on the finite escape phenomena for matrix riccati equations we show a necessary and sufficient condition for the solution of riccati differential equations to escape to the infinity in finite time this implies that we obtain a necessary and sufficient condition for the existence of the solution of all time t geq 0 this condition is given as a rank condition of the matrix which we refer to as the escape criterion matrix this matrix consists of the equation matrices and an equilibrium point of riccati differential equation

equivariant cohomology fock space and loop groups equivariant de rham cohomology is extended to the infinite dimensional setting of a loop subgroup acting on a loop group using hida supersymmetric fock space for the weil algebra and malliavin test forms on the loop group the mathai quillen isomorphism in the brst formalism of kalkman is defined so that the equivalence of various models of the equivariant de rham cohomology can be established

connections and curvature in the riemannian geometry of configuration spaces torsion free connections and a notion of curvature are introduced on the infinite dimensional nonlinear configuration space of a riemannian manifold m under a poisson measure this allows us to state identities of weitzenbock type and energy identities for anticipating stochastic integral operators the one dimensional poisson case itself gives rise to a non trivial geometry a de rham hodge kodaira operator and a notion of ricci tensor under the poisson measure the methods used in this paper have been thus far applied to d dimensional brownian path groups and rely on the introduction of a particular tangent bundle and associated damped gradient

stochastic anticipative integrals on a riemannian manifold we shall use the nualart pardoux approach to the skorohod integral on rd to construct the divergence operator on the path space over a riemannian manifold the cruzeiro malliavin structure equation on the path space is crucial for us to obtain the energy inequality

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

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

a quasistatic frictional problem with normal compliance a friction and contact problem will be considered for an elastic solid body which under the influence of volume and surface forces is in contact with a rigid foundation

a posteriori error estimation of h p finite element approximations of frictional contact problems abstract dynamic and static frictional contact problems are described using the normal compliance law on the contact boundary dynamic problems are recast into quasistatic problems by time discretization an a posteriori error estimator is developed for a nonlinear elliptic equation of corresponding static or quasistatic problems the a posteriori error estimator is applied to a frictionless case and extended to frictional contact problems also an adaptive strategy is introduced and h p finite element meshes are obtained through a procedure based on a priori and a posteriori error estimations numerical examples are given to support the theoretical results

modulus of continuity of the canonic brownian motion on the group of diffeomorphisms of the circle abstract an explicit modulus of holder continuity is given for the flow associated to the canonic brownian motion on the diffeomorphism group of the circle

on solution multiplicity in friction problems with normal compliance this work is concerned with the normal compliance model and friction in linear elastostatics we consider the two dimensional sliding problem and we seek those interface parameters leading to infinitely many solutions of the continuous and the finite element discretized problem the determination of the interface parameters uses a specific eigenvalue problem in the discrete case the framework is illustrated with an elementary example and some finite element computations

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

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

a cameron martin type quasi invariance theorem for pinned brownian motion on a compact riemannian manifold the results in driver 13 for quasi invariance of wiener measure on the path space of a compact riemannian manifold m are extended to the case of pinned wiener measure to be more explicit let h 0 1 tom be a c1 function where m is a compact riemannian manifold o e m is a base point and tom is the tangent space to m at o e m let w m be the space of continuous paths from 0 1 into m v be wiener measure on w m concentrated on paths starting at o e m and hs w denote the stochastic parallel translation operator along a path to e w m up to time s note hs w is only well defined up to v equivalence for c e w m let xh w denote the vector field along a given by xsh c hs cw h s for each s e 0 1 one should interpret xh as a vector field on w m the vector field xh induces a flow sh t w m w m which leaves wiener measure v quasi invariant see driver 13 it is shown in this paper that the same result is valid if h l 0 and the wiener measure v is replaced by a pinned wiener measure ive the measure ve is proportional to the measure v conditioned on the set of paths which start at 0 e m and end at a fixed end point e e m also as in 13 one gets an integration by parts formula for the vector fields xh defined above

theory and approximation of quasistatic frictional contact problems abstract a variational formulation of a class of quasistatic frictional contact problems is derived for which a compliant interface model of the contact surface is assumed proof of existence and uniqueness of solutions is given finite element approximations of a regularized version of the fractional problem are developed and a priori error estimates are derived the theory is implemented and used to solve representative test problems of frictional contact the numerical simulations show that frictional stress distributions can be very sensitive to load histories and surface properties of contact interfaces

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

approximation and numerical realization of 2d contact problems with coulomb friction and a solution dependent coefficient of friction the paper analyzes discrete contact problems with the coulomb law of friction which involves a solution dependent coefficient of friction f solutions to these problems are defined as fixed points of an auxiliary mapping it is shown that there exists at least one solution provided that f is bounded and continuous in r 1 further conditions guaranteeing uniqueness of the solution are studied the paper is completed by numerical results of several model examples

constrained anisotropic elastic materials in unilateral contact with or without friction we consider time independent problems of linear homogeneous anisotropic elasticiy for materials with internal constraints that are in frictional or frictionless contact with a rigid support

the divergence on submanifolds of the wiener space abstract using malliavin s calculus the divergence the covariant derivative and the riemann and ricci curvatures of a submanifold of the wiener space are defined it is shown that the ricci and riemann curvatures appear in the commutator of the divergence operator and covariant derivative operator capacities are used to restrict the commutator formula to a submanifold and to deduce that the curvature belongs to all l p da spaces where a is the area measure associated to the submanifold a version of weitzenbock s formula a bochner type formula and a criterion for the existence of the divergence are proved a generalization to higher dimensional tensor fields is stated and a connection with stochastic integration on the submanifold of the wiener space is worked out

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

an error estimate for the signorini problem with coulomb friction approximated by finite elements the present paper is concerned with the unilateral contact model and the coulomb friction law in linear elastostatics we consider a mixed formulation in which the unknowns are the displacement field and the normal and tangential constraints on the contact area the chosen finite element method involves continuous elements of degree one and continuous piecewise affine multipliers on the contact zone a convenient discrete contact and friction condition is introduced in order to perform a convergence study we finally obtain a first a priori error estimate under the assumptions ensuring the uniqueness of the solution to the continuous problem

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

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

certain gradient flows and submanifolds in wiener spaces abstract in malliavin calculus submanifolds in wiener spaces have been studied by many people instead of loop spaces for such submanifolds we show the existence of the tabular neighborhoods which are basic tools to study the properties of the differential operators on the submanifolds to this end it is crucial to show the existence and the regularity of the solutions to certain ordinary differential equations in wiener spaces as an application we show the continuity of the divergence operators on submanifolds

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

existence theorems for contact with friction of two linear elastic bodies abstract in this article i give an iteration method for solving the signorini problem by the idea of concentration cancellation the nonlinear quantity in the mathematical formulation of the problem caused by contact friction effect on the boundary is relaxed to a convex functional therefore the relaxed problem can be solved by using convex minimization techniques which are numerically stable finally good convergence property of the proposed method is proved

analytical construction of a negative semidefinite solution of algebraic riccati equations and its application a sufficient condition for the existence of a symmetric negative semidefinite solution of algebraic matrix riccati equations is given and at the same time the negative semidefinite solution is constructed by an analytical method a few properties of the negative semidefinite solution are also investigated it is shown that the negative semidefinite solution having those properties plays a more important role than a positive semidefinite solution in the existence problem of the solution of the corresponding riccati differential difference equations the results are applied to this existence problem

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

stochastic equivariant cohomologies and cyclic cohomology we give two stochastic diffeologies on the free loop space which allow us to define stochastic equivariant cohomology theories in the chen souriau sense and to establish a link with cyclic cohomology with the second one we can establish a stochastic fixed point theorem

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

non existence of infinitesimally invariant measures on loop groups let g be a compact lie group l g the associated loop group the canonical symplectic form on l g set h the hamiltonian function for which the associated hamiltonian vector field is the infinitesimal rotation then h generates a canonical semi definite riemannian structure on l g which induces a riemannian structure on the free loop group l g g l0 g this metric corresponds to the sobolev norm h1 using orthonormal frame methodology the positivity and finiteness of the ricci curvature of l0 g is proved by studying the dissipation towards high modes of a unitary group valued sde it is proved that the loop group does not have any infinitesimally invariant measure

on asymptotic properties for some parameter dependent variational inequalities abstract this paper is concerned with asymptotic and monotonicity properties of some parameter dependent variational inequalities the main part of the study deals with inequalities modelling friction problems with normal compliance or tresca s conditions in which the parameter stands for the friction coefficient the corresponding inequalities are generalizations of variational inequalities of the second kind we then study an inequality of the first kind representing the elastoplastic torsion problem where the parameter represents the plasticity yield

quotient of a loop group and witten genus we define a dirac ramond operator over the quotient lg h of a loop group by a subgroup h of the compact lie group g we state the conjecture that its equivariant index with respect of the natural circle action over the quotient of the loop group is equal to the witten genus of the homogeneous manifold g h we motivate the conjecture by a short time argument which allows to come back to a dirac ramond operator at the manner of taubes on a limit model where all the computations can be done by hand

explicit stochastic analysis of brownian motion and point measures on riemannian manifolds abstract the gradient and divergence operators of stochastic analysis on riemannian manifolds are expressed using the gradient and divergence of the flat brownian motion by this method we obtain the almost sure version of several useful identities that are usually stated under expectations the manifold valued brownian motion and random point measures on manifolds are treated successively in the same framework and stochastic analysis of the brownian motion on a riemannian manifold turns out to be closely related to classical stochastic calculus for jump processes in the setting of point measures we introduce a damped gradient that was lacking in the multidimensional case

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

on some existence and uniqueness results in contact problems with nonlocal friction the workwe present is a mathematical study of signorini s problem with friction i e the problem of equilibrium of a linearly elastic body in contact with a rigid foundation until recently the questions of existence and uniqueness of solutions to such problems were open recently two ideas have appeared which have led to some positive results the first presented by netas jarusek haslinger 11 involves a concept of small friction and provides for the development of an existence result for a very special case of a uniformly loaded infinite strip the second established by duvaut 5 is based on the idea of a nonlocal friction law which enables one to establish an existence result for any friction and also to establish uniqueness for small friction in this paper we present some new results in this direction and a new proof of a non local existence theory our approach differs considerably from the proof outlined in 5 instead of using fixed point arguments we develop a theory of perturbed variational inequalities the origin of which can be found in 9 and establish a generalization for the class of nonlinear variational inequalities characterizing the non local signorini problem this approach should prove to be very useful in studies of approximations of friction prohlems

some remarks on time dependent variational problems and their asymptotic behaviour abstract the subject matter of this paper is the asymptotic behaviour of quasi static variational inequalities as regards physical parameters like the friction coefficient compliance coefficient etc by convex duality the quasi static problems can be recast into the forms of standard evolution problems whose study relies on well known methods in this framework the stability with respect to small friction coefficients reduces to long time behaviour for evolution problems

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

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

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

existence theorem for the difference riccati equations to obtain a necessary and sufficient condition for the existence of a solution of discrete riccati equations a new approach is proposed and an interesting relation between the equilibrium points of riccati equations and the existence of the solution is shown the results can be applied to nonsymmetric riccati equations however the results in this paper do not improve our resources for the optimal control problem or filtering problem though roughly speaking the known results on those problems are included

variational inequality with almost history dependent operator for frictionless contact problems abstract we study two quasistatic contact problems which describe the frictionless contact between a body and deformable foundation on an infinite time interval the contact is modelled by the normal compliance condition with limited penetration and memory the first problem deals with evolution of a body made of a viscoplastic material and in the second problem the material is viscoelastic with long memory the constitutive functions of these materials have a non polynomial growth for each problem we derive a variational formulation that has the form of an almost history dependent variational inequality for the displacement field we demonstrate existence and uniqueness results of abstract almost history dependent inclusion and variational inequality in the reflexive orlicz sobolev space finally we apply the abstract results to prove existence of the unique weak solution to the contact problems

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

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

two results on solution uniqueness and multiplicity for the linear elastic friction problem with normal compliance abstract this study is concerned with the frictional contact problem governed by the normal compliance law in linear elasticity the paper presents two contributions dealing with the stationary problem we first obtain improved bounds ensuring the uniqueness of a solution second we exhibit examples in which infinitely many solutions to the problem exist

analysis of a viscoelastic contact problem with multivalued normal compliance and unilateral constraint abstract we consider a mathematical model which describes the quasistatic contact between a viscoelastic body and a foundation the material s behavior is modeled with a constitutive law with long memory the contact is frictionless and is modeled with a multivalued normal compliance condition and unilateral constraint we present the classical formulation of the problem list the assumptions on the data and derive a variational formulation of the model then we prove its unique solvability the proof is based on arguments of history dependent quasivariational inequalities we also study the dependence of the solution with respect to the data and prove a convergence result further we introduce a fully discrete scheme to solve the problem numerically under certain solution regularity assumptions we derive an optimal order error estimate finally we provide numerical validations both for the convergence and the error estimate results in the study of a two dimensional test problem