of a perturbed semigroup

measuring complexity nonextensivity and chaos in the dna sequence of the major histocompatibility complex abstract we analyze 4 mb sequences of the major histocompatibility complex mhc which is a dna segment on chromosome 6 with high gene density controlling many immunological functions and associated with many diseases the analysis is based on modern theoretical and mathematical tools of complexity theory such as nonlinear time series analysis and tsallis non extensive statistics the results revealed that the dna complexity and self organization can be related to fractional dynamical nonlinear processes with low dimensional deterministic chaotic and non extensive statistical character which generate the dna sequences under the extremization of tsallis q entropy principle while it still remains an open question as to whether the dna walk is a fractional brownian motion fbm a static anomalous diffusion process or a non gaussian dynamical fractional anomalous diffusion process the results of this study testify for the latter providing also a possible explanation for the previously observed long range power law correlations of nucleotides as well as the long range correlation properties of coding and non coding sequences present in dna sequences

random attractors for the stochastic coupled fractional ginzburg landau equation with additive noise this paper is concerned with the stochastic coupled fractional ginzburg landau equation with additive noise we first transform the stochastic coupled fractional ginzburg landau equation into random equations whose solutions generate a random dynamical system then we prove the existence of random attractor for random dynamical system

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

a generalized fractional trace identity and hamiltonian structure of the fractional soliton equation hierarchy a generalized fractional trace identity and hamiltonian structure of the fractional soliton equation hierarchy are presented by using the modified riemann liouville derivative and exterior derivatives of fractional orders example of the fractional kaup newell kn soliton equation hierarchy and hamiltonian system are considered which is a new integrable hierarchy and possesses hamiltonian structure

integration on loop groups ii heat equation for the wiener measure abstract let g be a compact semi simple lie group and l g the loop group over g on l g is defined a family t l t 0 of quasi invariant wiener measures we construct on l g a laplace operator l such that t l are the measures generated by the process associated to l modified by a feynman kac density

fractional ginzburg landau equation for fractal media abstract we derive the fractional generalization of the ginzburg landau equation from the variational euler lagrange equation for fractal media to describe fractal media we use the fractional integrals considered as approximations of integrals on fractals some simple solutions of the ginzburg landau equation for fractal media are considered and different forms of the fractional ginzburg landau equation or nonlinear schrodinger equation with fractional derivatives are presented the agrawal variational principle and its generalization have been applied

fractional generalization of gradient and hamiltonian systems we consider a fractional generalization of hamiltonian and gradient systems we use differential forms and exterior derivatives of fractional orders we derive fractional generalization of helmholtz conditions for phase space examples of fractional gradient and hamiltonian systems are considered the stationary states for these systems are derived

dynamics with low level fractionality abstract the notion of fractional dynamics is related to equations of motion with one or a few terms with derivatives of a fractional order this type of equation appears in the description of chaotic dynamics wave propagation in fractal media and field theory for the fractional linear oscillator the physical meaning of the derivative of order 2 is dissipation in systems with many spacially coupled elements oscillators the fractional derivative along the space coordinate corresponds to a long range interaction we discuss a method of constructing a solution using an expansion in n with small and positive integer n the method is applied to the fractional linear and nonlinear oscillators and to fractional ginzburg landau or parabolic equations

nonholonomic constraints with fractional derivatives we consider the fractional generalization of nonholonomic constraints defined by equations with fractional derivatives and provide some examples the corresponding equations of motion are derived using variational principle we prove that fractional constraints can be used to describe the evolution of dynamical systems in which some coordinates and velocities are related to velocities through a power law memory function

asymptotic behavior of stochastic fractional power dissipative equations on rn abstract we study a reaction fractional diffusion equation with additive noise on the entire space r n with particular interest in the asymptotic behavior of solutions we first transform the equation into a random equation whose solutions generate a random dynamical system a priori estimates for solutions are derived when the nonlinearity satisfies certain growth conditions using estimates for far field values of solutions and a cut off technique asymptotic compactness is proved thus the existence of a random attractor in l 2 r n is established

an efficient split step quasi compact finite difference method for the nonlinear fractional ginzburg landau equations abstract in this paper we propose a split step quasi compact finite difference method to solve the nonlinear fractional ginzburg landau equations both in one and two dimensions the original equations are split into linear and nonlinear subproblems the riesz space fractional derivative is approximated by a fourth order fractional quasi compact method furthermore an alternating direction implicit scheme is constructed for the two dimensional linear subproblem the unconditional stability and convergence of the schemes are proved rigorously in the linear case numerical experiments are performed to confirm our theoretical findings and the efficiency of the proposed method

on a fractional ginzburg landau equation and 1 2 harmonic maps into spheres this paper is devoted to the asymptotic analysis of a fractional version of the ginzburg landau equation in bounded domains where the laplacian is replaced by an integro differential operator related to the square root laplacian as defined in fourier space in the singular ginzburg landau limit we show that solutions with uniformly bounded energy converge weakly to sphere valued 1 2 harmonic maps i e the fractional analogues of the usual harmonic maps in addition the convergence holds in smooth functions spaces away from a n 1 rectifiable closed set of finite n 1 hausdorff measure the proof relies on the representation of the square root laplacian as a dirichlet to neumann operator in one more dimension and on the analysis of a boundary version of the ginzburg landau equation besides the analysis of the fractional ginzburg landau equation we also give a general partial regularity result for stationary 1 2 harmonic maps in arbitrary dimension

a cameron martin type quasi invariance theorem for brownian motion on a compact riemannian manifold abstract let m be a compact manifold without boundary o be a fixed base point in m g be a riemannian metric on m and be a g compatible covariant derivative on tm the tangent space to m assume the torsion t of satisfies the skew symmetry condition g t x y y 0 for all vector fields x and y on m for example take to be the levi civita covariant derivative on m g also let denote the wiener measure on w0 m c 0 1 m 0 o and let h s denote stochastic parallel translation relative to along the path w0 m up to time s 0 1 given a c1 function h 0 1 t0m it is shown that the differential equation t h t h with initial condition 0 id w m w m has a solution r maps w m w m the measurable maps from w m to w m this function is a flow on w m i e for all t r t t furthermore t has the quasi invariance property the law t of t with respect to the wiener measure is equivalent to for all t r this result is used to prove an integration by parts formula for the h derivative hf defined by h f d dt 0 f t where f is a c2 cylinder function on w m

dynamics of the 3 d fractional complex ginzburg landau equation abstract we study the initial boundary value problem of the fractional complex ginzburg landau equation in three spatial dimensions with the dissipative effect given by a fractional laplacian a priori estimates are derived when the nonlinearity satisfies certain growth conditions using galerkin s method the existence of a global smooth solution is established uniqueness is also proved furthermore the existence of a global attractor is proved and estimates of the hausdorff and fractal dimensions for the global attractor are obtained

an l 2 theory for differential forms on path spaces i an l 2 theory of differential forms is proposed for the banach manifold of continuous paths on a riemannian manifold m furnished with its brownian motion measure differentiation must be restricted to certain hilbert space directions the h tangent vectors to obtain a closed exterior differential operator the relevant spaces of differential forms the h forms are perturbed by the curvature of m a hodge decomposition is given for l 2 h one forms and the structure of h two forms is described the dual operator d is analysed in terms of a natural connection on the h tangent spaces malliavin calculus is a basic tool c 2007 elsevier inc all rights reserved

fractal burgers equations abstract the paper studies local and global in time solutions to a class of multidimensional generalized burgers type equations with a fractional power of the laplacian in the principal part and with general algebraic nonlinearity such equations naturally appear in continuum mechanics our results include existence uniqueness regularity and asymptotic behavior of solutions to the cauchy problem as well as a construction of self similar solutions the role of critical exponents is also explained

psi series solution of fractional ginzburg landau equation one dimensional ginzburg landau equations with derivatives of noninteger order are considered using psi series with fractional powers the solution of the fractional ginzburg landau fgl equation is derived the leading order behaviours of solutions about an arbitrary singularity as well as their resonance structures have been obtained it was proved that fractional equations of order with polynomial nonlinearity of order s have the noninteger power like behaviour of order 1 s near the singularity

linearized adi schemes for two dimensional space fractional nonlinear ginzburg landau equation abstract space and time approximations for two dimensional space fractional complex ginzburg landau equation are examined the schemes under consideration are discreted by the second order backward differential formula bdf2 in time and two classes of the fractional centered finite difference methods in space a linearized technique is employed by the extrapolation we prove the unique solvability and stability for both numerical methods the convergence of both numerical methods is analyzed at length utilizing the energy argument and the convergence orders under the optimal step size ratio are o 2 h 2 and o 2 h 4 in the sense of the discrete l 2 norm where is the time step size h max h x h y and h x h y are spatial grid sizes in the x direction and y direction respectively in addition we construct a multistep alternating direction implicit adi scheme and a multistep compact adi scheme based on bdf2 for the efficiently numerical implementation finally numerical examples are carried out to verify our theoretical results

on the ornstein uhlenbeck operators on wiener riemannian manifolds abstract in this paper we study the properties of infinite dimensional ornstein uhlenbeck operators on submanifolds in wiener spaces we establish the hypoellipticity of the laplacians acting on the space of tensors by using this we show the essential selfadjointness of the laplacians acting on the space of 0 forms i e function spaces

improved estimation in a contingency table independence structure abstract estimation of the cell probabilities in a two way contingency table is considered when it is plausible that the table might have an independence structure relating to the two traits in a classical nonparametric setup the unrestricted maximum likelihood estimators of the cell probabilities are the corresponding sample proportions under the assumption of independence the restricted estimators are the product of the respective row and column sample proportions the latter estimators behave better than the former when independence actually holds but a different picture may emerge for possible departure from the assumed independence structure the restricted estimators may be heavily biased inefficient and even inconsistent for this reason a preliminary test on independence based on the classical contingency chi squared statistic may be conveniently incorporated in the formulation of a preliminary test estimator of the matrix of cell probabilities since typically we have a multiparameter e

a general finite element formulation for fractional variational problems abstract this paper presents a general finite element formulation for a class of fractional variational problems fvps the fractional derivative is defined in the riemann liouville sense for fvps the euler lagrange and the transversality conditions are developed in the fractional finite element formulation ffef presented here the domain of the equations is divided into several elements and the functional is approximated in terms of nodal variables minimization of this functional leads to a set of algebraic equations which are solved using a numerical scheme three examples are considered to show the performance of the algorithm results show that as the number of discretization is increased the numerical solutions approach the analytical solutions and as the order of the derivative approaches an integer value the solution for the integer order system is recovered for unspecified boundary conditions the numerical solutions satisfy the transversality conditions this indicates that for the class of problems considered the numerical solutions can be obtained directly from the functional and there is no need to solve the fractional euler lagrange equations thus the formulation extends the traditional finite element approach to fvps

a multi domain spectral method for time fractional differential equations this paper proposes an approach for high order time integration within a multi domain setting for time fractional differential equations since the kernel is singular or nearly singular two main difficulties arise after the domain decomposition how to properly account for the history memory part and how to perform the integration accurately to address these issues we propose a novel hybrid approach for the numerical integration based on the combination of three term recurrence relations of jacobi polynomials and high order gauss quadrature the different approximations used in the hybrid approach are justified theoretically and through numerical examples based on this we propose a new multi domain spectral method for high order accurate time integrations and study its stability properties by identifying the method as a generalized linear method numerical experiments confirm hp convergence for both time fractional differential equations and time fractional partial differential equations

critical dynamics in systems controlled by fractional kinetic equations abstract the field theory renormalization group is used for analyzing the fractional langevin equation with the order of the temporal derivative 0 1 fractional laplacian of the order and gaussian noise correlator the case of non linearity m with odd m 3 is considered it is proved that the model is multiplicatively renormalizable propagators were found in the momentum and coordinate representation expressed in terms of fox s h functions existence of the dissipative scaling regime in the framework of the e expansion for 2 1 l l 1 2 is proved requirement of the continuous dependence of the critical exponents on imposes the condition m 3 the main quantitative result is the calculation of the dynamical critical exponent z for 1 2 up to e 2 we have obtained for it the expression z 1 2 4 0 1555 e 2 o e 3

asymptotic behavior for non autonomous fractional stochastic ginzburg landau equations on unbounded domains in this paper we first prove the existence and uniqueness of tempered pullback random attractors for a non autonomous stochastic fractional ginzburg landau equation driven by multiplicative noise with 0 1 in l2r3 then we obtain the upper semicontinuity of random attractors when the intensity of noise approaches zero due to the lack of the compactness of sobolev embeddings on unbounded domains we establish the pullback asymptotic compactness of solutions in l2 r3 by the tail estimates of solutions

analysis on wiener spaces i nonlinear maps abstract the author introduces several kinds of capacities and regularity of nonlinear maps on wiener spaces and studies their properties

stochastic analysis on the path space of a riemannian manifold i markovian stochastic calculus abstract on the brownian flow of a compact riemannian manifold an intrinsic stochastic calculus is defined this calculus is markovian the stochastic calculus of variation on the bundle of orthonormal frames appears only as an intermediate tool the bismut connection plays a paramount role in the final construction

renormalized differential geometry on path space structural equation curvature abstract the theory of integration in infinite dimensions is in some sense the backbone of probability theory on this backbone the stochastic calculus of variations has given rise to the flesh of differential calculus its first step is the construction at each point of the probability space of a cameron martin like tangent space in which the desired differential calculus can be developed this construction proceeds along the lines of first order differential geometry in this paper we address the following questions what could be the meaning of curvature of the probability space how and why how can curvatures be defined and computed why could a second order differential geometry be relevant to stochastic analysis we try to answer these questions for the probability space associated to the brownian motion of a compact riemannian manifold why a basicenergy identity for anticipative stochastic integralswill be obtained as a byproduct of our computation of curvature how there are essentially four bottlenecks in the development of differential geometry on wiener riemann manifolds i the difficulty of finding an atlas of local charts such that the changes of charts preserve the class of the wiener like measures together with their associated cameron martin like tangent spaces ii the difficulty of finding cylindrical approximations preserving the natural geometrical objects iii the difficulty of renormalizing the divergent series to which the summation operations of finite dimensional differential geometry give rise in the non intrinsic context of local charts iv the nonavailability of the computational procedures analogous to the local coordinates systems of the classical differential geometry in the context of path space the ito filtration provides a much richer structure than that available in the framework of an abstract wiener riemann manifold our work is a systematic attempt to replace the machinery of local charts with amethodology of moving frames in our context stochastic parallel transport provides a canonical moving frame on the path space the concept of a cylindrical approximation has to be reshaped in our new situation into some geometric limit theorems establishing that the riemannian geometric objects of the cylindrical approximations induce by a limiting procedure geometric objects on the path space those limit theorems are reminiscent of the classical theorems which say that a stratonovich sde is the limit of an appropriate sequence of ode the canonical coordinate system provided by the moving frame will make it possible to proceed to the needed renormalizations byintrinsicstochastic integrals in this context the anticipative stochastic integral theory of nualart and pardoux will play a decisive role finally the moving frame will provide aneffective algorithm of computation for this differential geometry in infinite dimensions in our study we encounter a new type of renormalization thehypoelliptic renormalization which corresponds to the fact that the bracket of smooth vector fields taking their values in the cameron martin space can get out of this cameron martin space this hypoelliptic problem induces the nonrenormalizability of some geometrical objects it leads also to a concept oftangent processesto probability spaces extending that based on the cameron martin theorem for tangent processes a formula of integration by parts still holds furthermore the tangent processes form a lie algebra under the bracket on the other hand tangent processes cannot be stochastically integrated this operation is well defined only for cameron martin type vector fields

fractional variational calculus in terms of riesz fractional derivatives this paper presents extensions of traditional calculus of variations for systems containing riesz fractional derivatives rfds specifically we present generalized euler lagrange equations and the transversality conditions for fractional variational problems fvps defined in terms of rfds we consider two problems a simple fvp and an fvp of lagrange results of the first problem are extended to problems containing multiple fractional derivatives functions and parameters and to unspecified boundary conditions for the second problem we present lagrange type multiplier rules for both problems we develop the euler lagrange type necessary conditions which must be satisfied for the given functional to be extremum problems are considered to demonstrate applications of the formulations explicitly we introduce fractional momenta fractional hamiltonian fractional hamilton equations of motion fractional field theory and fractional optimal control the formulations presented and the resulting equations are similar to the formulations for fvps given in agrawal 2002 j math anal appl 272 368 2006 j phys a math gen 39 10375 and to those that appear in the field of classical calculus of variations these formulations are simple and can be extended to other problems in the field of fractional calculus of variations

fast iterative solvers and simulation for the space fractional ginzburg landau equations abstract in this paper we propose a new splitting iteration method for solving the toeplitz like complex linear system arising from the space fractional ginzburg landau equations of which the coefficient matrix is equal to the sum of a complex scaled toeplitz plus diagonal matrix and a symmetric toeplitz plus diagonal matrix the new splitting method enjoys computational advantage since circulant preconditioner and fast fourier transform fft can be used for solving the involved linear subsystem under some conditions the convergence properties of the corresponding iteration method are derived numerical examples are given to illustrate the effectiveness of the proposed method

existence and upper semicontinuity of random attractors for non autonomous fractional stochastic ginzburg landau equations this paper deals with the dynamical behavior of solutions for nonautonomous stochastic fractional ginzburg landau equations driven by multiplicative noise with 0 1 we first transform the stochastic equation into a random equation the solutions of which generate a random dynamical system consequently we establish the existence and uniqueness of tempered pullback random attractors for the equations in a bounded domain finally we obtain the upper semicontinuity of random attractors when the intensity of noise approaches zero

sobolev spaces over loop groups abstract in this paper we study the properties of orstein uhlenbeck operators over loop groups which are naturally defined by the group structure we establish the hypoellipticity of the laplacians acting on the space of tensors by using this we show the essential self adjointness of the laplacians acting on the real valued function spaces

assessing robustness of intrinsic tests of independence in two way contingency tables for testing nested hypotheses from a bayesian standpoint a desirable condition is that the prior for the alternative model concentrates mass around the smaller or null model for testing independence in contingency tables the intrinsic priors satisfy this requirement furthermore the degree of concentration of the priors is controlled by a discrete parameter t the training sample size which plays an important role in the resulting answer in this article we report on the robustness of the tests of independence for small or moderate sample sizes in contingency tables with respect to intrinsic priors with different degrees of concentration around the null we compare these tests to frequentist tests and other robust bayes tests for large sample sizes robustness is achieved because the intrinsic bayesian tests are consistent examples using real and simulated data are given supplemental materials technical details and data sets are available online

frame bundle of riemannian path space and ricci tensor in adapted differential geometry abstract the vanishing of the renormalized ricci tensor of the path space above a ricci flat riemannian manifold is discussed

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

fractional dynamics of systems with long range space interaction and temporal memory abstract field equations with time and coordinate derivatives of noninteger order are derived from a stationary action principle for the cases of power law memory function and long range interaction in systems the method is applied to obtain a fractional generalization of the ginzburg landau and nonlinear schrodinger equations as another example dynamical equations for particle chains with power law interaction and memory are considered in the continuous limit the obtained fractional equations can be applied to complex media with without random parameters or processes

dynamical behavior of non autonomous fractional stochastic reaction diffusion equations abstract in this paper we investigate the asymptotic behavior of solutions for the non autonomous fractional stochastic reaction diffusion equations driven by multiplicative noise with s 0 1 we prove the existence and uniqueness of tempered pullback random attractors for the equations in a bounded domain u by the compactness of sobolev embedding h s u l 2 u which is different from the previous work 46

asymptotics of spectral gaps on loop spaces over a class of riemannian manifolds abstract we prove the existence of spectral gaps of ornstein uhlenbeck operators on loop spaces over a class of riemannian manifolds which include hyperbolic spaces this is an alternative proof and an extension of a result in chen et al 13 further we study the asymptotic behavior of the spectral gap

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

ito maps and analysis on path spaces we consider versions of malliavin calculus on path spaces of compact manifolds with diffusion measures defining gross sobolev spaces of differentiable functions and proving their intertwining with solution maps mathcal i of certain stochastic differential equations this is shown to shed light on fundamental uniqueness questions for this calculus including uniqueness of the closed derivative operator d and markov uniqueness of the associated dirichlet form a continuity result for the divergence operator by kree and kree is extended to this situation the regularity of conditional expectations of smooth functionals of classical wiener space given mathcal i is considered and shown to have strong implications for these questions a major role is played by the possibly sub riemannian connections induced by stochastic differential equations damped markovian connections are used for the covariant derivatives

bayesian estimation methods for 2 2 contingency tables using mixtures of dirichlet distributions abstract in the estimation of cell probabilities from a 2 2 table a prior distribution is developed that can reflect prior beliefs about the cross classification structure in the table the posterior means using this prior shrink the classical estimates towards the association structure specified a priori by the user closed form approximations to the posterior means and credible regions are developed in the special case where the two variables of the table are believed independent

a generalized fractional kn equation hierarchy and its fractional hamiltonian structure a generalized hamiltonian structure of the fractional soliton equation hierarchy is presented by using differential forms and exterior derivatives of fractional orders we construct the generalized fractional trace identity through the riemann liouville fractional derivative an example of the fractional kn soliton equation hierarchy and hamiltonian structure is presented which is a new integrable hierarchy and possesses hamiltonian structure