some examples of kinetic schemes whose diffusion limit is il in s exponential fitting this paper is concerned with diffusive approximations of some numerical schemes for several linear or weakly nonlinear kinetic models which are motivated by wide range applications including radiative transfer or neutron transport run and tumble models of chemotaxis dynamics and vlasov fokker planck plasma modeling the well balanced method applied to such kinetic equations leads to time marching schemes involving a scattering s matrix itself derived from a normal modes decomposition of the stationary solution one common feature these models share is the type of diffusive approximation their macroscopic densities solve drift diffusion systems for which a distinguished numerical scheme is il in scharfetter gummel s exponential fitting discretization we prove that these well balanced schemes relax within a parabolic rescaling towards such type of discretization by means of an appropriate decomposition of the s matrix hence are asymptotic preserving

the basic principles of irreversible thermodynamics in the context of an informational statistical approach abstract we consider on the light of information statistical thermodynamics the basic principles of phenomenological thermodynamic theories namely the principles of equipresence objectivity and memory as well as the characteristics of entropy production informational statistical thermodynamics arises as a particular application of maximum information entropy inference we discuss how in this perspective together with the property of contraction of description based on bogoliubov s hierarchy of relaxation times it is possible to account for the above mentioned principles particular analyses are carried out on the basis of a model system moreover selecting the basic set of thermodynamical variables in a closed and in truncated approaches we assess the validity of the approximations in particular we derive the limiting conditions under which one recovers the constitutive equations of linear irreversible thermodynamics the associated equations of evolution for the basic macrovariables and a particular h theorem are described and discussed

on l1 functions with a very singular behaviour in this article we construct special sequences of l1 functions with a very singular behaviour establish some of their limit properties and give certain applications of the results obtained

bootstrapping the student t statistic let x 1 x n n 1 be independent identically distributed random variables and consider the student t statistic t n based upon these random variables gine gotze and mason 1997 proved that t n converges in distribution to a standard normal random variable if and only if x is in the domain of attraction of a normal random variable and ex 0 we shall show that roughly the same holds true for the bootstrapped student t statistic t n in the process we shall disclose all the possible subsequential limiting laws of t n the proofs introduce a number of amusing tricks that may be of independent interest

the analytic solutions of some boundary layer problems in the theory of brownian motion the authors give analytic solutions of the generalised albedo and milne problems for the uhlenbeck ornstein process and discuss the relation between these and the original wang uhlenbeck 1945 problem of determining the distribution of first passage times

colored noise and a characteristic level crossing problem in this paper we discuss the activation of a dynamical system of one spatial variable forced by an external gaussian stationary colored noise the model is an analog of the smoluchowski equation when external fluctuations are present we assume that the underlying deterministic dynamics are derived from a potential forming a well we enlarge the state space by including the noise variable and then use half range expansion techniques singular perturbations and matched asymptotics to solve the boundary value problem for the steady state probability density function in the asymptotic limit of small correlation time and over the entire regime of noise strength we find a uniformly valid expression for the activation rate to the top of the potential barrier a characteristic point for the unforced dynamical system we show the important effects of the boundary behavior of the process on its activation rate if the intensity of the noise is small compared to the autocorrelation of the noise then the effective activation energy is modified by the noise autocorrelation we consider an example of the activation process in a double well potential and compare our uniformly valid results with results reported in the literature

stratified structure of the universe and burgers equation a probabilistic approach the model of the potential turbulence described by the 3 dimensional burgers equation with random initial data was developped by zeldovich and shandarin in order to explain the existing large scale structure of the universe most of the recent probabilistic investigations of large time asymptotics of the solution deal with the central limit type results the gaussian scenario under suitable moment assumptions on the initial velocity field these results and some open questions are discussed in sect 2 where we concentrate on the gaussian model and the shot noise model in sect 3 we construct a probabilistic model of strong initial fluctuations a zero range shot noise field with high amplitudes which reveals an intermittent large time behaviour with the velocity vec v t x determined by the position of the largest initial fluctuation discounted by the heat kernelg t x in a neighborhood ofx the asymptoties of such local maximum ast can be analyzed with the help of the theory of records sect 4 finally in sect 5 we introduce a global definition of a point process oft local maxima and show the weak convergence of the suitably rescaled process to a non trivial limit ast

characterization of lil behavior in banach space in a recent paper by the authors a general result characterizing two sided lil behavior for real valued random variables has been established in this paper we look at the corresponding problem in the banach space setting we show that there are analogous results in this more general setting in particularly we provide a necessary and sufficient condition for lil behavior with respect to sequences of the form p nh n where h is from a suitable subclass of the positive nondecreasing slowly varying functions to prove these results we have to use a different method one of our main tools is an improved fuk nagaev type inequality in banach space which should be of independent interest

the density profile for the klein kramers equation near an absorbing wall we derive asymptotic series for the expansion coefficients of a function in terms of the pagani functions which occur in the boundary layer solutions of the klein kramers equation the results enable us to determine the density profile in the stationary solution of this equation near an absorbing wall from the numerically determined velocity distribution at the wall with an accuracy of about 2 we also obtain information about the analytic behavior of the density profile this profile increases near the wall with the square root of the distance to the wall finally the asymptotic analysis leads to an understanding of the slow convergence of variational approximations to the solution of the absorbing wall problem and of the exponents that occur when one studies the variational approximations to various quantities of interest as functions of the number of terms in the variational ansatz this is used to obtain a better variational estimate for the density at the wall

semicontinuity problems in the calculus of variations soit f r 3 r une fonction deux fois continument differentiable et pour u w 1 1 a b f u a b a b f x u x u x dx si f r n r m r nm r satisfait 1 f est mesurable en x continue en cs et 2 0 f x s a x c s p p ou p 1 a est une fonction localement sommable non negative et c une constante non negative alors la fonctionnelle f u a b est sequentiellement faiblement semi continue inferieurement sur w 1 p r m si et seulement si f x s est quasi convexe pour chaque s et presque chaque x

the analytic structure of the stationary kinetic boundary layer for brownian particles near an absorbing wall a method for solving kinetic boundary layer problems for the 1d klein kramers equation the kinetic equation for brownian particles was proposed a few years ago by marshall and watson 1987 for the special case of particles moving under the influence of a constant or zero external force in this paper the authors apply this method to a number of classical stationary boundary layer problems for completely or partially absorbing walls these are the milne problem in which a current flows towards the wall from infinity and the albedo problem in which particles are injected into the system at the wall with a prescribed velocity distribution the solutions are known to be non analytic at the wall for zero normal velocity the authors pay particular attention to the nature of this singularity for several special cases the results are compared with results obtained by approximate methods

bootstrapping empirical functions in our previous sections we have established weak convergence of mean residual life total time of test lorenz and goldie processes to appropriate gaussian processes apart from a few special cases cf section 8 these limiting processes are functions of the underlying distributions consequently when testing for statistical hypotheses for example one would have to compute the resulting limiting distribution for each f of interest the same is true when trying to use our results for constructing confidence bands for the theoretical functionals of the said empirical processes this type of problems can be solved by adapting the bootstrap method proposed by efron 1979 1982 to the present situation as we will see the bootstrap method is simply a monte carlo simulation determined by the given observations

solution to the linear bgk equation via the principles of invariance the solution to the linearized bgk equation of kinetic theory is obtained via the principles of invariance as formulated by chandrasekhar the solution represents an alternative to the singular eigenfunction method featuring regular integral equations which are more easily evaluated laplace transforms are obtained for the net flow velocity and velocity distribution for both the albedo and milne problems with wall reflection the transforms are then numerically inverted to a high degree of accuracy results are compared to those of kainz and titulaer

non central limit theorems for random selections selection from finite sets is a basic procedure of statistics and the partial sum behavior of selected elements is completely known under the uniform asymptotic negligibility condition of central limit theory the purpose of the present paper is to determine the asymptotic behavior of partial sums when the central limit theorem fails as an application we describe the limiting properties of permutation and bootstrap statistics in case of infinite variance

indefinite sturm liouville problems and half range completeness where the function h changes sign on i such problems occur in certain physical models particularly in transport theory and statistical physics see 2 9 11 131 and references therein as for the standard eigenvalue problem it is true under quite general conditions and not difficult to prove that there is a set of eigenfunctions un which is complete in an appropriate hilbert space of functions this is known as full range completeness for the physical problems the question of interest is generally that of half range completeness to formulate this we set

the growth of small droplets from a gas mixture results and estimates for very small radii abstract to test and to complement the moment method for solving kinetic equations in the space outside of a sphere that absorbs one component out of a gas mixture we develop a solution method especially suitable for radii smaller or not much larger than a mean free path the method is used to calculate the stationary current towards an absorbing sphere suspended in a gas mixture that is in equilibrium far from the sphere for the simplest kinetic equation a variant of the linear bgk equation and for radii below two mean free paths this procedure gives the current towards the sphere to within an accuracy of at most a few tenths of a percent for radii around two mean free paths the new results agree with those of a variant of the moment method for lower radii discrepancies of up to a few percent are found our new method does not work for the klein kramers equation a kinetic equation for brownian particles for the latter equation we obtain an estimate for the current at very low radii from the free space solution by uhlenbeck and ornstein its numerical evaluation strongly indicates an analytic behavior for small radii different from the one found in the bgk case

the kinetic boundary layer for the linearized boltzmann equation around an absorbing sphere we construct approximate solutions of the linearized boltzmann equation for a gas outside of a completely absorbing sphere a simple model for a liquid droplet growing in a supersaturated vapor the solutions are linear combinations of two chapman enskog type solutions which carry heat and particle currents and boundary layer eigenfunctions that decay with increasing distance from the sphere on a distance of the order of a mean free path to construct the boundary layer eigenfunctions and the linear combination that satisfies the boundary condition at the sphere we expand the solution in burnett functions and truncate the resulting system of equations for the expansion coefficients for one particular truncation prescription which generalizes grad s 13 moment scheme good initial convergence with increasing order of truncation is obtained for both moderately small and large radii of the sphere the results for small radii extrapolate smoothly toward the known limit of zero radius we present results for the reaction rate the particle current arriving at the sphere divided by the density at infinity and for the density and temperature profiles in the boundary layer the explicit calculations are carried out for maxwell molecules but the method appears to be suitable for more general intermolecular potentials

half range solutions of indefinite sturm liouville problems abstract for diffusion equations of indefinite sturm liouville type we develop two equivalent methods of constructing explicit representations for the solutions the first method is based on an eigenfunction expansion whereas the second uses a wiener hopf type integral equation and factorization some illustrative examples are worked out

the influence of temperature gradients on the kinetic boundary layer problem for a condensing droplet we extend an earlier method for solving kinetic boundary layer problems to the case of particles moving in aspatially inhomogeneous background the method is developed for a gas mixture containing a supersaturated vapor and a light carrier gas from which a small droplet condenses the release of heat of condensation causes a temperature difference between droplet and gas in the quasistationary state the kinetic equation describing the vapor is the stationary klein kramers equation for brownian particles diffusing in a temperature gradient by means of an expansion in burnett functions this equation is transformed into a set of coupled algebrodifferential equations by numerical integration we construct fundamental solutions of this equation that are subsequently combined linearly to fulfill appropriate mesoscopic boundary conditions for particles leaving the droplet surface in view of the intrinsic numerical instability of the system of equations a novel procedure is developed to remove the admixture of fast growing solutions to the solutions of interest the procedure is tested for a few model problems and then applied to a slightly simplified condensation problem with parameters corresponding to the condensation of mercury in a background of neon the effects of thermal gradients and thermodiffusion on the growth rate of the droplet are small of the order of 1 but well outside of the margin of error of the method

dynamical system models of transport chaos characteristics the macroscopic limit and irreversibility the escape rate formalism and the thermostating algorithm describe relaxation towards a decaying state with absorbing boundaries and a steady state of periodic systems respectively it has been shown that the key features of the transport properties of both approaches if modeled by low dimensional dynamical systems can conveniently be described in the framework of multibaker maps in the present paper we discuss in detail the steps required to reach a meaningful macroscopic limit the limit involves a sequence of coarser and coarser descriptions projections until one reaches the level of irreversible macroscopic advection diffusion equations the influence of boundary conditions is studied in detail only a few of the chaos characteristics possess a meaningful macroscopic limit but none of these is sufficient to determine the entropy production in a general non equilibrium state 2003 elsevier b v all rights reserved

first passage times for the uhlenbeck ornstein process a numerical solution is obtained for the laplace transformed backward kramers equation from which the mean first passage time may be obtained the main difficulties are associated with a the parabolic nature of the time development operator and b the existence of a double structure in the solution near the absorbing barrier both of these difficulties are resolved by computational methods derived from boundary layer theory the reliability of the method is assessed by comparing its results with an earlier analytic solution for the case of a uniform force field the authors also present the results for a harmonic force field for which no analytic solution is yet known

a supplement to precise asymptotics in the law of the iterated logarithm abstract let x x n n 1 be a sequence of real valued i i d random variables with e x 0 and e x 2 1 and set s n i 1 n x i n 1 this paper studies the precise asymptotics in the law of the iterated logarithm for example using a result on convergence rates for probabilities of moderate deviations for s n n 1 obtained by li et al internat j math math sci 15 1992 481 497 we prove that for every b 1 2 1 lim 0 2 b 1 2 n 3 log log n b n p s n n 2 n log log n a n e 2 2 b 2 b 1 2 whenever lim n log log n n 1 2 a n where s 0 t s 1 e t d t s 0 2 t e x 2 i x t e x i x t 2 t 0 and n 2 2 n log log n n 3 this result generalizes and improves theorem 2 8 of li et al internat j math math sci 15 1992 481 497 and theorem 1 of gut and sp taru ann probab 28 2000 1870 1883

the kinetic boundary layer for the klein kramers equation a new numerical approach we explore a numerical technique for determining the structure of the kinetic boundary layer of the klein kramers equation for noninteracting brownian particles in a fluid near a wall that absorbs the brownian particles the equation is of interest in the theory of diffusion controlled reactions and of the coagulation of colloidal suspensions by numerical simulation of the langevin equation equivalent to the klein kramers equation we amass statistics of the velocities at the first return to the wall and of the return times for particles injected into the fluid at the wall with given velocities the data can be used to construct the solutions of the standard problems at an absorbing wall the milne and the albedo problem we confirm and extend earlier results by burschka and titulaer obtained by a variational method vexed by the slow convergence of the underlying eigenfunction expansion we briefly discuss some further boundary layer problems that can be attacked by exploiting the results reported here

half range completeness for the fokker planck equation half range completeness theorems are proved for eigenfunctions associated to the one dimensional fokker planck equation in a semi infinite medium existence and uniqueness results for perfectly absorbing partially absorbing and purely specularly reflecting boundary conditions are deduced for the stationary and time dependent problems similar results are obtained for a slab geometry

the kinetic boundary layer for the fokker planck equation with absorbing boundary the fokker planck equation for the distribution of position and velocity of a brownian particle is a particularly simple linear transport equation its normal solutions and an apparently complete set of stationary boundary layer solutions can be determined explicitly by a numerical algorithm we select linear combinations of them that approximately fulfill the boundary condition for a completely absorbing plane wall and that approach a linearly increasing position space density far from the wall various aspects of these approximate solutions are discussed in particular we find that the extrapolated asymptotic density reaches zero at a distance xm beyond the wall we find xm 1 46 in units of the velocity persistence length of the brownian particle this study was motivated by certain problems in the theory of diffusion controlled reactions and the results might be used to test approximate theories employed in that field

half range completeness for the fokker planck equation with an external force this paper deals with a generalization of a classical result obtained by r beals and v protopopescu for the fokker planck equations to the case in which a constant external force is present

the similarity problem for indefinite sturm liouville operators and the help inequality abstract we study two problems the first one is the similarity problem for the indefinite sturm liouville operator a sgn x d w d x d r d x acting in l w 2 b b it is assumed that w r l loc 1 b b are even and positive a e on b b the second object is the so called help inequality 0 b 1 r f d x 2 k 2 0 b f 2 w d x 0 b 1 w 1 r f 2 w d x where the coefficients w r l loc 1 0 b are positive a e on 0 b both problems are well understood when the corresponding sturm liouville differential expression is regular the main objective of the present paper is to give criteria for both the validity of the help inequality and the similarity to a self adjoint operator in the singular case namely we establish new criteria formulated in terms of the behavior of the corresponding weyl titchmarsh m functions at 0 and at as a biproduct of this result we show that both problems are closely connected namely the operator a is similar to a self adjoint one precisely if the help inequality with w r and r w is valid next we characterize the behavior of m functions in terms of coefficients and then these results enable us to reformulate the obtained criteria in terms of coefficients finally we apply these results for the study of the two way diffusion equation also known as the time independent fokker planck equation

the structure of the class of subexponential distributions letx 1 x 2 x n be a sequence of positive independent identically distributed random variables with the same distribution function d f f and denote byx 1 n x 2 n x n n the order statistics of the sample we characterize the class of d f f for which p x 1 n x 2 n ldots x n i n x sim p x n i n x as x to infty for fixedn andi i n 1 and we show that it is independent ofn this leads to the genesis of a new class of d f l i we show that the sequence l i 0 is strictly decreasing and we illustrate how the classesl i determine the probabilistic structure of the classl of subexponential distributions

large deviations and law of the iterated logarithm for partial sums normalized by the largest absolute observation let x n 1 leq n infty be a sequence of independent identically distributed random variables in the domain of attraction of a stable law with index 0 alpha 2 the limit of x n 1 log p s n max x i geq x n is found when x n to infty and x n n to 0 the large deviation result is used to prove the law of the iterated logarithm for the self normalized partial sums

asymptotic normality and subsequential limits of trimmed sums let x i be i i d and s n s n r n the sum of the first n x i with the r n largest and s n smallest excluded assume r n rightarrow infty s n rightarrow infty n 1 r n rightarrow 0 n 1 s n rightarrow 0 necessary and sufficient conditions are obtained for the existence of delta n gamma n such that gamma 1 n s n s n r n delta n converges weakly to a standard normal the set of all subsequential limit laws for these sequences is characterized and sufficient conditions are given for x i to be in the domain of partial attraction of a given law in the class these conditions are also necessary if a unique factorization result for characteristic functions is true

the structure of the stationary kinetic boundary layer for the linear bgk equation recently very accurate numerical methods for solving kinetic boundary layer problems for linear kinetic equations have been developed to test such methods detailed information about exactly solvable models can be very helpful to this purpose the authors present results for the stationary 1d linear bgk equation obtained by the singular eigenfunction method that extend results already known in the literature both in scope and in precision special attention is paid to the nature of the singularity in the distribution function for small velocities near the boundary this singularity cannot be reproduced exactly by the numerical methods available explicit expressions are presented for the milne problem and for the albedo problem with input particles of a single velocity they compare the results with those of a recently developed variant of the two stream moment method they find excellent agreement except close to the singularity for many quantities of interest the accuracy obtainable by the novel two stream moment method cannot be reproduced with comparable numerical effort by evaluation of the exact solution

full and partial range eigenfunction expansions for sturm liouville problems with indefinite weights this article is concerned with eigenvalue problems of the form au tu in a hilbert space h where ais a selfadjoint positive operator generated by a second order sturm liouville differential expression and t a selfadjoint indefinite multiplicative operator which is one to one emphasis is on the full range and partial range expansionproperties of the eigenfunctions

partial range completeness and existence of solutions to two way diffusion equations separating variables to solve a two way diffusion equation leads to a nonstandard eigenvalue problem in one variable it is shown that the eigenfunctions having negative eigenvalues are complete on the part of the domain where initial conditions are imposed while those with positive eigenvalues are complete where final conditions are imposed the corresponding exponentially growing and exponentially decaying solutions may be used to expand arbitrary solutions on semi infinite intervals in the time variable a natural iterative procedure for obtaining solutions on finite intervals is shown to converge in some cases a linearly growing solution must also be taken into account

phase space diffusion in a system with a partially permeable wall a new boundary condition for a diffusive system with a partially permeable wall described by the klein kramers equation is proposed the green functions for such a system are given and a generalization to the system with a series of walls is briefly discussed it is shown that the well known boundary condition for the case of a partially absorbing wall which gives the so called radiation boundary condition for the diffusion equation cannot be used for the system under consideration

an accurate two stream moment method for kinetic boundary layer problems of linear kinetic equations the authors develop a method for solving 1d kinetic boundary layer problems for linear equations that uses separate hermite moment expansions in the velocity variable for particles moving towards and away from a plane wall this so called two stream method is tested for two especially simple kinetic equations the linear bgk equation and the klein kramers equation the kinetic equation for brownian particles for both of these equations extensive exact information is available for two simple boundary layer problems the milne and the albedo problem a comparison of these exact results with those obtained with this version of the two stream moment method shows that the accuracy obtainable by this method exceeds that of earlier methods by several orders of magnitude in particular the method allows them to obtain accurate results not only for the moments of the distribution function but also for the distribution function itself

light induced slip abstract we derive the slip boundary conditions for the velocity and temperature of a one component gas irradiated by resonant light in contrast to standard gas kinetics the slip now already arises in zeroth order in the knudsen number this zeroth order slip is therefore induced by the resonant light explicit expressions are found in the case of monochromatic excitation after defining a set of accomodation coefficients that characterize the wall collision kernel we calculate the light induced particle flow pressure difference and heat flow which arise from the zeroth order slip

a drop of ink falls from my pen it comes to earth i know not when the authors obtain for a brownian particle in a uniform force field the mean and asymptotic first passage times as functions of the particle s initial position and velocity with the recurrence times given as a special case they discuss the region of phase space for which the diffusion model of brownian motion provides an adequate approximation and conclude that there is no possibility of obtaining the recurrence times within that model they find that the nature of the boundary value problem is profoundly altered when the motion is treated as a process in phase rather than configuration space because the time development operator is then parabolic rather than elliptic they argue that such a change in the treatment of brownian motion places it within the sphere of transport theory rather than diffusion theory and that consequently results such as ours have relevance to the study of phenomena such as radiative transfer and neutron transport

overview of theoretical models for reaction rates there is enormous recent interest in the development of models for rate processes because rates are an almost universal characterization in the physical and biological sciences in this paper we provide an introduction to several of the problems to be discussed in greater depth by other speakers at a symposium held at the national institutes of health on may 6 8 1985 this review will focus on 1 the smoluchowski model for reaction rates together with its extension by onsager 2 first passage time formalism for discrete and continuous master equations and fokker planck equations 3 the kramers model and its extensions 4 diffusion in the presence of trapping centers

non classical law of the iterated logarithm behaviour for trimmed sums we study the law of the iterated logarithm for the partial sum of i i d random variables when the r n largest summands are excluded where r n o log logn this complements earlier work in which the case log logn o rn was considered a law of the iterated logarithm is again seen to prevail for a wide class of distributions but for reasons quite different from previously

half range expansion analysis for langewin dynamics in the high friction limit with a singular absorbing boundary condition noncharacteristic case we consider the dynamics of a brownian particle given by the langevin equation in a strip under the effects of a deterministic force the trajectories of particles originate at a source whose spatial location in the phase space coincides with the location of adsorbing boundaries this leads to singular behavior of trajectories in the high friction limit we use the half range expansion technique and systematic asymptotics to solve a boundary value problem for the fokker planck operator and to calculate the steady state transition probability density the mean time to absorption and the distribution of exit points we do not make assumptions about other parameters in the problem except that they areo 1 relative to the friction coefficient we calculate explicitly the correct location of the milne type extrapolation for absorbing boundary conditions for the smoluchowski approximation to the langevin equation

hydrodynamic singular regimes in 1 1 kinetic models and spectral numerical methods classical results from spectral theory of stationary linear kinetic equations are applied to efficiently approximate two physically relevant weakly nonlinear kinetic models a model of chemotaxis involving a biased velocity redistribution integral term and a vlasov fokker planck vfp system both are coupled to an attractive elliptic equation producing corresponding mean field potentials spectral de compositions of stationary kinetic distributions are recalled based on a variation of case s elementary solutions for the first model and on a sturm liouville eigenvalue problem for the second one well balanced godunov schemes with strong stability properties are deduced moreover in the stiff hydrodynamical scaling an hybridized algorithm is set up for which asymptotic preserving properties can be established under mild restrictions on the computational grid several numerical validations are displayed including the consistency of the vfp model with burgers hopf dynamics on the velocity field after blowup of the macroscopic density into dirac masses

the vacuum einstein equations as closed ideals of matrix valued differential forms the vacuum einstein equations are written as a closed differential ideal of three dimensional matrices of differential forms allowing the weyl tensor to appear explicitly in the equations possible uses of this formulation are indicated

chapman enskog as an application of the method for eliminating fast variables although the chapman enskog treatment of the boltzmann equation is one of the first examples of the elimination of fast variables it is not usually presented consistently from that point of view here it is developed systematically as a special case of the general method for eliminating fast variables from nonlinear equations as a result certain ambiguities can be remedied first it is inconsistent with the separation of time scales to extend the phenomenological description of the gas by including some of the higher moments of the distribution function such as the heat flow in the application to the relativistic boltzmann equation the dilemma concerning the choice of the lowest order approximation is resolved in the final section it is demonstrated that unsystematic elimination of fast variables leads to secular terms

the lower limit of a normalized random walk soit sn une marche aleatoire et n une suite de constantes telles que n n est non decroissante on donne un test integral universel qui determine la limite inferieure de sn n jusqu a une echelle constante pour lim sup 2n n

stability of perpetuities for a series of randomly discounted terms we give an integral criterion to distinguish between almost sure absolute convergence and divergence in probability to oo these being the only possible forms of asymptotic behavior this solves the existence problem for a one dimensional perpetuity that remains from a 1979 study by vervaat and yields a complete characterization of the existence of distributional fixed points of a random affine map in dimension one