the true self repelling motion we construct and study a continuous real valued random process which is of a new type it is self interacting self repelling but only in a local sense it only feels the self repellance due to its occu pation time measure density in the immediate neighbourhood of the point it is just visiting we focus on the most natural process with these properties that we call true self repelling motion this is the continuous counterpart to the integer valued true self avoiding walk which had been studied among others by the first author one of the striking properties of true self repelling motion is that although the coupleoxt occupation time measure of x at time tu is a continuous markov process x is not driven by a stochastic diaerential equation and is not a semi martingale it turns out for instance that it has a finite variation of order 3 2 which contrasts with the finite quadratic variation of semi martingales one of the key tools in the construction of x is a continuous system of coalescing brownian motions similar to those that have been constructed by arratia a1 a2 we derive various properties of x existence and properties of the occupation time densities ltoxu local variation etc and an identity that shows that the dynamics of x can be very loosely speaking described as follows dxt is equal to the gradient in space of ltoxu in a gener alized sense even though x7 ltoxu is not diaerentiable

navier stokes equations for stochastic particle systems on the lattice we introduce a class of stochastic models of particles on the cubic lattice d with velocities and study the hydrodynamical limit on the diffusive spacetime scale assuming special initial conditions corresponding to the incompressible regime we prove that in dimensiond 3 there is a law of large numbers for the empirical density and the rescaled empirical velocity field moreover the limit fields satisfy the corresponding incompressible navier stokes equations with viscosity matrices characterized by a variational formula formally equivalent to the green kubo formula

aggregation blowup and collapse the abc s of taxis in reinforced random walks in many biological systems movement of an organism occurs in response to a diffusible or otherwise transported signal and in its simplest form this can be modeled by diffusion equations with advection terms of the form first derived by patlak bull of math biophys 15 1953 pp 311 338 however other systems are more accurately modeled by random walkers that deposit a nondiffusible signal that modifies the local environment for succeeding passages in these systems one example of which is the myxobacteria the question arises as to whether aggregation is possible under suitable hypotheses on the transition rules and the production of a control species that modulates the transition rates davis probab theory related fields 84 1990 pp 203 229 has studied this question for a certain class of random walks and here we extend this analysis to the continuum limit of such walks we first derive several general classes of partial differential equations that depend on how the movement rules are

equilibrium fluctuations for a system of harmonic oscillators with conservative noise we investigate the harmonic chain forced by a multiplicative noise the evolution is given by an infinite system of stochastic differential equations total energy and deformation are preserved the conservation of momentum is destroyed by the noise gaussian product measures are the extremal stationary states of this model equilibrium fluctuations of the conserved fields at a diffusive scaling are described by a couple of generalized ornstein uhlenbeck processes

a system of reaction diffusion equations arising in the theory of reinforced random walks we investigate the properties of solutions of a system of chemotaxis equations arising in the theory of reinforced random walks we show that under some circumstances finite time blow up of solutions is possible in other circumstances the solutions will decay to a spatially constant solution collapse we also give some intuitive arguments which demonstrate the possibility of the existence of aggregation piecewise constant solutions

approach to equilibrium of glauber dynamics in the one phase region i the attractive case various finite volume mixing conditions in classical statistical mechanics are reviewed and critically analyzed in particular somefinite size conditions are discussed together with their implications for the gibbs measures and for the approach to equilibrium of glauber dynamics inarbitrarily large volumes it is shown that dobrushin shlosman s theory ofcomplete analyticity and its dynamical counterpart due to stroock and zegarlinski cannot be applied in general to the whole one phase region since it requires mixing properties for regions ofarbitrary shape an alternative approach based on previous ideas of oliveri and picco is developed which allows to establish results on rapid approach to equilibrium deeply inside the one phase region in particular in the ferromagnetic case we considerably improve some previous results by holley and aizenman and holley our results are optimal in the sene that for example they show for the first time fast convergence of the dynamicsfor any temperature above the critical one for thed dimensional ising model with or without an external field in part ii we extensively consider the general case not necessarily attractive and we develop a new method based on renormalizations group ideas and on an assumption of strong mixing in a finite cube to prove hypercontractivity of the markov semigroup of the glauber dynamics

derivation of the leroux system as the hydrodynamic limit of a two component lattice gas the long time behavior of a couple of interacting asymmetric exclusion pro cesses of opposite velocities is investigated in one space dimension we do not allow two particles at the same site and a collision effect exchange takes place when particles of opposite velocities meet at neighboring sites there are two conserved quantities and the model admits hyperbolic euler scaling the hydrodynamic limit results in the classical leroux system of conservation laws even beyond the appearance of shocks actually we prove convergence to the set of entropy solutions the question of unique ness is left open to control rapid oscillations of lax entropies via logarithmic sobolev inequality estimates the symmetric part of the process is speeded up in a suitable way thus a slowly vanishing viscosity is obtained at the macroscopic level following 4 5 the stochastic version of tartar murat theory of compensated compactness is extended to two component stochastic models

a family of high order multistep methods with vanished phase lag and its derivatives for the numerical solution of the schrodinger equation many simulation algorithms chemical reaction systems differential systems arising from the modelling of transient behaviour in the process industries etc contain the numerical solution of systems of differential equations for the efficient solution of the above mentioned problems linear multistep methods or runge kutta single step methods are used for the simulation of chemical procedures the radial schrodinger equation is used frequently in the present paper we will study a class of linear multistep methods more specifically the purpose of this paper is to develop an efficient algorithm for the approximate solution of the radial schrodinger equation and related problems this algorithm belongs in the category of the multistep methods in order to produce an efficient multistep method the phase lag property and its derivatives are used hence the main result of this paper is the development of an efficient multistep method for the numerical solution of systems of ordinary differential equations with oscillating or periodical solutions the reason of their efficiency as the analysis proved is that the phase lag and its derivatives are eliminated another reason of the efficiency of the new obtained methods is that they have high algebraic order

microscopic position and structure of a shock in ca 184 we consider the time evolution of the cellular automaton ca 184 with random initial conditions we derive a partial differential equation that describes the macroscopic time evolution of the coarse grained local density and we study some solutions of this equation in particular shock solutions and rarefaction type solutions for initial step profiles in order to elucidate the emergence of the large scale hydrodynamic behaviour we define a microscopic position of a shock and find its mean velocity its diffusion coefficient and the microscopic structure of the shock as seen from this position

hydrodynamic limit of condensing two species zero range processes with sub critical initial profiles two species condensing zero range processes zrps are interacting particle systems with two species of particles and zero range interaction exhibiting phase separation outside a domain of sub critical densities we prove the hydrodynamic limit of nearest neighbour mean zero two species condensing zrp with bounded local jump rate for sub critical initial profiles i e for initial profiles whose image is contained in the region of sub critical densities the proof is based on h t yau s relative entropy method which relies on the existence of sufficiently regular solutions to the hydrodynamic equation in the particular case of the species blind zrp we prove that the solutions of the hydrodynamic equation exist globally in time and thus the hydrodynamic limit is valid for all times

high order closed newton cotes trigonometrically fitted formulae for the numerical solution of the schrodinger equation abstract in this paper we investigate the connection between closed newton cotes formulae trigonometrically fitted methods symplectic integrators and efficient integration of the schrodinger equation the study of multistep symplectic integrators is very poor although in the last decades several one step symplectic integrators have been produced based on symplectic geometry see the relevant literature and the references here in this paper we study the closed newton cotes formulae and we write them as symplectic multilayer structures based on the closed newton cotes formulae we also develop trigonometrically fitted symplectic methods an error analysis for the one dimensional schrodinger equation of the new developed methods and a comparison with previous developed methods is also given we apply the new symplectic schemes to the well known radial schrodinger equation in order to investigate the efficiency of the proposed method to these type of problems

entropy dissipation estimates in a zero range dynamics we study the exponential decay of relative entropy functionals for zero range processes on the complete graph for the standard model with rates increasing at infinity we prove entropy dissipation estimates uniformly over the number of particles and the number of vertices

spectral gap for zero range dynamics we give a lower bound on the spectral gap for symmetric zero range processes under some conditions on the rate function we show that the gap shrinks as n 2 independent of the density for the dynamics localized on a cube of size n d we follow the method outlined by lu and yau where a similar spectral gap is proved for kawasaki dynamics

spectral gap and logarithmic sobolev inequality for kawasaki and glauber dynamics we prove that the spectral gap of the kawasaki dynamics shrink at the rate of 1 l2 for cubes of sizel provided that some mixing conditions are satisfied we also prove that the logarithmic sobolev inequality for the glauber dynamics in standard cubes holds uniformly in the size of the cube if the dobrushin shlosman mixing condition holds for standard cubes

an extended numerov type method for the numerical solution of the schrodinger equation abstract a new method which is based on the new property of phase lag for computing eigenvalues of schrodinger equations with potentials is developed in this paper we investigate two cases 1 i the specific case in which the potential v x is an even function with respect to x it is assumed also that the wavefunctions tend to zero for x 2 ii the general case for positive and negative eigenvalues and for the well known cases of the morse potential and of the woods saxon or optical potential numerical and theoretical results show that this new method is more efficient compared with previously derived methods

mixing of the symmetric exclusion processes in terms of the corresponding single particle random walk we prove an upper bound for the e mixing time of the symmetric exclusion process on any graph g with any feasible number of particles our estimate is proportional to trw g ln v e where v is the number of vertices in g and trw g is the 1 4 mixing time of the corresponding single particle random walk this bound implies new results for symmetric exclusion on expanders percolation clusters the giant component of the erdos renyi random graph and poisson point processes in rd our technical tools include a variant of morris s chameleon process

the eigenfunction expansion of the solution for the nonhomogeneous sturm liouville problem containing white noise abstract this paper is concerned with the random problem containing white noise p t u t q t u t t u t f t t b t u 0 u 0 u 1 u 1 0 uniqueness and existence theorems are proved for solutions of this problem with different and the green s function and eigenfunction series are used to describe the solutions

a family of trigonometrically fitted partitioned runge kutta symplectic methods abstract we are presenting a family of trigonometrically fitted partitioned runge kutta symplectic methods of fourth order with six stages the solution of the one dimensional time independent schrodinger equation is considered by trigonometrically fitted symplectic integrators the schrodinger equation is first transformed into a hamiltonian canonical equation numerical results are obtained for the one dimensional harmonic oscillator and the exponential potential

the logarithmic sobolev constant of the lamplighter we give estimates on the logarithmic sobolev constant of some finite lamplighter graphs in terms of the spectral gap of the underlying base also we give examples of application

perturbation of singular equilibria of hyperbolic two component systems a universal hydrodynamic limit we consider one dimensional locally finite interacting particle systems with two conservation laws which under the eulerian hydrodynamic limit lead to two by two systems of conservation laws t x u 0 t u x u 0

onsager relations and eulerian hydrodynamic limit for systems with several conservation laws we present the derivation of the hydrodynamic limit under eulerian scaling for a general class of one dimensional interacting particle systems with two or more conservation laws following yau s relative entropy method it turns out that in case of more than one conservation laws in order that the system exhibit hydrodynamic behaviour some particular identities reminiscent of onsager s reciprocity relations must hold we check validity of these identities whenever a stationary measure with product structure exists it also follows that as a general rule the equilibrium thermodynamic entropy as function of the densities of the conserved variables is a globally convex lax entropy of the hyperbolic systems of conservation laws arising as hydrodynamic limit as concrete examples we also present a number of models modeling deposition or domain growth phenomena the onsager relations arising in the context of hydrodynamic limits under hyperbolic scaling seem to be novel the fact that equilibrium thermodynamic entropy is lax entropy for the arising euler equations was noticed earlier in the context of hamiltonian systems with weak noise see ref 7

a new method for the numerical solution of fourth order bvp s with oscillating solutions abstract a new finite difference scheme with minimal phase lag for the numerical solution of fourth order differential equations is developed in this paper numerical and theoretical results show that this new approach is more efficient compared with previously derived methods

nonlinear diffusion limit for a system with nearest neighbor interactions we consider a system of interacting diffusions the variables are to be thought of as charges at sites indexed by a periodic one dimensional lattice the diffusion preserves the total charge and the interaction is of nearest neighbor type with the appropriate scaling of lattice spacing and time a nonlinear diffusion equation is derived for the time evolution of the macroscopic charge density

logarithmic sobolev inequality for generalized simple exclusion processes let be a probability measure on the set 0 1 r for some r and l a cube of width l in d denote by gc l the grand canonical product measure on the configuration space on l with as the marginal measure here the superscript indicates the grand canonical ensemble the canonical ensemble denoted by c l n is defined by conditioning gc l given the total number of particles to be n consider the exclusion dynamics where each particle performs random walk with rates depending only on the number of particles at the same site the rates are chosen such that for every n and l fixed the measure c l n is reversible we prove the logarithmic sobolev inequality in the sense that flogfd c l n for any probability density f with respect to c l n here the constant is independent of n or l and d denotes the dirichlet form of the dynamics the dependence on l is optimal

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

runge kutta interpolants with minimal phase lag abstract we develop the basic theory for the construction of explicit scaled runge kutta methods srk for the integration of first order differential equations having an oscillatory solution based on this theory and on the theory of runge kutta methods with minimal phase lag we construct an embedded runge kutta 10 12 method and an explicit scaled runge kutta method numerical results indicate the efficiency of the new algorithms

a higher accuracy finite difference method for an elliptic equation of order four the elliptic equation 1 with boundary value conditions either 2 or 3 is solved by an o h4 convergent finite difference method combined with an implicit iterative method the method is tested by examples of boundary value problems with use of the enclosed mathematica module solve bhpeq the module solve bhpeq gives options either it implements the o h4 convergent nine mesh points scheme or the o h2 convergent standard five points scheme when the finite difference approximation of eq 1 is considered

the density function of the solution of a random initial value problem containing small stochastic processes this paper concerns the random initial value problem consisting of the differential equation bf x bf f t bf x boldsymbol xi t omega with the initial condition bf x t 0 boldsymbol alpha omega when boldsymbol xi t omega boldsymbol xi 0 t varepsilon boldsymbol xi 1 t omega varepsilon 2 boldsymbol xi 2 t omega cdots then the density function has the form p bf x t p 0 x t varepsilon p 1 x t varepsilon 2 p 2 bf x t cdots perturbation and decomposition methods lead to first order partial differential equations for p 0 p 1 and p 2 by means of an auxiliary random initial value problem the coefficients in the partial differential equations can be evaluated in terms of certain deterministic procedures

the distribution function of the solution of the random eigenvalue problem for differential equation abstract by means of the prufer substitution and the liouville equation this paper gives an efficient way to evaluate the distribution functions of the solutions of the random eigenvalue problem

processus des misanthropes we consider a system of identical interacting particles moving on the lattice d the rate at which a particle at the site x jumps to the site y is p y x b x y where p is an irreducible probability on d and b x y is an increasing resp decreasing function of the number x resp y of particles at site x resp y we study the convergence of the system to equilibrium and describe the invariant measures

between equilibrium fluctuations and eulerian scaling perturbation of equilibrium for a class of deposition models we investigate propagation of perturbations of equilibrium states for a wide class of 1d interacting particle systems the class of systems considered incorporates zero range k exclusion misanthropic bricklayers models and much more we do not assume attractivity of the interactions we apply yau s relative entropy method rather than coupling arguments the result is partial extension of t seppalainen s recent paper for 0 1 5 fixed we prove that rescaling microscopic space and time by n respectively n1 the macroscopic evolution of perturbations of microscopic order n of the equilibrium states is governed by burgers equation the same statement should hold for 0 1 2 as in seppalainen s cited paper but our method does not seem to work for 1 5

poincare and logarithmic sobolev inequality for ginzburg landau processes in random environment we consider reversible conservative ginzburg landau processes in a random environment whose potential are bounded perturbations of the gaussian potential evolving on a d dimensional cube of length l we prove in all dimensions that the spectral gap of the generator and the logarithmic sobolev constant are of order l 2 almost surely with respect to the environment