a limiting result for the structure of collisions between many independent diffusions we study a system of a large number n of independent diffusions undergoing collisions in a roughly constant mean free path regime we derive a propagation of chaos result when n goes to infinity concerning the limit behavior of the trajectory of a particle its first time of collision with all the other particles and the trajectory of the colliding particle

some generalizations of the trapping problem abstract we consider two effects on kinetics that have not generally been taken into account in analyses of classical trapping models for the reactions a t t and a t 0 the effects due to correlations in the initial positions of the a s and those due to the state of mobility of each of the species our analysis is formulated in terms of three dimensional brownian motion we give a heuristic treatment of the short time regime based on statistical properties of the wiener sausage the effects of initially correlated positions are modelled in terms of a set of a multiplicity of a particles located at the origin

a filtration of the modular representation functor abstract let f and k be algebraically closed fields of characteristics p 0 and 0 respectively for any finite group g we denote by k r f g k z g 0 f g the modular representation algebra of g over k where g 0 f g is the grothendieck group of finitely generated f g modules with respect to exact sequences the usual operations induction inflation restriction and transport of structure with a group isomorphism between the finitely generated modules of group algebras over f induce maps between modular representation algebras making k r f an inflation functor we show that the composition factors of k r f are precisely the simple inflation functors s c v i where c ranges over all nonisomorphic cyclic p groups and v ranges over all nonisomorphic simple k out c modules moreover each composition factor has multiplicity 1 we also give a filtration of k r f

evanescent continuous time random walks we study how an evanescence process affects the number of distinct sites visited by a continuous time random walker in one dimension we distinguish two very different cases namely when evanescence can only occur concurrently with a jump and when evanescence can occur at any time the first is characteristic of trapping processes on a lattice whereas the second is associated with spontaneous death processes such as radioactive decay in both of these situations we consider three different forms of the waiting time distribution between jumps namely exponential long tailed and ultraslow

effects of an absorbing boundary on the average volume visited by n spherical brownian particles abstract the number of distinct sites visited by a lattice random walk and its continuum analog the volume swept out by a diffusing spherical particle are used to model different phenomena in physics chemistry and biology therefore the problem of finding statistical properties of these random variables is of importance there have been several studies of the more general problem of the volume of the region explored by n random walks or brownian particles in an unbounded space we here study the effects of a planar absorbing boundary on the average of this volume the boundary breaks the translational invariance of the space and introduces an additional spatial parameter the initial distance of the brownian particles from the surface we derive expressions for the average volume visited in three dimensions and the average span in one dimension as functions of the time for given values of the initial distance to the absorbing boundary and n the results can be transformed to those for n lattice random walks by appropriately choosing the radius and diffusion constant of the spheres

a large deviation result for the range of random walk and for the wiener sausage let s n be a random walk on d and let r n be the number of different points among 0 s1 s n 1 we prove here that if d 2 then x lim n 1 n logp r n nx exists for x 0 and establish some convexity and monotonicity properties of x the one dimensional case will be treated in a separate paper

brownian motion in a poissonian potential we study here ad dimensional brownian motion in a random potentialv obtained as the sum of translations of a given fixed non negative shape function at the points of a poisson cloud of constant intensityv we are interested in the larget behavior for typical cloud configurations of the brownian path in timet under the influence of the natural feynman kac weight associated tov in particular we show that the location at timet of the process tends to be concentrated near points of suitably low local eigenvalue of 1 2 v which lie almost at distancet from the origin near these points one can find in the cloud a big hole or clearing of size const logt 1 d with volume like a ball of radiusr 0 d v logt 1 d

the functor of rational representations for p groups abstract let k be a field of characteristic p 0 let c p k be the category whose objects are the finite p groups morphisms are the k linear combinations of bisets and composition of morphisms is obtained by k linear extension from the usual product of bisets let f p k denote the category of k linear functors from c p k to the category of k vector spaces this paper investigates the structure of the functor kr q mapping a p group p to kr q p k z r q p as an object of f p k where r q p is the grothendieck group of the category of finite dimensional q p modules the main result is an explicit description of the lattice of all subfunctors of kr q in particular for p odd it is shown that kr q is a uniserial object in f p k for p 2 the lattice of subfunctors of kr q can be described as the lattice of closed subsets of a graph whose vertices are the 2 groups of normal 2 rank 1 in both cases a composition series of kr q is obtained which leads to a formula giving the dimension of the evaluations sq k p of the simple objects sq k of f p k associated to p groups q of normal p rank 1 different from cp this formula can be phrased in terms of rational representations of p but also in terms of the geometry of the lattice of subgroups of p using the notions of basic subgroups and origins for example if p 2 the dimension of s 1 k p is equal to the number of absolutely irreducible q p modules

composition factors of functors abstract let x be a family of finite groups satisfying certain conditions and k be a field we study composition factors radicals and socles of biset and related functors defined on x over k for such a functor m and for a group h in x we construct bijections between some classes of maximal respectively simple subfunctors of m and some classes of maximal respectively simple k out h submodules of m h we use these bijections to relate the multiplicity of a simple functor s h v in m to the multiplicity of v in a certain k out h module related to m h we then use these general results to study the structure of one of the important biset and related functors namely the burnside functor b k which assigns to each group g in x its burnside algebra b k g k z b g where b g is the burnside ring of g we find the radical and the socle of b k in most cases of x and k for example if k is of characteristic p 0 and x is a family of finite abelian p groups we find the radical and the socle series of b k considered as a biset functor on x over k we finally study restrictions of functors to nonfull subcategories for example we find some conditions forcing a simple deflation functor to remain simple as a mackey functor for an inflation functor m defined on abelian groups over a field of characteristic zero we also obtain a criterion for m to be semisimple in terms of the images of inflation and induction maps on m

foncteurs d ensembles munis d une double action resume sommaire 1 introduction 2 notations generalites 3 ensembles munis d une double action 3 1 notations 3 2 formule de mackey 3 3 decomposition d un morphisme 3 4 sous categories 3 4 1 ensembles p q libres 3 4 2 groupes reductifs 4 foncteurs simples 5 une theorie de vortex et de source 6 une seconde theorie de vortex et source 7 le foncteur de burnside 7 1 un homomorphisme deb g dans end g 7 2 les sous foncteurs du foncteur de burnside en caracteristique 0 7 2 1 premier exemple 7 2 2 les constantesmg n 7 2 3 lesb groupes 7 2 4 les sous foncteurs deb 7 2 5 exemples deb groupes 8 exemples de foncteurs simples 8 1 le cas des foncteurs de mackey globaux 8 2 les foncteurssh k 8 3 certains foncteurs simples en caracteristique 0 9 l algebreegg end g egg 9 1 une base naturelle 9 2 multiplication 9 3 semi simplicite 10 le foncteur des representations complexes 10 1 caracteres 10 2 sous foncteurs et decomposition 10 3 semi simplicite

moderate deviations for the volume of the wiener sausage for a 0 let wa t be the a neighbourhood of standard brownian motion in rd starting at 0 and observed until time t it is well known that elwa t l ikat t oc for d 3 with kea the newtonian capacity of the ball with radius a we prove that lm t d 2 d log p wa t l oc this is markedly different from the optimal strategy for large deviations iwa t i 5 is nonanalytic at some critical point in 0 kia above which it follows a pure power law this crossover is associated with a collapse transition in the optimal strategy we also derive the analogous moderate deviation result for d 2 in this case eiwa t 2wrt logt t oc and we prove that

volume of the domain visited by n spherical brownian particles the average value and variance of the volume of the domain visited in timet byn spherical brownian particles starting initially at the same point are presented as quadratures of the solutions of simple diffusion problems of the survival of a point brownian particle in the presence of one and two spherical traps as an illustration explicit time dependences are obtained for the average volume in one and three dimensions

asymptotics of the generating function for the volume of the wiener sausage we consider the generating function mathbb e exp lambda c varepsilon t of the voltime of the wiener sausagec t which is the e neighbourhood of the wiener path in the time interval 0 t for 0 it had been investigated by van den berg and toth but in contrast to the case 0 there is no simple expression for the exponential rate known we determine the asymptotic behaviour of this rate for small and large

expected number of distinct sites visited by n random walks in the presence of an absorbing boundary in earlier work we have studied the expected number of distinct sites ends visited by n random walkers in time t on a translationally invariant lattice optical applications suggest the interest in analysing the same problem for a semi infinite lattice in three dimensions bounded by a plane of absorbing sites we here study this problem showing a multiplicity of time regimes and at the longest times showing that the ends is proportional to n t where t is the time in the absence of a boundary the comparable result is proportional to nt thus the boundary effect eliminates approximately t random walks

hydrodynamic friction and the capacitance of arbitrarily shaped objects the translational friction coefficient and the capacitance of a variety of objects are calculated with a probabilistic method involving hitting the probed objects with random walks launched from an enclosing spherical surface this method is applied to exactly solvable examples to test the program accuracy and to physically important and analytically intractable examples cube chain of spheres at the vertices of self avoiding and random walks etc large fluctuations in the friction of polymer chains with a random coil structure are found to give large deviations from the mean field kirkwood riseman theory and hydrodynamic fluctuation effects are found to diminish with the chain swelling accompanying excluded volume interaction capacity applications are reviewed and our probabilistic estimates of polymer friction are compared with previous calculations using alternative methods transients to the capacity and related properties are expressed in terms of fluctuations in the wiener sausage volume volume swept out by a brownian particle where a repeated visit to a spatial region does not contribute to the volume increase in time

brownian asymptotics in a poissonian environment we consider a brownian motion moving in a random potential obtained by translating a given fixed non negative shape function at the points of a poisson cloud we derive the almost sure principal long time behavior of the expectation of the natural feynman kac functional which is insensitive to the detail of the shape function we also study the situation of hard obstacles where brownian motion is killed once it comes within distancea of a point of the cloud the nature of the results then changes between the case whena is small or large in connection with the presence or absence of an infinite component in the complement of the obstacles

wiener sausage volume moments the statistical characteristics of a spatial region visited by a spherical brownian particle during timet wiener sausage are investigated the expectation value and dispersion of this quantity are obtained for a space of arbitrary dimension in the one dimensional case the distribution of probability density and the moments of any order are determined for this quantity

solution of the 1d schrodinger equation in semiconductor heterostructures using the immersed interface method due to the enormous progress in computer technology and numerical methods that has been achieved over the past several decades the use of numerical simulation methods in all scientific disciplines gain more and more importance in the physics field these methods have provided remarkable numerical solutions to problems considered analytically intractable the solution of the schrodinger equation in semiconductor heterostructures is a good example however many of these numerical schemes are cumbersome to implement for nonexperts in numerical computing with this reason as motivation a novel method simple enough to implement yet powerful enough to solve schrodinger equation in semiconductor devices with high accuracy is presented herein

variance of the range of a random walk tn the expectation of the square of the number of distinct sites occupied by a random walk in steps 1 throughn is obtained from its relation to the dual first occupancy probabilityfij x x and the latter quantity is obtained from a recursion with the first occupancy probabilityf k x the variancevn of the number of distinct sites occupied is calculated directly from tn the procedure is illustrated by the calculation ofvn 4096 n and the derivation of asymptotic expansions forvn for a particular random walk in dimensions 1 through 3

high order eigenstate calculation of arbitrary quantum structures quantum engineering of electronic energy states using nanoscale layers of semiconductor compounds allows the design and the observation of quantum phenomena which are typically observed in atomic structures furthermore semiconductors are present in nearly all modern electronic devices and are a crucial component of integrated circuits due to the relatively high rate of manufacturing defects it is crucial to have a method for testing new semiconductor formations without requiring a sample to be fabricated a simple fast and very accurate numerical technique is presented to calculate the eigenstates of such arbitrary quantum structures the method is based on a high order finite difference scheme which allows the use of sparse matrix algebra thus significantly reducing computational time and allowing for high precision results even for the high energy states

the immersed interface method for elliptic equations with discontinuous coefficients and singular sources the authors develop finite difference methods for elliptic equations of the form nabla cdot beta x nabla u x kappa x u x f x in a region omega in one or two space dimension

analysis on lie groups theorie locale diffusion de la chaleur sur des groupes de lie connexes la theorie globale les inegalites de sobolev bases graduees sur des corps lisses construction des bases graduees estimation locale du noyau diffusion de la chaleur demonstration de l inegalite de sobolev locale formes de dirichlet sur des groupes le semi groupe de convolution les conditions de sobolev modifiees

universal fluctuations in the support of the random walk a random walk starts from the origin of ad dimensional lattice the occupation numbern x t equals unity if aftert steps site x has been visited by the walk and zero otherwise we study translationally invariant sumsm t of observables defined locally on the field of occupation numbers examples are the numbers t of visited sites the areae t of the appropriately defined surface of the set of visited sites and in dimension d 3 the euler index of this surface ind 3 theaverages m t all increase linearly witht ast we show that in d 3 to leading order in an asymptotic expansion int thedeviations from average m t m t m t are up to a normalization allidentical to a single universal random variable this result resembles an earlier one in dimensiond 2 we show that this universality breaks down ford 3

on a random process interpolating between markovian and non markovian random walks the authors give the exact solution in one dimension of the continuum analogue of a model of stanley et al 1983 which interpolates between markovian polya and self avoiding random walks on lattices and make some heuristic comments on the nature of phase transitions in the model in higher dimensions

lifschitz tail on hyperbolic space neumann conditions on considere un espace hyperbolique h de dimension n et l operateur laplacien hyperbolique h etant donne une distribution aleatoire de points a l aide d un processus ponctuel de poisson tel que chaque point se situe dans une boule hyperbolique de rayons on etudie la question suivante si est un petit nombre reel combien y a t il de valeurs propres inferieures a pour l operateur 1 2 h dans une boule b de centre o avec des conditions de neumann sur la frontiere et avec des conditions aux limites de dirichlet sur les boules aleatoires formant des intersections avec b si le rayon de b est suffisamment grand

statistical properties of the set of sites visited by the two dimensional random walk we study the support i e the set of visited sites of a t step random walk on a two dimensional square lattice in the large t limit a broad class of global properties m t of the support is considered including for example the number s t of its sites the length of its boundary the number of islands of unvisited sites that it encloses the number of such islands of given shape size and orientation and the number of occurrences in space of specific local patterns of visited and unvisited sites on a finite lattice we determine the scaling functions that describe the averages on appropriate lattice size dependent time scales on an infinite lattice we first observe that the all increase with t as where k is an m dependent positive integer we then consider the class of random processes constituted by the fluctuations around average we show that to leading order as t gets large these fluctuations are all proportional to a single universal random process normalized to for the probability law of tends to that of varadhan s renormalized local time of self intersections an implication is that in the long time limit all are proportional to

on the volume of the intersection of two wiener sausages for a 0 let w a 1 t and w a 2 t be the a neighbourhoods of two independent standard brownian motions in r d starting at 0 and observed until time t we prove that for d 3 and c 0 lim t 1 1 t d 2 d logp jw a 1 ct w a 2 ct j t i a d c and derive a variational representation for the rate constant i a d c here a is the newtonian capacity of the ball with radius a we show that the optimal strategy to realise the above large deviation is for w a 1 ct and w a 2 ct to form a swiss cheese the two wiener sausages cover part of the space leaving random holes whose sizes are of order 1 and whose density varies on scale t 1 d according to a certain optimal prole we study in detail the function c 7 i a d c it turns out that i a d c

sur une conjecture de m kac we consider the following heat conduction problem let k be a compact set in euclidean space 3 suppose that k is held at the temperature 1 while the surrounding medium is at the temperature 0 at time 0 following spitzer we investigate the asymptotic behaviour of the integral e k t which represents the total energy flow in time t from the set k to the surrounding medium 3 k an asymptotic expansion is given for e k t which refines a theorem due to spitzer this expansion also verifies and improves a formal calculation of kac similar results are proved in higher dimensions up to the constant m k the quantity e k t can be interpreted as the expected value of the volume of the wiener sausage associated with k and a d dimensional brownian motion this point of view both plays a major role in the proofs and leads to a probabilistic interpretation of the different terms of the expansion

trapping reactions with randomly moving traps exact asymptotic results for compact exploration trapping a1b b and recombination a1b 0 reactions tr and rr involving randomly moving a and b particles which react when they meet at a certain distance b are ubiquitous in nature a few stray examples include quenching of delocalized excitations coagulation recombination of radicals charge carriers or defects or biological processes related to population survival 1 in recent years there has been much interest in the longtime behavior of these processes following a remarkable discovery 2 10 of many particle effects which induce essential departures from the conventionally expected behavior 1 a pronounced deviation from the text book predictions was found for the diffusion controlled rr in case when initially the particles of the a and b species are all distributed at random with strictly equal mean densities n0 it has been first shown 2 and subsequently proven 3 4 that as t the mean density n t follows n t a d n 0 dt 2d 4 where d is the space dimensionality a d is a constant and d5d a 1db is the sum of particles diffusion coefficients this law contradicts the decay law obtained within the smoluchowski approach sa n t 1 f bd t 1 where as t

a new proof of moser s parabolic harnack inequality using the old ideas of nash in 1958 nash published his fundamental work on the local holder continuity of solutions of second order parabolic equations with coefficients which need not be smooth 8 the primary purpose of that work was to study the properties of the fundamental solution corresponding to the parabolic operator and from these properties to derive regularity for a general solution though the work is often cited in the literature about weak solutions of elliptic and parabolic equations one feels that nash s ideas were never fully understood and maybe still are not and that because of this the more understandable and seemingly more fruitful ideas of de giorgi 4 and moser 6 7 were subsequently adopted

exponential estimates for the wiener sausage letc e t be the wiener sausage of radius e inr d up to timet we obtain bounds on the asymptotics ofe exp c e t ast for all 0

on the span of brownian motion in a field in one dimension abstract we study the effect of a field on the span of a particle diffusing on a line i e the length covered by a brownian particle which moves on a line for time t in the presence of a constant field this is the one dimensional analog of the wiener sausage volume exact expressions are found for the probability density for the span together with the first two moments our results indicate that at very short times the dominant effect is diffusion while at very long times the field plays the dominant role

socles and radicals of mackey functors we study the socle and the radical of a mackey functor m for a finite group g over a field k usually of characteristic p 0 for a subgroup h of g we construct bijections between some classes of the simple subfunctors of m and some classes of the simple kn g h submodules of m h we relate the multiplicity of a simple mackey functor sh vg in the socle of m to the multiplicity of v in the socle of a certain kn g h submodule of m h we also obtain similar results for the maximal subfunctors of m we then apply these general results to some special mackey functors for g including the functors obtained by inducing or restricting a simple mackey functor mackey functors for a p group the fixed point functor and the burnside functor bkg for instance we find the first four top factors of the radical series of bkg for a p group g and assuming further that g is an abelian p group we find the radical series of bkg

on the density of states of schrodinger operators with a random potential using very recent results on ergodic theorems for superadditive processes on rd the authors prove the existence of the density of states for a wide class of random schrodinger operators in particular new non asymptotic estimates on the density of states are obtained and examples are discussed

lifschitz tail in a magnetic field coexistence of classical and quantum behavior in the borderline case we establish the exact low energy asymptotics of the integrated density of states lifschitz tail in a homogeneous magnetic field and poissonian impurities with a repulsive single site potential of gaussian decay it has been known that the gaussian potential tail discriminates between the so called classical and quantum regimes and precise asymptotics are known in these cases for the borderline case the coexistence of the classical and quantum regimes was conjectured here we settle this last remaining open case to complete the full picture of the magnetic lifschitz tails

the socle of a projective mackey functor for a p group we study the simple subfunctors of indecomposable projective mackey functors for a p group p unlike the case of group algebras the mackey algebra is not in general self injective thus the socle of an indecomposable projective functor is not in general simple we first show that the simple subfunctors of a projective functor p h k where h p are indexed by the normalizer in h of a subgroup of h we then study the socle of a specific projective mackey functor namely the burnside functor b p and we focus on the case where p is abelian in particular our study enables us to determine the socle of an indecomposable projective mackey functor indexed by a cyclic p group an abelian p group of rank 2 and an elementary abelian p group of rank 3

lifschitz tail and wiener sausage ii the present paper is a continuation of 23 we present here two examples of situations where the method developed in 23 applies the first example is that of a periodic diffusion z in r n 1 with generator l isia dj where a is a c2 z periodic elliptic matrix we consider the wiener sausage w u0 c u i z c where c is a nonpolar compact subset of r for instance a closed ball or disc in r if a denotes the constant matrix of homogenized coefficients of the periodic diffusion see 1 i we show that

rhetorical biset functors rational p biset functors and their semisimplicity in characteristic zero rhetorical biset functors can be defined for any family of finite groups that is closed under subquotients up to isomorphism the rhetorical p biset functors almost coincide with the rational p biset functors we show that over a field with characteristic zero the rhetorical biset functors are semisimple and furthermore they admit a character theory involving primitive characters of automorphism groups of cyclic groups

the group of endo permutation modules the group d p of all endo permutation modules for a finite p group p is a finitely generated abelian group we prove that its torsion free rank is equal to the number of conjugacy classes of non cyclic subgroups of p we also obtain partial results on its torsion subgroup we determine next the structure of q otimes d viewed as a functor which turns out to be a simple functor s e q indexed by the elementary group e of order p 2 and the trivial out e module q finally we describe a rather strange exact sequence relating q otimes d p q otimes b p and q otimes r p where b p is the burnside ring and r p is the grothendieck ring of qp modules

a field theoretic approach to the wiener sausage the wiener sausage the volume traced out by a sphere attached to a brownian particle is a classical problem in statistics and mathematical physics initially motivated by a range of field theoretic technical questions we present a single loop renormalised perturbation theory of a stochastic process closely related to the wiener sausage which however proves to be exact for the exponents and some amplitudes the field theoretic approach is particularly elegant and very enjoyable to see at work on such a classic problem while we recover a number of known classical results the field theoretic techniques deployed provide a particularly versatile framework which allows easy calculation with different boundary conditions even of higher momenta and more complicated correlation functions at the same time we provide a highly instructive non trivial example for some of the technical particularities of the field theoretic description of stochastic processes such as excluded volume lack of translational invariance and immobile particles the aim of the present work is not to improve upon the well established results for the wiener sausage but to provide a field theoretic approach to it in order to gain a better understanding of the field theoretic obstacles to overcome