semiclassical analysis of low lying eigenvalues ii tunneling we discuss the leading asymptotics of eigenvalue splittings of 2 x2v in the limit as x o and where v is a non negative potential with several zeros for example if eo x e1 x are the two lowest eigenvalues in a situation where v has precisely two zeros a and b related by a symmetry then limx x lin e1 x eo x is given as the distance from a to b in a certain riemann metric

a liouville type theorem for schrodinger operators in this paper we prove a sufficient condition in terms of the behavior of a ground state of a symmetric critical operator p 1 such that a nonzero subsolution of a symmetric nonnegative operator p 0 is a ground state particularly if p j v j for j 0 1 are two nonnegative schrodinger operators defined on omega subseteq mathbb r d such that p 1 is critical in with a ground state the function psi nleq 0 is a subsolution of the equation p 0 u 0 in and satisfies psi leq c varphi in then p 0 is critical in and psi is its ground state in particular psi is up to a multiplicative constant the unique positive supersolution of the equation p 0 u 0 in similar results hold for general symmetric operators and also on riemannian manifolds

the equivalence of certain heat kernel and green function bounds abstract we show that certain upper and lower bounds on the green function and heat kernel of a second order elliptic operator in a bounded region are equivalent and imply a number of apparently stronger bounds we also show that these bounds are equivalent to the harnack inequality except for peculiar regions and are therefore almost always valid

intrinsic ultracontractivity and eigenfunction estimates for schrodinger operators let d be a domain in rn n 2 and let h h0 v where h0 is a divergence form uniformly elliptic operator with dirichlet boundary conditions and v is in the kato class intrinsic ultracontractivity is proved for the semigroup of h when d is holder domain of order 0 or a uniformly holder domain of order for 0 2 for every 2 there exists a uniformly holder domain of order for which the dirichlet laplacian is not intrinsically ultracontractive for a large class of domains it is shown that the heat kernels for h0 and h decay at the same rate at the boundary applications are given to the lifetime of conditioned brownian motion some of our results seem to be new even for smooth domains

some classes of non analytic markov semigroups we deal with markov semigroups tt corresponding to second order elliptic operators au u du f where f is an unbounded locally lipschitz vector field on rn we obtain new conditions on f under which tt is not analytic in cb rn in particular we prove that the one dimensional operator au u x3u with domain u c2 r u u x3u cb r is not sectorial in cb r under suitable hypotheses on the growth of f we introduce a class of non analytic markov semigroups in lp rn where is an invariant measure for tt

existence uniqueness and behavior of solutions for a class of nonlinear parabolic problems abstract we prove existence uniqueness regularity results and estimates describing the behavior both for large and small times of a solution u of some nonlinear parabolic equations of leray lions type including the p laplacian in particular we show how the summability of the initial datum u 0 and the value of p influence the behavior of the solution u producing ultracontractive or supercontractive estimates or extinction in finite time or different kinds of decay estimates

intrinsic ultracontractivity of a schrodinger semigroup in rn abstract we give a possibly sharp sufficient condition on the electric potential q r n 0 in the schrodinger operator a q x on l 2 r n that guarantees that the schrodinger heat semigroup e a t t 0 on l 2 r n generated by a is intrinsically ultracontractive moreover if q x q x is radially symmetric we show that our condition on q is also necessary i e truly sharp it reads r 0 q r 1 2 d r for some r 0 0 our proofs make essential use of techniques based on a logarithmic sobolev inequality rosen s inequality proved via a new fenchel young inequality and a very precise asymptotic formula due to hartman and wintner

random graphs associated to some discrete and continuous time preferential attachment models we give a common description of simon barabasi albert ii pa and price growth models by introducing suitable random graph processes with preferential attachment mechanisms through the ii pa model we prove the conditions for which the asymptotic degree distribution of the barabasi albert model coincides with the asymptotic in degree distribution of the simon model furthermore we show that when the number of vertices in the simon model with parameter alpha goes to infinity a portion of them behave as a yule model with parameters lambda beta 1 alpha 1 and through this relation we explain why asymptotic properties of a random vertex in simon model coincide with the asymptotic properties of a random genus in yule model as a by product of our analysis we prove the explicit expression of the in degree distribution for the ii pa model given without proof in newman contemp phys 46 323 351 2005 references to traditional and recent applications of the these models are also discussed

spectral properties of metastable markov semigroups abstract it is shown that if a symmetric markov semigroup e ht on the hilbert space l 2 x is hypercontractive then the approximate degeneracy of the ground state has several consequences concerning other parts of the spectrum of h and concerning the unitary group e iht in particular in the presence of a space inversion symmetry all the eigenvalues occur in pairs with gaps comparable to the gap between the bottom two eigenvalues

intrinsic ultracontractivity of nonsymmetric diffusions with measure valued drifts and potentials recently in preprint 2006 we extended the concept of intrinsic ultracontractivity to nonsymmetric semigroups in this paper we study the intrinsic ultracontractivity of nonsymmetric diffusions with measure valued drifts and measure valued potentials in bounded domains our process y is a diffusion process whose generator can be formally written as l with dirichlet boundary conditions where l is a uniformly elliptic second order differential operator and 1 d is such that each component i i 1 d is a signed measure belonging to the kato class k d and v is a nonnegative measure belonging to the kato class k d 2 we show that scale invariant parabolic and elliptic harnack inequalities are valid for y in this paper we prove the parabolic boundary harnack principle and the intrinsic ultracontractivity for the killed diffusion y d with measure valued drift and potential when d is one of the following types of bounded domains twisted holder domains of order a e 1 3 1 uniformly holder domains of order a 0 2 and domains which can be locally represented as the region above the graph of a function this extends the results in j funct anal 100 1991 181 206 and probab theory related fields 91 1992 405 443 as a consequence of the intrinsic ultracontractivity we get that the supremum of the expected conditional lifetimes of y d is finite

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

intrinsic ultracontractivity and the dirichlet laplacian we show that if d x y 0 x 1 f x y 1 where f is negative upper semicontinuous and bounded below then the semigroup associated with the dirichlet laplacian on d is intrinsically ultracontractive analogs in higher dimensions are proved we construct domains of infinite area with this property

nonuniqueness of the positive dirichlet problem for parabolic equations in cylinders abstract we investigate uniqueness of nonnegative solutions to parabolic equations in cylinders of r n 1 with zero dirichlet boundary condition where bases of cylinders are unbounded domains we give necessary and or sufficient conditions for the uniqueness in terms of geometric properties of the bases

improved log sobolev inequalities hypercontractivity and uncertainty principle on the hypercube abstract log sobolev inequalities lsis upper bound entropy via a multiple of the dirichlet form i e norm of a gradient in this paper we prove a family of entropy energy inequalities for the binary hypercube which provide a non linear comparison between the entropy and the dirichlet form and improve on the usual lsis for functions with small support these non linear lsis in turn imply a new version of the hypercontractivity for such functions as another consequence we derive a sharp form of the uncertainty principle for the hypercube a function whose energy is concentrated on a set of small size and whose fourier energy is concentrated on a small hamming ball must be zero the tradeoff between the sizes that we derive is asymptotically optimal this new uncertainty principle implies a new estimate on the size of fourier coefficients of sparse boolean functions we observe that an analogous asymptotically optimal uncertainty principle in the euclidean space follows from the sharp form of young s inequality due to beckner this hints that non linear lsis augment young s inequality which itself is sharp for finite groups

sharp estimates for dirichlet eigenfunctions in horn shaped regions we prove sharp exponential decay for the dirichlet eigenfunctions in horn shaped regions in the plane the estimate is obtained using a method of carleman which is widely used in the study of harmonic measure such estimates can be applied to study the intrinsic ultracontractivity properties of the heat semigroup for such regions

hardy littlewood theory for semigroups soit t t t 0 un semi groupe symetrique sousmarkovien et soit n 2 alors on a les proprietes equivalentes a dim t t n b il existe c 2 0 s t t t f c 2 t n 2 f 1 f l 1 x t 0 en fait a b meme dans le cas n 2

two sided heat kernel estimates for censored stable like processes in this paper we study the precise behavior of the transition density func tions of censored resurrected stable like processes in c 1 1 open sets in r d where d 1 and 1 2 we first show that the semigroup of the censored stable like process in any bounded lipschitz open set is intrinsically ultracontractive we then establish sharp two sided estimates for the transition density functions of a large class of censored stable like processes in c 1 1 open sets we further obtain sharp two sided estimates for the green functions of these censored stable like processes in bounded c 1 1 open sets

progressive intrinsic ultracontractivity and heat kernel estimates for non local schrodinger operators abstract we study the long time asymptotic behaviour of semigroups generated by non local schrodinger operators of the form h l v the free operator l is the generator of a symmetric levy process in r d d 1 with non degenerate jump measure and v is a sufficiently regular confining potential we establish sharp two sided estimates of the corresponding heat kernels for large times and identify a new general regularity property which we call progressive intrinsic ultracontractivity to describe the large time evolution of the corresponding schrodinger semigroup we discuss various examples and applications of these estimates for instance we characterize the heat trace and heat content our examples cover a wide range of processes and we have to assume only mild restrictions on the growth resp decay of the potential and the jump intensity of the free process our approach is based on a combination of probabilistic and analytic methods our examples include fractional and quasi relativistic schrodinger operators

on dynamic random graphs with degree homogenization via anti preferential attachment probabilities abstract we analyze a dynamic random undirected graph in which newly added vertices are connected to those already present in the graph either using with probability p an anti preferential attachment mechanism or with probability 1 p a preferential attachment mechanism we derive the asymptotic degree distribution in the general case and study the asymptotic behavior of the expected degree process in the general and that of the degree process in the pure anti preferential attachment case degree homogenization mainly affects convergence rates for the former case and also the limiting degree distribution in the latter lastly we perform a simulative study of a variation of the introduced model allowing for anti preferential attachment probabilities given in terms of the current maximum degree of the graph

gaussian upper bounds for the heat kernels of some second order operators on riemannian manifolds abstract we describe a method of obtaining gaussian upper bounds on heat kernels which unifies and improves recent results for hypoelliptic operators in divergence form and for laplace beltrami operators on complete riemannian manifolds

generalized nonlinear yule models with the aim of considering models related to random graphs growth exhibiting persistent memory we propose a fractional nonlinear modification of the classical yule model often studied in the context of macroevolution here the model is analyzed and interpreted in the framework of the development of networks such as the world wide web nonlinearity is introduced by replacing the linear birth process governing the growth of the in links of each specific webpage with a fractional nonlinear birth process with completely general birth rates among the main results we derive the explicit distribution of the number of in links of a webpage chosen uniformly at random recognizing the contribution to the asymptotics and the finite time correction the mean value of the latter distribution is also calculated explicitly in the most general case furthermore in order to show the usefulness of our results we particularize them in the case of specific birth rates giving rise to a saturating behaviour a property that is often observed in nature the further specialization to the non fractional case allows us to extend the yule model accounting for a nonlinear growth

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

uniform bounds for solutions to quasilinear parabolic equations abstract we consider a class of quasilinear parabolic equations whose model is the heat equation corresponding to the p laplacian operator u pu di 1 i u p 2 iu with p 2 d on a domain d r d of finite measure we prove that u t x c d t u0 r for all t 0 x d and for all initial data u0 lr d provided r is not smaller than a suitable r0 where are positive constants explicitly computed in terms of d p r the nonlinear cases associated with the case p 2 display exactly the same contractivity properties which hold for the linear heat equation we also show that the nonlinear evolution considered is contractive on any lq space for any q 2

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

heat kernel eigenfunctions and conditioned brownian motion in planar domains abstract some recent results concerning uniform convergence of the shape of the heat kernel to that of the first eigenfunction and the lifetime of h processes in bounded lipschitz domains in r n are extended to all planar domains of finite area

a priori estimates for fractional nonlinear degenerate diffusion equations on bounded domains we investigate quantitative properties of the nonnegative solutions u t x geq 0 to the nonlinear fractional diffusion equation partial t u mathcal l u m 0 posed in a bounded domain x in omega subset mathbb r n with m 1 for t 0 as mathcal l we use one of the most common definitions of the fractional laplacian delta s 0 s 1 in a bounded domain with zero dirichlet boundary conditions we consider a general class of very weak solutions of the equation and obtain a priori estimates in the form of smoothing effects absolute upper bounds lower bounds and harnack inequalities we also investigate the boundary behaviour and we obtain sharp estimates from above and below in addition we obtain similar estimates for fractional semilinear elliptic equations either the standard laplacian case s 1 or the linear case m 1 are recovered as limits the method is quite general suitable to be applied to a number of similar problems

on the asymptotic decay of l2 solutions of one body schrodinger equations in unbounded domains the asymptotic decay of l 2 solutions of schrodinger equations v 0 in r x er n x r r is investigated where v x v 1 r v 2 x with v 1 for r and with some e 0 for large r under additional assumptions on the decay of v 1 pointwise upper bounds to and lower bounds to the spherical average of are given showing the same asymptotics for r for the case v const 0 for r investigated in 8 a simplified treatment is given

semismall perturbations semi intrinsic ultracontractivity and integral representations of nonnegative solutions for parabolic equations we consider nonnegative solutions of a parabolic equation in a cylinder d i where d is a noncompact domain of a riemannian manifold and i 0 t with 0 t or i 0 under the assumption ssp i e the constant function 1 is a semismall perturbation of the associated elliptic operator on d we establish an integral representation theorem of nonnegative solutions in the case i 0 t any nonnegative solution is represented uniquely by an integral on d 0 md 0 t where md is the martin boundary of d for the elliptic operator and in the case i 0 any nonnegative solution is represented uniquely by the sum of an integral on md 0 and a constant multiple of a particular solution we also show that ssp implies the condition siu i e the associated heat kernel is semi intrinsically ultracontractive

lifetimes of conditioned diffusions we investigate when an upper bound on expected lifetimes of conditioned diffusions associated with elliptic operators in divergence and non divergence form can be found the critical value of the parameter is found for each of the following classes of domains l p domains p n 1 uniformly regular twistedl p domains p n 1 and twisted holder domains 1 3 a related parabolic boundary harnack principle is proved

traps for reflected brownian motion consider an open set d r d d 2 and a closed ball b d let e x tb denote the expectation of the hitting time of b for reflected brownian motion in d starting from x d we say that d is a trap domain if supx e x tb a domain d is not a trap domain if and only if the reflecting brownian motion in d is uniformly ergodic we fully characterize the simply connected planar trap domains using a geometric condition and give a number of less complete results for d 2

the boundary behavior of heat kernels of dirichlet laplacians we prove the following complete and qualitatively sharp description of heat kernels g of dirichlet laplacians on bounded c1 1 domains d there exist positive constants c1 c2 and t 0 depending on d such that for x dist x d x y t 1c1tn 2e c2 x y 2 t g x t y 0 x y t 11c1tn 2e x y 2 c2t for all x y d and 0 t are known bounds when d is unbounded are also given

martin boundaries of elliptic skew products semismall perturbations and fundamental solutions of parabolic equations abstract we consider positive solutions of elliptic partial differential equations on non compact domains of riemannian manifolds we explicitly determine martin boundaries and martin kernels for a class of elliptic equations in skew product form by exploiting and developing perturbation theory for elliptic equations and short long time estimates for fundamental solutions of parabolic equations

scale free behavior of networks with the copresence of preferential and uniform attachment rules abstract complex networks in different areas exhibit degree distributions with a heavy upper tail a preferential attachment mechanism in a growth process produces a graph with this feature we herein investigate a variant of the simple preferential attachment model whose modifications are interesting for two main reasons to analyze more realistic models and to study the robustness of the scale free behavior of the degree distribution we introduce and study a model which takes into account two different attachment rules a preferential attachment mechanism with probability 1 p that stresses the rich get richer system and a uniform choice with probability p for the most recent nodes i e the nodes belonging to a window of size w to the left of the last born node the latter highlights a trend to select one of the last added nodes when no information is available the recent nodes can be either a given fixed number or a proportion n of the total number of existing nodes in the first case we prove that this model exhibits an asymptotically power law degree distribution the same result is then illustrated through simulations in the second case when the window of recent nodes has a constant size we herein prove that the presence of the uniform rule delays the starting time from which the asymptotic regime starts to hold the mean number of nodes of degree k and the asymptotic degree distribution are also determined analytically finally a sensitivity analysis on the parameters of the model is performed

central limit theorems for supercritical branching markov processes abstract in this paper we establish spatial central limit theorems for a large class of supercritical branching markov processes with general spatial dependent branching mechanisms these are generalizations of the spatial central limit theorems proved in 1 for branching ou processes with binary branching mechanisms compared with the results of 1 our central limit theorems are more satisfactory in the sense that the normal random variables in our theorems are non degenerate

deciding game invariance abstract in a previous paper duchene and rigo introduced the notion of invariance for take away games on heaps roughly speaking these are games whose rulesets do not depend on the position given a sequence s of positive tuples of integers the question of whether there exists an invariant game having s as set of p positions is relevant in particular it was recently proved by larsson et al that if s is a pair of complementary beatty sequences then the answer to this question is always positive in this paper we show that for a fairly large set of sequences expressed by infinite words the answer to this question is decidable

ergodicity of stochastic plates we prove the stabilization of plates subject to a stochastic evolution of ginzburg landa type we study the asymptotic behaviour of the rate of stabilization when the intensity of the noise vanishes an approximate simulated annealing property is also given

boundary harnack principle for symmetric stable processes in this paper we study potential theoretic properties of the symmetric stable processes 0 2 establishing the boundary harnack principle for ratios of harmonic functions on any open sets identifying the martin boundary with the euclidean boundary for open sets with a certain interior fatness property and extending earlier results on intrinsic ultracontractivity and the conditional gauge theorem to certain open sets

on the sequentiality of the successor function abstract letube a strictly increasing sequence of integers by a greedy algorithm every nonnegative integer has a greedyu representation the successor function maps the greedyu representation ofnonto the greedyu representation ofn 1 we characterize the sequencesusuch that the successor function associated withuis a left resp a right sequential function we also show that the odometer associated touis continuous if and only if the successor function is right sequential

a harnack type inequality for non symmetric markov semigroups abstract for a strong feller and irreducible markov semigroup on a locally compact polish space the harnack type inequality 1 1 holds if and only if the semigroup has a unique invariant probability measure and is ultracontractive moreover new sufficient conditions for this inequality to hold as well as upper bound estimates of the underlying constant are presented for diffusion semigroups on riemannian manifolds

l1 properties of intrinsic schrodinger semigroups abstract we investigate the l 1 properties of the intrinsic markov semigroup associated with a schrodinger operator on r n which possesses a positive ground state we discover cases for which this semigroup is norm analytic for positive times and others for which the semigroup is norm discontinuous in the strongest possible sense in the case of the harmonic oscillator we show that the generator of the intrinsic semigroup has totally different spectrum depending on whether one works in l 1 r e x 2 dx or l 2 r e x 2 dx in more general cases we show that the equality of the l 1 and l 2 spectrum is closely related to whether the schrodinger semigroup is intrinsically ultracontractive

on the continuous time limit of the barabasi albert random graph abstract we prove that via an appropriate scaling the degree of a fixed vertex in the barabasi albert model appeared at a large enough time converges in distribution to a yule process using this relation we explain why the limit degree distribution of a vertex chosen uniformly at random as the number of vertices goes to infinity coincides with the limit distribution of the number of species in a genus selected uniformly at random in a yule model as time goes to infinity to prove this result we do not assume that the number of vertices increases exponentially over time linear rates on the contrary we retain their natural growth with a constant rate superimposing to the overall graph structure a suitable set of processes that we call the planted model and introducing an ad hoc sampling procedure

sharp bounds for ground state eigenfunctions on domains with horns and cusps abstract we prove a lower bound on the ground state eigenfunction of the dirichlet laplacian for horn shaped and cusped domains the novelty of the method is that for horn shaped domains it implies a new result for dimension two and more importantly is valid for higher dimensions moreover the same technique applies just as easily to cusps finally we construct an example to show that the estimates are sharp

the global behavior of heat kernels in exterior domains abstract we establish the global boundary behavior of dirichlet heat kernels on some unbounded domains these combined with the interior estimate in the recent paper by grigoryan and saloff coste comm pure appl math 55 1 2002 93 provide a complete and qualitatively sharp description of heat kernels g of dirichlet laplacians on some unbounded c1 1 domains d m here m is certain complete noncompact manifold with nonnegative ricci curvature for example there exist positive constants c1 c2 depending on the unbounded domain d m such that for x dist x d x t 1 1 y t 1 1 c 1 e c 2 d x y 2 t b x t g x t y 0 x t 1 1 y t 1 1 e d x y 2 c 2 t c 1 b x t for all x y d and t 0 existence of these global bounds had been an unresolved problem even in the euclidean case

large time behavior of the lp norm of schrodinger semigroups abstract letv 0 v c0 rv with v 3 be such that h 1 2 v 0 but for any e 0 1 2 1 e v is not positive we determine the exact rate of divergence of the norm of e th as a map from l to l a number of related problems are discussed

an invariance principle for branching diffusions in bounded domains we study branching diffusions in a bounded domain d of mathbb r d in which particles are killed upon hitting the boundary partial d it is known that any such process undergoes a phase transition when the branching rate beta exceeds a critical value a multiple of the first eigenvalue of the generator of the diffusion we investigate the system at criticality and show that the associated genealogical tree when the process is conditioned to survive for a long time converges to aldous continuum random tree under appropriate rescaling the result holds under only a mild assumption on the domain and is valid for all branching mechanisms with finite variance and a general class of diffusions

gibbs measures for brownian paths under the effect of an external and a small pair potential we consider brownian motion in the presence of an external and a weakly coupled pair interaction potential and show that its stationary measure is a gibbs measure uniqueness of the gibbs measure for two cases is shown also the typical path behaviour the degree of mixing and some further properties are derived we use cluster expansion in the small coupling parameter

deterministic synchronization of automations with bounded delay resume nous montrons qu un automate a delai borne sous sequentiel i e deterministe sur les entrees est equivalent a un automate en ligne c est a dire lettre a lettre apres une periode transitoire fixe ou l automate lit le debut de l entree sans rien sortir et sous sequentiel les resultats de synchronisation etablis anterieurement pour les automates a delai borne ne preservaient pas la sous sequentialite

on the lp independence of the spectrum of the hodge laplacian on non compact manifolds abstract the central aim of this paper is the study of the spectrum of the hodge laplacian on differential forms of any order k in l p the underlying space is a c smooth open manifold m n with ricci curvature bounded below and uniformly subexponential volume growth it will be demonstrated that on such manifolds the l p spectrum of the hodge laplacian on differential k forms is independent of p for 1 p whenever the weitzenbock tensor on k forms is also bounded below it follows as a corollary that the isolated eigenvalues of finite multiplicity are l p independent the proof relies on the existence of a gaussian upper bound for the heat kernel of the hodge laplacian by considering the l p spectra on the hyperbolic space h n 1 we conclude that the subexponential volume growth condition is necessary in the case of one forms as an application we will show that the spectrum of the laplacian on one forms has no gaps on certain manifolds with a pole or that are in a warped product form this will be done under less strict curvature restrictions than what has been known so far and it was achieved by finding the l 1 spectrum of the laplacian

maximum and anti maximum principlesand eigenfunctions estimates via perturbation theory of positive solutions of elliptic equations in this paper we discuss some new results concerning perturbation theory for second order elliptic partial differential equations related to positivity properties of such equations we continue the study of some different notions of small perturbations and discuss their relations to comparisons of green s functions refined maximum and anti maximum principles ground state and the decay of eigenfunctions in particular we show that if v is a positive function which is a semismall perturbation of a subcritical schrodinger operator h defined on a domain omega subset mathbb r d and phi k k geq 0 are the dirichlet eigenfunctions of the equation hu lambda vu then for any k geq 0 the function phi k phi 0 is bounded and has a continuous extension up to the martin boundary of the pair omega h where phi 0 is the ground state of h with a principal eigenvalue lambda 0

dirichlet heat kernel estimates for subordinate brownian motions with gaussian components in this paper we derive explicit sharp two sided estimates for the dirichlet heat kernels in c 1 1 open sets d in r d of a large class of subordinate brownian motions with gaussian components when d is bounded our sharp two sided dirichlet heat kernel estimates hold for all t 0 integrating the heat kernel estimates with respect to the time variable t we obtain sharp two sided estimates for the green functions in bounded c 1 1 open sets of such subordinate brownian motions with gaussian components

generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains using three different notions of the generalized principal eigenvalue of linear second order elliptic operators in unbounded domains we derive necessary and sufficient conditions for the validity of the maximum principle as well as for the existence of positive eigenfunctions for the dirichlet problem relations between these principal eigenvalues their simplicity and several other properties are further discussed 2015 wiley periodicals inc

poincare s inequalities and talagrand s concentration phenomenon for the exponential distribution we present a simple proof based on modified logarithmic sobolev inequalities of talagrand s concentration inequality for the exponential distribution we actually observe that every measure satisfying a poincare inequality shares the same concentration phenomenon we also discuss exponential integrability under poincare inequalities and its consequence to sharp diameter upper bounds on spectral gaps

on diagonal estimates on schrodinger semigroup kernels and reduced heat kernels we prove various estimates for the kernels of semigroups generated by schrodinger operators with magnetic field and potential of polynomial growth we also investigate the reduced heat kernels

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

integral representations of nonnegative solutions for parabolic equations and elliptic martin boundaries abstract we consider nonnegative solutions of a parabolic equation in a cylinder d 0 t where d is a noncompact domain of a riemannian manifold under the assumption iu i e the associated heat kernel is intrinsically ultracontractive we establish an integral representation theorem any nonnegative solution is represented uniquely by an integral on d 0 m d 0 t where m d is the martin boundary of d for the associated elliptic operator we apply it in a unified way to several concrete examples to explicitly represent nonnegative solutions we also show that iu implies the condition sp i e the constant function 1 is a small perturbation of the elliptic operator on d

on modified logarithmic sobolev inequalities for bernoulli and poisson measures we show that for any positive functionfon the discrete cube 0 1 n ent np f pqe np1f df 2where npis the product measure of the bernoulli measure with probability of successp as well as related inequalities which may be shown to imply in the limit the classical gaussian logarithmic sobolev inequality as well as a logarithmic sobolev inequality for poisson measure we further investigate modified logarithmic sobolev inequalities to analyze integrability properties of lipschitz functions on discrete spaces in particular we obtain under modified logarithmic sobolev inequalities some concentration results for product measures that extend the classical exponential inequalities for sums of independent random variables

decay of correlations and uniqueness of gibbs lattice systems with nonquadratic interaction the aim of this paper is to develop the classical lattice models with unbounded spin to the case of nonquadratic polynomial interaction we demonstrate that the distinct relation between the growths of potentials leads to the uniqueness and the fast decay of correlations for gibbs measure

heat kernel bounds on manifolds with cusps abstract we describe a method of obtaining pointwise upper bounds for the heat kernel of a riemannian manifold with cusps we apply our results to a class of approximately hyperbolic manifolds by which we mean manifolds which have bounded geometry with respect to a hyperbolic structure with cusps these manifolds include all asymptotically hyperbolic manifolds for such manifolds we obtain upper bounds on the heat kernels which we believe to be nearly optimal

numeration systems linear recurrences and regular sets a numeration system based on a strictly increasing sequence of positive integers u0 1 u1u2 expresses a non negative integer n as a sum n j 0 i ajuj in this case we say the string a i a i 1 a1 a0 is a representation for n

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