almost additive ergodic theorems for amenable groups abstract in this paper we prove a general convergence theorem for almost additive set functions on unimodular amenable groups these mappings take their values in some banach space by extending the theory of e quasi tiling techniques we set the ground for far reaching applications in the theory of group dynamics in particular we verify the almost everywhere convergence of abstract bounded additive processes as well as a banach space approximation result for the spectral distribution function integrated density of states for random operators on discrete structures in a metric space further we include a banach space valued version of the lindenstrauss ergodic theorem for amenable groups

convergence rates of the truncated euler maruyama method for stochastic differential equations influenced by higham et al 2002 several numerical methods have been developed to study the strong convergence of the numerical solutions to stochastic differential equations sdes under the local lipschitz condition these numerical methods include the tamed euler maruyama em method the tamed milstein method the stopped em the backward em the backward forward em etc recently we developed a new explicit method in mao 2015 called the truncated em method for the nonlinear sde d x t f x t d t g x t d b t and established the strong convergence theory under the local lipschitz condition plus the khasminskii type condition x t f x p 1 2 g x 2 k 1 x 2 however due to the page limit there we did not study the convergence rates for the method which is the aim of this paper we will under some additional conditions discuss the rates of l q convergence of the truncated em method for 2 q p and show that the order of l q convergence can be arbitrarily close to q 2

on bernoulli decompositions for random variables concentration bounds and spectral localization as was noted already by a n kolmogorov any random variable has a bernoulli component this observation provides a tool for the extension of results which are known for bernoulli random variables to arbitrary distributions two applications are provided here i an anti concentration bound for a class of functions of independent random variables where probabilistic bounds are extracted from combinatorial results and ii a proof based on the bernoulli case of spectral localization for random schrodinger operators with arbitrary probability distributions for the single site coupling constants for a general random variable the bernoulli component may be defined so that its conditional variance is uniformly positive the natural maximization problem is an optimal transport question which is also addressed here

moment analysis for localization in random schrodinger operators we study localization effects of disorder on the spectral and dynamical properties of schrodinger operators with random potentials the new results include exponentially decaying bounds on the transition amplitude and related projection kernels including in the mean these are derived through the analysis of fractional moments of the resolvent which are finite due to the resonance diffusing effects of the disorder the main difficulty which has up to now prevented an extension of this method to the continuum can be traced to the lack of a uniform bound on the lifshitz krein spectral shift associated with the local potential terms the difficulty is avoided here through the use of a weak l 1 estimate concerning the boundary value distribution of resolvents of maximally dissipative operators combined with standard tools of relative compactness theory

an optimal wegner estimate and its application to the global continuity of the integrated density of states for random schrodinger operators we prove that the integrated density of states ids of random schrodinger operators with anderson type potentials on l 2 r d for d geq1 is locally holder continuous at all energies with the same holder exponent 0 alpha leq1 as the conditional probability measure for the single site random variable as a special case we prove that if the probability distribution is absolutely continuous with respect to lebesgue measure with a bounded density then the ids is lipschitz continuous at all energies the single site potential u in l 0 infty r d must be nonnegative and compactly supported the unperturbed hamiltonian must be periodic and satisfy a unique continuation principle we also prove analogous continuity results for the ids of random anderson type perturbations of the landau hamiltonian in two dimensions all of these results follow from a new wegner estimate for local random hamiltonians with rather general probability measures

a remark on the dirichlet neumann decoupling and the integrated density of states abstract we prove an estimate on the difference of the number of eigenvalues for schrodinger operators with dirichlet and neumann boundary conditions in large boxes in the proof we use krein s theory of the spectral shift function as an application we show that the integrated density of states is independent of the choice of boundary conditions

the uniqueness of the integrated density of states for the schrodinger operators with magnetic fields the integrated density of states ids for the schrodinger operators is defined in two ways by using the counting function of eigenvalues of the operator restricted to bounded regions with appropriate boundary conditions or by using the spectral projection of the whole space operator a sufficient condition for the coincidence of the two definitions above is given moreover a sufficient condition for the coincidence of the ids for the dirichlet boundary conditions and the ids for the neumann boundary conditions is given the proof is based only on the fundamental items in functional analysis such as the min max principle etc

the parabolic anderson model on riemann surfaces we show well posedness for the parabolic anderson model on 2 dimensional closed riemannian manifolds to this end we extend the notion of regularity structures to curved space and explicitly construct the minimal structure required for this equation a central ingredient is the appropriate re interpretation of the polynomial model which we build up to any order

the absolute continuity of the integrated density of states for magnetic schrodinger operators with certain unbounded random potentials the object of the present study is the integrated density of states of a quantum particle in multi dimensional euclidean space which is characterized by a schrodinger operator with magnetic field and a random potential which may be unbounded from above and below in case that the magnetic field is constant and the random potential is ergodic and admits a so called one parameter decomposition we prove the absolute continuity of the integrated density of states and provide explicit upper bounds on its derivative the density of states this local lipschitz continuity of the integrated density of states is derived by establishing a wegner estimate for finite volume schrodinger operators which holds for rather general magnetic fields and different boundary conditions examples of random potentials to which the results apply are certain alloy type and gaussian random potentials besides we show a diamagnetic inequality for schrodinger operators with neumann boundary conditions

strong convergence of the semi implicit euler method for nonlinear stochastic volterra integral equations with constant delay abstract this paper is mainly concerned with the strong convergence analysis of the semi implicit euler method for nonlinear stochastic volterra integral equations svies with constant delay the solvability and the boundedness of the numerical solution are established it is proved that the strong convergence order of the semi implicit euler method is 0 5 under lipschitz conditions moreover the strong superconvergence order is 1 0 if further the kernel of the stochastic term satisfies 0 0 the theoretical results are illustrated by extensive numerical examples

microlocalization percolation and anderson localization for the magnetic schrodinger operator with a random potential abstract we study the spectral properties of the magnetic schrodinger operator with a random potential using results from microlocal analysis and percolation we show that away from the landau levels the spectrum is almost surely pure point with at least exponentially decaying eigenfunctions moreover it is shown that the rate of decay is proportional to b

mean square stability of stochastic volterra integro differential equations abstract the mean square stability of a non linear stochastic volterra integro differential equation is studied non convolution volterra terms arise in both the drift and the dispersion term moreover for the convolution case we determine the rate of convergence in terms of an integrability condition on the volterra kernels

a lower wegner estimate and bounds on the spectral shift function for continuum random schrodinger operators abstract we prove a strictly positive locally uniform lower bound on the density of states dos of continuum random schrodinger operators on the entire spectrum i e we show that the dos does not have a zero within the spectrum this follows from a lower wegner estimate for finite volume continuum random schrodinger operators we assume throughout iid random variables and the single site distribution having a lebesgue density bounded from below on its support the main mathematical novelty in this paper are pointwise in energy bounds on the expectation of the spectral shift function at all energies for these operators where we mainly focus on perturbations corresponding to a change from dirichlet to neumann boundary conditions along the boundary of a cube we show that the bound scales with the area of the hypersurface where the boundary conditions are changed we also prove bounds on the averaged spectral shift function for perturbations by bounded and compactly supported multiplication operators

existence of the density of states for multi dimensional continuum schrodinger operators with gaussian random potentials a wegner estimate is proved for quantum systems in multi dimensional euclidean space which are characterized by one particle schrodinger operators with random potentials that admit a certain one parameter decomposition in particular the wegner estimate applies to systems with rather general gaussian random potentials as a consequence these systems possess an absolutely continuous integrated density of states whose derivative the density of states is locally bounded an explicit upper bound is derived

wegner type bounds for a multi particle continuous anderson model with an alloy type external potential we consider an n particle quantum systems in d with interaction and in presence of a random external alloy type potential a continuous n particle anderson model we establish wegner type bounds inequalities for such models giving upper bounds for the probability that random spectra of hamiltonians in finite volumes intersect a given set

landau hamiltonians with random potentials localization and the density of states we prove the existence of localized states at the edges of the bands for the two dimensional landau hamiltonian with a random potential of arbitrary disorder provided that the magnetic field is sufficiently large the corresponding eigenfunctions decay exponentially with the magnetic field and distance we also prove that the integrated density of states is lipschitz continuous away from the landau energies the proof relies on a wegner estimate for the finite area magnetic hamiltonians with random potentials and exponential decay estimates for the finitearea green s functions the proof of the decay estimates for the green s functions uses fundamental results from two dimensional bond percolation theory

dynamical delocalization in random landau hamiltonians we prove the existence of dynamical delocalization for random landau hamiltonians near each landau level since typically there is dynamical localization at the edges of each disordered broadened landau band this implies the existence of at least one dynamical mobility edge at each landau band namely a boundary point between the localization and delocalization regimes which we prove to converge to the corresponding landau level as either the magnetic field goes to infinity or the disorder goes to zero in this article we prove the existence of dynamical delocalization for random landau hamiltonians near each landau level more precisely we prove that for these two dimensional hamiltonians there exists at least one energy e near each landau level such that e 1 where e the local transport exponent introduced in gk5 is a measure of the rate of transport in wave packets with spectral support near e since typically there is dynamical localization at the edges of each disordered broadened landau band this implies the existence of at least one dynamical mobility edge at each landau band namely a boundary point between the localization and delocalization regimes which we prove to converge to the corresponding landau level as either the magnetic field goes to infinity or the disorder goes to zero random landau hamiltonians are the subject of intensive study due to their links with the integer quantum hall e ect kli for which von klitzing received the 1985 nobel prize in physics they describe an electron moving in a very thin flat conductor with impurities under the influence of a constant magnetic field perpendicular to the plane of the conductor and play an important role in the understanding of the quantum hall e ect l aoa t h nt ku be avss bees laughlin s argument l as pointed out by halperin h uses the assumption that under weak disorder and strong magnetic field the energy spectrum consists of bands of extended states separated

convergence and stability of the semi implicit euler method for a linear stochastic differential delay equation the paper deals with convergence and stability of the semi implicit euler method for a linear stochastic differential delay equation it is proved that the semi implicit euler method is convergent with strong order p the conditions under which the method is ms stable and gms stable are determined and the numerical experiments are given

sampling and equidistribution theorems for elliptic second order operators lifting of eigenvalues and applications abstract we consider elliptic second order partial differential operators with lipschitz continuous leading order coefficients on finite cubes and the whole euclidean space we prove quantitative sampling and equidistribution theorems for eigenfunctions the estimates are scale free in the sense that for a sequence of growing cubes we obtain uniform estimates these results are applied to prove lifting of eigenvalues as well as the infimum of the essential spectrum and an uncertainty relation aka spectral inequality for short energy interval spectral projectors several applications including random operators are discussed in the proof we have to overcome several challenges posed by the variable coefficients of the leading term

linear wegner estimate for alloy type schrodinger operators on metric graphs we study spectra of alloy type random schrodinger operators on metric graphs for finite edge subsets we prove a wegner estimate which is linear in the volume i e the total length of the edges and the length of the energy interval the single site potential needs to have fixed sign the metric graph does not need to have a periodic structure a further result is the existence of the integrated density of states for ergodic random hamiltonians on metric graphs with a z structure for certain models the two above results together imply the lipschitz continuity of the integrated density of states

strong superconvergence of the eulermaruyama method for linear stochastic volterra integral equations the eulermaruyama method is presented for linear stochastic volterra integral equations then the strong convergence property is analyzed for convolution kernels and general kernels respectively it is well known that for stochastic ordinary differential equations the strong convergence order of the eulermaruyama method is 12 however the strong superconvergence order of 1 is obtained for linear stochastic volterra integral equations with convolution kernels if the kernel k2 of the diffusion term satisfies k2 0 0 and this strong superconvergence property is inherited by linear stochastic volterra integral equations with general kernels if the kernel k2 of the diffusion term satisfies k2 t t 0 the theoretical results are illustrated by extensive numerical examples

numerical integration of stochastic differential equations with nonglobally lipschitz coefficients we propose a new concept which allows us to apply any numerical method of weak approximation to a very broad class of stochastic differential equations sdes with nonglobally lipschitz coefficients following this concept we discard the approximate trajectories which leave a sufficiently large sphere we prove that accuracy of any method of weak order p is estimated by varepsilon o h p where varepsilon can be made arbitrarily small with increasing radius of the sphere the results obtained are supported by numerical experiments

heat semigroup and singular pdes we provide in this work a semigroup approach to the study of singular pdes in the line of the paracontrolled approach developed recently by gubinelli imkeller and perkowski starting from a heat semigroup we develop a functional calculus and introduce a paraproduct based on the semigroup for which commutator estimates and schauder estimates are proved together with their paracontrolled extensions this machinery allows us to investigate singular pdes in potentially unbounded riemannian manifolds under mild geometric conditions as an illustration we study the generalized parabolic anderson model equation and prove under mild geometric conditions its well posed character in small time on a potentially unbounded 2 dimensional riemannian manifold for an equation driven by a coloured noise and for all times for the linear parabolic anderson model equation in 2 dimensional unbounded manifolds this machinery can be extended to an even more singular setting and deal with sobolev spaces rather than holder spaces

almost sure and moment exponential stability in the numerical simulation of stochastic differential equations relatively little is known about the ability of numerical methods for stochastic differential equations sdes to reproduce almost sure and small moment stability here we focus on these stability properties in the limit as the timestep tends to zero our analysis is motivated by an example of an exponentially almost surely stable nonlinear sde for which the euler maruyama em method fails to reproduce this behavior for any nonzero timestep we begin by showing that em correctly reproduces almost sure and small moment exponential stability for sufficiently small timesteps on scalar linear sdes we then generalize our results to multidimensional nonlinear sdes we show that when the sde obeys a linear growth condition em recovers almost surely exponential stability very well under the less restrictive condition that the drift coefficient of the sde obeys a one sided lipschitz condition where em may break down we show that the backward euler method maintains almost surely exponential stability

localization of classical waves ii electromagnetic waves we consider electromagnetic waves in a medium described by a position dependent dielectric constant we assume that is a random perturbation of a periodic function and that the periodic maxwell operator has a gap in the spectrum where we prove the existence of localized waves i e finite energy solutions of maxwell s equations with the property that almost all of the wave s energy remains in a fixed bounded region of space at all times localization of electromagnetic waves is a consequence of anderson localization for the self adjoint operators we prove that in the random medium described by the random operator exhibits anderson localization inside the gap in the spectrum of this is shown even in situations when the gap is totally filled by the spectrum of the random operator we can prescribe random environments that ensure localization in almost the whole gap

a diamagnetic inequality for semigroup differences the diamagnetic inequality for the magnetic schrodinger semigroup is extended to the difference of the semigroups of magnetic schrodinger operators with neumann and dirichlet boundary conditions on arbitrary open domains and rather general magnetic vector potentials a and potentials v in particular this bound renders moot all the technical issues in the recent proofs of the independence of the boundary conditions for the integrated density of states for magnetic schrodinger operators independence of the boundary conditions for the free case that is for vanishing potentials and vector potentials immediately implies independence of the boundary conditions of the integrated density of states for a large class of magnetic schrodinger operators

almost sure exponential stability in the numerical simulation of stochastic differential equations this paper is mainly concerned with whether the almost sure exponential stability of stochastic differential equations sdes is shared with that of a numerical method under the global lipschitz condition we first show that the sde is p th moment exponentially stable for p in 0 1 if and only if the stochastic theta method is p th moment exponentially stable for a sufficiently small step size we then show that the p th moment exponential stability of the sde or the stochastic theta method implies the almost sure exponential stability of the sde or the stochastic theta method respectively hence our new theory enables us to study the almost sure exponential stability of the sdes using the stochastic theta method instead of the method of the lyapunov functions that is we can now carry out careful numerical simulations using the stochastic theta method with a sufficiently small step size delta t if the stochastic theta method is p th moment exponentially stable for a sufficiently small

the lp theory of the spectral shift function the wegner estimate and the integrated density of states for some random operators we develop the l p theory of the spectral shift function for p 1 applicable to pairs of self adjoint operators whose difference is in the trace ideal p for 0 p 1 this result is a key ingredient of a new short proof of the wegner estimate applicable to a wide variety of additive and multiplicative random perturbations of deterministic background operators the proof yields the correct volume dependence of the upper bound this implies the local holder continuity of the integrated density of states at energies in the unperturbed spectral gap under an additional condition of the single site potential local holder continuity is proved at all energies this new wegner estimate together with other standard results establishes exponential localization for a new family of models for additive and multiplicative perturbations

theoretical and numerical analysis of the euler maruyama method for generalized stochastic volterra integro differential equations abstract in this paper we concern with the theoretical and numerical analysis of the generalized stochastic volterra integro differential equations svides the existence uniqueness boundedness and holder continuity of the analytic solutions for generalized svides are investigated the euler maruyama method for generalized svides is presented the boundedness of the numerical solution is proved and the strong convergence order is obtained the theoretical results are illustrated by some numerical examples

a theory of regularity structures we introduce a new notion of regularity structure that provides an algebraic framework allowing to describe functions and or distributions via a kind of jet or local taylor expansion around each point the main novel idea is to replace the classical polynomial model which is suitable for describing smooth functions by arbitrary models that are purpose built for the problem at hand in particular this allows to describe the local behaviour not only of functions but also of large classes of distributions we then build a calculus allowing to perform the various operations multiplication composition with smooth functions integration against singular kernels necessary to formulate fixed point equations for a very large class of semilinear pdes driven by some very singular typically random input this allows for the first time to give a mathematically rigorous meaning to many interesting stochastic pdes arising in physics the theory comes with convergence results that allow to interpret the solutions obtained in this way as limits of classical solutions to regularised problems possibly modified by the addition of diverging counterterms these counterterms arise naturally through the action of a renormalisation group which is defined canonically in terms of the regularity structure associated to the given class of pdes our theory also allows to easily recover many existing results on singular stochastic pdes kpz equation stochastic quantisation equations burgers type equations and to understand them as particular instances of a unified framework one surprising insight is that in all of these instances local solutions are actually smooth in the sense that they can be approximated locally to arbitrarily high degree as linear combinations of a fixed family of random functions distributions that play the role of polynomials in the theory as an example of a novel application we solve the long standing problem of building a natural markov process that is symmetric with respect to the finite volume measure describing the phi 4 3 euclidean quantum field theory it is natural to conjecture that the markov process built in this way describes the glauber dynamic of 3 dimensional ferromagnets near their critical temperature

unique continuation principle for spectral projections of schrodinger operators and optimal wegner estimates for non ergodic random schrodinger operators we prove a unique continuation principle for spectral projections of schrodinger operators we consider a schrodinger operator h v on rm l 2 mathbb r d and let h denote its restriction to a finite box with either dirichlet or periodic boundary condition we prove unique continuation estimates of the type i h w i h i h with 0 for appropriate potentials w 0 and intervals i as an application we obtain optimal wegner estimates at all energies for a class of non ergodic random schrodinger operators with alloy type random potentials crooked anderson hamiltonians we also prove optimal wegner estimates at the bottom of the spectrum with the expected dependence on the disorder the wegner estimate improves as the disorder increases a new result even for the usual ergodic anderson hamiltonian these estimates are applied to prove localization at high disorder for anderson hamiltonians in a fixed interval at the bottom of the spectrum

localization in general one dimensional random systems we discuss two ways of extending the recent ideas of localization from discrete schrodinger operators jacobi matrices to the continuum case one case allows us to prove localization in the goldshade molchanov pastur model for a larger class of functions than previously the other method studies the model v wherev is a random constant in each hyper cube we extend wegner s result on the lipschitz nature of the ids to this model

strong convergence of euler type methods for nonlinear stochastic differential equations traditional finite time convergence theory for numerical methods applied to stochastic differential equations sdes requires a global lipschitz assumption on the drift and diffusion coefficients in practice many important sde models satisfy only a local lipschitz property and since brownian paths can make arbitrarily large excursions the global lipschitz based theory is not directly relevant in this work we prove strong convergence results under less restrictive conditions first we give a convergence result for euler maruyama requiring only that the sde is locally lipschitz and that the pth moments of the exact and numerical solution are bounded for some p 2 as an application of this general theory we show that an implicit variant of euler maruyama converges if the diffusion coefficient is globally lipschitz but the drift coefficient satisfies only a one sided lipschitz condition this is achieved by showing that the implicit method has bounded moments and may be viewed as an euler maruyama approximation to a perturbed sde of the same form second we show that the optimal rate of convergence can be recovered if the drift coefficient is also assumed to behave like a polynomial

convergence and stability of balanced methods for stochastic delay integro differential equations abstract this paper deals with a family of balanced implicit methods for the stochastic delay integro differential equations it is shown that the balanced methods which own the implicit iterative scheme in the diffusion term give strong convergence rate of at least 1 2 it proves that the mean square stability for the stochastic delay integro differential equations is inherited by the strong balanced methods and the weak balanced methods with sufficiently small stepsizes several numerical experiments are given for illustration and show that the fully implicit methods are superior to those of the explicit methods in terms of mean square stabilities

localization for some continuous random hamiltonians in d dimensions abstract we prove the existence with probability one of an interval of pure point spectrum for some families of continuous random schrodinger operators in d dimensions for anderson like models with positive short range single site potentials we also prove that the corresponding eigenfunctions decay exponentially and that the integrated density of states is lipschitz continuous for the other families of random potentials that we study we show that the corresponding eigenfunctions decay faster than an inverse power of x which depends upon the decay rate of the single site potential to obtain these results we develop an extension of the classical aronszajn donoghue theory for a class of relatively compact perturbations and a spectral averaging method which extends kotani s trick to these more general families of operators

localization of classical waves i acoustic waves we consider classical acoustic waves in a medium described by a position dependent mass density x we assume that x is a reandom perturbation of a periodic function 0 x and that the periodic acoustic operator a 0 nabla cdot tfrac 1 varrho 0 x nabla has a gap in the spectrum we prove the existence of localized waves i e finite energy solutions of the acoustic equations with the property that almost all of the wave s energy remains in a fixed bounded region of space at all times with probability one localization of acoustic waves is a consequence of anderson localization for the self adjoint operators a nabla cdot tfrac 1 varrho 0 x nabla onl2 d we prove that in the random medium described by x the random operatora exhibits anderson localization inside the gap in the spectrum ofa0 this is shown even in situations when the gap is totally filled by the spectrum of the random opertor we can prescribe random environments that ensure localization in almost the whole gap

inverse tunnelingestimates and applications to the study of spectral statistics ofrandom operators on the real line we present a proof of a minami type estimate for one dimensional random schrodinger operators valid at all energies in the localization regime provided a wegner estimate is known to hold the minami type estimate is then applied to two models to obtain results on their spectral statistics the heuristics underlying our proof of minami type estimates is that close by eigenvalues of a one dimensional schrodinger operator correspond either to eigenfunctions that live far away from each other in space or they come from some tunneling phenomena in the second case one can undo the tunneling and thus construct quasi modes that live far away from each other in space resume nous demontrons une inegalite de type minami pour des operateurs de schrodinger aleatoires uni dimensionnel dans toute la region localisee si une estimee de wegner est connue cette estimee de type de minami est alors appliquee pour obtenir les statistiques spectrales pour deux modeles l heuristique qui guide ce travail est que des valeurs propres proches pour un operateur de schrodinger sur un intervalle sont soient localisees loin l une de l autre soit sont la consequence d un phenomene d effet tunnel dans le second cas on peut en defaisant cet effet tunnel construire des quasi modes qui sont localises loin l un de l autre 0 introduction consider the following two random operators on the real line the anderson model 0 1 h d2 dx2 w

strong convergence rates for backward euler on a class of nonlinear jump diffusion problems we generalise the current theory of optimal strong convergence rates for implicit euler based methods by allowing for poisson driven jumps in a stochastic differential equation sde more precisely we show that under one sided lipschitz and polynomial growth conditions on the drift coefficient and global lipschitz conditions on the diffusion and jump coefficients three variants of backward euler converge with strong order of one half the analysis exploits a relation between the backward and explicit euler methods

bounds on the spectral shift function and the density of states we study spectra of schrodinger operators on d first we consider a pair of operators which differ by a compactly supported potential as well as the corresponding semigroups we prove almost exponential decay of the singular values n of the difference of the semigroups as n and deduce bounds on the spectral shift function of the pair of operators

eigenvalue statistics for random schrodinger operators with non rank one perturbations we prove that certain natural random variables associated with the local eigenvalue statistics for generalized lattice anderson models constructed with finite rank perturbations are compound poisson distributed this distribution is characterized by the fact that the levy measure is supported on at most a finite set of positive integers determined by the rank the proof relies on a minami type estimate for finite rank perturbations for anderson type continuum models on mathbb r d we prove a similar result for certain natural random variables associated with the local eigenvalue statistics we prove that the compound poisson distribution associated with these random variables has a levy measure whose support is at most the set of positive integers