asymptotics for euclidean minimal spanning trees on random points asymptotic results for the euclidean minimal spanning tree onn random vertices inrd can be obtained from consideration of a limiting infinite forest whose vertices form a poisson process in allrd in particular we prove a conjecture of robert bland the sum of thed th powers of the edge lengths of the minimal spanning tree of a random sample ofn points from the uniform distribution in the unit cube ofrd tends to a constant asn

uniform positivity improving property sobolev inequalities and spectral gaps abstract we introduce a certain kind of strong ergodicity condition to study the existence of spectral gap for markov generator we can estimate the spectral gap using the ergodicity condition and a sobolev type inequality we apply our results to the dirichlet form on wiener spaces riemannian manifolds and loop spaces

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

spectral properties of laplacians on an abstract wiener space with a weighted wiener measure abstract we study the hodge kodaira laplacians on an abstract wiener space with a weighted wiener measure under some exponential integrability conditions for the density function we determine the kernels establish the relation between spectrum sets and essential spectrum sets and prove the hodge kodaira decomposition theorem

holomorphic dirichlet forms on complex manifolds a complete description of all holomorphic dirichlet forms on complex hyperbolic space is given they are classified in terms of the lie algebra of the automorphism group which we take to act in the unit ball of n we show that the measure defining the holomorphic dirichlet form is finite if and only if the real part of the associated holomorphic vector field points outward at the boundary of the unit ball a logarithmic sobolev inequality is proven to hold whenever this measure is finite all holomorphic dirichlet forms over complex hyperbolic space complex projective space and n are now known

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

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

navier stokes equations for almost incompressible flow the isentropic navier stokes equations at low mach numbers describe almost incompressible flow by a variable transformation the equations take a symmetric form suitable for computation estimates of the solution and its derivatives are obtained for the linearized equations on domains with open boundaries this is a necessary basis for the application of any numerical method whenever some part of the computational boundary is not a solid wall a semi implicit difference approximation is applied to the nonlinear equations for computation of flow over a backward facing step

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

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

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

comparison theorems for the gap of schrodinger operators on cherche des bornes inferieures aux decompositions de valeur propre en fonction des proprietes geometriques de l ensemble avec v x e ainsi que sur la taille de v sur cet ensemble

a simple randomized scheme for constructing low weight k connected spanning subgraphs with applications to distributed algorithms the main focus of this paper is the analysis of a simple randomized scheme for constructing low weight k connected spanning subgraphs in this paper we focus on the metric graph we use the term metric graph for a complete graph with metric weights we first show that our scheme gives a simple approximation algorithm to construct a minimum weight k connected spanning subgraph in a metric graph an np hard problem we show that our algorithm gives an approximation ratio of o klogn for a metric graph o k for a random graph with nodes uniformly randomly distributed in 0 1 2 and for a complete graph with random edge weights u 0 1 we show that our scheme is optimal with respect to the amount of local information needed to compute any connected spanning subgraph we then show that our scheme can be applied to design an efficient distributed algorithm for constructing such an approximate k connected spanning subgraph for any k 1 in a point to point distributed model where the processors form a complete network our algorithm takes time and an expected number of messages our result in conjunction with a result of korach et al e korach s moran s zaks the optimality of distributive constructions of minimum weight and degree restricted spanning trees in a complete network of processors siam journal on computing 16 2 1987 231 236 implies that the expected message complexity of our algorithm is significantly better than the best distributed algorithm that finds an optimal k connected subgraph we also show that for geometric instances our randomized scheme constructs low degree k connected spanning subgraphs which have o klogn maximum degree with high probability

the optimality of distributive constructions of minimum weight and degree restricted spanning trees in a complete network of processors in a previous paper we showed that the distributive construction of a spanning tree in a complete network of processors can be done in o n log n messages we show in this work that if the spanning tree is required to satisfy certain properties then the complexity of its construction increases first we show that the construction of a minimum weight spanning tree requires in the worst case at least omega n 2 messages and then we show that the construction of a spanning tree where the maximum degree is at most k may require at least omega n 2 k messages in the worst case actually in both cases the lower bounds are shown for the number of edges used in the worst case moreover the results are valid for both asynchronous and synchronous networks and are independent of the lengths of the messages on the other hand there are algorithms for the above tasks which achieve these lower bounds up to a constant factor and use messages of o log n length

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

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

krylov subspace methods for solving large unsymmetric linear systems some algorithms based upon a projection process onto the krylov subspace k sub m span r sub 0 ar sub 0 a sup m 1 r sub 0 are developed generalizing the method of conjugate gradients to unsymmetric systems these methods are extensions of arnoldi s algorithm for solving eigenvalue problems the convergence is analyzed in terms of the distance of the solution to the subspace k sub m and some error bounds are established showing in particular a similarity with the conjugate gradient method for symmetric matrices when the eigenvalues are real several numerical experiments are described and discussed

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

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

uniformly integrable operators and large deviations for markov processes abstract in this paper we introduce and investigate the notion of uniformly integrable operators on l p e its relations to classical compactness and hypercontractivity are exhibited several consequences of this notion are established such as perron frobenius type theorems independence on p of the spectral radius in l p p continuity of spectral radius and especially the existence of spectral gap in the irreducible case we also present some infinitesimal criteria ensuring the uniform integrability of a positive semigroup or of its resolvent for a essentially irreducible markov process we show that the uniform integrability in l p of some type of resolvent associated with the transition semigroup implies the large deviation principle of level 3 with some rate function given by a modified donsker varadhan entropy functional we also prove that the uniform integrability condition becomes even necessary in the symmetric case finally we present several applications of our results to feynman kac semigroups to the thermodynamical limits of grand ensembles and to non symmetric markov processes given by girsanov s formula

optimal lower bounds for some distributed algorithms for a complete network of processors abstract lower bounds for distributed algorithms for complete networks of processors i e networks where each pair of processors is connected by a communication line are discussed we first show an n log n lower bound for the number of messages required by any algorithm in a given class of distributed algorithms for such networks this class includes algorithms for problems like finding a leader or constructing a spanning tree we then show an n 2 lower bound for other problems like constructing a maximal matching or a hamiltonian circuit in proving the lower bounds we are counting the edges which carry messages during the executions of the algorithms ignoring the actual number of messages carried by each edge interestingly this number is shown to be of the same order of magnitude as the total number of messages needed by these algorithms the proofs of the lower bounds apply for synchronous networks and for arbitrarily long messages

eigenanalysis of some preconditioned helmholtz problems in this work we calculate the eigenvalues obtained by preconditioning the discrete helmholtz operator with sommerfeld like boundary conditions on a rectilinear domain by a related operator with boundary conditions that permit the use of fast solvers the main innovation is that the eigenvalues for two and three dimensional domains can be calculated exactly by solving a set of one dimensional eigenvalue problems this permits analysis of quite large problems for grids fine enough to resolve the solution for a given wave number preconditioning using neumann boundary conditions yields eigenvalues that are uniformly bounded located in the first quadrant and outside the unit circle in contrast dirichlet boundary conditions yield eigenvalues that approach zero as the product of wave number with the mesh size is decreased these eigenvalue properties yield the first insight into the behavior of iterative methods such as gmres applied to these preconditioned problems

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

logarithmic sobolev inequalities on noncompact riemannian manifolds this paper presents a dimension free harnack type inequality for heat semigroups on manifolds from which a dimension free lower bound is obtained for the logarithmic sobolev constant on compact manifolds and a new criterion is proved for the logarithmic sobolev inequalities abbrev lsi on noncompact manifolds as a result it is shown that lsi may hold even though the curvature of the operator is negative everywhere

logarithmic sobolev inequalities and exponential integrability we prove the exponential integrability of lipschitz functions in abstract wiener spaces by using logarithmic sobolev inequalities we introduce two notions of lipschitz continuity and apply the method of davis and simon j funct anal 59 1984 335 395 to prove our main theorem moreover we give several conditions under which the function is exponentially integrable

fourier analysis of frequency filtering decomposition preconditioners in this paper frequency filtering decomposition ffd preconditioner is analyzed by the approach of fourier analysis the condition number estimation of a preconditioned 2 d model problem is presented analysis reveals that condition number of the preconditioned matrix grows like o h 1 with h be the mesh size by using the framework of ffd a stabilized frequency filtering decomposition sffd method is proposed and analyzed by fourier method results show that sffd preconditioner is superior to ffd preconditioner in the sense that k m s f f d 1a k m f f d 1a numerical tests are performed to illustrate the theoretical results and the superiority of sffd preconditioner

modified tangential frequency filtering decomposition and its fourier analysis in this paper a modified tangential frequency filtering decomposition mtffd preconditioner is proposed the optimal order of the modification and the optimal relaxation parameter is determined by fourier analysis with the choice of optimal order of modification the fourier results show that the condition number of the preconditioned matrix is mathcal o h frac 2 3 and the spectrum distribution of the preconditioned matrix can be predicted by the fourier results the performance of mtffd preconditioner is compared with tangential frequency filtering tffd preconditioner on a variety of large sparse matrices arising from the discretization of pdes with discontinuous coefficients the numerical results show that the mtffd preconditioner is much more efficient than the tffd preconditioner

spectral gaps for hyperbounded operators abstract we consider a positive and power bounded linear operator t on l p over a finite measure space and prove that if t l p l q for some q p then the essential spectral radius of t is strictly smaller than 1 as a special case we obtain a recent result of miclo who proved this assertion for self adjoint ergodic markov operators in the case p 2 and thereby solved a long open problem of simon and hoegh krohn our methods draw a connection between spectral theory and the geometry of banach spaces they rely on a result going back to groh that encodes spectral gap properties via ultrapowers and on the fact that an infinite dimensional l p space cannot be isomorphic to an l q space for q p we also prove a number of variations of our main result i it follows from theorems of lotz and martinez that the condition t l p l q can be replaced with the weaker assumption that t maps the positive part of the l p unit ball into a uniformly p integrable set ii while it is known that the positivity assumption on t cannot in general be omitted we show that we can replace it with the assumption that t is contractive both on l p and on l q iii we prove a version of the theorem which allows us under appropriate circumstances to also consider non finite measures spaces iv our result also has a uniform version there exists an upper bound c 0 1 for the essential spectral radius of t where c depends on certain quantitative properties of t l p and l q

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

hypercontractivity and spectral gap of symmetric diffusions with applications to the stochastic ising models abstract we describe some results relating the spectral gap and logarithmic sobolev constants we work initially in the context of symmetric diffusions on a finite dimensional manifold and later apply our results to the analysis of certain infinite systems of mildly interacting diffusions

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 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

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

spectral analysis of nonsymmetric quasi toeplitz matrices with applications to preconditioned multistep formulas the eigenvalue spectrum of a class of nonsymmetric preconditioned matrices arising in time dependent partial differential equations is analyzed and discussed the matrices generated by the underlying numerical integrators are small rank perturbations of block toeplitz matrices circulant like preconditioners based on the former are considered the eigenvalue distribution of the preconditioned matrix influences often crucially the convergence of krylov iterative accelerators due to several reasons lack of symmetry band structure and coefficients depending on the size the classical approach based on smooth generating functions gives very little insight here therefore to characterize the eigenvalues a difference equation approach exploiting the band toeplitz and circulant patterns generalizing the well known results of trench is proposed

curvature dimension estimates for the laplace beltrami operator of a totally geodesic foliation we study bakry emery type estimates for the laplace beltrami operator of a totally geodesic foliation in particular we are interested in situations for which the 2 operator may not be bounded from below but the horizontal bakry emery curvature is as we prove it under a bracket generating condition this weaker condition is enough to imply several functional inequalities for the heat semigroup including the wang harnack inequality and the log sobolev inequality we also prove that under proper additional assumptions the generalized curvature dimension inequality introduced by baudoin garofalo is uniformly satisfied for a family of riemannian metrics that converge to the sub riemannian one

a semi implicit method for hyperbolic problems with different time scales hyperbolic systems with two different time scales are considered where the solutions vary on the slow scale only for this type of problem semi implicit difference methods are very natural and in this paper we analyze the leap frog backwards euler scheme in particular it is shown that when the ratio varepsilon between the slow and the fast scale tends to zero the solutions of the approximation converge to solutions of the reduced differential equation numerical experiments are included for illustration of the theoretical results

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

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

harnack inequality and strong feller property for stochastic fast diffusion equations abstract as a continuation to f y wang harnack inequality and applications for stochastic generalized porous media equations ann probab 35 2007 1333 1350 where the harnack inequality and the strong feller property are studied for a class of stochastic generalized porous media equations this paper presents analogous results for stochastic fast diffusion equations since the fast diffusion equation possesses weaker dissipativity than the porous medium one does some technical difficulties appear in the study as a compensation to the weaker dissipativity condition a sobolev nash inequality is assumed for the underlying self adjoint operator in applications some concrete examples are constructed to illustrate the main results

the fractional fourier transform and applications this paper describes the fractional fourier transform which admits computation by an algorithm that has complexity proportional to the fast fourier transform algorithm whereas the discrete fourier transform dft is based on integral roots of unity e 2 pi i n the fractional fourier transform is based on fractional roots of unity e 2 pi i alpha where alpha is arbitrary the fractional fourier transform and the corresponding fast algorithm are useful for such applications as computing dfts of sequences with prime lengths computing dfts of sparse sequences analyzing sequences with noninteger periodicities performing high resolution trigonometric interpolation detecting lines in noisy images and detecting signals with linearly drifting frequencies in many cases the resulting algorithms are faster by arbitrarily large factors than conventional techniques

lower estimates of transition densities and bounds on exponential ergodicity for stochastic pde s a formula for the transition density of a markov process defined by an infinite dimensional stochastic equation is given in terms of the ornstein uhlenbeck bridge and a useful lower estimate on the density is provided as a consequence uniform exponential ergodicity and v ergodicity are proved for a large class of equations we also provide computable bounds on the convergence rates and the spectral gap for the markov semigroups defined by the equations the bounds turn out to be uniform with respect to a large family of nonlinear drift coefficients examples of finite dimensional stochastic equations and semilinear parabolic equations are given

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

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 and spectral gaps perturbation theory abstract we prove the logarithmic sobolev inequality for a diffusion operator on the wiener space the operator we consider is a perturbation of the ornstein uhlenbeck operator we use the logarithmic sobolev inequality for the ornstein uhlenbeck operator our method is based on dirichlet forms and it can be applied to more general diffusions

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

poincare inequality for weighted first order sobolev spaces on loop spaces abstract let e be the loop space over a compact connected riemannian manifold with a torsion skew symmetric connection let l d be the ornstein uhlenbeck operator on a nonempty connected component d of the loop space e and let v d r be the restriction on d of the potential in the logarithmic sobolev inequality found by l gross on the loop group by s aida and by f z gong and z m ma on the loop space respectively we prove that the schrodinger operator l v l d v always has a spectral gap at the bottom 0 v of its spectrum and thus has its ground state transformed operator 1 l v 0 v where is the unique ground state of l v in particular our result proves l gross s conjecture about the existence of a spectral gap for the ground state transform of the schrodinger operator studied by him on the loop group in addition in all the above cases we identify the domain of the dirichlet forms associated with the ground state transforms as weighted first order sobolev spaces with weight given by 2 thus establishing a poincare inequality for them all these results are consequences from some new results in this paper on dirichlet forms characterizing certain classes with spectral gaps and from results by s aida and m hino

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

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

hitting times functional inequalities lyapunov conditions and uniform ergodicity the use of lyapunov conditions for proving functional inequalities was initiated in 5 it was shown in 4 30 that there is an equivalence between a poincare inequality the existence of some lyapunov function and the exponential integrability of hitting times in the present paper we close the scheme of the interplay between lyapunov conditions and functional inequalities by showing that strong functional inequalities are equivalent to lyapunov type conditions showing that these lyapunov conditions are characterized by the finiteness of generalized exponential moments of hitting times we also give some complement concerning the link between lyapunov conditions and in tegrability property of the invariant probability measure and as such transportation inequalities and we show that some unbounded lyapunov conditions can lead to uniform ergodicity and coming down from infinity property

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

randomized distributed edge coloring via an extension of the chernoff hoeffding bounds certain types of routing scheduling and resource allocation problems in a distributed setting can be modeled as edge coloring problems we present fast and simple randomized algorithms for edge coloring a graph in the synchronous distributed point to point model of computation our algorithms compute an edge coloring of a graph g with n nodes and maximum degree delta with at most 1 6 delta o log 1 delta n colors with high probability arbitrarily close to 1 for any fixed delta 0 they run in polylogarithmic time the upper bound on the number of colors improves upon the 2 delta 1 coloring achievable by a simple reduction to vertex coloring to analyze the performance of our algorithms we introduce new techniques for proving upper bounds on the tail probabilities of certain random variables the chernoff hoeffding bounds are fundamental tools that are used very frequently in estimating tail probabilities however they assume stochastic independence among certain random variables which may not always hold our results extend the chernoff hoeffding bounds to certain types of random variables which are not stochastically independent we believe that these results are of independent interest and merit further study

analysis of preconditioners for hyperbolic partial differential equations preconditioners for conjugate gradient cg like iterative methods are analyzed the systems of equations arise from discretizations of time dependent hyperbolic partial differential equations pdes in two space dimensions incomplete lu ilu and block ilu preconditioners are considered for a model problem subject to periodic boundary conditions the spectra of the preconditioned coefficient matrices are determined by fourier analysis it is found that the condition number is not improved by the ilu preconditioners furthermore the number of distinct eigenvalues is not decreased a semicirculant preconditioner applied to a problem subject to dirichlet boundary conditions at the inflow boundaries is also examined analytical formulas for the eigenvalues and the eigenvectors are derived for this purpose a special analysis based on spectral decomposition and residue theory is developed when the grid ratio in space is less than one the spectrum asymptotically becomes two finite curve segments which are independent of the number of gridpoints for the restarted generalized minimal residual gmres iteration a slight reduction of the grid ratio from one substantially improves the convergence rate this is also predicted by an asymptotic analysis

fourier analysis of iterative methods for elliptic boundary value problems his paper presents a fourier method for analyzing stationary iterative methods and preconditioners for discretized elliptic boundary value problems as in the von neumann stability analysis of hyperbolic and parabolic problems the approach is easier to apply reveals more details about convergence properties than about standard techniques and can be applied in a systematic way to a wide class of numerical methods although the analysis is applicable only to periodic problems the results essentially reproduce those of classical convergence and condition number analysis for problems with other boundary conditions such as the dirichlet problem in addition they give suggestive new evidence of the strengths and weaknesses of methods such as incomplete factorization preconditioners in the dirichlet case

scan first search and sparse certificates an improved parallel algorithm for k vertex connectivity given a graph g v e a certificate of k vertex connectivity is an edge subset e subset e such that the subgraph v e is k vertex connected if and only if g is k vertex connected let n and m denote the number of vertices and edges a certificate is called sparse if it contains o kn edges for undirected graphs this paper introduces a graph search called the scan first search and shows that a certificate with at most k n 1 edges can be computed by executing scan first search k times in sequence on subgraphs of g for each of the parallel distributed and sequential models of computation the complexity of scan first search matches the best complexity of any graph search on that model in particular the parallel scan first search runs in o log n time using c n m processors on a crcw pram where c n m is the number of processors needed to find a spanning tree in each connected component in o log n time and the parallel certificate algorithm runs in o k log n time us

a stability analysis of incomplete lu factorizations abstract the combination of iterative methods with preconditionings based on incomplete lu factorizations constitutes an effective class of methods for solving the sparse linear systems arising from the discretization of elliptic partial differential equations in this paper we show that there are some settings in which the incomplete lu preconditioners are not effective and we demonstrate that their poor performance is due to numerical instability our analysis consists of an analytic and numerical study of a sample two dimensional non self adjoint elliptic problem discretized by several finite difference schemes author

functional inequalities for empty essential spectrum abstract in terms of the equivalence of poincare inequality and the existence of spectral gap the super poincare inequality is suggested in the paper for the study of essential spectrum it is proved for symmetric diffusions that such an inequality is equivalent to empty essential spectrum of the corresponding diffusion operator this inequality recovers known sobolev and nash type ones it is also equivalent to an isoperimetric inequality provided the curvature of the operator is bounded from below some results are also proved for a more general setting including symmetric jump processes moreover estimates of inequality constants are also presented which lead to a proof of a result on ultracontractivity suggested recently by d stroock finally concentration of reference measures for super poincare inequalities is studied the resulting estimates extend previous ones for poincare and log sobolev inequalities

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

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

on hyperboundedness and spectrum of markov operators consider an ergodic markov operator m reversible with respect to a probability measure mu on a general measurable space it is shown that if m is bounded from mathbb l 2 mu to mathbb l p mu where p 2 then it admits a spectral gap this result answers positively a conjecture raised by hoegh krohn and simon j funct anal 9 121 80 1972 in the more restricted semi group context the proof is based on isoperimetric considerations and especially on cheeger inequalities of higher order for weighted finite graphs recently obtained by lee et al proceedings of the 2012 acm symposium on theory of computing 1131 1140 acm new york 2012 it provides a quantitative link between hyperboundedness and an eigenvalue different from the spectral gap in general in addition the usual cheeger inequality is extended to the higher eigenvalues in the compact riemannian setting and the exponential behaviors of the small eigenvalues of witten laplacians at small temperature are recovered

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

on the stochastic p laplace equation the p laplace equation with random perturbation is studied for the singular case 1 p 2 in this paper some properties of the invariant measure and transition semigroups are obtained the main tool is the dimension free harnack inequality which is established by using the coupling argument as consequences some ergodicity compactness and contractive properties are derived for the associated transition semigroups the main results are applied to stochastic reaction diffusion equations and the stochastic p laplace equation in hilbert space

estimates of logarithmic sobolev constant an improvement of bakry emery criterion this paper is mainly devoted to estimate the logarithmic sobolev abbrev l s constant for diffusion operators on manifold or in rd in most cases we study the lower bounds but a generalization to a korzeniowski j funct anal 71 1987 363 370 theorem 1 for the upper bound is also presented theorem 1 5 based on a simple observation due to j d deuschel and d w stroock j funct anal 92 1990 30 48 of the comparison between the l s constants for different potentials the powerful bakry emery criterion for the l s inequality is improved considerably in the paper especially for the manifolds with non positive sectional curvatures theorem 1 3 1 in terms of our notation r inf x p r infx tx m x 1 ricc hessv x x where x p is the distance betweenxand an arbitrary fixed pointp m the improvement can be roughly stated as follows the condition infr 0 r 0 for which the criterion is available is now replaced by supr 0 r 0