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

hamilton s gradient estimates and a monotonicity formula for heat flows on metric measure spaces abstract in this paper we extend the hamilton s gradient estimates hamilton 1993 and a monotonicity formula of entropy ni 2004 for heat flows from smooth riemannian manifolds to non smooth metric measure spaces with appropriate riemannian curvature dimension condition

generalized liouville property for schrodinger operator on riemannian manifolds in this paper we prove that the dimension of the space of positive bounded respectively mathcal l harmonic functions on a complete riemannian manifold with mathcal l regular ends is equal to the number of ends mathcal l nonparabolic ends respectively this result is a solution of an open problem of grigor yan related to the liouville property for the schrodinger operator mathcal l we also prove that if a given complete riemannian manifold is roughly isometric to a complete riemannian manifold satisfying the volume doubling condition the poincare inequality and the finite covering condition on each end then the dimension of the space of positive bounded respectively solutions for the schrodinger operator with a potential satisfying a certain decay rate on the manifold is equal to the number of ends mathcal l nonparabolic ends respectively this is a partial answer of the question suggested by li related to the regularity of ends of a complete riemannian manifold especially our results directly generalize various earlier results of yau of li and tam of grigor yan and of present authors but with different techniques that the peculiarity of the schrodinger operator demands

ornstein uhlenbeck processes with singular drifts integral estimates and girsanov densities we consider a perturbation of a hilbert space valued ornstein uhlenbeck process by a class of singular nonlinear non autonomous maximal monotone time dependent drifts the only further assumption on the drift is that it is bounded on balls in the hilbert space uniformly in time first we introduce a new notion of generalized solutions for such equations which we call pseudo weak solutions and prove that they always exist and obtain pathwise estimates in terms of the data of the equation then we prove that their laws are absolutely continuous with respect to the law of the original ornstein uhlenbeck process in particular we show that pseudo weak solutions always have continuous sample paths in addition we obtain integrability estimates of the associated girsanov densities some of our results concern non random equations as well while probabilistic results are new even in finite dimensional autonomous settings

global stability for an sei model of infectious disease with immigration we study an sei model of disease transmission with immigration into all three classes for incidence we allow for a nonlinear response to the number of infectives including mass action and saturating incidence as special cases there is no disease free equilibrium and therefore no basic reproduction number for all parameter values the only equilibrium is an endemic equilibrium using a lyapunov function we show that this equilibrium is globally asymptotically stable

analytic and geometric background of recurrence and non explosion of the brownian motion on riemannian manifolds we provide an overview of such properties of the brownian motion on complete non compact riemannian manifolds as recurrence and non explosion it is shown that both properties have various analytic characterizations in terms of the heat kernel green function liouville properties etc on the other hand we consider a number of geometric conditions such as the volume growth isoperimetric inequalities curvature bounds etc which are related to recurrence and non explosion

a non standard finite difference scheme for a delayed and diffusive viral infection model with general nonlinear incidence rate abstract a non standard finite difference scheme is proposed to solve a delayed and diffusive viral infection model with general nonlinear incidence rate the results show that the discrete model preserves the positivity and boundedness of solutions in order to ensure the well posedness of the problem moreover this method preserves all equilibria of the original continuous model by constructing lyapunov functionals we show that the global stability of equilibria is completely determined by the basic reproduction number 0 which implies that the proposed discrete model can efficiently blue preserve the global stability of equilibria of the corresponding continuous model numerical experiments are carried out to support the theoretical results

on the upper estimate of the heat kernel of a complete riemannian manifold let m be a complete non compact riemannian manifold whose sectional curvature is bounded between two constants k and k then one expects that the heat diffusion in such a manifold behaves like the heat diffusion in euclidean space the purpose of this paper is to give a justification of such a statement in 5 j cheeger and the third author found a lower estimate of the heat kernel by comparing it with the heat kernel of the space form whose curvature is the lower bound of the curvature of the manifold this lower estimate is sharp if we insist the dependence should be on the lower bound of the ricci curvature alone it remains to give an upper estimate of the heat kernel one does not however expect to have a comparison theorem for the upper bound because it is more sensitive to the geometry of the manifold in fact the heat kernel of the upper hialf space and the heat kernel of the complete hyperbolic manifold with finite volume have quite different behavior this is reflected by the fact that the laplacian has no discrete spectrum in the first case while infinite number of discrete eigenvalues may exist in the latter case what we will prove here is that in any case the heat kernel has to decay in a manner similar to the euclidean heat kernel thus we will prove that for any constant c 4 there exists c1 depending on c t the bound of the curvature of m and x so that for all t e 0 t

geometry and analysis of dirichlet forms let e be a regular strongly local dirichlet form on l2 x m and d the associated intrinsic distance assume that the topology induced by d coincides with the original topology on x and that x is compact satisfies a doubling property and supports a weak 1 2 poincare inequality we first discuss the non coincidence of the intrinsic length structure and the gradient structure under the further assumption that the ricci curvature of x is bounded from below in the sense of lott sturm villani the following are shown to be equivalent i the heat flow of e gives the unique gradient flow of u ii e satisfies the newtonian property iii the intrinsic length structure coincides with the gradient structure moreover for the standard resistance dirichlet form on the sierpinski gasket equipped with the kusuoka measure we identify the intrinsic length structure with the measurable riemannian and the gradient structures we also apply the above results to the coarse ricci curvatures and asymptotics of the gradient of the heat kernel

absolutely continuous flows generated by sobolev class vector fields in finite and infinite dimensions abstract we prove the existence of the global flow ut generated by a vector field a from a sobolev class w1 1 on a finite or infinite dimensional space x with a measure provided is sufficiently smooth and that a a and a where a is the divergence with respect to are exponentially integrable in addition the measure is shown to be quasi invariant under ut in the case x r n and p dx where p w1 1loc r n is a locally uniformly positive probability density a sufficient condition is exp c a exp c a p p l1 for all c in the infinite dimensional case we get analogous results for measures differentiable along sufficiently many directions examples of measures which fit our framework important for applications are symmetric invariant measures of infinite dimensional diffusions and gibbs measures typically in both cases such measures are essentially non gaussian our result in infinite dimensions significantly extends previously studied cases where was a gaussian measure finally we study flows generated by vector fields whose values are not necessarily in the cameron martin space

large time behavior of schrodinger heat kernels and applications we obtain global in time and qualitatively sharp bounds for the heat kernel g of the schrodinger operator v the potential v satisfies v x c d x b near infinity with b 0 2 when v 0 our result can be described as follows g is bounded from above and below by the multiples of standard gaussians with a weight function if b 2 then the weight is bounded between two positive it constants if b 2 the weight is bounded between two positive functions of t d x and d y which have polynomial decay if b 2 or b 0 an application to a semilinear elliptic problem is also given

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

on the parabolic kernel of the schrodinger operator etude des equations paraboliques du type q x t t u x t 0 sur une variete riemannienne generale introduction estimations de gradients inegalites de harnack majorations et minorations des solutions fondamentales equation de la chaleur et noyau de green operateur de schrodinger

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

local monotonicity and mean value formulas for evolving riemannian manifolds we derive identities for general flows of riemannian metrics that may be regarded as local mean value monotonicity or lyapunov formulae these generalize pre vious work of the first author for mean curvature flow and other nonlinear diusions our results apply in particular to ricci flow where they yield a local monotone quantity di rectly analogous to perelman s reduced volume v and a local identity related to perelman s average energy f

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

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

renormalization group analysis of spectral problems in quantum field theory abstract in this paper we present a self contained and detailed exposition of the new renormalization group technique proposed in 1 2 its main feature is that the renormalization group transformation acts directly on a space of operators rather than on objects such as a propagator the partition function or correlation functions we apply this renormalization transformation to a hamiltonian describing the physics of an atom interacting with the quantized electromagnetic field and we prove that excited atomic states turn into resonances when the coupling between electrons and field is nonvanishing

heat kernel on smooth metric measure spaces with nonnegative curvature we derive a local gaussian upper bound for the f heat kernel on complete smooth metric measure space m g e f dv with nonnegative bakry emery ricci curvature as applications we obtain a sharp l f 1 liouville theorem for f subharmonic functions and an l f 1 uniqueness property for nonnegative solutions of the f heat equation assuming f is of at most quadratic growth in particular any l f 1 integrable f subharmonic function on gradient shrinking and steady ricci solitons must be constant we also provide explicit f heat kernel for gaussian solitons

short time behavior of hermite functions on compact lie groups abstract let p t x be the gaussian heat kernel on r n at time t the classical hermite polynomials at time t may be defined by a rodriguez formula given by h x t p t x p t x where is a constant coefficient differential operator on r n recent work of gross 1993 and hijab 1994 has led to the study of a new class of functions on a general compact lie group g in analogy with the r n case these hermite functions on g are obtained by the same formula wherein p t x is now the heat kernel on the group x is replaced by x 1 and is a right invariant differential operator let g be the lie algebra of g composing a hermite function on g with the exponential map produces a family of functions on g we prove that these functions scaled appropriately in t approach the classical hermite polynomials at time 1 as t tends to 0 both uniformly on compact subsets of g and in l p g where 1 p is a gaussian measure on g similar theorems are established when g is replaced by g k where k is some closed connected subgroup of g

stochastic completeness for graphs with curvature dimension conditions abstract we prove pointwise gradient bounds for heat semigroups associated to general possibly unbounded laplacians on infinite graphs satisfying the curvature dimension condition cd k using gradient bounds we show stochastic completeness for graphs satisfying the curvature dimension condition

traces of heat operators on riemannian foliations we consider the basic heat operator on functions on a riemannian foliation of a compact riemannian manifold and we show that the trace k b t of this operator has a particular asymptotic expansion as t 0 the coefficients of t and of t log t in this expansion are obtainable from local transverse geometric invariants functions computable by analyzing the manifold in an arbitrarily small neighborhood of a leaf closure using this expansion we prove some results about the spectrum of the basic laplacian such as the analogue of weyl s asymptotic formula also we explicitly calculate the first two nontrivial coefficients of the expansion for special cases such as codimension two foliations and foliations with regular closure

the log sobolev inequality on loop space over a compact riemannian manifold abstract we obtain a log sobolev inequality with a neat and explicit potential for the gradient on a based loop space over a compact riemannian manifold the potential term relies only on the curvature of the manifold and the hessian of the heat kernel and is l p integrable for all p 1 the log sobolev inequality is derived by a martingale representation theorem for the differentiable functions on loop space which is a variation of the clark ocone haussmann formula

boundary behavior of harmonic functions on manifolds in this paper we mainly set up a kind of representation theorem of harmonic functions on manifolds with ricci curvature bounded below and study non tangential limits of harmonic functions

a new topological approach to the l uniqueness of operators and the l1 uniqueness of fokker planck equations abstract the usual semigroups of kernels on a polish space e are in general not strongly continuous on l e with respect to the norm topology we introduce a new topology on l e such that they become c 0 semigroups for which we can establish a simplified hille yosida theorem the new topology will allow us to introduce the uniqueness of pre generator on l e which turns out to be equivalent to the l 1 uniqueness of the associated fokker planck equation among many others and it is intimately related with the liouville properties for l 1 harmonic functions the uniqueness of several second order elliptic differential operators in l are studied 1 one dimensional diffusion operators a x f b x f 2 schrodinger operators 1 2 v 3 multi dimensional diffusion generator 1 2

strong p completeness of stochastic differential equations and the existence of smooth flows on noncompact manifolds here we discuss the regularity of solutions of sde s and obtain conditions under which a sde on a complete riemannian manifoldm has a global smooth solution flow in particular improving the usual global lipschitz hypothesis whenm r n there are also results on non explosion of diffusions

stochastic differential equations on noncompact manifolds moment stability and its topological consequences in this paper we discuss the stability of stochastic differential equations and the interplay between the moment stability of a sde and the topology of the underlying manifold sufficient and necessary conditions are given for the moment stability of a sde in terms of the coefficients finally we prove a vanishing result for the fundamental group of a complete riemannian manifold in terms of purely geometrical quantities

the heat kernel of a schrodinger operator with inverse square potential we consider the schrodinger operator h v x with radial potential v which may have singularity at 0 and a quadratic decay at infinity first we study the structure of positive harmonic functions of h and give their precise behavior second under quite general conditions we prove an upper bound for the correspond heat kernel p x y t of the type 0 p x y t ct n 2u min x t u min y t u t 2exp x y 2ct for all x y rn and t 0 where u is a positive harmonic function of h third if u2 is an a2 weight on rn then we prove a lower bound of a similar type

differential harnack inequalities on riemannian manifolds i linear heat equation abstract in the first part of this paper we get new li yau type gradient estimates for positive solutions of heat equation on riemannian manifolds with ricci m k k r as applications several parabolic harnack inequalities are obtained and they lead to new estimates on heat kernels of manifolds with ricci curvature bounded from below in the second part we establish a perelman type li yau hamilton differential harnack inequality for heat kernels on manifolds with ricci m k which generalizes a result of l ni 2004 2006 20 21 as applications we obtain new harnack inequalities and heat kernel estimates on general manifolds we also obtain various entropy monotonicity formulas for all compact riemannian manifolds

global dynamics of a diffusive and delayed viral infection model with cellular infection and nonlinear infection rate abstract a diffusive and delayed viral dynamics model which incorporates cell to cell transmission cell mediated immune responses and general nonlinear incidence is investigated by constructing lyapunov functionals it is shown that the global threshold dynamics of the model is fully determined by the basic reproduction numbers of the virus and the immune response r 0 and r 1 the result implies that both the basic reproduction number of the virus and the immune response might be underevaluated without considering either the virus to cell transmission or cell to cell transmission the main results obtained in this paper extend some known results

uniqueness for the heat equation in riemannian manifolds abstract we investigate uniqueness in suitable weighted lebesgue spaces of solutions to the heat equation in geodesically complete riemannian manifolds

discretization and dynamic consistency of a delayed and diffusive viral infection model abstract a diffusive and delayed viral infection model with nonlinear incidence has been studied and the global dynamical behaviors of the original model is investigated by constructing lyapunov functionals furthermore the analysis is carried out for the discrete model which is obtained by applying the nonstandard finite difference nsfd scheme to the original continuous model the global stability for the corresponding equilibria is investigated by constructing discrete lyapunov functionals as well as the positivity and boundedness of solutions of the corresponding continuous model the results imply that the discretization scheme can efficiently preserves the qualitative properties of solutions for the original continuous model numerical experiments are carried out to support the theoretical results

jacobi s last geometric statement extends to a wider class of liouville surfaces numerical evidence is presented which strongly suggests that jacobi s last geometric statement that the conjugate locus from a point has exactly four cusps and the corresponding cut locus consists of only one topological segment holds for compact real analytic liouville surfaces diffeomorphic to s 2 if the gaussian curvature is everywhere positive and has exactly six critical points these being two saddles two global minima and two global maxima as is the case for an ellipsoid our experiments suggest that this is a sufficient rather than a necessary condition furthermore for compact real analytic liouville surfaces diffeomorphic to s 2 upon which the gaussian curvature can be negative but has exactly six critical points these being two saddles two global minima and two global maxima it appears that the cut locus is always a subarc of a line given by x 1 const or x 2 const where x 1 x 2 are canonical coordinates with respect to which the metric has the form f 1 x 1 f 2 x 2 dx 2 1 dx 2 2 in the case of an ellipsoid these curves are lines of curvature

a host vector model for malaria with infective immigrants abstract this paper considers a host vector mathematical model for the spread of malaria that incorporates recruitment of human population through a constant immigration with a fraction of infective immigrants the model analysis is carried out to find the steady states and their stability it is found that in the presence of infective immigrant humans there is no disease free equilibrium point however the model exhibits a unique endemic equilibrium state if the fraction of the infective immigrants is positive when the fraction of infective immigrants approaches a small value there is sharp threshold for which the disease can be reduced in the community the unique endemic equilibrium for which there is a fraction of infective immigrants is globally asymptotically stable

global dynamics of an sveir epidemic model with distributed delay and nonlinear incidence a novel delay sveir epidemic model is formulated the uniform persistence and global stability of the system are discussed some good results and efficient approach are presented an sveir epidemic model with imperfect vaccination and nonlinear incidence and a general latent distribution is formulated by constructing lyapunov functionals it is shown that the disease will die out if the vaccination reproduction number r v a c 1 and the disease becomes endemic if r v a c 1 furthermore vaccination effects are analyzed two special forms the probability of remaining in latent class are discussed when the probability is negatively exponentially distributed we present an efficient approach of proving global stability of the endemic equilibrium of the sveir system of ordinary differential equations odes which may improve some known approaches when the probability is a step function the delay differential equation dde system derived is used to study the impacts of vaccination and saturated incidence on the mumps transmission

surgery of the faber krahn inequality and applications to heat kernel bounds 2 cutting faber krahn inequalities 3 2 1 fk functions 3 2 2 cutting a manifold 6 2 3 non parabolic case 7 2 4 fk functions in balls 9 2 5 relative fk function 16

logarithmic derivatives of heat kernels and logarithmic sobolev inequalities with unbounded diffusion coefficients on loop spaces abstract in this paper we will give a sufficient condition on the logarithmic derivative of the heat kernel under which a logarithmic sobolev inequality lsi in abbreviation on a loop space holds as an application we prove an lsi on a pinned path space over the hyperbolic space h n with constant sectional curvature a a 0 the diffusion coefficient of the dirichlet form is an unbounded but exponentially integrable function applying to the case when a 0 we can prove an lsi with a logarithmic sobolev constant 18 in the case of standard pinned brownian motion using the lsi on the pinned path space on h n we will prove an lsi on each homotopy class of the loop space over a constant negative curvature compact riemannian manifold

implications of energy conditions on standard static space times abstract in the framework of standard static space times we state a family of sufficient or necessary conditions for a set of physically reasonable energy and convergence conditions in relativity and related theories we concentrate our study on questions about the sub harmonicity of the warping function the scalar curvature map conformal hyperbolicity conjugate points and the time like diameter of this class of space times

extremal of log sobolev inequality and w entropy on noncompact manifolds abstract let m be a complete connected noncompact manifold with bounded geometry under a condition near infinity we prove that the log sobolev functional 1 1 has an extremal function decaying exponentially near infinity we also prove that an extremal function may not exist if the condition is violated this result has the following consequences 1 it seems to give the first example of connected complete manifolds with bounded geometry where a standard log sobolev inequality does not have an extremal 2 it gives a negative answer to the open question on the existence of extremal of perelman s w entropy in the noncompact case which was stipulated by perelman 2002 22 p 9 3 2 remark 3 it helps to prove in some cases that noncompact shrinking breathers of ricci flow are gradient shrinking solitons

square integrable holomorphic functions on infinite dimensional heisenberg type groups we introduce a class of non commutative complex infinite dimensional heisenberg like lie groups based on an abstract wiener space the holomorphic functions which are also square integrable with respect to a heat kernel measure on these groups are studied in particular we establish a unitary equivalence between the square integrable holomorphic functions and a certain completion of the universal enveloping algebra of the lie algebra of this class of groups using quasi invariance of the heat kernel measure we also construct a skeleton map which characterizes globally defined functions from the l 2 closure of holomorphic polynomials by their values on the cameron martin subgroup

non self adjoint jacobi matrices with a rank one imaginary part we develop direct and inverse spectral analysis for finite and semi infinite non self adjoint jacobi matrices with a rank one imaginary part it is shown that given a set of n not necessarily distinct nonreal numbers in the open upper lower half plane uniquely determines an n n jacobi matrix with a rank one imaginary part having those numbers as its eigenvalues counting algebraic multiplicity algorithms of reconstruction for such finite jacobi matrices are presented a new model complementing the well known livsic triangular model for bounded linear operators with a rank one imaginary part is obtained it turns out that the model operator is a non self adjoint jacobi matrix we show that any bounded prime non self adjoint linear operator with a rank one imaginary part acting on some finite dimensional respectively separable infinite dimensional hilbert space is unitarily equivalent to a finite respectively semi infinite non self adjoint jacobi matrix this obtained theorem strengthens a classical result of stone established for self adjoint operators with simple spectrum we establish the non self adjoint analogs of the hochstadt and gesztesy simon uniqueness theorems for finite jacobi matrices with nonreal eigenvalues as well as an extension and refinement of these theorems for finite non self adjoint tri diagonal matrices to the case of mixed eigenvalues real and nonreal a unique jacobi matrix unitarily equivalent to the operator of integration ff x 2i xlf t dt in the hilbert space l2 0 l is found as well as spectral properties of its perturbations and connections with the well known bernoulli numbers we also give the analytic characterization of the weyl functions of dissipative jacobi matrices with a rank one imaginary part

on a homogenization problem we study the average green s function of stochastic uniformly elliptic operators of divergence form on zd mathbb z d when the randomness is independent and has small variance we prove regularity of the fourier transform of the self energy the proof relies on the schur complement formula and the analysis of singular integral operators combined with a steinhaus system

some liouville type theorems for harmonic functions on finsler manifolds abstract in this paper we give some liouville type theorems for l p p r harmonic resp subharmonic superharmonic functions on forward complete finsler manifolds moreover we derive a gradient estimate for harmonic functions on a closed finsler manifold as an application one obtains that any harmonic function on a closed finsler manifold with nonnegative weighted ricci curvature ric n n n must be constant

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

heat kernel on smooth metric measure spaces and applications we derive a harnack inequality for positive solutions of the f heat equation and gaussian upper and lower bound estimates for the f heat kernel on complete smooth metric measure spaces with bakry emery ricci curvature bounded below both upper and lower bound estimates are sharp when the bakry emery ricci curvature is nonnegative the main argument is the de giorgi nash moser theory as applications we prove an l f 1 liouville theorem for f subharmonic functions and an l f 1 uniqueness theorem for f heat equations when f has at most linear growth we also obtain eigenvalues estimates and f green s function estimates for the f laplace operator

gradient estimate for the degenerate parabolic equation ut δf u h u on manifolds abstract in this paper we study the harnack differential inequality and the hamilton type gradient estimate for the positive solutions of the ecumenic degenerate parabolic equation u t f u h u on a complete riemannian manifold with f u 0 we show that the harnack quantity trick introduced by li and yau is still useful in our case however the arguments and computations are much more involved and skilled and new harnack quantities have to be constructed besides the harnack differential inequality we also derive the hamilton type gradient estimate with the help of the harnack quantity trick which improves the previous related work to more general cases some interesting applications of our new results are also presented

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

a concrete estimate for the weak poincare inequality on loop space the aim of the paper is to study the pinned wiener measure on the loop space over a simply connected compact riemannian manifold together with a hilbert space structure and the ornstein uhlenbeck operator d d we give a concrete estimate for the weak poincare inequality assuming positivity of the ricci curvature of the underlying manifold the order of the rate function is s for any 0

curvature dimension inequalities on sub riemannian manifolds obtained from riemannian foliations part ii using the curvature dimension inequality proved in part i we look at consequences of this inequality in terms of the interaction between the sub riemannian geometry and the heat semigroup p t corresponding to the sub laplacian we give bounds for the gradient entropy a poincare inequality and a li yau type inequality these results require that the gradient of p t f remains uniformly bounded whenever the gradient of f is bounded and we give several sufficient conditions for this to hold

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

rough metrics on manifolds and quadratic estimates we study the persistence of quadratic estimates related to the kato square root problem across a change of metric on smooth manifolds by defining a class of rough riemannian like metrics that are permitted to be of low regularity and degenerate on sets of measure zero we also demonstrate how to transmit quadratic estimates between manifolds which are homeomorphic and locally bi lipschitz as a consequence we demonstrate the invariance of the kato square root problem under lipschitz transformations and obtain solutions to this problem on functions and forms on compact manifolds with a rough metric furthermore we show that a lower bound on the injectivity radius is not a necessary condition to solve the kato square root problem

on the averaged green s function of an elliptic equation with random coefficients we consider a divergence form elliptic difference operator on the lattice mathbb z d with a coefficient matrix that is an i i d perturbation of the identity matrix recently bourgain introduced novel techniques from harmonic analysis to prove the convergence of the feshbach schur perturbation series related to the averaged green s function of this model our main contribution is a refinement of bourgain s approach which improves the key decay rate from 2d epsilon to 3d epsilon the optimal decay rate is conjectured to be 3d as an application we derive estimates on higher derivatives of the averaged green s function which go beyond the second derivatives considered by delmotte deuschel and related works

holder estimates and regularity for holomorphic and harmonic functions in this paper we proved that if a singular manifold satisfies a weak mean value property for positive subharmonic functions then one can derive an oscillation bound for bounded holomorphic functions moreover if we further assume that the volume decays at most polynomially at a singular point

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

uniqueness of l 1 solutions for the laplace equation and the heat equation on riemannian manifolds on deduit une hypothese geometrique optimale sur une variete de riemann complete qui assure des proprietes d unicite pour des solutions l 1 de l equation de laplace et de la chaleur

large time behavior of the heat equation on complete manifolds with non negative ricci curvature sur une variete de riemann complete m on considere le noyau de la chaleur h x y t qui resout l equation de la chaleur t f x t 0 sur mx 0 avec la donnee initiale f x 0 f x en posant f x t m h x y t f y dy

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