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

quantum algebraic structures compatible with the harmonic oscillator newton equation we study some of the algebraic structures that are compatible with the quantization of the harmonic oscillator through its newton equation examples of such structures are given they include undeformed and q deformed oscillators as well as the su 2 and the deformed suq 2 lie algebras which appear in a variety of physical models

a poincare inequality on loop spaces we show that the laplacian on the loop space over a class of riemannian manifolds has a spectral gap the laplacian is defined using the levi civita connection the brownian bridge measure and the standard bismut tangent spaces

transportation cost inequalities on path and loop groups let g be a connected lie group with the lie algebra g the action of cameron martin space h g on the path space pe g introduced by l gross illinois j math 36 1992 447 is free using this fact we define the h distance on pe g which enables us to establish a transportation cost inequality on pe g this method will be generalized to the path space over the loop group le g so that we obtain a transportation cost inequality for heat measures on le g

theory of extended solutions for fast diffusion equations in optimal classes of data radiation from singularities this paper is devoted to constructing a general theory of nonnegative solutions for the equation

invariant inner products on spaces of solutions of the klein gordon and helmholtz equations we construct sesquilinear forms which are invariant under the similarity groups of the klein gordon and helmholtz equations these give rise to positive definite inner products on subspaces of solutions of these equations for the klein gordon case the known poincare invariant inner product for positive energy solutions is recovered for the helmholtz case a new euclidean invariant inner product is presented which involves the function and its normal derivative to a line integrated nonlocally over that line

smoothness of heat kernel measures on infinite dimensional heisenberg like groups abstract we study measures associated to brownian motions on infinite dimensional heisenberg like groups in particular we prove that the associated path space measure and heat kernel measure satisfy a strong definition of smoothness

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

small time gaussian behavior of symmetric diffusion semi groups this work is involved with the short time asymptotics of diffusion semigroups in a general setting a generalization of fang s version of varadhan s formula is proven for general dirichlet spaces that are local and conservative the intrinsic metric appearing in the formula is characterized by pointwise distance for canonical dirichlet spaces on loop groups

continuous vs discrete fractional fourier transforms abstract we compare the finite fourier exponential and fourier kravchuk transforms both are discrete finite versions of the fourier integral transform the latter is a canonical transform whose fractionalization is well defined we examine the harmonic oscillator wavefunctions and their finite counterparts mehta s basis functions and the kravchuk functions the fractionalized fourier kravchuk transform was proposed in j opt soc amer a 14 1997 1467 1477 and is here subject of numerical analysis in particular we follow the harmonic motions of coherent states within a finite discrete optical model of a shallow multimodal waveguide

boundary asymptotic and uniqueness of solutions to the p laplacian with infinite boundary values we prove boundary asymptotics to solutions of weighted p laplacian equations that take infinite value on the boundary of a bounded domain uniqueness of such solutions would then follow as a consequence our results extend previously known results by allowing weights that are unbounded in the domain

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

blow up phenomena for polytropic equation with inhomogeneous density and source the subject of this investigation is the blow up phenomena of the positive solutions of the mixed problem for the one dimensional polytropic filtration equation with inhomogeneous density and source it is shown that under certain conditions on the nonlinearities and data blow up will occur at some finite time note that the technique applied for the proof does not use the zel dovich kompaneets barenblatt solutions since the construction of such type of function is more complicated in our case therefore we obtain a result by multiplying on a special factor which has convenient properties in particular by choosing the parameters of the factor and using the properties of the solution we obtain the inequality which allows us to show the blow up phenomena

global existence and blow up of solutions to a class of doubly degenerate parabolic equations coupled via nonlinear boundary flux the paper deals with the critical curve for a degenerate parabolic system coupled via nonlinear boundary flux by constructing the self similar super solution and subsolution we obtain the critical global existence curve and the critical curve of fujita type is conjectured with the aid of some new results an interesting feature of our results is that the critical global existence curve and the critical fujita curve are determined by a matrix and by the solution of a linear algebraic system respectively

wigner distribution function for euclidean systems euclidean systems include poly and monochromatic wide angle optics acoustics and also infinite discrete data sets we use a recently defined wigner operator and quasiprobability distribution function to set up and study the phase space evolution of these models subject to differential and difference equations respectively infinite data sets and two dimensional monochromatic helmholtz fields are thus shown by their wigner function on a cylinder of direction and location the wigner function for polychromatic wavefields has c number coordinates of two dimensional wavenumber and position

wigner distributions and quantum mechanics on lie groups the case of the regular representation we consider the problem of setting up the wigner distribution for states of a quantum system whose configuration space is a lie group the basic properties of wigner distributions in the familiar cartesian case are systematically generalized to accommodate new features which arise when the configuration space changes from n dimensional euclidean space rn to a lie group g the notion of canonical momentum is carefully analyzed and the meanings of marginal probability distributions and their recovery from the wigner distribution are clarified for the case of compact g an explicit definition of the wigner distribution is proposed possessing all the required properties geodesic curves in g which help introduce a notion of the midpoint of two group elements play a central role in the construction

quantizations on the circle and coherent states we present a possible construction of coherent states on the unit circle as configuration space our approach is based on borel quantizations on s1 including the aharonov bohm type quantum description coherent states are constructed by perelomov s method as group related coherent states generated by weyl operators on the quantum phase space because of the duality of canonical coordinates and momenta i e the angular variable and the integers this formulation can also be interpreted as coherent states over an infinite periodic chain for the construction we use the analogy with our quantization and coherent states over a finite periodic chain where the quantum phase space was the coherent states constructed in this work are shown to satisfy the resolution of unity to compare them with canonical coherent states some of their further properties are also studied demonstrating similarities as well as substantial differences this article is part of a special issue of journal of physics a mathematical and theoretical devoted to coherent states mathematical and physical aspects

discrete repulsive oscillator wavefunctions for the study of infinite discrete systems on phase space the three dimensional lorentz algebra and group so 2 1 and so 2 1 provide a discrete model of the repulsive oscillator its eigenfunctions are found in the principal irreducible representation series where the compact generator that we identify with the position operator has the infinite discrete spectrum of the integers z while the spectrum of energies is a double continuum the right and left moving wavefunctions are given by hypergeometric functions that form a dirac basis for 2 z under contraction the discrete system limits to the well known quantum repulsive oscillator numerical computations of finite approximations raise further questions on the use of dirac bases for infinite discrete systems

critical exponents and blow up rate for a nonlinear diffusion equation with logarithmic boundary flux abstract in this paper we establish the critical global existence exponent and the critical fujita exponent for the nonlinear diffusion equation u t log 1 u u x x in r 0 subject to a logarithmic boundary flux log 1 u u x 0 t 1 u log p 1 u 0 t t 0 furthermore give the blow up rate for the nonglobal solutions

critical exponents and non extinction for a fast diffusive polytropic filtration equation with nonlinear boundary sources in this paper we discuss the critical exponents and non extinction property for a nonlinear boundary value problem of a fast diffusive polytropic filtration equation

critical curves for fast diffusive non newtonian equations coupled via nonlinear boundary flux this paper deals with the critical curve for a nonlinear boundary value problem of a fast diffusive non newtonian system we first obtain the critical global existence curve by constructing the self similar supersolution and subsolution and then the critical fujita curve is conjectured with the aid of some new results

the taylor map on complex path groups let w g denote the path group of an arbitrary complex connected lie group the existence of a heat kernel measure t on w g has been shown in m cecil b k driver heat kernel measure on loop and path groups preprint http www math uconn edu cecil papers p2 pdf infin dimens anal quantum probab relat top submitted for publication the present work establishes an isometric map the taylor map from the space of l2 t holomorphic functions on w g to a subspace of the dual of the universal enveloping algebra of lie h g where h g is the lie subgroup of finite energy paths this map is shown to be surjective in the case where g is a simply connected graded lie group

integration by parts formula and logarithmic sobolev inequality on the path space over loop groups the geometric stochastic analysis on the riemannian path space developed recently gives rise to the concept of tangent processes roughly speaking it is the infinitesimal version of the girsanov theorem using this concept we shall establish a formula of integration by parts on the path space over a loop group following the martingale method developed in capitaine hsu and ledoux we shall prove that the logarithmic sobolev inequality holds on the full paths as a particular case of our result we obtain the driver lohrenz s heat kernel logarithmic sobolev inequalities over loop groups the stochastic parallel transport introduced by driver will play a crucial role

metrics hs and behaviours as s 1 2 on loop groups abstract for all metric h s with s 1 2 the heat measures on free loop groups have been constructed by p malliavin the geometric stochastic analysis is well developed the metric h 1 2 is critical and involves naturally in mathematical physics the ricci tensors ric s play the fundamental role in these situations in this note we shall study the behaviour of ric s as s 1 2

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

heat kernel analysis on semi infinite lie groups this paper studies brownian motion and heat kernel measure on a class of infinite dimensional lie groups we prove a cameron martin type quasi invariance theorem for the heat kernel measure and give estimates on the lp norms of the radon nikodym derivatives we also prove that a logarithmic sobolev inequality holds in this setting

hilbert schmidt groups as infinite dimensional lie groups and their riemannian geometry abstract we describe the exponential map from an infinite dimensional lie algebra to an infinite dimensional group of operators on a hilbert space notions of differential geometry are introduced for these groups in particular the ricci curvature which is understood as the limit of the ricci curvature of finite dimensional groups is calculated we show that for some of these groups the ricci curvature is

linear transformations and aberrations in continuous and finite systems in geometric optics there is a natural distinction between the paraxial and aberration regimes which contain respectively the linear and nonlinear canonical transformations of position and momentum in the phase space in the lie theoretical presentation linear inhomogeneous transformations are generated by linear and quadratic functions of the phase space while aberrations of increasing order are generated by homogeneous functions with higher powers of these coordinates in a way parallel but distinct from the schrodinger quantization of continuous classical systems we quantize the geometric optical model into discrete finite dimensional systems based on the lie algebra whose wavefunctions are n point signals phase space is a sphere and transformations are represented by the n n unitary matrices that form the group we factor this group into linear and nonlinear unitary transformations of phase space and classify all its n2 4 aberrations this offers a new parametrization of based on a chosen subgroup

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

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 nonlinear diffusion system coupled via nonlinear boundary flux abstract in this paper we deal with the global existence and nonexistence of solutions to a nonlinear diffusion system coupled via nonlinear boundary flux by constructing various kinds of sub and super solutions and using the basic properties of m matrix we give the necessary and sufficient conditions for global existence of nonnegative solutions the critical curve of fujita type is conjectured with the aid of some new results which extend the recent results of wang et al nonlinear anal 71 2009 2134 2140 and li et al j math anal appl 340 2008 876 883 to more general equations

logarithmic sobolev inequalities for pinned loop groups abstract letgbe a connected compact type lie group equipped with anadg invariant inner product on the lie algebra g ofg given this data there is a well known left invariant h1 riemannian structure on l l g the infinite dimensional group of continuous based loops ing using this riemannian structure we define and construct a heat kernel t g0 associated to the laplace beltrami operator on l g heret 0 g0 l g and t g0 is a certain probability measure on l g for fixedg0 l g andt 0 we use the measure t g0 and the riemannian structure on l g to construct a classical pre dirichlet form the main theorem of this paper asserts that this pre dirichlet form admits a logarithmic sobolev inequality

tomograms and other transforms a unified view a general framework is presented which unifies the treatment of wavelet like quasidistribution and tomographic transforms explicit formulae relating the three types of transforms are obtained the case of transforms associated with the symplectic and affine groups is treated in some detail special emphasis is given to the properties of the scale time and scale frequency tomograms tomograms are interpreted as a tool to sample the signal space by a family of curves or as the matrix element of a projector

critical exponents for a fast diffusive polytropic filtration equation with nonlinear boundary flux abstract in this paper we deal with a fast diffusive polytropic filtration equation u t x u m x p 2 u m x 1 p 1 1 m in r 0 subject to a nonlinear boundary flux u m x p 2 u m x 0 t u q 0 t t 0 we first get the behavior of the solution at infinity and establish the critical global existence exponent and critical fujita exponent for the fast diffusive polytropic filtration equation furthermore give the blow up set and upper bound of the blow up rate for the nonglobal solutions

position and momentum bases for the monochromatic maxwell fish eye and the sphere in geometric optics the maxwell fish eye is a medium where light rays follow circles while in scalar wave optics this medium can only trap fields of certain discrete frequencies in the monochromatic case characterized by a positive integer l there are independent fields we identify two bases of functions one known as the sherman volobuyev functions is characterized as of most definite momenta the other is new and composed of most definite positions and normal derivatives for the fish eye scalar wavefields their construction uses the stereographic projection of the sphere and their identification is corroborated in the contraction limit to the homogeneous helmholtz medium

wigner distribution function for finite systems we construct a wigner distribution function for finite data sets it is based on a finite optical system a linear wave guide where the finite number of discrete sensors is equal to the number of modes which the guide can carry the dynamical group for this model is su 2 and the wave functions are sets of n 2l 1 data points the wigner distribution function assigns classical c numbers to the operators of position momentum and wave guide mode

precise asymptotics of certain wiener functionals abstract in this article we present a method for developing certain wiener integrals in an asymptotic expansion as the variance of the wiener goes to 0 the basic ideas are a combination of methods borrowed from the theory of large deviations together with techniques from the calculus of wiener space

optimal transport maps for monge kantorovich problem on loop groups let g be a compact lie group and consider the loop group leg l c 0 1 g l 0 l 1 e let be the heat kernel measure at the time 1 for any density function f on leg such that ent f we shall prove that there exists a unique optimal transportation map t leg leg which pushes forward to f

a correction to the paper integration by parts and quasi invariance for heat kernel measures on loop groups abstract it is asserted in definition 4 2 in 1 that the random operators u t defined there are unitary as was pointed out to the author by shizan fang it is clear that u t is an isometry but it is not obvious that u t is surjective the purpose of this note is to fill this gap

logarithmic sobolev inequality on free loop groups for heat kernel measures associated with the general sobolev spaces abstract in this paper we will prove the logarithmic sobolev inequality on free loop groups for various heat kernel measures which p malliavin 1989 1991 in diffusion process and related problems in analysis m a pinsley ed vol i birkhauser basel constructed those measures are associated with the sobolev spaces of order s s 1 2 of the free loops in the lie algebra we will equipp the free loop groups with those metrics and will show that a formula of weitzenbock type holds which enables us to apply the method of driver and lohrenz 1996 j funct anal 146 381 448

absolute continuity of heat kernel measure with pinned wiener measure on loop groups let t 0 k be a connected compact lie group equipped with an ad k invariant inner product on the lie algebra of k associated to this data are two measures 0 t and t 0 on l k the space of continuous loops based at e e k the measure 0 t is pinned wiener measure with variance t while the measure 0 t is a heat kernel measure on l k the measure 0 t is constructed using a k valued brownian motion while the measure 0 t is constructed using a l k valued brownian motion in this paper we show that 0 t is absolutely continuous with respect to 0 t and the radon nikodym derivative d 0 t d 0 t is bounded

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

quasi invariance of brownian measures on the group of circle homeomorphisms and infinite dimensional riemannian geometry abstract a cameron martin type theorem is proved for the canonical brownian motion on the group of homeomorphisms of the circle

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

heat kernel analysis on infinite dimensional heisenberg groups abstract we introduce a class of non commutative heisenberg like infinite dimensional lie groups based on an abstract wiener space the ricci curvature tensor for these groups is computed and shown to be bounded brownian motion and the corresponding heat kernel measures t t 0 are also studied we show that these heat kernel measures admit 1 gaussian like upper bounds 2 cameron martin type quasi invariance results 3 good l p bounds on the corresponding radon nikodym derivatives 4 integration by parts formulas and 5 logarithmic sobolev inequalities the last three results heavily rely on the boundedness of the ricci tensor

on blow up of solution of nonhomogeneous polytropic equation with source abstract in this paper we discuss the blow up property of the positive solutions of the mixed problem for a one dimensional polytropic filtration equation with source

convergence of finite dimensional distributions of heat kernel measures on loop groups abstract in this paper we consider heat kernel measure on loop groups associated to the h 1 2 metric unlike h s case s 1 2 there is a difficulty that h 1 2 is not contained in the space of continuous loops so we take limits there are two limiting methods one is to use delta functions and to let s go down to 1 2 the other is to fix s at 1 2 and to approximate the delta functions for the second approach a generalization of heat kernel measures is needed then the first approach can be obtained as a special case of the second one the limit in the sense of finite dimensional distribution is the fictitious infinite dimensional haar measure

the critical curve for a non newtonian polytropic filtration system coupled via nonlinear boundary flux this paper deals with the critical curve of the non newtonian polytropic filtration equation coupled via nonlinear boundary flux the critical global existence curve is obtained by constructing various self similar supersolutions and subsolutions furthermore we get some new results on the critical fujita curve

the geometry of loop groups l espace g des boucles basees sur un groupe de lie compact admet une metrique de kahler on exprime sa courbure en termes d operateurs de toeplitz et on definit des classes de chern par analogie avec la theorie de chern weil aux dimensions finies il y a une structure de fredhohn naturelle sur g induite a partir de la famille des operateurs de toeplitz on obtient des formules explicites pour su n

a model of porous catalyst accounting for incipiently non isothermal effects abstract an approximate model accounting for incipiently non isothermal effects is derived from a well known model of porous catalyst for appropriate realistic limiting values of the parameters in this limit the original model is a singularly perturbed m d reaction diffusion system and the approximate model is given by the m d heat equation with nonlinear boundary condition coupled with infinitely many ifm 2 1 d semilinear parabolic equations one for each point of the boundary of the spatial domain some limiting cases are still considered in the approximate model that lead to further simplifications

heat equation derivative formulas for vector bundles we use martingale methods to give bismut type derivative formulas for differentials and co differentials of heat semigroups on forms and more generally for sections of vector bundles the formulas are mainly in terms of weitzenbock curvature terms in most cases derivatives of the curvature are not involved in particular our results improve b k driver s formula in 1997 j math pures appl 9 76 703 737 for logarithmic derivatives of the heat kernel measure on a riemannian manifold our formulas also include the formulas of k d elworthy and x m li 1998 c r acad sci paris ser i math 327 87 92

wigner weyl isomorphism for quantum mechanics on lie groups the wigner weyl isomorphism for quantum mechanics on a compact simple lie group g is developed in detail several features are shown to arise which have no counterparts in the familiar cartesian case notable among these is the notion of a semiquantized phase space a structure on which the weyl symbols of operators turn out to be naturally defined and figuratively speaking located midway between the classical phase space t g and the hilbert space of square integrable functions on g general expressions for the star product for weyl symbols are presented and explicitly worked out for the angle angular momentum case

a quasi invariance theorem for measures on banach spaces we show that for a measure on a banach space directional differentiability implies quasi translation invariance this result is shown to imply the cameron martin theorem a second application is given in which ty is the image of a gaussian measure under a suitably regular map