local behavior of solutions of quasilinear elliptic equations with general structure this paper is motivated by the observation that solutions to certain variational inequalities involving partial differential operators of the form diva x u vu b x u vu where a and b are borel measurable are solutions to the equation diva x u vu b x u vu u for some nonnegative radon measure i among other things it is shown that if u is a holder continuous solution to this equation then the measure u satisfies the growth property j b x r 1 is given by the structure of the differential operator conversely if u is assumed to satisfy this growth condition then it is shown that u satisfies a harnack type inequality thus proving that u is locally bounded under the additional assumption that a is strongly monotonic it is shown that u is holder continuous

gibbs measures on brownian currents motivated by applications to quantum field theory we consider gibbs measures for which the reference measure is wiener measure and the interaction is given by a double stochastic integral and a pinning external potential in order properly to characterize these measures through dlr equations we are led to lift wiener measure and other objects to a space of configurations where the basic observables are not only the position of the particle at all times but also the work done by test vector fields we prove existence and basic properties of such gibbs measures in the small coupling regime by means of cluster expansion

probabilistic solution of pauli type equations extends the feynman kac formula to the case of imaginary time schrodinger equations heat equations for multicomponent wavefunctions the approach covers in particular the usual pauli equation for a spin 1 2 particle in an arbitrary magnetic field the formula contains besides the expectation with respect to a wiener process in ordinary space an expectation with respect to a jump process over the discrete indices describing the internal degrees of freedom to illustrate the method the authors evaluate the formula in some special cases and they derive also various inequalities a detailed comparison is made with a recent work by gaveau and vauthier 1981

riemannian symmetric superspaces and their origin in random matrix theory gaussian random matrix ensembles defined over the tangent spaces of the large families of cartan s symmetric spaces are considered such ensembles play a central role in mesoscopic physics as they describe the universal ergodic limit of disordered and chaotic single particle systems the generating function for the spectral correlations of each ensemble is reduced to an integral over a riemannian symmetric superspace in the limit of large matrix dimension such a space is defined as a pair g h m r where g h is a complex analytic graded manifold homogeneous with respect to the action of a complex lie supergroup g and m r is a maximal riemannian submanifold of the support of g h

exact monte carlo time dynamics in many body lattice quantum systems on the basis of a feynman kac type formula involving poisson stochastic processes a monte carlo algorithm has recently been introduced which describes exactly the real or imaginary time evolution of many body lattice quantum systems we extend this algorithm to the exact simulation of time dependent correlation functions the techniques generally employed in monte carlo simulations to control fluctuations namely reconfigurations and importance sampling are adapted to the present algorithm and their validity is rigorously proved we complete the analysis by several examples for the hard core boson hubbard model and for the heisenberg model

large deviations from classical paths hamiltonian flows as classical limits of quantum flows we prove that in the limit 0 the probability for the paths of the stochastic jump process associated to the quantum time evolution to be in a tublet around the classical trajectory is of order 1 exp a we give some applications of this result to the study of the classical limit of wigner functions

non linear elliptic and parabolic equations involving measure data in this paper we prove the existence of solutions for equations of the type div a du f in a bounded open set u 0 on where a is a possibly non linear function satisfying some coerciveness and monotonicity assumptions and f is a bounded measure we also consider the equation div a du g u f in u 0 on with f l1 or f m u u 0 and the parabolic equivalent of the first elliptic equation

on the weak continuity of elliptic operators and applications to potential theory in this paper we establish weak continuity results for quasilinear elliptic and subellip tic operators of divergence form acting on corresponding classes of subharmonic functions these results are analogous to our earlier results for fully nonlinear fc hessian operators from the weak continuity we derive various potential theoretic results including capacity estimates potential esti mates and the wiener criterion for regular boundary points our methods make substantial use of harnack inequalities for solutions 1 introduction there are numerous papers on the local behavior of solu tions subsolutions and supersolutions of second order quasilinear elliptic diver gence structure partial differential equations as well as extensions to degenerate elliptic and subelliptic equations in this paper we show that many properties including the quasi continuity of subharmonic functions asymptotic behavior of fundamental solutions removability of singularities wiener criteria for regular boundary points can be derived via harnack inequalities for solutions the two main ingredients in our approach which follow from our treatment of the fully nonlinear hessian operators in 45 46 47 together with that of labutin 30 are a weak convergence result and a pointwise potential estimate for subharmonic functions the weak convergence result is analogous to that for hessian operators 45 46 while the potential estimate extends that found for quasilinear elliptic operators by kilpelainen and maly 27 weak convergence results in the theory of nonlinear partial differential equa tions have been addressed in the survey 13 in this paper we treat nonlinear elliptic divergence structure operators for which harnack type inequalities are available to state our result we need an appropriate notion of subharmonic func tion in the fully nonlinear case 46 we employed the viscosity notion based on differential inequalities for smooth test functions for quasilinear divergence structure equations the traditional definition using the homogeneous equation is more convenient accordingly we shall say that a continuous function w in a domain q in euclidean space ir is harmonic with respect to an operator or c harmonic in short if it is a solution of the homogeneous equation c u 0

fermionic path integration and grassmann brownian motion a grassmann probability theory with anticommuting random variables and stochastic processes is developed using an extension of berezin integration to infinite dimensional spaces a kolmogorov type consistency condition allows integration on spaces of paths in anticommuting space one particular stochastic process grassmann brownian motion is described and the associated measure used to give a path integral formula for the kernel of the evolution operator in fermionic quantum mechanics the fourier mode expansion of grassmann brownian motion is derived

on k 1 of a self product of a curve we present elements of h 1 c c 62 for certain specific curves c the image of the element under the boundary map arising from the localization sequence of k theory is the graph of frobenius endomorphism of the reduction of the curve modulo 3

zero cycles on hilbert blumenthal surfaces introduction the main object of study in this paper with respect to zero cycles is a special class of hilbert blumenthal surfaces x which are defined over q as smooth compactifications of quasi projective varieties s q more precisely of coarse moduli schemes s that represent the moduli stack of polarized abelian surfaces with real multiplication by the ring of integers in a real quadratic field f q d we assume that d q is a prime 1 4 and that the class number of f is 1 then s c the complex points of s can be described as h h sl2 of where h is the upper halfplane in the early seventies hirzebruch and zagier hz defined for each integer n a curve tn on s called hirzebruch zagier cycles and showed that their intersection numbers occur as fourier coefficients of modular forms of level q with nebentypes eq the quadratic character of f q in this connection with modular forms hirzebruchzagier cycles reveal very similar properties to hecke correspondences on the selfproduct of the modular curve x0 q this crucial observation of hirzebruch and zagier together with tunnell s proof of the tate conjecture for a product of modular curves inspired harder langlands and rapoport hlr to prove the tate conjecture for divisors on hilbert blumenthal surfaces over abelian number fields the proof of the tate conjecture was then accomplished by klingenberg kl and murty and ramakrishnan mr in the general case from the new strategy to study torsion zero cycles on algebraic surfaces as developed in ls and used in l1 l2 compare also lr it is clear that a crucial point is the tate conjecture in characteristic p at good reduction primes as one of our main results we prove the tate conjecture in characteristic p for primes p that split in f for a certain class of hilbert blumenthal surfaces for this we recall that to each modular cusp form f of weight 2 level q and nebentypes eq that is f s2 0 q eq w e associate a hilbert modular cusp form f s2 sl2 of under the doi naganuma

riesz potentials and nonlinear parabolic equations the spatial gradient of solutions to nonlinear degenerate parabolic equations can be pointwise estimated by the caloric riesz potential of the right hand side datum exactly as in the case of the heat equation heat kernels type estimates persist in the nonlinear case

berezin integrals and poisson processes we show that the calculation of berezin integrals over anticommuting variables can be reduced to the evaluation of expectations of functionals of poisson processes via an appropriate feynman kac formula in this way the tools of ordinary analysis can be applied to berezin integrals and as an example we prove a simple upper bound possible applications of our results are briefly mentioned

universal potential estimates abstract we prove a class of endpoint pointwise estimates for solutions to quasilinear possibly degenerate elliptic equations in terms of linear and nonlinear potentials of wolff type of the source term such estimates allow to bound size and oscillations of solutions and their gradients pointwise and entail in a unified approach virtually all kinds of regularity properties in terms of the given datum and regularity of coefficients in particular local estimates in holder lipschitz morrey and fractional spaces as well as calderon zygmund estimates follow as a corollary in a unified way moreover estimates for fractional derivatives of solutions by mean of suitable linear and nonlinear potentials are also implied the classical wolff potential estimate by kilpelainen mal and trudinger wang as well as recent wolff gradient bounds for solutions to quasilinear equations embed in such a class as endpoint cases

gradient estimates via non linear potentials we present pointwise gradient bounds for solutions to p laplacean type non homogeneous equations employing non linear wolff type potentials and then prove similar bounds via suitable caloric potentials for solutions to parabolic equations

calculating principal eigen functions of non negative integral kernels particle approximations and applications often in applications such as rare events estimation or optimal control it is required that one calculates the principal eigenfunction and eigenvalue of a nonnegative integral kernel except in the finite dimensional case usually neither the principal eigenfunction nor the eigenvalue can be computed exactly in this paper we develop numerical approximations for these quantities we show how a generic interacting particle algorithm can be used to deliver numerical approximations of the eigenquantities and the associated so called twisted markov kernel as well as how these approximations are relevant to the aforementioned applications in addition we study a collection of random integral operators underlying the algorithm address some of their mean and pathwise properties and obtain error estimates finally numerical examples are provided in the context of importance sampling for computing tail probabilities of markov chains and computing value functions for a class of stochastic optimal control pro

probabilistic solutions of generalised birth and death equations and applications to field theory the author gives probabilistic solutions to the equations describing non relativistic quantum electrodynamical systems these solutions involve besides the usual diffusion processes also birth and death processes corresponding to the photon number variables some inequalities are stated and in particular bounds to the ground state energy of systems composed of a non relativistic particle interacting with a field are established the result is general and it is applied as an example to the polaron problem

gradient estimates below the duality exponent we show sharp local a priori estimates and regularity results for possibly degenerate non linear elliptic problems with data not lying in the natural dual space we provide a precise non linear potential theoretic analog of classical potential theory results due to adams duke math j 42 765 778 1975 and adams and lewis studia math 74 169 182 1982 concerning morrey spaces imbedding regularity properties for this we introduce a technique allowing for a non local representation of solutions via riesz potentials in turn yielding optimal local estimates simultaneously in both rearrangement and non rearrangement invariant function spaces in fact we also derive sharp estimates in lorentz spaces covering borderline cases which remained open for some while

nonlinear calderon zygmund theory in the limiting case we prove a maximal differentiability and regularity result for solutions to nonlinear measure data problems specifically we deal with the limiting case of the classical theory of calderon and zygmund in the setting of nonlinear possibly degenerate equations and we show a complete linearization effect with respect to the differentiability of solutions a prototype of the results obtained here tells for instance that if rm div du p 2 du mu quad mbox in omega subset mathbb r n with mu being a borel measure with locally finite mass on the open subset omega subset mathbb r n and p 2 1 n then du p 2 du in w sigma 1 rm loc omega quad mbox for every sigma in 0 1 the case sigma 1 is obviously forbidden already in the classical linear case of the poisson equation triangle u mu

singular quasilinear and hessian equations and inequalities we solve the existence problem in the renormalized or viscosity sense and obtain global pointwise estimates of solutions for quasilinear and hessian equations with measure coefficients and data including the following model problems pu uq fk u uq u 0 on rn or on a bounded domain rn here p is the p laplacian defined by pu div u u p 2 and fk u is the k hessian i e the sum of the k k principal minors of the hessian matrix d2u k 1 2 n and are general nonnegative measurable functions or measures on

diffusion monte carlo methods with a fixed number of walkers in this paper we discuss various aspects of diffusion monte carlo methods using a fixed number of walkers first a rigorous proof of the divergence of pure diffusion monte carlo pdmc methods dmc without branching in which the weights are carried along trajectories is given second a bias free monte carlo method combining dmc and pdmc approaches and based on a minimal stochastic reconfiguration of the population is discussed finally some illustrative calculations for a system of coupled quantum rotators are presented

the singular set of minima of integral functionals in this paper we provide upper bounds for the hausdorff dimension of the singular set of minima of general variational integrals open image in new window where f is suitably convex with respect to dv and holder continuous with respect to x v in particular we prove that the hausdorff dimension of the singular set is always strictly less than n where open image in new window

quantum dynamics without the wavefunction when suitably generalized and interpreted the path integral offers an alternative to the more familiar quantal formalism based on state vectors self adjoint operators and external observers mathematically one generalizes the path integral as propagator to a quantal measure on the space of all conceivable worlds and this generalized measure expresses the dynamics or law of motion of the theory much as wiener measure expresses the dynamics of brownian motion within such histories based schemes new and more realistic possibilities open up for resolving the philosophical problems of the state vector formalism in particular one can dispense with the need for external agents by locating the predictive content of in its sets of measure zero such sets are to be precluded but unrestricted application of this rule engenders contradictions one possible response would remove the contradictions by circumscribing the application of the preclusion concept another response more in the tradition of quantum logic would accommodate the contradictions by dualizing to a space of co events and effectively identifying reality with an element of this dual space

path sampling with stochastic dynamics some new algorithms we propose here some new sampling algorithms for path sampling in the case when stochastic dynamics are used in particular we present a new proposal function for equilibrium sampling of paths with a monte carlo dynamics the so called brownian tube proposal this proposal is based on the continuity of the dynamics with respect to the random forcing and generalizes all previous approaches when stochastic dynamics are used the efficiency of this proposal is demonstrated using some measure of decorrelation in path space we also discuss a switching strategy that allows to transform ensemble of paths at a finite rate while remaining at equilibrium in contrast with the usual jarzynski like switching this switching is very interesting to sample constrained paths starting from unconstrained paths or to perform simulated annealing in a rigorous way

c1 α local regularity of weak solutions of degenerate elliptic equations abstract it is demonstrated the local c 1 alpha nature of weak solutions of elliptic equations of the type 1 1 in the introduction under the degeneracy or singularity assumptions a sub 1 a sub 3

elliptic equations with measurable nonlinearities in nonsmooth domains abstract we study elliptic equations with measurable nonlinearities in nonsmooth domains we establish an optimal global w 1 p estimate under the condition that the associated nonlinearity is allowed to be merely measurable in one variable but has a sufficiently small bmo semi norm in the other variables while the underlying domain is sufficiently flat in the reifenberg sense that the boundary of the domain is locally trapped between two narrow strips

exact solution of the ising model on group lattices of genus g 1 we discuss how to apply the dimer method to ising models on group lattices having nontrivial topological genus g we find that the use of group extension and the existence of both external and internal group isomorphisms greatly reduces the number of distinct pfaffians and leads to explicit topological formulas for their sign and weight in the expansion of the partition function the complete solution for the ising model on the klein lattice group l 2 7 with g 3 is given

magnetic lieb thirring inequalities we study the generalizations of the well known lieb thirring inequality for the magnetic schrodinger operator with nonconstant magnetic field our main result is the naturally expected magnetic lieb thirring estimate on the moments of the negative eigenvalues for a certain class of magnetic fields including even some unbounded ones we develop a localization technique in path space of the stochastic feynman kac representation of the heat kernel which effectively estimates the oscillatory effect due to the magnetic phase factor

riesz potential type estimates for nonlinear elliptic equations abstract for p 2 we study nonlinear elliptic equations of p laplacian type with measure data div a d u u x in when the nonlinearity depends on the variable of the possibly unbounded solution itself we obtain the point wise gradient bound and the gradient continuity in terms of a certain riesz potential of the right hand side measure

thermodynamic large fluctuations from uniformized dynamics large fluctuations have received considerable attention as they encode information on the fine scale dynamics large deviation relations known as fluctuation theorems also capture crucial nonequilibrium thermodynamical properties here we report that using the technique of uniformization the thermodynamic large deviation functions of continuous time markov processes can be obtained from markov chains evolving in discrete time this formulation offers theoretical and numerical approaches to explore large deviation properties in particular the time evolution of autonomous and nonautonomous processes can be expressed in terms of a single poisson rate in this way the uniformization procedure leads to a simple and efficient way to simulate stochastic trajectories that reproduce the exact fluxes statistics we illustrate the formalism for the current fluctuations in a stochastic pump model

linear potentials in nonlinear potential theory pointwise gradient bounds via riesz potentials such as those available for the linear poisson equation actually hold for general quasilinear degenerate equations of p laplacean type the regularity theory of such equations completely reduces to that of the classical poisson equation up to the c1 level

boundary riesz potential estimates for elliptic equations with measurable nonlinearities abstract we consider elliptic equations with measurable nonlinearities in a half cube q 2 r x 1 x r n x 1 0 where the boundary data is given on q 2 r x 1 x r n x 1 0 we obtain a point wise estimate of the gradient in terms of riesz potential of the right hand side data and the oscillation of the gradient of the boundary data under the assumption that the nonlinearity is only allowed to be measurable in x 1 variable

riesz potential estimates for a class of double phase problems abstract a double phase problem with a bounded borel measure on the right hand side is studied we prove an optimal pointwise gradient estimate for such a measure data problem via riesz potentials of the measure under log dini continuity assumption on the modulating coefficients as a consequence we find an optimized c 1 regularity criterion

gradient estimates via linear and nonlinear potentials we prove new potential and nonlinear potential pointwise gradient estimates for solutions to measure data problems involving possibly degenerate quasilinear operators whose prototype is given by pu in particular no matter the nonlinearity of the equations considered we show that in the case p 2 a pointwise gradient estimate is possible using standard linear riesz potentials the proof is based on the identification of a natural quantity that on one hand respects the natural scaling of the problem and on the other allows to encode the weaker coercivity properties of the operators considered in the case p 2 in the case p 2 we prove a new gradient estimate employing nonlinear potentials of wolff type

global gradient estimates for parabolic equations with measurable nonlinearities abstract we establish a global calderon zygmund theory of nonlinear parabolic equations with measurable nonlinearities in divergence form by proving that the spatial gradient of a weak solution is as integrable as the inhomogeneous term nonlinearity a x t is assumed to be only measurable in one spatial variable and has locally small bmo semi norm in the other spatial and time variables uniformly with respect to variable the boundary of the bounded domain can be beyond the lipschitz category but it is well trapped in two narrow strips at each point and at each scale

boundary regularity in variational problems we prove that if u omega subset mathbb r n to mathbb r n is a solution to the dirichlet variational problem mathop rm min limits w int omega f x w dw rm d x quad rm subject to quad w equiv u 0 rm on partial omega involving a regular boundary datum u 0 and a regular integrand f x w dw strongly convex in dw and satisfying suitable growth conditions then mathcal h n 1 almost every boundary point is regular for u in the sense that du is holder continuous in a relative neighborhood of the point the existence of even one such regular boundary point was previously not known except for some very special cases treated by jost meier math ann 262 549 561 1983 our results are consequences of new up to the boundary higher differentiability results that we establish for minima of the functionals in question the methods also allow us to improve the known boundary regularity results for solutions to non linear elliptic systems and in some cases to improve the known interior singular sets estimates for minimizers moreover our approach allows for a treatment of systems and functionals with rough coefficients belonging to suitable sobolev spaces of fractional order

brownian path integral from dirac equation a probabilistic approach to the foldy wouthuysen transformation the spectrum of the dirac hamiltonian hd has a positive and a negative part the first corresponds to the energy levels of the electron the second to minus the energy levels of the positron the foldy wouthuysen transformation is a tool to obtain a decomposition of hd in a direct sum hd he he where he is the hamiltonian of the electron and he is that of the positron unfortunately this decomposition has been obtained in a closed form only when the external field is purely magnetic while in the presence of an electric field only a perturbative expansion is available the authors give a path integral representation of the semigroup phi x t exp t he phi 0 x in an external electromagnetic field their formula is the relativistic version for dirac particles of the well known feynman kac ito formula

functional integral representations of the pauli fierz model with spin 1 2 a feynman kac type formula for a levy and an infinite dimensional gaussian random process associated with a quantized radiation field is derived in particular a functional integral representation of e thpf generated by the pauli fierz hamiltonian with spin 1 2 in non relativistic quantum electrodynamics is constructed when no external potential is applied hpf turns translation invariant and it is decomposed as a direct integral hpf r3 hpf p dp the functional integral representation of e thpf p is also given although all these hamiltonians include spin nevertheless the kernels obtained for the path measures are scalar rather than matrix expressions as an application of the functional integral representations energy comparison inequalities are derived

global gradient estimates for nonlinear equations of p laplace type from composite materials abstract we study global gradient estimates for the weak solution to divergence form nonlinear elliptic equations with p growth nonlinearities from composite materials we assume that the domain is bounded and is composed of disjoint reifenberg flat domains and the nonlinearities have small bmo seminorms in each subdomain under these assumptions based on our new geometric observation for disjoint reifenberg domains and gradient estimates for nonlinear equations with measurable p growth nonlinearities we establish global w 1 q estimates for p q