emergent dynamics of winfree oscillators on locally coupled networks abstract the winfree model is the first mathematical model for synchronization of weakly coupled oscillators compared to the well known kuramoto model the winfree model does not conserve the total phase this leads to rich dynamic features compared to those produced by other phase models in this paper we study the emergent dynamics of the winfree model on a locally coupled static network a randomly chosen phase configuration undergoes several dynamic phase transitions such as incoherence partial locking complete locking partial oscillator death and complete oscillator death as the coupling strength increases we provide several rigorous analytical results on the emergence of these dynamic features we also provide several numerical simulations and compare their results to the analytical results

current fluctuations of the stationary asep and six vertex model our results in this article are twofold first we consider current fluctuations of the stationary asymmetric simple exclusion process asep run for some long time t and show that they are of order t1 3 along a characteristic line upon scaling by t1 3 we establish that these fluctuations converge to the long time height fluctuations of the stationary kardar parisi zhang kpz equation that is to the baik rains distribution this result has long been predicted under the context of kpz universality and in particular extends upon a number of results in the field including the work of ferrari and spohn from 2005 when they established the same result for the tasep and the work of balazs and seppalainen from 2010 when they established the t1 3 scaling for the general asep second we introduce a class of translation invariant gibbs measures that characterizes a one parameter family of slopes for an arbitrary ferroelectric symmetric six vertex model this family of slopes corresponds to what is known as the conical singularity or tricritical point in the free energy profile for the ferroelectric six vertex model we consider fluctuations of the height function of this model on a large grid of size t and show that they too are of order t1 3 along a certain characteristic line this confirms a prediction of bukman and shore from 1995 stating that the ferroelectric six vertex model should exhibit kpz growth at the conical singularity upon scaling the height fluctuations by t1 3 we again recover the baik rains distribution in the large t limit recasting this statement in terms of the asymmetric stochastic six vertex model confirms a prediction of gwa and spohn from 1992

a stochastic burgers equation from a class of microscopic interactions we consider a class of nearest neighbor weakly asymmetric mass conservative particle systems evolving on mathbb z which includes zero range and types of exclusion processes starting from a perturbation of a stationary state when the weak asymmetry is of order o n gamma for 1 2 gamma leq1 we show that the scaling limit of the fluctuation field as seen across process characteristics is a generalized ornstein uhlenbeck process however at the critical weak asymmetry when gamma 1 2 we show that all limit points satisfy a martingale formulation which may be interpreted in terms of a stochastic burgers equation derived from taking the gradient of the kpz equation the proofs make use of a sharp boltzmann gibbs estimate which improves on earlier bounds

fluid dynamic limits of kinetic equations i formal derivations the connection between kinetic theory and the macroscopic equations of fluid dynamics is described in particular our results concerning the incompressible navier stokes equations are based on a formal derivation in which limiting moments are carefully balanced rather than on a classical expansion such as those of hilbert or chapman enskog the moment formalism shows that the limit leading to the incompressible navier stokes equations like that leading to the compressible euler equations is a natural one in kinetic theory and is contrasted with the systematics leading to the compressible navier stokes equations some indications of the validity of these limits are given more specifically the connection between the diperna lions renormalized solution of the classical boltzmann equation and the leray solution of the navier stokes equations is discussed

regularity of the moments of the solution of a transport equation let u u x v satisfy the transport equation u v xu f x rn v rnwhere f belongs to some space of type lp dx d v where is a positive bounded measure on rn we study the resulting regularity of the moment u x v d v in terms of fractional sobolev spaces for example counter examples are given in order to test the optimality of our results

global existence in l1 for the enskog equation and convergence of the solutions to solutions of the boltzmann equation for the enskog equation in a box an existence theorem is proved for initial data with finite mass energy and entropy then by letting the diameter of the molecules go to zero the weak convergence of solutions of the enskog equation to solutions of the boltzmann equation is proved

the stochastic heat equation feynman kac formula and intermittence we study in one space dimension the heat equation with a random potential that is a white noise in space and time this equation is a linearized model for the evolution of a scalar field in a space time dependent random medium it has also been related to the distribution of two dimensional directed polymers in a random environment to the kpz model of growing interfaces and to the burgers equation with conservative noise we show how the solution can be expressed via a generalized feynman kac formula we then investigate the statistical properties the two point correlation function is explicitly computed and the intermittence of the solution is proven this analysis is carried out showing how the statistical moments can be expressed through local times of independent brownian motions

fluid dynamic description of flocking via the povzner boltzmann equation we introduce and discuss the possible dynamics of groups of indistinguishable agents which are interacting according to their relative positions with the aim of deriving hydrodynamic equations these models are developed to mimic the collective motion of groups of species such as bird flocks fish schools herds of quadrupeds or bacteria colonies our starting model for these interactions is the povzner equation 21 which describes a dilute gas in which binary collisions of elastic spheres depend on their relative positions following the cucker and smale model 9 we will consider binary interactions between agents that are dissipative collisions in which the coefficient of restitution depends on their relative distance under the assumption of weak dissipation it is shown that the povzner equation is modified through a correction in the form of a nonlinear friction type operator using this correction we formally obtain from the povzner equation in a direct way a fluid dynamic description of a system of agents with weak dissipative interactions with a coefficient of restitution that depends on their relative distance

statistical mechanics of nonlinear wave equations 4 cubic schrodinger the cubic schrodinger equation is considered on the circle both in the de focussing and the focussing case the existence of the flow is proved together with the invariance of the appropriate gibbsian measure namely the petit canonical measure in the defocussing case and the micro canonical measure in the focussing case

large scale behavior of three dimensional continuous phase coexistence models we study a class of three dimensional continuous phase coexistence models and show that under different symmetry assumptions on the potential the large scale behavior of such models near a bifurcation point is described by the dynamical urn x wiley 00103640 media cpa21738 cpa21738 math 0001 models for p 2 3 4 this result is specific to space dimension 3 and does not hold in dimension 2

the conical point in the ferroelectric six vertex model we examine the last unexplored regime of the asymmetric six vertex model the low temperature phase of the so called ferroelectric model the original publication of the exact solution by sutherland yang and yang and various derivations and reviews published afterward do not contain many details about this regime we study the exact solution for this model by numerical and analytical methods in particular we examine the behavior of the model in the vicinity of an unusual coexistence point that we call the conical point this point corresponds to additional singularities in the free energy that were not discussed in the original solution we show analytically that at this point many polarizations coexist and that unusual scaling properties hold in its vicinity

algebraic renormalisation of regularity structures we give a systematic description of a canonical renormalisation procedure of stochastic pdes containing nonlinearities involving generalised functions this theory is based on the construction of a new class of regularity structures which comes with an explicit and elegant description of a subgroup of their group of automorphisms this subgroup is sufficiently large to be able to implement a version of the bphz renormalisation prescription in this context this is in stark contrast to previous works where one considered regularity structures with a much smaller group of automorphisms which lead to a much more indirect and convoluted construction of a renormalisation group acting on the corresponding space of admissible models by continuous transformations our construction is based on bialgebras of decorated coloured forests in cointeraction more precisely we have two hopf algebras in cointeraction coacting jointly on a vector space which represents the generalised functions of the theory two twisted antipodes play a fundamental role in the construction and provide a variant of the algebraic birkhoff factorisation that arises naturally in perturbative quantum field theory

existence of piecewise weak solutions of a discrete cucker smale s flocking model with a singular communication weight abstract we prove existence of global c 1 piecewise weak solutions for the discrete cucker smale s flocking model with a non lipschitz communication weight s s 0 1 we also discuss the possibility of finite in time alignment of the velocities of the particles

singular spdes in domains with boundaries we study spaces of modelled distributions with singular behaviour near the boundary of a domain that in the context of the theory of regularity structures allow one to give robust solution theories for singular stochastic pdes with boundary conditions the calculus of modelled distributions established in hairer invent math 198 2 269 504 2014 https doi org 10 1007 s00222 014 0505 4 is extended to this setting we formulate and solve fixed point problems in these spaces with a class of kernels that is sufficiently large to cover in particular the dirichlet and neumann heat kernels these results are then used to provide solution theories for the kpz equation with dirichlet and neumann boundary conditions and for the 2d generalised parabolic anderson model with dirichlet boundary conditions in the case of the kpz equation with neumann boundary conditions we show that depending on the class of mollifiers one considers a boundary renormalisation takes place in other words there are situations in which a certain boundary condition is applied to an approximation to the kpz equation but the limiting process is the hopf cole solution to the kpz equation with a different boundary condition

fluctuations of one dimensional ginzburg landau models in nonequilibrium we study the fluctuation of one dimensional ginzburg landau models in nonequilibrium along the hydrodynamic diffusion limit the hydrodynamic limit has been proved to be a nonlinear diffusion equation by fritz guo papanicolaou varadhan etc we proved that if the potential is uniformly convex then the fluctuation process is governed by an ornstein uhlenbeck process whose drift term is obtained by formally linearizing the hydrodynamic equation

multidimensional sdes with singular drift and universal construction of the polymer measure with white noise potential we study the existence and uniqueness of solution for stochastic differential equations with distributional drift by giving a meaning to the stroock varadhan martingale problem associated to such equations the approach we exploit is the one of paracontrolled distributions introduced in forum math pi 3 2015 e6 as a result we make sense of the three dimensional polymer measure with white noise potential

on the bootstrap saddlepoint approximations we compare saddlepoint approximations to the exact distributions of a studentized mean and to its bootstrap approximation we show that on bounded sets these empirical saddlepoint approximations achieve second order relative errors uniformly we also consider the relative errors for larger deviations it follows that the studentized t bootstrap p value and the coverage of the bootstrap confidence interval have second order relative errors

three dimensional navier stokes equations driven by space time white noise abstract in this paper we prove existence and uniqueness of local solutions to the three dimensional 3d navier stokes n s equation driven by space time white noise using two methods first the theory of regularity structures introduced by martin hairer in 16 and second the paracontrolled distribution proposed by gubinelli imkeller perkowski in 12 we also compare the two approaches

on the size of chaos in the mean field dynamics we consider the error arising from the approximation of an n particle dynamics with its description in terms of a one particle kinetic equation we estimate the distance between the j marginal of the system and the factorized state obtained in a mean field limit as n our analysis relies on the evolution equation for the correlation error rather than on the usual bbgky hierarchy the rate of convergence is shown to be o j 2 n in any bounded interval of time size of chaos as expected from heuristic arguments our formalism applies to an abstract hierarchical mean field model with bounded collision operator and a large class of initial data covering a stochastic jump processes converging to the homogeneous boltzmann and the povzner equation and b quantum systems giving rise to the hartree equation

on distributions of ratios a large number of exact inferential procedures in statistics and econometrics involve the sampling distribution of ratios of random variables if the denominator variable is positive then tail probabilities of the ratio can be expressed as those of a suitably defined difference of random variables if in addition the joint characteristic function of numerator and denominator is known then standard fourier inversion techniques can be used to reconstruct the distribution function from it most research in this field has been based on this correspondence but which breaks down when both numerator and denominator are supported on the entire real line the present manuscript derives inversion formulae and saddlepoint approximations that remain valid in this case and reduce to known results when the denominator is almost surely positive applications include the iv est imator of a structural parameter in a just identified equation

solving the kpz equation we introduce a new concept of solution to the kpz equation which is shown to extend the classical cole hopf solution this notion provides a factorisation of the cole hopf solution map into a universal measurable map from the probability space into an explicitly described auxiliary metric space composed with a new solution map that has very good continuity properties the advantage of such a formulation is that it essentially provides a pathwise notion of a solution together with a very detailed approximation theory in particular our construction completely bypasses the cole hopf transform thus laying the groundwork for proving that the kpz equation describes the uctuations of systems in the kpz universality class as a corollary of our construction we obtain very detailed new regularity results about the solution as well as its derivative with respect to the initial condition other byproducts of the proof include an explicit approximation to the stationary solution of the kpz equation a well posedness result for the fokker planck equation associated to a particle diusing in a rough space time dependent potential and a new periodic homogenisation result for the heat equation with a space time periodic potential one ingredient in our construction is an example of a non gaussian rough path such that the area process of its natural approximations needs to be renormalised by a diverging term for the approximations to converge

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

discretisations of rough stochastic pdes we develop a general framework for spatial discretisations of parabolic stochastic pdes whose solutions are provided in the framework of the theory of regularity structures and which are functions in time as an application we show that the dynamical phi 4 3 model on the dyadic grid converges after renormalisation to its continuous counterpart this result in particular implies that as expected the phi 4 3 measure with a sufficiently small coupling constant is invariant for this equation and that the lifetime of its solutions is almost surely infinite for almost every initial condition

fluid dynamic limits of kinetic equations ii convergence proofs for the boltzmann equation using relative entropy estimates about an absolute maxwellian it is shown that any properly scaled sequence of diperna lions renormalized solutions of some classical boltzmann equations has fluctuations that converge to an infinitesimal maxwellian with fluid variables that satisfy the incompressibility and boussinesq relations moreover if the initial fluctuations entropically converge to an infinitesimal maxwellian then the limiting fluid variables satisfy a version of the leray energy inequality if the sequence satisfies a local momentum conservation assumption the momentum densities globaly converge to a solution of the stokes equation a similar discrete time version of this result holds for the navier stokes limit with an additional mild weak compactness assumption the continuous time navier stokes limit is also discussed 01993 john wiley sons inc

the schrodinger equation with spatial white noise the average wave function abstract we prove a representation for the average wave function of the schrodinger equation with a white noise potential in d 1 2 in terms of the renormalized self intersection local time of a brownian motion

a stochastic system of particles modelling the euler equations we consider a system of n spheres interacting through elastic collisions at a stochastic distance in the limit n for a suitable rescaling of the interaction parameters we prove that the one particle distribution function converges to a local maxwellian whose gross density velocity and temperature satisfy the euler equation

the asep and determinantal point processes we introduce a family of discrete determinantal point processes related to orthogonal polynomials on the real line with correlation kernels defined via spectral projections for the associated jacobi matrices for classical weights we show how such ensembles arise as limits of various hypergeometric orthogonal polynomial ensembles we then prove that the q laplace transform of the height function of the asep with step initial condition is equal to the expectation of a simple multiplicative functional on a discrete laguerre ensemble a member of the new family this allows us to obtain the large time asymptotics of the asep in three limit regimes a for finitely many rightmost particles b gue tracy widom asymptotics of the height function c kpz asymptotics of the height function for the asep with weak asymmetry we also give similar results for two instances of the stochastic six vertex model in a quadrant the proofs are based on limit transitions for the corresponding determinantal point processes

saddlepoint approximations for likelihood ratio like statistics with applications to permutation tests we obtain two theorems extending the use of a saddlepoint approximation to multiparameter problems for likelihood ratio like statistics which allow their use in permutation and rank tests and could be used in bootstrap approximations in the first we show that in some cases when no density exists the integral of the formal saddlepoint density over the set corresponding to large values of the likelihood ratio like statistic approximates the true probability with relative error of order l n in the second we give multivariate generalizations of the lugannani rice and barndorff nielsen or r formulas for the approximations these theorems are applied to obtain permutation tests based on the likelihood ratio like statistics for the k sample and the multivariate two sample cases numerical examples are given to illustrate the high degree of accuracy and these statistics are compared to the classical statistics in both cases

time delay effect on the flocking in an ensemble of thermomechanical cucker smale particles abstract we study the communication time delay effect on the emergent dynamics of thermodynamic cucker smale tcs ensemble with temperature as an internal state in the presence of time delay conservation laws for momentum and energy do not hold so that the analytic tool based on the energy method cannot be applied for the tcs model as it is we show that temperature stays positive in the presence of small time delay as long as temperatures stay positive the tcs model is well posed by direct application of cauchy lipschitz theory when the size of time delay and initial data are sufficiently small we can derive differential inequalities for velocity and temperature diameters using these differential inequalities we show that temperature and velocity diameters tend to zero exponentially fast whereas the position diameter is uniformly bounded we also present several numerical simulations and compare them with analytic results

a coupled kpz equation its two types of approximations and existence of global solutions abstract this paper concerns the multi component coupled kardar parisi zhang kpz equation and its two types of approximations one approximation is obtained as a simple replacement of the noise term by a smeared noise with a proper renormalization while the other one introduced in 6 is suitable for studying the invariant measures by applying the paracontrolled calculus introduced by gubinelli et al 8 9 we show that two approximations have the common limit under the properly adjusted choice of renormalization factors for each of these approximations in particular if the coupling constants of the nonlinear term of the coupled kpz equation satisfy the so called trilinear condition the renormalization factors can be taken the same in two approximations and the difference of the limits of two approximations are explicitly computed moreover under the trilinear condition the wiener measure twisted by the diffusion matrix becomes stationary for the limit and we show that the solution of the limit equation exists globally in time when the initial value is sampled from the stationary measure this is shown for the associated tilt process combined with the strong feller property shown by hairer and mattingly 12 this result can be extended for all initial values

the nonaccretive radiative transfer equations existence of solutions and rosseland approximation we present an existence theory and an asymptotic analysis for the radiative transfer equations ue t xuee ue e2 ue ue 0 inx 1 ue x sn k ue t 0 u0 where u u t x t r x x rn 1 sn and ue t x 1 sn ue t x d we prove that even if has a singularity 0 1 has a solution ue l r x sn as e 0 we show that ue converges pointwise to a function u l r x solution of the degenerate parabolic equation u t f u 0 in x u x k u t 0 u0 this is achieved without any monotonicity assumption on and therefore one cannot use the theory of nonlinear contraction semigroups

a global unique solvability of entropic weak solution to the one dimensional pressureless euler system with a flocking dissipation abstract we present a unique global solvability and flocking estimate of an entropic weak solution to the one dimensional pressureless euler system with a flocking dissipation in all to all coupling setting this model appears naturally as a quasi equilibrium model for hydrodynamic description of cucker smale flocking for the unique global solvability we adopt the variation approach from ding and huang s work 19 on the inhomogeneous pressureless gas dynamic model when initial mass and velocity are locally integrable and bounded measurable functions respectively we give explicit representations for the global entropic weak solutions our results do not require any smallness of initial data except that initial mass density is almost everywhere positive

controlling rough paths abstract we formulate indefinite integration with respect to an irregular function as an algebraic problem which has a unique solution under some analytic constraints this allows us to define a good notion of integral with respect to irregular paths with holder exponent greater than 1 3 e g samples of brownian motion and study the problem of the existence uniqueness and continuity of solution of differential equations driven by such paths we recover young s theory of integration and the main results of lyons theory of rough paths in holder topology

existence and uniqueness for nonlinear boundary value problems in kinetic theory a boundary value problem for the stationary nonlinear boltzmann equation in a slab has been examined in a weightedl space it has been proved that the problem possesses a unique solution for boundary data small enough the proof is based on the implicit function theorem it has also been shown that for the linearized problem the fredholm alternative applies

from discrete cucker smale model to continuous cucker smale model in a temperature field we study emergent dynamics of the discrete and continuous cucker smale c s models in a self consistent temperature field and provide refined monocluster flocking estimates more precisely we present a sufficient framework leading to the monocluster flocking toward the constant common temperature and velocity exponentially fast this improves the authors earlier work where existence of common temperature and velocity are not identified our proposed sufficient framework is formulated in terms of the initial data and system parameters using the refined flocking we also provide a uniform in time convergence from the discrete model to the continuous model under some suitable compatibility condition on the asymptotic velocities for the discrete and continuous model as the time step tends to zero we also provide several numerical examples and compare them with analytical results we study emergent dynamics of the discrete and continuous cucker smale c s models in a self consistent temperature field and provide refined monocluster flocking estimates more precisely we present a sufficient framework leading to the monocluster flocking toward the constant common temperature and velocity exponentially fast this improves the authors earlier work where existence of common temperature and velocity are not identified our proposed sufficient framework is formulated in terms of the initial data and system parameters using the refined flocking we also provide a uniform in time convergence from the discrete model to the continuous model under some suitable compatibility condition on the asymptotic velocities for the discrete and continuous model as the time step tends to zero we also provide several numerical examples and compare them with analytical results

kinetic models for topological nearest neighbor interactions we consider systems of agents interacting through topological interactions these have been shown to play an important part in animal and human behavior precisely the system consists of a finite number of particles characterized by their positions and velocities at random times a randomly chosen particle the follower adopts the velocity of its closest neighbor the leader we study the limit of a system size going to infinity and under the assumption of propagation of chaos show that the limit kinetic equation is a non standard spatial diffusion equation for the particle distribution function we also study the case wherein the particles interact with their k closest neighbors and show that the corresponding kinetic equation is the same finally we prove that these models can be seen as a singular limit of the smooth rank based model previously studied in 10 the proofs are based on a combinatorial interpretation of the rank as well as some concentration of measure arguments

saddlepoint approximations for marginal and conditional probabilities of transformed variables the lugannani and rice formula for tail areas in the univariate case has recently been extended to tail areas for marginal distributions and for conditional distributions in certain multivariate settings however the results on relative order of errors are given only formally or under the strong continuity assumptions necessary to obtain density approximations this paper attempts to give a unified treatment of these results for smooth transformations of multivariate means under weaker conditions appropriate to indirect edgeworth approximations for probabilities

glauber dynamics of 2d kac blume capel model and their stochastic pde limits abstract we study the glauber dynamics of a two dimensional blume capel model or dilute ising model with kac potential parametrized by the inverse temperature and the chemical potential we prove that the locally averaged spin field rescales to the solution of the dynamical 4 equation near a curve in the plane and to the solution of the dynamical 6 equation near one point on this curve our proof relies on a discrete implementation of da prato debussche method 13 as in 33 but an additional coupling argument is needed to show convergence of the linearized dynamics

space time discrete kpz equation we study a general family of space time discretizations of the kpz equation and show that they converge to its solution the approach we follow makes use of basic elements of the theory of regularity structures hairer in invent math 198 2 269 504 2014 as well as its discrete counterpart hairer and matetski in discretizations of rough stochastic pdes 2015 arxiv 1511 06937 since the discretization is in both space and time and we allow non standard discretization for the product the methods mentioned above have to be suitably modified in order to accommodate the structure of the models under study

conformal invariance of spin correlations in the planar ising model we rigorously prove the existence and the conformal invariance of scaling limits of the magnetization and multi point spin correlations in the critical ising model on arbitrary simply connected planar domains this solves a number of conjectures coming from the physical and the mathematical literature the proof relies on convergence results for discrete holomorphic spinor observables and probabilistic techniques

heat semigroup and singular pdes we provide in this work a semigroup approach to the study of singular pdes in the line of the paracontrolled approach developed recently by gubinelli imkeller and perkowski starting from a heat semigroup we develop a functional calculus and introduce a paraproduct based on the semigroup for which commutator estimates and schauder estimates are proved together with their paracontrolled extensions this machinery allows us to investigate singular pdes in potentially unbounded riemannian manifolds under mild geometric conditions as an illustration we study the generalized parabolic anderson model equation and prove under mild geometric conditions its well posed character in small time on a potentially unbounded 2 dimensional riemannian manifold for an equation driven by a coloured noise and for all times for the linear parabolic anderson model equation in 2 dimensional unbounded manifolds this machinery can be extended to an even more singular setting and deal with sobolev spaces rather than holder spaces

the dynamic phi 4_3 φ 3 4 model comes down from infinity we prove an a priori bound for the dynamic phi 4 3 model on the torus which is independent of the initial condition in particular this bound rules out the possibility of finite time blow up of the solution it also gives a uniform control over solutions at large times and thus allows one to construct invariant measures via the krylov bogoliubov method it thereby provides a new dynamic construction of the euclidean phi 4 3 field theory on finite volume our method is based on the local in time solution theory developed recently by gubinelli imkeller perkowski and catellier chouk the argument relies entirely on deterministic pde arguments such as embeddings of besov spaces and interpolation which are combined to derive energy inequalities

the grad limit for a system of soft spheres a model describing the dynamics of a system of soft spheres is introduced the model contains a random mechanism which mollifies the deterministic model of hard sphere molecules it allows a rigorous deduction of the corresponding boltzmann equation for a wide class of initial data and arbitrarily long times

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

biometrika centenary nonparametrics summary contributions in biometrika to nonparametric statistics are surveyed with particular emphasis on more recent work it is argued that even in the journal s very early years it had interests in smoothing methods for density estimation and in techniques for nonparametric regression in addition to its more obvious focus on classical nonparametric statistics such as ranking methods biometrika s early involvement with permutation testing has links to its contemporary work on bootstrap methods emphases on different types of nonparametric statistics have evolved over time however a rapidly increasing proportion of the journal s pages is devoted to papers that include some use of nonparametric ideas or methods

strong solutions to the stochastic quantization equations we prove the existence and uniqueness of a strong solution of the stochastic quantization equation in dimension 2 for almost all initial data with respect to the invariant measure the method is based on a fixed point result in suitable besov spaces

stochastic higher spin vertex models on the line we introduce a four parameter family of interacting particle systems on the line which can be diagonalized explicitly via a complete set of bethe ansatz eigenfunctions and which enjoy certain markov dualities using this for the systems started in step initial data we write down nested contour integral formulas for moments and fredholm determinant formulas for laplace type transforms taking various choices or limits of parameters this family degenerates to many of the known exactly solvable models in the kardar parisi zhang universality class as well as leads to many new examples of such models in particular asymmetric simple exclusion process the stochastic six vertex model q totally asymmetric simple exclusion process and various directed polymer models all arise in this manner our systems are constructed from stochastic versions of the r matrix related to the six vertex model one of the key tools used here is the fusion of r matrices and we provide a probabilistic proof of this procedure

a theory of regularity structures we introduce a new notion of regularity structure that provides an algebraic framework allowing to describe functions and or distributions via a kind of jet or local taylor expansion around each point the main novel idea is to replace the classical polynomial model which is suitable for describing smooth functions by arbitrary models that are purpose built for the problem at hand in particular this allows to describe the local behaviour not only of functions but also of large classes of distributions we then build a calculus allowing to perform the various operations multiplication composition with smooth functions integration against singular kernels necessary to formulate fixed point equations for a very large class of semilinear pdes driven by some very singular typically random input this allows for the first time to give a mathematically rigorous meaning to many interesting stochastic pdes arising in physics the theory comes with convergence results that allow to interpret the solutions obtained in this way as limits of classical solutions to regularised problems possibly modified by the addition of diverging counterterms these counterterms arise naturally through the action of a renormalisation group which is defined canonically in terms of the regularity structure associated to the given class of pdes our theory also allows to easily recover many existing results on singular stochastic pdes kpz equation stochastic quantisation equations burgers type equations and to understand them as particular instances of a unified framework one surprising insight is that in all of these instances local solutions are actually smooth in the sense that they can be approximated locally to arbitrarily high degree as linear combinations of a fixed family of random functions distributions that play the role of polynomials in the theory as an example of a novel application we solve the long standing problem of building a natural markov process that is symmetric with respect to the finite volume measure describing the phi 4 3 euclidean quantum field theory it is natural to conjecture that the markov process built in this way describes the glauber dynamic of 3 dimensional ferromagnets near their critical temperature

energy solutions of kpz are unique the kardar parisi zhang kpz equation is conjectured to universally describe the fluctuations of weakly asymmetric interface growth here we provide the first intrinsic well posedness result for the kpz equation on the real line by showing that its energy solutions as introduced by goncalves and jara in gj14 and refined in gj13 are unique together with the convergence results of gj14 and many follow up papers this establishes the weak kpz universality conjecture for a wide class of models our proof builds on an observation of funaki and quastel fq15 a remarkable consequence is that the energy solution to the kpz equation is not equal to the cole hopf solution but it involves an additional drift t 12

weakly asymmetric non simple exclusion process and the kardar parisi zhang equation we analyze a class of non simple exclusion processes and the corresponding growth models by generalizing the discrete cole hopf transformation of gartner stoch process appl 27 233 260 1987 we identify the main non linearity and eliminate it by imposing a gradient type condition for hopping range at most 3 using the generalized transformation we prove the convergence of the exclusion process toward the kardar parisi zhang kpz equation this is the first universality result under the weak asymmetry concerning interacting particle systems while this class of exclusion processes are not explicitly solvable under the weak asymmetry we obtain the exact one point limiting distribution for the step initial condition by using the previous result of amir et al commun pure appl math 64 4 466 537 2011 and our convergence result

probability distribution of the free energy of the continuum directed random polymer in 1 1 dimensions we consider the solution of the stochastic heat equation tz d 1 2 x zz p w with delta function initial condition z t d0 x dixd0 whose logarithm with appropriate normalization is the free energy of the con tinuum directed polymer or the hopf cole solution of the kardar parisi zhang equation with narrow wedge initial conditions we obtain explicit formulas for the one dimensional marginal distributions the crossover distributions which interpolate between a standard gaussian dis tribution small time and the gue tracy widom distribution large time the proof is via a rigorous steepest descent analysis of the tracy widom formula for the asymmetric simple exclusion process with antishock initial data which is shown to converge to the continuum equations in an appropriate weakly asymmetric limit the limit also describes the crossover behavior between the symmetric and asymmetric exclusion processes 2010 wiley periodicals inc

self normalized cramer type large deviations for independent random variables let x 1 x 2 be independent random variables with zero means and finite variances it is well known that a finite exponential moment assumption is necessary for a cramer type large deviation result for the standardized partial sums in this paper we show that a cramer type large deviation theorem holds for self normalized sums only under a finite 2 th moment 0 1 in particular we show p s n v n x 1 x 1 o 1 1 x 2 d 2 n for 0 x d n where d n n i 1 ex 2 i 1 2 n i 1 e x i 2 1 2 and v n n i 1 x 2 i 1 2 applications to the studentized bootstrap and to the self normalized law of the iterated logarithm are discussed

paracontrolled quasilinear spdes we introduce a non linear paracontrolled calculus and use it to renor malise a class of singular spdes including certain quasilinear variants of the periodic two dimensional parabolic anderson model

a central limit theorem for the kpz equation we consider the kpz equation in one space dimension driven by a stationary centred space time random field which is sufficiently integrable and mixing but not necessarily gaussian we show that in the weakly asymmetric regime the solution to this equation considered at a suitable large scale and in a suitable reference frame converges to the hopf cole solution to the kpz equation driven by space time gaussian white noise while the limiting process depends only on the integrated variance of the driving field the diverging constants appearing in the definition of the reference frame also depend on higher order moments

global existence to the bgk model of boltzmann equation abstract we present an existence and a stability proof for solutions to the bgk model of boltzmann equation f v x f f m f t 0 x r n v r n m f p 2 t n 2 exp v u 2 2t p pu p u 2 pt 1 x r n 1 v v 2 f t x v dv it relies on the strong compactness of u t and on a new estimate on the third moment of v dv we also prove the entropy relation for 1

open asep in the weakly asymmetric regime we consider asep on a bounded interval and on a half line with sources and sinks on the full line bertini and giacomin in 1997 proved convergence under weakly asymmetric scaling of the height function to the solution of the kpz equation we prove here that under similar weakly asymmetric scaling of the sources and sinks as well the bounded interval asep height function converges to the kpz equation on the unit interval with neumann boundary conditions on both sides different parameter for each side and likewise for the half line asep to kpz on a half line this result can be interpreted as showing that the kpz equation arises at the triple critical point maximal current high density low density of the open asep 2018 wiley periodicals inc

time dependent critical fluctuations of a one dimensional local mean field model one dimensional stochastic ising systems with a local mean field interaction kac potential are investigated it is shown that near the critical temperature of the equilibrium gibbs distribution the time dependent process admits a scaling limit given by a nonlinear stochastic pde the initial conditions of this approximation theorem are then verified for equilibrium states when the temperature goes to its critical value in a suitable way earlier results of bertini presutti rudiger saada are improved the proof is based on an energy inequality obtained by coupling the glauber dynamics to its voter type linear approximation

weak universality for a class of 3d stochastic reaction diffusion models we establish the large scale convergence of a class of stochastic weakly nonlinear reaction diffusion models on a three dimensional periodic domain to the dynamic phi 3 4 model within the framework of paracontrolled distributions our work extends previous results of hairer and xu to nonlinearities with a finite amount of smoothness in particular c 9 is enough we use the malliavin calculus to perform a partial chaos expansion of the stochastic terms and control their l p norms in terms of the graphs of the standard phi 3 4 stochastic terms

kpz equation limit of higher spin exclusion processes we prove that under a particular weak scaling the 4 parameter interacting particle system introduced by corwin and petrov comm math phys 343 2016 651 700 converges to the kardar parisi zhang kpz equation this expands the relatively small number of systems for which weak universality of the kpz equation has been demonstrated

the kpz limit of asep with boundary it was recently proved in corwin and shen cpam cs16 that under weakly asymmetric scaling the height functions for asep with sources and sinks converges to the hopf cole solution of the kpz equation with inhomogeneous neumann boundary conditions in their assumptions cs16 chose positive values for the neumann boundary condition and they assumed initial data which is close to stationarity by developing more extensive heat kernel estimates we clarify and extend their results to negative values of the neumann boundary parameters and we also show how to generalize their results to empty initial data which is very far from stationarity combining our result with barraquand et al duke math j bbcw17 we obtain the laplace transform of the one point distribution for half line kpz and use this to confirm t1 3 scale goe tracy widom long time fluctuations at the origin

on the cauchy problem for boltzmann equations global existence and weak stability we study the large data cauchy problem for boltzmann equations with general collision kernels we prove that sequences of solutions which satisfy only the physically natural a priori bounds converge weakly in l to a solution from this stability result we deduce global existence of a solution to the cauchy problem our method relies upon recent compactness results for velocity averages a new formulation of the boltzmann equation which involves nonlinear normalization and an analysis of subsolutions and supersolutions it allows us to overcome the lack of strong a priori estimates and define a meaningful collision operator for general configurations

nonlinear fluctuations of weakly asymmetric interacting particle systems we introduce what we call the second order boltzmann gibbs principle which allows one to replace local functionals of a conservative one dimensional stochastic process by a possibly nonlinear function of the conserved quantity this replacement opens the way to obtain nonlinear stochastic evolutions as the limit of the fluctuations of the conserved quantity around stationary states as an application of this second order boltzmann gibbs principle we introduce the notion of energy solutions of the kpz and stochastic burgers equations under minimal assumptions we prove that the density fluctuations of one dimensional stationary weakly asymmetric conservative particle systems are sequentially compact and that any limit point is given by energy solutions of the stochastic burgers equation we also show that the fluctuations of the height function associated to these models are given by energy solutions of the kpz equation in this sense unfortunately we lack a uniqueness result for these energy solutions we conjecture these solutions to be unique and we show some regularity results for energy solutions of the kpz burgers equation supporting this conjecture

stochastic burgers and kpz equations from particle systems we consider two strictly related models a solid on solid interface growth model and the weakly asymmetric exclusion process both on the one dimensional lattice it has been proven that in the diffusive scaling limit the density field of the weakly asymmetric exclusion process evolves according to the burgers equation and the fluctuation field converges to a generalized ornstein uhlenbeck process we analyze instead the density fluctuations beyond the hydrodynamical scale and prove that their limiting distribution solves the non linear burgers equation with a random noise on the density current for the solid on solid model we prove that the fluctuation field of the interface profile if suitably rescaled converges to the kardar parisi zhang equation this provides a microscopic justification of the so called kinetic roughening i e the non gaussian fluctuations in some non equilibrium processes our main tool is the cole hopf transformation and its microscopic version we also develop a mathematical theory for the macroscopic equations

equations de type de boltzmann spatialement homogenes we study the construction of a nonhomogeneous n valued markov process whose laws at fixed times evolve according to a boltzmann type equation spatially homogeneous we then consider the problem of the asymptotic propagation of chaos for a system of a large number of interacting particles

two dimensional navier stokes equations driven by a space time white noise abstract we study the two dimensional navier stokes equations with periodic boundary conditions perturbed by a space time white noise it is shown that although the solution is not expected to be smooth the nonlinear term can be defined without changing the equation we first construct a stationary martingale solution then we prove that for almost every initial data with respect to a measure supported by negative spaces there exists a unique global solution in the strong probabilistic sense

complete classification of the asymptotical behavior for singular c s model on the real line abstract in this paper we study the singular cucker smale c s model on the real line for long range case i e 1 we prove the uniqueness of the solution in the sense of definition 2 1 and the unconditional flocking emergence moreover the sufficient and necessary condition for collision and sticking phenomenon will be provided for short range case i e 1 we construct the uniform in time lower bound of the relative distance between particles and provide the sufficient and necessary condition for the emergence of multi cluster formation for critical case i e 1 we show the uniform lower bound of the relative distance and unconditional flocking emergence these results provide a complete classification of the collective behavior for c s model on the real line

lattice approximation to the dynamical phi_ 3 4 model we study the lattice approximations to the dynamical 43 34 model by paracontrolled distributions proposed in forum math pi 3 2015 e6 we prove that the solutions to the lattice systems converge to the solution to the dynamical 43 34 model in probability locally uniformly in time since the dynamical 43 34 model is not well defined in the classical sense and renormalisation has to be performed in order to define the nonlinear term a corresponding suitable drift term is added in the stochastic equations for the lattice systems

discontinuous galerkin method for a nonlocal hydrodynamic model of flocking dynamics abstract in this paper we devise an efficient and robust numerical method for a nonlocal nonlinear model of flocking dynamics the governing equations are a hydrodynamic limit of the model of cucker and smale which consists of the compressible euler equations with added nonlinear nonlocal interaction terms the numerical scheme is based on the discontinuous galerkin method a semi implicit scheme is used in the time discretization which requires only the solution of one linear system per time level while retaining the stability of an implicit scheme a crucial point is the construction of a suitable linearization of the nonlocal terms which does not result in fill in of the system matrices element wise and inter element artificial diffusion is added to the scheme along with a postprocessing procedure to deal with near vacuum states that typically arise in the solution we demonstrate the efficiency and robustness of the scheme on numerical experiments in 1d and 2d

emergent behavior in flocks we provide a model for both continuous and discrete time describing the evolution of a flock our model is parameterized by a constant beta capturing the rate of decay which in our model is polynomial of the influence between birds in the flock as they separate in space our main result shows that when beta 1 2 convergence of the flock to a common velocity is guaranteed while for betages1 2 convergence is guaranteed under some condition on the initial positions and velocities of the birds only

invariant measures for the 2d defocusing nonlinear schrodinger equation consider the2d defocusing cubic nlsiu t u u u 2 0 with hamiltonian smallint left left nabla phi right 2 tfrac 1 2 left phi right 4 right it is shown that the gibbs measure constructed from the wick ordered hamiltonian i e replacing 4 by 4 is an invariant measure for the appropriately modified equationiu t u u u 2 2 u 2 dx u 0 there is a well defined flow on thesupport of the measure in fact it is shown that for almost all data the solutionu u 0 satisfiesu t e it c hs for somes 0 first a result local in time is established and next measure invariance considerations are used to extend the local result to a global one cf b2

biometrika centenary theory and general methodology note reprinted in biometrika one hundred years edited by d m titterington and d r cox oxford university press 11 50 reference stat article 2001 002doi 10 1093 biomet 88 1 13view record in web of science record created on 2006 04 21 modified on 2017 05 12