dirichlet form approach to infinite dimensional wiener processes with singular interactions we construct infinite dimensional wiener processes with interactions by constructing specific quasi regular dirichlet forms our assumptions are very mild accordingly our results can be applied to singular interactions such as hard core potentials lennard jones type potentials and dyson s model we construct nonequilibrium dynamics

fractal reinforcement of elastic membranes we describe a homogenization model of an elastic membrane reinforced by the inclusion of a fractal string we follow a variational approach consisting in proving the convergence of certain energy functionals this leads to the spectral convergence of a sequence of weighted second order elliptic partial differential operators to a singular elliptic operator with a fractal term

mathematical and numerical modeling of solute dynamics in blood flow and arterial walls the numerical modeling of solutes absorption processes by the arterial wall is of paramount interest for the understanding of the relationships between the local features of blood flow the nourishing of the inner arterial wall by the blood solutes and the pathologies that can appear when this process is for some reason perturbed in the present work two models for the solutes dynamics are investigated in the first model which is essentially based on the one introduced by rappitsch and perktold j biomech engrg 118 1996 pp 511 519 and rappitsch perktold and pernkopf internat j numer methods fluids 25 1997 pp 847 857 the navier stokes equations for an incompressible fluid describing the blood velocity and pressure fields are coupled with an advection diffusion equation for the solute concentration the wellposedness of this model is discussed the second model considers also the solutes dynamics inside the arterial wall described by a pure diffusion equation actually this is a heterogeneous model coupling different equations in different parts of the domain at hand its wellposedness is proven moreover in view of the numerical study an iterative finite element method by subdomains is proposed and its convergence properties are analyzed finally several numerical results comparing the different models in situations of physiologic interest are illustrated

composite media and asymptotic dirichlet forms we introduce a metric topology on the space of dirichlet forms and study compactness and closure properties of families of local and non local forms

ultradiffusion on the fractal branching koch curves ultradiffusion is defined on a family of fractal branching koch curves by an exact renormalization decimation transformation the anomalous long time behaviour of the autocorrelation function is obtained for the particular hierarchy the authors find a non universal crossover c1 m 1 2m and a universal one c2 1 2 as the effective temperature c is increased where m is the maximum ramification number of the fractal they also find that for large m the dynamic behaviour of ultradiffusion on the fractal is independent of the effective temperature c

uniqueness of dirichlet forms associated with systems of infinitely many brownian balls in ℝd dirichlet forms associated with systems of infinitely many brownian balls in d are studied introducing a linear operator l0 defined on a space of smooth local functions we show the uniqueness of dirichlet forms associated with self adjoint markovian extensions of l0 we also discuss the ergodicity of the reversible process associated with the dirichlet form

the equivalence of ensembles for classical systems of particles for systems of particles in classical phase space with standard hamiltonian we consider spatially averaged microcanonical gibbs distributions in finite boxes we show that infinite volume limits along suitable subsequences exist and are grand canonical gibbs measures on the way we establish a variational formula for the thermodynamic entropy density as well as a variational characterization of grand canonical gibbs measures

coupling biot and navier stokes equations for modelling fluid poroelastic media interaction the interaction between a fluid and a poroelastic structure is a complex problem that couples the navier stokes equations with the biot system the finite element approximation of this problem is involved due to the fact that both subproblems are indefinite in this work we first design residual based stabilization techniques for the biot system motivated by the variational multiscale approach then we state the monolithic navier stokes biot system with the appropriate transmission conditions at the interface for the solution of the coupled system we adopt both monolithic solvers and heterogeneous domain decomposition strategies different domain decomposition methods are considered and their convergence is analyzed for a simplified problem we compare the efficiency of all the methods on a test problem that exhibits a large added mass effect as it happens in hemodynamics applications

interacting brownian motions in infinite dimensions with logarithmic interaction potentials we investigate the construction of diffusions consisting of infinitely numerous brownian particles moving in mathbb r d and interacting via logarithmic functions two dimensional coulomb potentials these potentials are very strong and act over a long range in nature the associated equilibrium states are no longer gibbs measures we present general results for the construction of such diffusions and as applications thereof construct two typical interacting brownian motions with logarithmic interaction potentials namely the dyson model in infinite dimensions and ginibre interacting brownian motions the former is a particle system in mathbb r while the latter is in mathbb r 2 both models are translation and rotation invariant in space and as such are prototypes of dimensions d 1 2 respectively the equilibrium states of the former diffusion model are determinantal or pfaffian random point fields with sine kernels they appear in the thermodynamical limits of the spectrum of the ensembles of gaussian random matrices such as goe gue and gse the equilibrium states of the latter diffusion model are the thermodynamical limits of the spectrum of the ensemble of complex non hermitian gaussian random matrices known as the ginibre ensemble

γ convergence of equi coercive nonlinear energies defined on vector valued functions with non uniformly bounded coefficients abstract the present paper deals with the asymptotic behavior of equi coercive sequences n of nonlinear functionals defined over vector valued functions in w 0 1 p m where p 1 m 1 and is a bounded open set of r n n 2 the strongly local energy density f n d u of the functional n satisfies a lipschitz condition with respect to the second variable which is controlled by a positive sequence a n which is only bounded in some suitable space l r we prove that the sequence n converges for the strong topology of l p m to a functional which has a strongly local density f d u for sufficiently regular functions u this compactness result extends former results on the topic which are based either on maximum principle arguments in the nonlinear scalar case or adapted div curl lemmas in the linear case here the vectorial character and the nonlinearity of the problem need a new approach based on a careful analysis of the asymptotic minimizers associated with the functional n the relevance of the conditions which are imposed to the energy density f n d u is illustrated by several examples including some classical hyperelastic energies

homogenization of non uniformly bounded operators critical barrier for nonlocal effects this paper deals with the homogenization of a class of highly oscillating monotonic operators which are uniformly elliptic but non uniformly bounded we determine an asymptotic barrier below which we obtain a classical behaviour and above which nonlocal effects appear we also prove that this condition is optimal in the case of a fibre reinforced medium

mosco convergence of dirichlet forms in infinite dimensions with changing reference measures let e be an infinite dimensional locally convex space let n be a weakly convergent sequence of probability measures on e and let en be a sequence of dirichlet forms on e such that en is defined on l2 n general sufficient conditions for mosco convergence of the gradient dirichlet forms are obtained applications to gibbs states on a lattice and to the gaussian case are given weak convergence of the associated processes is discussed

analysis of degenerate elliptic operators of grusin type we analyse degenerate second order elliptic operators h in divergence form on l 2 r n r m we assume the coefficients are real symmetric and a 1 h h a 2 h for some a 1 a 2 0 where h delta nabla x 1 cdot c delta 1 delta 1 x 1 nabla x 1 c delta 2 delta 2 x 1 nabla x 2 2 here x 1 r n x 2 r m and c delta i delta i are positive measurable functions such that c delta i delta i x behaves like x delta i as x 0 and x delta i as x to infty with delta 1 delta 1 in 0 1 rangle and delta 2 delta 2 geq 0 our principal results state that the submarkovian semigroup s t e th is conservative and its kernel k t satisfies bounds 0 leq k t x y leq a b x t 1 2 b y t 1 2 1 2 where b x r denotes the volume of the ball b x r centred at x with radius r measured with respect to the riemannian distance associated with h the proofs depend on detailed subelliptic estimations on h a precise characterization of the riemannian distance and the corresponding volumes and wave equation techniques which exploit the finite speed of propagation we discuss further implications of these bounds and give explicit examples that show the kernel is not necessarily strictly positive nor continuous

multifractality and the kauffman model the author simulates the spread of an initial damage in the kauffman cellular automata at its critical point on the square lattice he calculates various moments of the probability that a site has been damaged n times and checks for multifractality in the critical exponents for different ensembles specifically he finds no evidence for multifractality when the moments of the probabilities are evaluated with lattice size l but multifractal behaviour occurs when the moments are monitored as a function of time t

equilibrium fluctuations for interacting brownian particles we consider an infinitely extended system of brownian particles interacting by a pair force gradv their initial distribution is stationary and given by the gibbs measure associated with the potentialv with fugacityz we assume thatv is symmetric finite range three times continuously differentiable superstable and positive and that the fugacity is small in the sense that 0 z 0 28 eedq 1 e v q in addition a certain essential self adjointness property is assumed we prove then that the time dependent fluctuations in the density on a spatial scale of order 1 and on a time scale of order 2 converge as 0 to a gaussian field with covariance edqg q e 2 t f q withp the density and the compressibility

analysis and geometry on configuration spaces the gibbsian case using a natural riemannian geometry like structure on the configuration space over rd we prove that for a large class of potentials the corresponding canonical gibbs measures on 1 can be completely characterized by an integration by parts formula that is if is the gradient of the riemannian structure on one can define a corresponding divergence div such that the canonical gibbs measures are exactly those measures for which and div are dual operators onl2 one consequence is that for such the corresponding dirichlet forms e are defined in addition each of them is shown to be associated with a conservative diffusion process on with invariant measure the corresponding generators are extensions of the operator div the diffusions can be characterized in terms of a martingale problem and they can be considered as a brownian motion on perturbed by a singular drift another main result of this paper is the following if is a canonical gibbs measure then it is extreme or a pure phase if and only if the corresponding weak sobolev spacew1 2 on is irreducible as a consequence we prove that for extreme canonical gibbs measures the above mentioned diffusions are time ergodic in particular this holds for tempered grand canonical gibbs measures ruelle measures provided that the activity constant is small enough we also include a complete discussion of the free case i e 0 where the underlying space rdis even replaced by a riemannian manifoldx

an algebraic and geometric approach to non bijective quadratic transformations this work is concerned with an extension of the so called levi civita and kustaanheimo stiefel transformations the extension is achieved along two lines firstly the authors examine the latter transformations in other dimensions than the ones originally considered by on the one hand levi civita and on the other hand by kustaanheimo and stiefel secondly they pass from the compact to the non compact case this leads to quadratic non bijective transformations that they refer to as hurwitz or kustaanheimo stiefel like and quasihurwitz or levi civita like transformations the hurwitz and quasihurwitz transformations are introduced and studied in an algebraic framework which relies on the use of eight dimensional cayley dickson algebras an explicit formulation of the hurwitz and quasihurwitz transformations is also given in terms of clifford algebras the hurwitz transformations are investigated from a geometrical viewpoint indeed they are connected to hopf and pseudohopf fibrations finally some differential aspects of the hurwitz and to a lesser extent quasihurwitz transformations are developed in view of future physical applications

multifractal nature of diffusion on hierarchical structures the authors study the moments of the first passage time t of random walks on hierarchical structures using the exact enumeration method they find that the different moments scale with the size of the system with an infinite hierarchy of exponents characterising a multifractal behaviour

fractal snowflake domain diffusion with boundary and interior drifts abstract we study a parabolic ventsell problem for a second order differential operator in divergence form and with interior and boundary drift terms on the snowflake domain we prove that under standard conditions a related cauchy problem possesses a unique classical solution and explain in which sense it solves a rigorous formulation of the initial ventsell problem as a second result we prove that functions that are intrinsically lipschitz on the snowflake boundary admit euclidean lipschitz extensions to the closure of the entire domain our methods combine the fractal membrane analysis the vector analysis for local dirichlet forms and pde on fractals coercive closed forms and the analysis of lipschitz functions

variations on a strange semi continuity result abstract this paper is devoted to the study of some lower semi continuity results whose model example is lim inf n u n v n u v for any sequence u n v n which weakly converges to u v in h 0 1 2 and satisfies the positivity assumptions u n 0 and v n 0 extensions of this semi continuity result are obtained in the following cases two fixed linear operators two monotone operators two varying linear operators varying domains and non uniformly bounded operators in these various cases the operators under consideration are second order elliptic operators in divergence form and truncation arguments play an essential role finally the case of higher order operators is treated by a quite different approach

weak convergence of diffusion processes on wiener space let be a gaussian measure on a suslin space x h be the corresponding cameron martin space and e i h be an orthonormal basis of h suppose that n n is a sequence of probability measures which converges weakly to a probability measure consider a sequence of dirichlet forms varepsilon n where varepsilon n f int x nabla h f 2 h rho n rm d gamma and sqrt rho n in w 2 1 gamma we prove some sufficient conditions for mosco convergence varepsilon n to varepsilon where varepsilon f int x nabla h f 2 h rho rm d gamma in particular if x is a hilbert space sup n sqrt rho n w 2 1 gamma infty and partial e i rho n rho n can be uniformly approximated by finite dimensional conditional expectations rm ie mathcal f n mu n big partial e i rho n rho n big for every fixed e i then under broad assumptions varepsilon n to varepsilon mosco and the distributions of the associated stochastic processes converge weakly

positivity of the self diffusion matrix of interacting brownian particles with hard core we prove the positivity of the self diffusion matrix of interacting brownian particles with hard core when the dimension of the space is greater than or equal to 2 here the self diffusion matrix is a coefficient matrix of the diffusive limit of a tagged particle we will do this for all activities z 0 of gibbs measures in particular for large z the case of high density particles a typical example of such a particle system is an infinite amount of hard core brownian balls

construction of infinite dimensional interacting diffusion processes through dirichlet forms by the theory of quasi regular dirichletforms and the associated special standard processes the existence of symmetric diffusion processes taking values in the space of non negative integer valued radon measures on mbox boldmath r nu and having gibbs invariant measures associated with some given pair potentials is considered the existence of such diffusions can be shown for a wide class of potentials involving some singular ones also as a consequence of an application of stochastic calculus a representation for the diffusion by means of a stochastic differential equation is derived

scaling of the first passage time and the survival probability on exact and quasi exact self similar structures the authors calculate the asymptotic behaviour of the moments of the first passage time and survival probability for random walks on an exactly self similar tree and on a quasi self similar comb by applying an exact decimation approach to the master equations for the hierarchical comb a transition from ordinary to anomalous diffusion occurs at r 2 where r is the ratio of teeth length in successive iterations of the structure in the anomalous regime r 2 the positive integer moments of the first passage time tq scale as ltau q with tau q 1 2q 1 ln r ln 2 where l is the linear distance from input to output the asymptotic behaviour of the survival probability is studied using both scaling theory and by a direct solution of the master equations they find that the characteristic time t in the asymptotic exponential decay of the survival probability exp t t scales as t approximately ltau with tau ln r2 ln 2 i e tau is distinct from tau 1 however substantial corrections to this asymptotic form for tau exist and these are needed to account for the recent simulation data of havlin and matan 1988

infinite dimensional stochastic differential equations and tail σ fields we present general theorems solving the long standing problem of the existence and pathwise uniqueness of strong solutions of infinite dimensional stochastic differential equations isdes called interacting brownian motions these isdes describe the dynamics of infinitely many brownian particles moving in mathbb r d with free potential varphi and mutual interaction potential varpsi we apply the theorems to essentially all interaction potentials of ruelle s class such as the lennard jones 6 12 potential and riesz potentials and to logarithmic potentials appearing in random matrix theory we solve isdes of the ginibre interacting brownian motion and the sine beta interacting brownian motion with beta 1 2 4 we also use the theorems in separate papers for the airy and bessel interacting brownian motions one of the critical points for proving the general theorems is to establish a new formulation of solutions of isdes in terms of tail sigma fields of labeled path spaces consisting of trajectories of infinitely many particles these formulations are equivalent to the original notions of solutions of isdes and more feasible to treat in infinite dimensions

non local dirichlet forms and symmetric jump processes we consider the non local symmetric dirichlet form e f given by with f the closure with respect to e 1 of the set of c 1 functions on r d with compact support where e 1 f f e f f f rd f x 2 dx and where the jump kernel j satisfies for 0 2 x y 1 this assumption allows the corresponding jump process to have jump intensities whose sizes depend on the position of the process and the direction of the jump we prove upper and lower estimates on the heat kernel we construct a strong markov process corresponding to e f we prove a parabolic harnack inequality for non negative functions that solve the heat equation with respect to e finally we construct an example where the corresponding harmonic functions need not be continuous

particle approximation of the wasserstein diffusion abstract we construct a system of interacting two sided bessel processes on the unit interval and show that the associated empirical measure process converges to the wasserstein diffusion von renesse and sturm 2009 25 assuming that markov uniqueness holds for the generating wasserstein dirichlet form the proof is based on the variational convergence of an associated sequence of dirichlet forms in the generalized mosco sense of kuwae and shioya 2003 19

a system of infinitely many mutually reflecting brownian balls in ℝd an infinite system of skorohod type equations is studied the unique solution of the system is obtained from a finite case by passing to the limit it is a diffusion process describing a system of infinitely many brownian hard balls and has a gibbs state associated with the hard core pair potential as a reversible measure

infinite dimensional stochastic differential equations related to random matrices we solve infinite dimensional stochastic differential equations isdes describing an infinite number of brownian particles interacting via two dimensional coulomb potentials the equilibrium states of the associated unlabeled stochastic dynamics are the ginibre random point field and dyson s measures which appear in random matrix theory to solve the isdes we establish an integration by parts formula for these measures because the long range effect of two dimensional coulomb potentials is quite strong the properties of brownian particles interacting with two dimensional coulomb potentials are remarkably different from those of brownian particles interacting with ruelle s class interaction potentials as an example we prove that the interacting brownian particles associated with the ginibre random point field satisfy plural isdes

dirichlet forms and diffusion processes for fermion random point fields abstract we construct the dirichlet forms and the associated diffusion processes on the configuration space of particles moving on the euclidean space r d for which certain fermion random point fields are invariant

nonlocal effects in two dimensional conductivity the paper deals with the asymptotic behaviour as e 0 of a two dimensional conduction problem whose matrix valued conductivity ae is e periodic and not uniformly bounded with respect to e we prove that only under the assumptions of equi coerciveness and l1 boundedness of the sequence ae the limit problem is a conduction problem of same nature this new result points out a fundamental difference between the two dimensional conductivity and the three dimensional one indeed under the same assumptions of periodicity equi coerciveness and l1 boundedness it is known that the high conductivity regions can induce nonlocal effects in three or greater dimensions

nonequilibrium dynamics of infinite particle systems with infinite range interactions we discuss the existence and uniqueness of nonequilibrium dynamics of infinitely many particles interacting via superstable pair interactions in one and two dimensions the interaction is allowed to be of infinite range and singular at the origin under suitable regularity conditions on the interaction potential we show that if the potential decreases polynomially as the distance between interacting two particles increases then the tempered solution to the system of hamiltonian equations exists moreover if the potential satisfies further that either it has a subexponential decreasing rate or it is everywhere two times continuously differentiable then we show that the tempered solution is unique the results extend those of dobrushin and fritz obtained for finite range interactions

on singularity of energy measures on self similar sets we provide general criteria for energy measures of regular dirichlet forms on self similar sets to be singular to bernoulli type measures in particular every energy measure is proved to be singular to the hausdorff measure for canonical dirichlet forms on 2 dimensional sierpinski carpets

insulating layers and robin problems on koch mixtures abstract this paper deals with a reinforcement problem for a plane domain whose boundary is a deterministic or random mixture of self similar koch curves we construct an e thin polygonal 2 dimensional fiber e n n n 0 e 1 around pre fractal approximating domains n and related suitable energy functionals the aim of this paper is to study the asymptotic behavior of the reinforced energy functionals while simultaneously the thickness of the fibers and the conductivity of the functionals on the fibers converges to 0 as n

n v limit for stochastic dynamics in continuous particle systems we provide an n v limit for the infinite particle infinite volume stochastic dynamics associated with gibbs states in continuous particle systems on d d 1 starting point is an n particle stochastic dynamic with singular interaction and reflecting boundary condition in a subset d with finite volume lebesgue measure v the aim is to approximate the infinite particle infinite volume stochastic dynamic by the above n particle dynamic in as n and v such that n v where is the particle density first we derive an improved ruelle bound for the canonical correlation functions under an appropriate relation between n and v then tightness is shown by using the lyons zheng decomposition the equilibrium measures of the accumulation points are identified as infinite volume canonical gibbs measures by an integration by parts formula and the accumulation points themselves are identified as infinite particle infinite volume stochastic dynamics via the associated martingale problem assuming a property closely related to markov uniqueness and weaker than essential self adjointness via mosco convergence techniques we can identify the accumulation points as markov processes and show uniqueness i e all accumulation corresponding to one invariant canonical gibbs measure coincide the proofs work for general repulsive interaction potentials of ruelle type and all temperatures densities and dimensions d 1 respectively may have a nontrivial negative part and infinite range as e g the lennard jones potential additionally our result provides as a by product an approximation of grand canonical gibbs measures by finite volume canonical gibbs measures with empty boundary condition

scaling properties of diffusion on comb like structures the authors study the scaling properties of the probability density of random walks on comb like structures they find that in contrast to the first passage time in which several exponents are needed to characterise their distribution the probability density can be described by a single exponent

on diffusion dynamics for continuous systems with singular superstable interaction we consider the time evolution of states for continuous infinite particle systems which corresponds to nonequilibrium diffusion dynamics for initial states 0 which are perturbations of the equilibrium we obtain a bound for finite volume nonequilibrium correlation functions and their continuity in time uniformly in volume for any finite time interval this gives the possibility to construct the time evolution of correlation functions and corresponding states in the thermodynamic limit

homogenization of stiff plates and two dimensional high viscosity stokes equations the paper deals with the homogenization of stiff heterogeneous plates assuming that the coefficients are equi bounded in l 1 we prove that the limit of a sequence of plate equations remains a plate equation which involves a strongly local linear operator acting on the second gradients this compactness result is based on a div curl lemma for fourth order equations on the other hand using an intermediate stream function we deduce from the plates case a similar result for high viscosity stokes equations in dimension two so that the nature of the stokes equation is preserved in the homogenization process finally we show that the l 1 boundedness assumption cannot be relaxed indeed in the case of the stokes equation the concentration of one very rigid strip on a line induces the appearance of second gradient terms in the limit problem which violates the compactness result obtained under the l 1 boundedness condition

binary jumps in continuum i equilibrium processes and their scaling limits let denote the space of all locally finite subsets configurations in rd a stochastic dynamics of binary jumps in continuum is a markov process on in which pairs of particles simultaneously hop over rd in this paper we study an equilibrium dynamics of binary jumps for which a poisson measure is a symmetrizing and hence invariant measure the existence and uniqueness of the corresponding stochastic dynamics are shown we next prove the main result of this paper a big class of dynamics of binary jumps converge in a diffusive scaling limit to a dynamics of interacting brownian particles we also study another scaling limit which leads us to a spatial birth and death process in continuum a remarkable property of the limiting dynamics is that its generator possesses a spectral gap a property which is hopeless to expect from the initial dynamics of binary jumps

determination of the closure of the set of elasticity functionals we determine the closure for mosco convergence in l2 3 of the set of elasticity functionals we prove that this closure coincides with the set of all non negative lower semicontinuous quadratic functionals which are objective i e which vanish for rigid motions the result is still valid if we consider only the set of isotropic elasticity functionals which have a prescribed poisson coefficient this shows that a very large family of materials can be reached when homogenizing a composite material with highly contrasted rigidity coefficients

poincare inequality for weighted first order sobolev spaces on loop spaces abstract let e be the loop space over a compact connected riemannian manifold with a torsion skew symmetric connection let l d be the ornstein uhlenbeck operator on a nonempty connected component d of the loop space e and let v d r be the restriction on d of the potential in the logarithmic sobolev inequality found by l gross on the loop group by s aida and by f z gong and z m ma on the loop space respectively we prove that the schrodinger operator l v l d v always has a spectral gap at the bottom 0 v of its spectrum and thus has its ground state transformed operator 1 l v 0 v where is the unique ground state of l v in particular our result proves l gross s conjecture about the existence of a spectral gap for the ground state transform of the schrodinger operator studied by him on the loop group in addition in all the above cases we identify the domain of the dirichlet forms associated with the ground state transforms as weighted first order sobolev spaces with weight given by 2 thus establishing a poincare inequality for them all these results are consequences from some new results in this paper on dirichlet forms characterizing certain classes with spectral gaps and from results by s aida and m hino

homogenization on nested fractals we study the homogenization problem on nested fractals letxt be the continuous time markov chain on the pre nested fractal given by puttingi i d random resistors on each cell it is proved that under some conditions alpha n x t e n t converges in law to a constant time change of the brownian motion on the fractal asn where is the contraction rate andte is a time scale constant as the brownian motion on fractals is not a semi martingale we need a different approach from the well developed martingale method

scaling limit of stochastic dynamics in classical continuous systems we investigate a scaling limit of gradient stochastic dynamics associated with gibbs states in classical continuous systems on rd d 1 the aim is to derive macroscopic quantities from a given microscopic or mesoscopic system the scaling we consider has been investigated by brox in 1980 rost in 1981 spohn in 1986 and guo and papanicolaou in 1985 under the assumption that the underlying potential is in c30 and positive we prove that the dirichlet forms of the scaled stochastic dynamics converge on a core of functions to the dirichlet form of a generalized ornstein uhlenbeck process the proof is based on the analysis and geometry on the configuration space which was developed by albeverio kondratiev and rockner in 1998 and works for general gibbs measures of ruelle type hence the underlying potential may have a singularity at the origin only has to be bounded from below and may not be compactly supported therefore singular interactions of physical interest are covered as for example the one given by the lennard jones potential which is studied in the theory of fluids furthermore using the lyons zheng decomposition we give a simple proof for the tightness of the scaled processes we also prove that the corresponding generators however do not converge in the l2 sense this settles a conjecture formulated by brox by rost and by spohn

nonlocal limits in the study of linear elliptic systems arising in periodic homogenization in the present paper we obtain the two scale limit system of a sequence of linear elliptic periodic problems with varying coefficients we show that this system has not the same structure than the classical one obtained when the coefficients are fixed this is due to the apparition of nonlocal effects our results give an example showing that the homogenization of elliptic problems with varying coefficients depending on one parameter gives in general a nonlocal limit problem

semilinear fractal problems approximation and regularity results abstract we prove existence and uniqueness of the weak solution for a second order semilinear transmission problem in a bounded domain in r 3 in the case of a koch type fractal layer s or the corresponding prefractal layer approximating s we study the asymptotic behavior of the solutions of the corresponding prefractal problem and we prove regularity results for the weak solution via a generalized green formula for domains with fractal boundaries

noncolliding brownian motion and determinantal processes a system of one dimensional brownian motions bms conditioned never to collide with each other is realized as i dyson s bm model which is a process of eigenvalues of hermitian matrix valued diffusion process in the gaussian unitary ensemble gue and as ii the h transform of absorbing bm in a weyl chamber where the harmonic function h is the product of differences of variables the vandermonde determinant the karlin mcgregor formula gives determinantal expression to the transition probability density of absorbing bm we show from the karlin mcgregor formula if the initial state is in the eigenvalue distribution of gue the noncolliding bm is a determinantal process in the sense that any multitime correlation function is given by a determinant specified by a matrix kernel by taking appropriate scaling limits spatially homogeneous and inhomogeneous infinite determinantal processes are derived we note that the determinantal processes related with noncolliding particle systems have a feature in common such that the matrix kernels are expressed using spectral projections of appropriate effective hamiltonians on the common structure of matrix kernels continuity of processes in time is proved and general property of the determinantal processes is discussed

analysis and geometry on configuration spaces abstract in this paper foundations are presented to a new systematic approach to analysis and geometry for an important class of infinite dimensional manifolds namely configuration spaces more precisely a differential geometry is introduced on the configuration space x over a riemannian manifold x this geometry is non flat even if x r d it is obtained as a natural lifting of the riemannian structure on x in particular a corresponding gradient divergence div and laplace beltrami operator h div gamma are constructed the associated volume elements i e all measures on x w r t which and div gamma become dual operators on l 2 x are identified as exactly the mixed poisson measures with mean measure equal to a multiple of the volume element dx on x in particular all these measures obey an integration by parts formula w r t vector fields on x the corresponding dirichlet forms e on l 2 x are therefore defined each is shown to be associated with a diffusion process which is thus the brownian motion on x and which is sub sequently identified as the usual independent infinite particle process on x the associated heat semigroup t t t 0 is calculated explicitly it is also proved that the diffusion process when started with is time ergodic or equivalently e gamma is irreducible or equivalently t t t 0 is ergodic if and only if is poisson measure z dx with intensity z dx for some z 0 furthermore it is shown that the laplace beltrami operator h div gamma gamma on l 2 x z dx is unitary equivalent to the second quantization of the laplacian x on x on the corresponding fock space n 0 l 2 x z dx n as another direct consequence of our results we obtain a representation of the lie algebra of compactly supported vector fields on x on poisson space finally generalizations to the case where dx is replaced by an absolutely continuous measure and also to interacting particle systems on x are described in particular the case where the mixed poisson measures are replaced by gibbs measures of ruelle type on x

continuity of eigenvalues of subordinate processes in domains let x x t t 0 be a symmetric markov process in a state space e and d an open set of e let s n s n t t 0 be a subordinator with laplace exponent n and s s t t 0 a subordinator with laplace exponent suppose that x is independent of s and s n in this paper we consider the subordinate processes open image in new window and open image in new window and their subprocesses open image in new window and x d killed upon leaving d suppose that the spectra of the semigroups of open image in new window and x d are all discrete with open image in new window being the eigenvalues of the generator of open image in new window and open image in new window being the eigenvalues of the generator of x d we show that if lim n n for every 0 then

on the construction and convergence of traces of forms abstract we elaborate a new method for constructing traces of quadratic forms in the framework of hilbert and dirichlet spaces our method relies on monotone convergence of quadratic forms and the canonical decomposition into regular and singular part we give various situations where the trace can be described more explicitly and compute it for some illustrating examples we then show that mosco convergence of dirichlet forms implies mosco convergence of a subsequence of their approximating traces

a domain decomposition method for advection diffusion processes with application to blood solutes in the present paper we consider a heterogeneous model for the dynamics of a blood solute both in the vascular lumen and inside the arterial wall in the lumen we consider an advection diffusion equation where the convective field is provided by the velocity of blood which is in turn obtained by solving the navier stokes equations inside the arterial wall we consider a pure diffusive dynamics since the endothelial layer at the interface between the lumen and the wall acts as a permeable membrane whose permeability depends on the shear rate exerted by the blood the solute concentration is discontinuous across this membrane a possible approach for the numerical study of this kind of problem is inspired by domain decomposition techniques in particular we introduce a splitting in the computation and alternate the solution of the advection diffusion equation in the lumen with that of the diffusion equation in the wall we set up an efficient iterative method based on a suitable reformulation of the problem in terms of a steklov poincare interface equation this formulation is a nonstandard one because of the concentration discontinuity at the lumen wall interface and plays a key role in the proof of convergence of our method in particular we prove that the convergence rate performed by the proposed method is independent of the finite element space discretization and provides a criterion for the selection of an acceleration parameter several numerical results referred to as biomedical applications support our theoretical conclusions and illustrate the efficiency of this algorithm

extensions of levy khintchine formula and beurling deny formula in semi dirichlet forms setting abstract the levy khintchine formula or more generally courrege s theorem characterizes the infinitesimal generator of a levy process or a feller process on r d for more general markov processes the formula that comes closest to such a characterization is the beurling deny formula for symmetric dirichlet forms in this paper we extend these celebrated structure results to include a general right process on a metrizable lusin space which is supposed to be associated with a semi dirichlet form we start with decomposing a regular semi dirichlet form into the diffusion jumping and killing parts then we develop a local compactification and an integral representation for quasi regular semi dirichlet forms finally we extend the formulae of levy khintchine and beurling deny in semi dirichlet forms setting through introducing a quasi compatible metric

mathematical and numerical models for transfer of low density lipoproteins through the arterial walls a new methodology for the model set up with applications to the study of disturbed lumenal flow in this work we introduce and discuss several mathematical models based on partial differential equations devised to study the coupled transport of macromolecules as low density lipoproteins in the blood stream and in the arterial walls these models are accurate provided that a suitable set of physical parameters characterizing the physical properties of the molecules and of the wall layers are available here we turn our attention on this aspect and propose a new methodology to compute the physical parameters needed for the model set up starting from available in vivo measurements then we focus on the study of the accumulation of low density lipoproteins in vascular districts featuring a highly disturbed flow our results demonstrate that mathematical models whose set up procedure benefits from an experimental feedback provide reliable information not only qualitatively but also quantitatively their application to geometrically perturbed vascular districts as for example a severe stenosis shows that geometrical parameters such as curvature and variations of the lumenal section strongly influence the accumulation of low density lipoproteins within the wall for instance in a stenotic segment with 75 area constriction the ldl concentration at the lumenal side of the wall is about 10 higher than for the undisturbed segment

scaling of the first passage time of biased diffusion on hierarchical comb structures biased diffusion on hierarchical comb structures is studied within an exact renormalisation group scheme the scaling exponents of the moments of the first passage time for random walks are obtained it is found that the scaling properties of the diffusion depend only on the direction of bias in a particular case the presence of bias may give rise to a new multifractality

mosco convergence and wiener measures for conductive thin boundaries abstract the mosco convergence of energy functionals and the convergence of associated wiener measures are proved for a domain with highly conductive thin boundary we obtain those results for matrix valued conductivities and a family of speed measures measures of the underlying domain in particular this family includes the lebesgue measure and the one which makes the energy functional superposition the expectation of the displacement of the associated processes close to the boundary goes to due to the explosion of the conductivity at the limit

a family of diffusion processes on sierpinski carpets we construct a family of diffusions p px on the d dimensional sierpinski carpet f the parameter ranges over d h where d h log 3 d 1 log 3 is the hausdorff dimension of the d dimensional sierpinski carpet f these diffusions p are reversible with invariant measures here are radon measures whose topological supports are equal to f and satisfy self similarity in the sense that 3a 3 a for all a f in addition the diffusion is self similar and invariant under local weak translations cell translations of the sierpinski carpet the transition density p p t x y is locally uniformly positive and satisfies a global gaussian upper bound in spite of these well behaved properties the diffusions are different from barlow bass brownian motions on the sierpinski carpet

random point fields associated with certain fredholm determinants i fermion poisson and boson point processes we introduce certain classes of random point fields including fermion and boson point processes which are associated with fredholm determinants of certain integral operators and study some of their basic properties limit theorems correlation functions palm measures etc also we propose a conjecture on an analogue of the determinant and permanent

quenched invariance principles for random walks and elliptic diffusions in random media with boundary via a dirichlet form extension theorem and making full use of two sided heat kernel estimates we establish quenched invariance principles for random walks in random environments with a boundary in particular we prove that the random walk on a supercritical percolation cluster or amongst random conductances bounded uniformly from below in a half space quarter space etc converges when rescaled diffusively to a reflecting brownian motion which has been one of the important open problems in this area we establish a similar result for the random conductance model in a box which allows us to improve existing asymptotic estimates for the relevant mixing time furthermore in the uniformly elliptic case we present quenched invariance principles for domains with more general boundaries

holomorphic bogoliubov functionals for interacting particle systems in continuum combinatorial harmonic analysis techniques are used to develop new analytical methods for the study of interacting particle systems in continuum based on a bogoliubov functional approach concrete applications of the methods are presented namely conditions for the existence of bogoliubov functionals a uniqueness result for gibbs measures in the high temperature regime we also propose a new approach to the study of non equilibrium stochastic dynamics in terms of evolution equations for bogoliubov functionals

non lipschitz uniform domain shape optimization in linear acoustics we introduce new parametrized classes of shape admissible domains in r n n 2 and prove that they are compact with respect to the convergence in the sense of characteristic functions the hausdorff sense the sense of compacts and the weak convergence of their boundary volumes the domains in these classes are bounded e domains with possibly fractal boundaries that can have parts of any non uniform hausdorff dimension greater or equal to n 1 and less than n we prove the existence of optimal shapes in such classes for maximum energy dissipation in the framework of linear acous tics a by product of our proof is the result that the class of bounded e domains with fixed e is stable under hausdorff convergence an additional and related result is the mosco convergence of robin type energy functionals on converging domains

some diffusion processes associated with two parameter poisson dirichlet distribution and dirichlet process the two parameter poisson dirichlet distribution pd is the distribution of an infinite dimensional random discrete probability it is a generalization of kingman s poisson dirichlet distribution the two parameter dirichlet process pi alpha theta nu 0 is the law of a pure atomic random measure with masses following the two parameter poisson dirichlet distribution in this article we focus on the construction and the properties of the infinite dimensional symmetric diffusion processes with respective symmetric measures pd and pi alpha theta nu 0 the methods used come from the theory of dirichlet forms

discrete approximation of symmetric jump processes on metric measure spaces in this paper we give general criteria on tightness and weak convergence of discrete markov chains to symmetric jump processes on metric measure spaces under mild conditions as an application we investigate discrete approximation for a large class of symmetric jump processes we also discuss some application of our results to the scaling limit of random walk in random conductance

uniqueness and regularity for a system of interacting bessel processes via the muckenhoupt condition we study the regularity of a diusion on a simplex with singular drift and reflecting boundary condition which describes a finite system of particles on an interval with coulomb interaction and reflection between nearest neighbors as our main result we establish the strong feller property for the process in both cases of repulsion and attraction in particular the system can be started from any initial state including multiple point configurations moreover we show that the process is a euclidean semi martingale if and only if the interaction is repulsive hence contrary to classical results about reflecting brownian motion in smooth domains in the attractive regime a construction via a system of skorokhod sdes is impossible finally we establish exponential heat kernel gradient estimates in the repulsive regime the main proof for the attractive case is based on potential theory in sobolev spaces with muckenhoupt weights

homogenization for conductive thin layers of pre fractal type the aim of this work is to obtain convergence results for the solutions of transmission problems across highly conductive layers of pre fractal type from the point of view of homogenization we prove the m convergence of the energy functionals to an energy functional which incorporates a singular term supported within the layer from the convergence of the energy functionals we deduce the convergence of the approximating solutions to the limit solution in a suitable sense

gradient dynamics of infinite point systems nonequilibrium gradient dynamics of d dimensional particle systems is investigated the interaction is given by a superstable pair potential of finite range solutions are constructed in the well defined set of locally finite configurations with a logarithmic order of energy fluctuations if the system is deterministic and d 2 then singular potentials are also allowed for stochastic models with a smooth interaction we need d 4 in order to develop some prerequisites for the theory of hydrodynamical fluctuations in equilibrium we investigate smoothness of the markov semigroup and describe some properties of its generator 0 introduction the purpose of this paper is to study existence and regularity properties of solutions to the following infinite system of stochastic differential equations consider a countable set s of d dimensional particles suspended in a fluid where the interaction is given by a pair potential u rgd 00 oo in a quasi microscopic description of such systems the effect of collisions with the particles of the fluid can be represented by uncorrelated stochastic forces and the soft resistance of the liquid medium reduces the order of the equations of motion from two to one see 13 23 and 24 for some further references configurations of these systems are countable subsets of rd such that any bounded domain contains a finite number of points only particles of a configuration will be identified by labelling points by elements of s thus a

discontinuous galerkin methods for mass transfer through semipermeable membranes a discontinuous galerkin dg method for the numerical solution of initial boundary value multicompartment partial differential equation models interconnected with interface conditions is presented and analyzed the study of interface problems is motivated by models of mass transfer of solutes through semipermeable membranes more specifically a model problem consisting of a system of semilinear parabolic advection diffusion reaction partial differential equations in each compartment equipped with respective initial and boundary conditions is considered nonlinear interface conditions modeling selective permeability congestion and partial reflection are applied to the compartment interfaces an interior penalty dg method is presented for this problem and it is analyzed in the space discrete setting the a priori analysis shows that the method yields optimal a priori bounds provided the exact solution is sufficiently smooth numerical experiments indicate agreement with the theoretical bounds and highlight the stability of the numerical method in the advection dominated regime

dirichlet forms and dirichlet operators for infinite particle systems essential self adjointness we study the dirichlet forms and the associated dirichlet operators for gibbs measures on infinite particle configuration space the dirichlet forms are defined to be gradient type forms by introducing a measurable field of rigged hilbert spaces on the configuration space under mild conditions on the interaction including singular potentials we show that the pre dirichlet operator is symmetric and that the closure of the pre dirichlet form satisfies the markovian property when the interaction is three times differentiable and decreasing sub exponentially we show that the dirichlet operator is essentially self adjoint on a domain consisting of bounded smooth local functions