on flows associated to sobolev vector fields in wiener spaces an approach a la diperna lions in this paper we extend the diperna lions theory of flows associated to sobolev vector fields to the case of cameron martin valued vector fields in wiener spaces e having a sobolev regularity the proof is based on the analysis of the continuity equation in e and on uniform gaussian commutator estimates in finite dimensional spaces

equations differentielles sur l espace de wiener et formules de cameron martin non lineaires proprietes algebriques des approximations sur r n une formule de perturbation des flots et convergence des approximations formule de changement de variable sur x et equation differentielle

on the uniqueness of solutions to continuity equations abstract we obtain sufficient conditions for the uniqueness of solutions to the cauchy problem for the continuity equation in classes of measures that need not be absolutely continuous

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

on continuity equations in infinite dimensions with non gaussian reference measure let be a gaussian measure on a locally convex space and h be the corresponding cameron martin space it has been recently shown by l ambrosio and a figalli that the linear rst order pde

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

a generalized two dimensional analytical solution for hydrodynamic dispersion in bounded media with the first type boundary condition at the source a generalized two dimensional analytical solution is developed for hydrodynamic dispersion in a unidirectional bounded surface and subsurface media flow field from time and space dependent sources with the first type boundary condition at the source longitudinal and transverse dispersion radioactive decay and linear adsorption are considered in the mathematical analysis laplace transform and fourier analysis techniques are used simultaneously and a general equation in infinite series form for the unsteady state concentration distribution has been obtained solution for a single strip source is obtained as a special case from the general solution the equations for convective dispersive flux components are also derived the solution of bruch and street 1967 for a symmetrically located strip source is shown to be a special case of the solution expressions for concentration distribution and convective dispersive flux components for the steady state solute transport case are also presented the results are compared with two different finite element codes with a good agreement the results of this model may be used for solute transport in a unidirectional porous media flow field as well as in rivers and canals with some additional assumptions the solutions can also be used for numerical model validation

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

the 2d euler equations and the statistical transport equations we prove the existence of weak solutions for the forward and backward statistical transport equations associated with the 2d euler equations such solutions can be interpreted respectively as a statistical lagrangian and a statistical eulerian description of the motion of the fluid

differential calculus on finite codimensional submanifolds of the wiener space the divergence operator abstract on finite codimensional submanifolds of the wiener space the ricci gaussian curvature is defined and used to prove a weitzenbock bochner identity from this identity sobolev estimates of the divergence of a vector field are obtained moreover using the method of the malliavin calculus covariant differentiation christoffel symbols and brackets of two vector fields are defined

quantum birkhoff gustavson normal form in some completely resonant cases abstract the goal of this paper is to present an explicit computation of the birkhoff gustavson normal form in the 1 1 1 2 and 1 3 resonances by using the concept of weyl quantization especially when it is about the first terms we use in particular the weyl symbolic calculus and the abstract theorem of the quantum birkhoff gustavson normal form in infinite dimension

a class of vector fields on path spaces abstract in this paper we show that the vector fieldx hon a based path spacewo m over a riemannian manifoldmdefined by parallel translating a curvehin the initial tangent spacetomvia an affine connection induces a solution flow which preserves the wiener measure on the based path spacewo m provided the affine connection is adjoint skew symmetric in the case when is a metric connection then is adjoint skew symmetric if and only if is torsion skew symmetric

anticipating flows on the wiener space generated by vector fields of low regularity abstract sobolev spaces for banach space valued wiener functionals are constructed using the operator norm for the h derivatives the corresponding sobolev norms are shown to be strictly weaker than the usual malliavin sobolev norms existing results on flows on an abstract wiener space generated by a vector field are generalized to the case where the vector field is a sum of a not necessarily bounded skew adjoint operator and a non linear part whose regularity is measured by the new sobolev norms this is equivalent to a non linear perturbation of a semi group of rotations

flow equations on spaces of rough paths abstract given an ito vector fieldm there is a unique solution t h to the differential equation d t h dt m t h 0 h h for any continuous and piece wisely smooth pathh we show that for anyt r the maph t h is continuous in thep variation topology for anyp 1 so that it uniquely extends to a solution flow on the space of all geometric rough paths applying this result to the driver s geometric flow equation on the path space over a closed riemannian manifold d t dt x h t 0 id wherexhis the vector field defined by parallel translatinghvia a connection our result especially yields a deterministic construction of the driver s flow

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

spectral asymptotics via the semiclassical birkhoff normal form this article gives a simple treatment of the quantum birkhoff normal form for semiclassical pseudo differential operators with smooth coefficients the normal form is applied to describe the discrete spectrum in a generalised non degenerate potential well yielding uniform estimates in the energy e this permits a detailed study of the spectrum in various asymptotic regions of the parameters e h and gives improvements and new proofs for many of the results in the field in the completely resonant case we show that the pseudo differential operator can be reduced to a toeplitz operator on a reduced symplectic orbifold using this quantum reduction new spectral asymptotics concerning the fine structure of eigenvalue clusters are proved in the case of polynomial differential operators a combinatorial trace formula is obtained

flows associated to tangent processes on the wiener space we prove under certain regularity assumptions on the coefficients that tangent processes namely semimartingales d a dx b d where a is an antisymmetric matrix generate flows on the classical wiener space main applications of the result can be found in the study of the geometry of path spaces

unicite de solutions d equations differentielles sur l espace de wiener soit x l espace de wiener et a un champ de vecteurs sur x on definit la notion de solution de l equation differentielle du dt 1 u t u 0 x on demontre un theoreme d unicite

integrated harnack inequalities on lie groups we show that the logarithmic derivatives of the convolution heat kernels on a uni modular lie group are exponentially integrable this result is then used to prove an integrated harnack inequality for these heat kernels it is shown that this integrated harnack inequality is equivalent to a version of wang s harnack inequality a key feature of all of these inequalities is that they are dimension independent finally we show these inequalities imply quasi invariance properties of heat kernel measures for two classes of infinite dimensional lie groups

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

the quantum normal form and its equivalents a quantum analog called the quantum normal form of the classical birkhoff gustavson normal form is presented the algebraic relationship between the quantum and birkhoff gustavson normal forms has been established by developing the latter using lie transforms it is shown that the birkhoff gustavson normal form can be obtained from the quantum normal form using an anharmonic oscillator and a henon heiles system as test cases the equivalence between the quantum normal form and the rayleigh schrodinger perturbation method is shown this equivalence provides an algebraic connection between the birkhoff gustavson normal form and the rayleigh schrodinger perturbation approach the question of weyl and torus quantizations of the birkhoff gustavson normal form is discussed in the light of the quantum normal form

on the diagonalization of quantum birkhoff gustavson normal form an application to quantum mechanics of one of classical perturbation theory methods the birkhoff gustavson normal form bgnf is described in the quantum case it results in the van vleck perturbation theory performed upon wick normal ordered operators algebraic aspects of this procedure and formal construction of invariants integrals of motion for a perturbed system are considered it turned out that a larger set of such operators existed in the quantum mechanics rather than in the classical one it is demonstrated that according to general results of the quantum mechanical perturbation theory the quantum bgnf may always be diagonalized and two formal processes for such diagonalization are constructed in the opposite case the classical bgnf is in general nondiagonalizable this reflects the fact that the classical perturbation theory cannot handle a system with two or more resonances possible reasons for such different behavior of two very close in spirit perturbation procedures are discussed results of the described procedure entirely performed upon the wick normal ordered operators are equivalent to those of rayleigh schrodinger perturbation expansion

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

the schrodinger equation and canonical perturbation theory lett0 ev be the schrodinger operator corresponding to the classical hamiltonianh0 ev whereh0 is thed dimensional harmonic oscillator with non resonant frequencies 1 d and the potentialv q1 qd is an entire function of order d 1 1 we prove that the algorithm of classical canonical perturbation theory can be applied to the schrodinger equation in the bargmann representation as a consequence each term of the rayleigh schrodinger series near any eigenvalue oft0 admits a convergent expansion in powers of of initial point the corresponding term of the classical birkhoff expansion moreover ifv is an even polynomial the above result and the kam theorem show that all eigenvalues n e oft0 ev such thatn coincides with a kam torus are given up to order e by a quantization formula which reduces to the bohr sommerfeld one up to first order terms in

an alternating direction galerkin technique for simulation of contaminant transport in complex groundwater systems a general contaminant transport simulation technique based on a galerkin finite element representation in space and an alternating direction time stepping scheme is developed the formulation which is in terms of natural coordinates following the principal directions of hydraulic conductivity of the medium decouples the spatial components of the equations leading to tridiagonal matrices the numerical accuracy can be easily controlled in each principal direction through the well established peclet and courant numbers results are found to be comparable to those from the classic finite element scheme but the cost of computation is generally much less because the computational effort varies only linearly with the grid size the technique is thus well suited for transport simulations with complex layered anisotropic groundwater systems requiring large and detailed grids

birkhoff gustavson normal form in classical and quantum mechanics a quantum analogue of the classical transformations to the birkhoff gustavson normal form is derived with unitary transformations taking over the role of canonical transformations the author finds a striking similarity to lie transformations in the classical case the author suggests that the quantum normal form is identical to the rayleigh schrodinger perturbation series and thus usually only asymptotic to the eigenvalues the convergence question of both classical and quantum normal forms is discussed in some detail

nonergodicity of euler fluid dynamics on tori versus positivity of the arnold ricci tensor abstract brownian motions above the group g of volume preserving diffeomorphisms of the torus t d d 2 are constructed the asymptotic behaviour for large time of those processes shows the nonexistence of a probability measure invariant under the deterministic incompressible fluid dynamics the energy induces on the group of volume preserving diffeomorphisms of t 2 a riemannian structure which has a positive renormalized ricci tensor

integration on loop groups i quasi invariant measures on a riemannian manifold m the brownian motion defines a wiener measure on the loops over m in the case of r quasi invariance means the cameron martin theorem s which fully describes the quasi invariance by translation of the wiener measure in order to state a generalization of this theorem we suppose that m is an homogeneous space under the action of a compact lie group the noncommutativity of this situation introduces a completely new phenomenon the hypoellipticity associated to the lie bracket of the basic vector fields curiously enough hypoellipticity which helps to get regularity in finite dimensions works against quasi invariance in infinite dimensions 18 we denote by g a compact connected lie group by h a connected subgroup of g and by m the homogeneous space g h we denote by y and 2 the corresponding lie algebras we choose a euclidean metric on 9 which is ad g invariant this metric induces a riemannian metric on g and on m we denote p g the paths space on g that is the space of continuous maps y from 0 l into g such that y o e we denote by l g the space of loops on g that is

integration by parts and quasi invariance for the horizontal wiener measure on foliated compact manifolds abstract we prove several sub riemannian versions of driver s integration by parts formula which first appeared in 17 namely our results are for the horizontal wiener measure on a totally geodesic riemannian foliation equipped with a sub riemannian structure it is also shown that the horizontal wiener measure is quasi invariant under the action of flows generated by suitable tangent processes

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

flows associated to adapted vector fields on the wiener space the existence and uniqueness of a flow associated to an adapted vector field on the wiener space with d a d b d are proved by a modified picard s iteration method mainly under the conditions of exponential integrability concerning b as well as the first order malliavin gradients of a and b a newton leibnitz type inequality for a kind of malliavin differentiable functionals is also proved which is the key point to prove the above result and has an independent interest

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

lagrangian flows driven by bv fields in wiener spaces we establish the renormalization property for essentially bounded solutions of the continuity equation associated to bounded variation bv fields in wiener spaces with values in the associated cameron martin space thus obtaining by standard arguments new uniqueness and stability results for correspondent lagrangian flows

integration by parts and quasi invariance for heat kernel measures on loop groups abstract integration by parts formulas are established both for wiener measure on the path space of a loop group and for the heat kernel measures on the loop group the wiener measure is defined to be the law of a certain loop group valued brownian motion and the heat kernel measures are time t t 0 distributions of this brownian motion a corollary of either of these integrations by parts formulas is the closability of the pre dirichlet form considered by b k driver and t lohrenz 1996 j functional anal 140 381 448 we also show that the heat kernel measures are quasi invariant under right under right and left translations by finite energy loops

asymptotic behavior of divergences and cameron martin theorem on loop spaces we first prove the lp convergence p 1 and a fernique type exponential integrability of divergence functionals for all cameron martin vector fields with respect to the pinned wiener measure on loop spaces over a compact riemannian manifold we then prove that the driver flow is a smooth transform on path spaces in the sense of the malliavin calculus and has an quasi continuous modification which can be quasi surely well defined on path spaces this leads us to construct the driver flow on loop spaces through the corresponding flow on path spaces combining these two results with the cruzeiro lemma j funct anal 54 1983 206 227 we give an alternative proof of the quasi invariance of the pinned wiener measure under driver s flow on loop spaces which was established earlier by driver trans amer math soc 342 1994 375 394 and enchev and stroock adv math 119 1996 127 154 by doob s h processes approach together with the short time estimates of the gradient and the hessian of the logarithmic heat kernel on compact riemannian manifolds we also establish the lp convergence p 1 and a fernique type exponential integrability theorem for the stochastic anti development of pinned brownian motions on compact riemannian manifold with an explicit exponential exponent our results generalize and sharpen some earlier results due to gross j funct anal 102 1991 268 313 and hsu math ann 309 1997 331 339 our method does not need any heat kernel estimate and is based on quasi sure analysis and sobolev estimates on path spaces

effective stability for a hamiltonian system near an elliptic equilibrium point with an application to the restricted three body problem abstract we consider an n degrees of freedom hamiltonian system near an elliptic equilibrium point the system is put in normal form up to an arbitrary order and with respect to some resonance module and estimates are obtained for both the size of the remainder and for the domain of convergence of the transformation leading to normal form a bound to the rate of diffusion is thus found and by optimizing the order of normalization exponential estimates of nekhoroshev s type are obtained this provides explicit estimates for the stability properties of the elliptic point and leads in some cases to effective stability i e stability up to finite but long times an application to the stability of the triangular libration points in the spatial restricted three body is also given

a generalized two dimensional analytical solute transport model in bounded media for flux type finite multiple sources this paper presents a generalized two dimensional analytical solute transport model with unidirectional flow field from time and space dependent sources in a bounded homogeneous medium using the flux type or third type or cauchy boundary condition at the inlet location of the medium the solute transport equation incorporates terms accounting for advection dispersion linear equilibrium sorption and first order decay general solutions were determined for arbitrary input boundary conditions with the help of fourier analysis and laplace transform techniques after presenting a generally applicable solution special solutions are given for a single and two finite sources having different inlet solute flux distributions the expressions for convective dispersive flux components are also derived expressions for concentration distribution and convective dispersive flux components for steady state transport case are also presented comparison of the results of this analytical model and a finite difference numerical model showed very good agreement the solutions may be used for predicting solute concentrations in a unidirectional porous media flow field as well as in rivers and canals the solutions may also be used for verification of more comprehensive numerical models and laboratory or field determination of solute transport parameters

the divergence of banach space valued random variables on wiener space the domain of definition of the divergence operator on an abstract wiener space w h is extended to include w valued and open image in new window valued integrands the main properties and characterizations of this extension are derived and it is shown that in some sense the added elements in s extended domain have divergence zero these results are then applied to the analysis of quasiinvariant flows induced by w valued vector fields and among other results it turns out that these divergence free vector fields are responsible for generating measure preserving flows

wasserstein space over the wiener space the goal of this paper is to study optimal transportation problems and gradient flows of probability measures on the wiener space based on and extending fundamental results of feyel ustunel carrying out the program of ambrosio gigli savare we present a complete characterization of the derivative processes for certain class of absolutely continuous curves we prove existence of the gradient flow curves for the relative entropy w r t the wiener measure and identify these gradient flow curves with solutions of the ornstein uhlenbeck evolution equation

quasi invariant gaussian measures for the cubic fourth order nonlinear schrodinger equation we consider the cubic fourth order nonlinear schrodinger equation on the circle in particular we prove that the mean zero gaussian measures on sobolev spaces h s mathbb t s frac 3 4 are quasi invariant under the flow

integration on loop group iii asymptotic peter weyl orthogonality using the wiener measure on the continuous loop group l g of a compact connected group g whose properties were studied in integration on loop group i we construct for g simply laced a stochastic central extension l g of l g despite the fact that this object has a multiplication defined almost everywhere we are able to define for it a convolution algebra

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

uniqueness of ground states for schrodinger operators over loop groups abstract over the space l of based loops in a compact connected lie group g there is a natural gradient operator induced by the left action on l of the hilbert lie group k 0 which consists of finite energy elements of l denote by pinned brownian motion measure on l the schrodinger operator v acting in l 2 l is shown to have a unique ground state over each homotopy class in l the proof of uniqueness is reduced to the proof of ergodicity of the left action of k 0 on l for a simply connected group g of compact type ergodicity in turn is proved by first characterizing the ito expansion coefficients of k 0 invariant functions and then regularizing such functions via a smooth approximation of multiple stratonovich integrals as a byproduct of this method one obtains an isometric isomorphism of l 2 g heat kernel measure at time one with a natural completion of the universal enveloping algebra of g thus extending to the noncommutative case the well known isometry between l 2 r d gauss measure and the space of symmetric tensors over r d

degree theory for wiener maps abstract degree theory is developed for a class of wiener maps as well as a few interesting tools such as partitions of unity and sard s theorem