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

essential spectral radius for markov semigroups i discrete time case using two new measures of non compactness p and w p for a positive kernel p on a polish space e we obtain a new formula of nussbaum gelfand type for the essential spectral radius ress p on b using that formula we show that different known sufficient conditions for geometric ergodicity such as doeblin s condition drift condition by means of lyapunov function geometric recurrence etc lead to variational formulas of the essential spectral radius all those can be easily transported on the weighted space bu some related results on l2 are also obtained especially in the symmetric case moreover we prove that for a strongly feller and topologically transitive markov kernel the large deviation principle of donsker varadhan for occupation measures of the associated markov process holds if and only if the essential spectral radius is zero this result allows us to show that the sufficient condition of donsker varadhan for the large deviation principle is in fact necessary the knowledge of ress p allows us to estimate eigenvalues of p in l2 in the symmetric case and to estimate the geometric convergence rate by means of that in the metric of wasserstein applications to different concrete models are provided for illustrating those general results

feynman kac semigroups ground state diffusions and large deviations we study the generalized schrodinger operator l v where l is the generator of a symmetric markov semigroup pt on l2 e m and the corresponding dirichlet form ev by means of the cramer functional v we give necessary and sufficient conditions for ev to be lower bounded and for the feynman kac semigroup pvt to be bounded some sufficient conditions for the essential self adjointness of l v are also given by means of large deviations we find a new condition which ensures the existence of ground state of l v and we construct the ground state process q t whose generator is given in the diffusion case by l l 1 where is the square field operator associated to l the self adjointness of l is discussed as applications we consider perturbation of the semigroups of second quantization on an abstract wiener space the time evolution of euclidean quantum fields and stochastic quantization

lp uniqueness of schrodinger operators and the capacitary positive improving property abstract we prove several lp uniqueness results for schrodinger operators l v by means of the feynman kac formula using the m p capacity theory for general markov semigroups we show that the associated feynman kac semigroup is positive improving in the sense of m p capacity improving the well known one in the sense of measure using that capacitary positive improving property and two new inequalities for generalized ornstein uhlenbeck generators we show the essential self adjointness of the ground state diffusion generator l l 2 associated with two dimensional euclidean quantum fields

superconvergence of the velocity in mimetic finite difference methods on quadrilaterals superconvergence of the velocity is established for mimetic finite difference approximations of second order elliptic problems over h 2 uniform quadrilateral meshes the superconvergence result holds for a full tensor coefficient the analysis exploits the relation between mimetic finite differences and mixed finite element methods via a special quadrature rule for computing the scalar product in the velocity space the theoretical results are confirmed by numerical experiments

asymptotics of sample entropy production rate for stochastic differential equations using the dimension free harnack inequality and the integration by parts formula for the associated diffusion semigroup we prove the central limit theorem the moderate deviation principle and the logarithmic iteration law for the sample entropy production rate of a family of stochastic differential equations

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

ultracontractivity and the heat kernel for schrodinger operators and dirichlet laplacians abstract abstract connections between integral kernels of positivity preserving semigroups and suitable lp contractivity properties are established then these questions are studied for the semigroups generated by v and h the dirichlet laplacian for an open connected region as an application under a suitable hypothesis sobolev estimates are proved valid up to of the form x c 0 x h k 2 where 0 is the unique positive l2 eigenfunction of h

lyapunov exponents of hybrid stochastic heat equations in this paper we investigate a class of hybrid stochastic heat equations by explicit formulae of solutions we not only reveal the sample lyapunov exponents but also discuss the pth moment lyapnov exponents moreover several examples are established to demonstrate that unstable deterministic or stochastic dynamical systems can be stabilized by markovian switching

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

composition operators belonging to operator ideals abstract with each composition operator c f f acting between classical bergman spaces ap and aq 1 p q a if and only if the diagonal operator d lp lq given by ak ak k belongs to a conversely we show that for any max 2 q 2 p 0 and r there exists a composition operator c ap aq whose associated sequence is equivalent to k log k k 2 this provides an effective tool to separate quasi banach ideals by means of composition operators the results can be generalized to standard weighted bergman spaces and beyond

perturbation of dirichlet forms lower semiboundedness closability and form cores we study perturbations e e qu of dirichlet forms e on some l2 space l2 m given by quadratic forms q g g d with a signed borel measure whose positive and negative parts are smooth measures with respect to the given dirichlet form we give necessary and sufficient conditions for e to be a lower semibounded closed quadratic form with a unique associated continuous semigroup pt on l2 m we also study the associated resolvent semigroup as well as the relation between e and the largest closed quadratic form which is smaller than e in particular we exhibit form cores

harnack and functional inequalities for generalized mehler semigroups abstract harnack inequalities are established for a class of generalized mehler semigroups which in particular imply upper bound estimates for the transition density moreover poincare and log sobolev inequalities are proved in terms of estimates for the square field operators furthermore under a condition well known in the gaussian case we prove that generalized mehler semigroups are strong feller the results are illustrated by concrete examples in particular we show that a generalized mehler semigroup with an stable part is not hyperbounded but exponentially ergodic and that the log sobolev constant obtained by our method in the special gaussian case can be sharper than the one following from the usual curvature condition moreover a harnack inequality is established for the generalized mehler semigroup associated with the dirichlet heat semigroup on 0 1 we also prove that this semigroup is not hyperbounded

the logarithmic sobolev inequality for continuous spin systems on a lattice abstract in this article we give cf theorem 1 6 a sufficient condition on a local specification for the associated gibbs state to satisfy a logarithmic sobolev inequality in the final section cf theorem 3 10 we relate our condition to the type of mixing condition introduced by dobrushin and shlosman in connection with their work on complete analyticity

harnack inequality and applications for stochastic generalized porous media equations by using coupling and girsanov transformations the dimension free harnack inequality and the strong feller property are proved for transition semigroups of solutions to a class of stochastic generalized porous media equations as applications explicit upper bounds of the l p norm of the density as well as hypercontractivity ultracontractivity and compactness of the corresponding semigroup are derived

markov additive processes i eigenvalue properties and limit theorems on etudie d un point de vue general les proprietes d existence et de regularite des valeurs propres et des fonctions propres de l operateur de feynman kac

large deviations for vector valued functionals of a markov chain lower bounds on obtient des bornes inferieures que l on exprime en termes des parametres de convergence de certains noyaux on donne une application aux mesures empiriques de chaines de markov

the equivalence of the logarithmic sobolev inequality and the dobrushin shlosman mixing condition given a finite range lattice gas with a compact continuous spin space it is shown cf theorem 1 2 that a uniform logarithmic sobolev inequality cf 1 4 holds if and only if the dobrushin shlosman mixing condition cf 1 5 holds as a consequence of our considerations we also show cf theorems 3 2 and 3 6 that these conditions are equivalent to a statement about the uniform rate at which the associated glauber dynamics tends to equilibrium in this same direction we show cf theorem 3 19 that these ideas lead to a surprisingly strong large deviation principle for the occupation time distribution of the glauber dynamics

superconvergence of mixed finite element approximations over quadrilaterals a superconvergence result is established in this article for approximate solutions of second order elliptic equations by mixed finite element methods over quadrilaterals the superconvergence indicates an accuracy of cal o h k 2 for the mixed finite element approximation if the raviart thomas or brezzi douglas fortin marini elements of order k are employed with optimal error estimate of cal o h k 1 numerical experiments are presented to illustrate the theoretical result

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

identifying a large deviation rate function assume a sequence of probabilities pn has a large deviation rate function i it is proved that i takes a form analogous to a convex conjugate if i is also assumed convex then i is a convex conjugate of an explicitly defined function ft the results are applied to the empirical law of a markov chain yielding universal bounds on i examples are given of markov chains in which the empirical law has a large deviation rate strictly between the given bounds

trace ideal criteria for toeplitz operators for a complex measure on the open unit disk u define an operator t on a hilbert space h of analytic functions with reproducing kernel k z w by u for a certain scale of hilbert spaces h 1 which includes the hardy space h2 and weighted bergman spaces a2 conditions are obtained which imply t belongs to a schatten ideal sp if is a positive measure then these conditions are necessary and sufficient application to composition operators and restriction operators are indicated

quasi compactness and absolutely continuous kernels we show how the essential spectral radius re q of a bounded positive kernel q acting on bounded functions is linked to the lower approximation of q by certain absolutely continuous kernels the standart doeblin s condition can be interpreted in this context and when suitably reformulated it leads to a formula for re q this results may be used to characterize the markov kernels having a quasi compact action on a space of measurable functions bounded with respect to some test function when no irreducibilty and aperiodicity are assumed

a mortar mimetic finite difference method on non matching grids we consider mimetic finite difference approximations to second order elliptic problems on non matching multiblock grids mortar finite elements are employed on the non matching interfaces to impose weak flux continuity optimal convergence and in certain cases superconvergence is established for both the scalar variable and its flux the theory is confirmed by computational results

criteria of spectral gap for markov operators let e f be a probability space and let p be a markov operator on l2 with 1 a simple eigenvalue such that p i e is an invariant probability measure of p then p 12 p p has a spectral gap i e 1 is isolated in the spectrum of p if and only if p limr sup f2 1 f pf r 1 this strengthens a conjecture of simon and hoegh krohn on the spectral gap for hyperbounded operators solved recently by l miclo in 10 consequently for a symmetric conservative irreducible dirichlet form on l2 a poincare log sobolev type inequality holds if and only if so does the corresponding defective inequality extensions to sub markov operators and non conservative dirichlet forms are also presented

functional inequalities and spectrum estimates the infinite measure case abstract as a continuation of 13 where a poincare type inequality was introduced to study the essential spectrum on the l 2 space of a probability measure this paper provides a modification of this inequality so that the infimum of the essential spectrum is well described even if the reference measure is infinite high order eigenvalues as well as the corresponding semigroup are estimated by using this new inequality criteria of the inequality and estimates of the inequality constants are presented finally some concrete examples are considered to illustrate the main results in particular estimates of high order eigenvalues obtained in this paper are sharp as checked by two examples on the euclidean space

a constructive approach to a class of ergodic hjb equations with unbounded and nonsmooth cost we consider a class of ergodic hamilton jacobi bellman hjb equations related to long time asymptotics of nonsmooth multiplicative functional of diffusion processes under suitable ergodicity assumptions on the underlying diffusion we show existence of these asymptotics and that they solve the related hjb equation in the viscosity sense

forward and reverse carleson inequalities for functions in bergman spaces and their derivatives is satisfied in the process a simple proof of certain inequalities on derivatives of functions in hp is obtained section 3 as well as some information on interpolation sequences for lp section 5 inequalities like 1 1 have already been used to provide information on bergman toeplitz operators l1 and are useful for obtaining representations of functions in lp section 4 theorem 4 6 the inequality which reverses the roles of a and m in 1 1 is relatively easy even if an appropriate weighting factor is included with m theorem a below it was obtained by oleinik and pavlov op and independently when p q 1 by stegenga s and with c 0 hastings ha the results of oleinik and pavlov which are more general than theorem a were extended somewhat by oleinik in 0 cima and wogen cw using stegenga s methods proved the analogue of theorem a in several variables the methods of l3 more or less supercede all these results

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

transportation information inequalities for markov processes in this paper one investigates the transportation information tci inequalities tc i for all probability measures on a metric space mathcal x d where is a given probability measure tc is the transportation cost from to with respect to the cost function c x y on mathcal x 2 i is the fisher donsker varadhan information of with respect to and 0 0 is a left continuous increasing function using large deviation techniques it is shown that tci is equivalent to some concentration inequality for the occupation measure of a reversible ergodic markov process related to i the tensorization property of tci and comparisons of tci with poincare and log sobolev inequalities are investigated several easy to check sufficient conditions are provided for special important cases of tci and several examples are worked out

functional inequalities on abstract hilbert spaces and applications we study the essential spectrum and the semigroup property for self adjoint operators on abstract hilbert spaces by using functional inequalities some known results obtained on the l 2 space w r t a measure space are generalized the functional inequality is also used to study non symmetric semigroups

hypercontractivity and applications for stochastic hamiltonian systems abstract the hypercontractivity is proved for the markov semigroup associated with a class of stochastic hamiltonian systems on hilbert spaces consequently the markov semigroup converges exponentially to the invariant probability measure in entropy and is compact for large time these strengthen the hypocoercivity results derived in the literature since the log sobolev inequality is invalid we introduce a new argument to prove the hypercontractivity using coupling and dimension free harnack inequality the main results are illustrated by concrete examples of the kinetic fokker planck equation and highly degenerate diffusion processes

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