measurability of optimal transportation and convergence rate for landau type interacting particle systems in this paper we consider nonlinear diffusion processes driven by space time white noises which have an interpretation in terms of partial differential equations for a specific choice of coefficients they correspond to the landau equation arising in kinetic theory the main goal of the paper is to construct an easily simulable diffusive interacting particle system converging towards this nonlinear process and to obtain an explicit pathwise rate this requires to find a significant coupling between finitely many brownian motions and the infinite dimensional white noise process the key idea will be to construct the right brownian motions by pushing forward the white noise processes through the brenier map realizing the optimal transport between the law of the nonlinear process and the empirical measure of independent copies of it a crucial problem will then be to establish the joint measurability of this optimal transport map with respect to the space variable and the parameter time randomness that makes the marginals vary to overcome this point we shall prove a general measurability result for the mass transportation problem and for the supports of the optimal transfer plans in the sense of set valued mappings this will allow us to construct the coupling and to obtain explicit convergence rates

on flows associated to sobolev vector fields in wiener spaces an approach a la diperna lions in this paper we extend the diperna lions theory of flows associated to sobolev vector fields to the case of cameron martin valued vector fields in wiener spaces e having a sobolev regularity the proof is based on the analysis of the continuity equation in e and on uniform gaussian commutator estimates in finite dimensional spaces

existence and uniqueness of weak solutions to ginzburg landau equation with external noise and stochastic perturbation abstract in this paper the ginzburg landau and fractional ginzburg landau equations with external noise and stochastic perturbations are studied and we obtain the global existence and uniqueness of weak l 1 solutions our proof is supported by the generalized ito formula and a careful analysis of the nonlinear transport equation after founding the well posedness for the nonlinear transport problem in the class of l p 1 p we also gain the well posedness of the ginzburg landau and fractional ginzburg landau equations in the same spaces correspondingly

extreme semilinear copulas abstract we study the extreme points in the krein milman sense of the class of semilinear copulas and provide their characterization related results into the more general setting of conjunctive aggregation functions i e semi copulas and quasi copulas are also presented

a metric space of subcopulas an approach via hausdorff distance abstract in this work we define a distance function on the set of bivariate subcopulas to generate a compact metric space moreover the copula space equipped with the uniform distance is essentially a metric subspace of this subcopula space we also characterize the convergence in this space and provide the interrelationship with the convergence of the distribution functions

dissipation statistics of a passive scalar in a multidimensional smooth flow we compute analytically the probability distribution function p e of the dissipation field e 2 of a passive scalar advected by a d dimensional random flow in the limit of large peclet and prandtl numbers batchelor kraichnan regime the tail of the distribution is a stretched exponential for e ln p e d2e 1 3

rearrangement inequalities for functionals with monotone integrands the inequalities of hardy littlewood and riesz say that certain integrals involving products of two or three functions increase under symmetric decreasing rearrangement it is known that these inequalities extend to integrands of the form f u1 um where f is supermodular in particular they hold when f has nonnegative mixed second derivatives i jf for all i j this paper concerns the regularity assumptions on f and the equality cases it is shown here that extended hardy littlewood and riesz inequalities are valid for supermodular integrands that are just borel measurable under some nondegeneracy conditions all equality cases are equivalent to radially decreasing functions under transformations that leave the functionals invariant i e measure preserving maps for the hardy littlewood inequality translations for the riesz inequality the proofs rely on monotone changes of variables in the spirit of sklar s theorem

characterization of all copulas associated with non continuous random variables we introduce a constructive method by means of a doubly stochastic measure to describe all the copulas that in view of sklar s theorem are able to connect a bivariate distribution to its marginals we use this to give the lower and upper optimal bounds for all the copulas that extend a given subcopula

quantitative concentration inequalities for empirical measures on non compact spaces we establish quantitative concentration estimates for the empirical measure of many independent variables in transportation distances as an application we provide some error bounds for particle simulations in a model mean field problem the tools include coupling arguments as well as regularity and moment estimates for solutions of certain diffusive partial differential equations

well posedness of the spatially homogeneous landau equation for soft potentials we consider the spatially homogeneous landau equation of kinetic theory and provide a differential inequality for the wasserstein distance with quadratic cost between two solutions we deduce some well posedness results the main difficulty is that this equation presents a singularity for small relative velocities our uniqueness result is the first one in the important case of soft potentials furthermore it is almost optimal for a class of moderately soft potentials that is for a moderate singularity indeed in such a case our result applies for initial conditions with finite mass energy and entropy for the other moderatley soft potentials we assume additionnally some moment conditions on the initial data for very soft potentials we obtain only a local in time well posedness result under some integrability conditions our proof is probabilistic and uses a stochastic version of the landau equation in the spirit of tanaka

a certain class of diffusion processes associated with nonlinear parabolic equations we introduce a martingale problem to associate diffusion processes with a kind of nonlinear parabolic equation then we show the existence and uniqueness theorems for solutions to the martingale problem

hydrodynamical limit for a hamiltonian system with weak noise starting from a general hamiltonian system with superstable pairwise potential we construct a stochastic dynamics by adding a noise term which exchanges the momenta of nearby particles we prolve that in the scaling limit the time conserved quantities energy momenta and density satisfy the euler equation of conservation laws up to a fixed timet provided that the euler equation has a smooth solution with a given initial data up to timet the strength of the noise term is chosen to be very small but nonvanishing so that it disappears in the scaling limit

copulas with given track and opposite track sections solution to a problem on diagonals abstract in this note we provide conditions for the existence of a copula with given track and opposite track sections generalizing conditions for the existence of copulas with given diagonal and opposite diagonal sections in de amo et al 2016 3 we also use this construction to characterize a class of copulas with a fixed diagonal section providing as a consequence a solution to the problem 2 4 posed by c sempi in mesiar and sarkoci 2010 16

renormalized solutions to the vlasov equation with coefficients of bounded variation we prove that weak bounded solutions to the vlasov equation with bv coefficients have the renormalization property and we show that when the renormalization property holds for a general transport equation it also holds for only lipschitz nonlinearities

on degrees of asymmetry of a copula with respect to a track abstract in this paper we provide lower and upper bounds on the degree of asymmetry of a copula with respect to a track a continuous and strictly increasing curve in the unit square generalizing the problem with respect to the main diagonal we also study an average asymmetry in sections for given tracks finally we also give an asymmetry map with respect to the opposite diagonal

on the rate of convergence in the entropic central limit theorem we study the rate at which entropy is produced by linear combinations of independent random variables which satisfy a spectral gap condition

the behavior of the specific entropy in the hydrodynamic scaling limit the paper studies the behavior of the specific entropyfor onedimensional simple exclusion processes under the hydrodynamic scaling of time and space it is shown that if the initial configurations possess a macroscopic profile then for each positive macroscopic time the specific microscopic entropyconverges to the macroscopic entropy the latter is defined in terms of the solution of the corresponding hydrodynamic equation 1 introduction the paper concerns simple exclusion models on the periodic integer lattice the symmetric simple exclusion process is reversible and we studyit under diffusive parabolic scaling of time and space the corresponding hydrodynamic equation describing the behavior of the limiting macroscopic densityis the heat equation it has smooth solutions uniquely defined bythe initial data asy mmetric simple exclusion models are nonreversible and require euler hyperbolic scaling which leads to inviscid burgers type equations solutions of such equations are not unique and may develop shocks relevant solutions are known to be the entropic solutions of these equations we studythe behavior of the microscopic entropyrelative to uniform measures and to local gibbs measures in words a version of the main result can be stated as follows let the initial configurations be deterministic and possess a macroscopic profile then under the appropriate scaling of time and space for anypositive macroscopic time the specific microscopic entropyconverges to the macroscopic entropycalculated from the relevant solution of the hydrodynamic equation the main emphasis is placed on the asymmetric case when the solution of the hydrodynamic equation develops shocks and the macroscopic entropy is lost at each shock our results show that in the scaling limit the particle system closely follows the hydrodynamic equation and loses the same amount of the specific entropyat the same macroscopic time the keyidea of this paper is presented in lemma a 1 it is a rather general statement which could be adopted for applications to other models one of the important ingredients of the proof of lemma a 1 is the logarithmic sobolev inequality theorem a 1 let us stress that we do not need an estimate on the constant in the logarithmic sobolev inequality the main difficulty in apply ing

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

some applications of mass transport to gaussian type inequalities as discovered by brenier mapping through a convex gradient gives the optimal transport in n in the present article this map is used in the setting of gaussian like measures to derive an inequality linking entropy with mass displacement by a straightforward argument as a consequence logarithmic sobolev and transport inequalities are recovered finally a result of caffarelli on the brenier map is used to obtain gaussian correlation inequalities

rate of convergence of the nanbu particle system for hard potentials and maxwell molecules we consider the numerically motivated nanbu stochastic particle system associated to the spatially homogeneous boltzmann equation for true hard potentials we establish a rate of propagation of chaos of the particle system to the unique solution of the boltzmann equation more precisely we estimate the expectation of the squared wasserstein distance with quadratic cost between the empirical measure of the particle system and the solution the rate we obtain is almost optimal as a function of the number of particles but is not uniform in time

three superposition principles currents continuity equations and curves of measures abstract we establish a general superposition principle for curves of measures solving a continuity equation on metric spaces without any smooth structure nor underlying measure representing them as marginals of measures concentrated on the solutions of the associated ode defined by some algebra of observables we relate this result with decomposition of acyclic normal currents in metric spaces as an application a slightly extended version of a probabilistic representation for absolutely continuous curves in kantorovich wasserstein spaces originally due to s lisini is provided in the metric framework this gives a hierarchy of implications between superposition principles for curves of measures and for metric currents

strong solutions of stochastic equations with singular time dependent drift we prove existence and uniqueness of strong solutions to stochastic equations in domains with unit diffusion and singular time dependent drift b up to an explosion time we only assume local l q l p integrability of b in g with d p 2 q 1 we also prove strong feller properties in this case if b is the gradient in x of a nonnegative function blowing up as g x g we prove that the conditions 2d t k 2d t ke 0 2 imply that the explosion time is infinite and the distributions of the solution have sub gaussian tails

the lyapunov spectrum of a continuous product of random matrices we present a functional integration method for the averaging of continuous productsp t ofn n random matrices as an application we compute exactly the statistics of the lyapunov spectrum ofp t this problem is relevant to the study of the statistical properties of various disordered physical systems and specifically to the computation of the multipoint correlators of a passive scalar advected by a random velocity field apart from these applications our method provides a general setting for computing statistical properties of linear evolutionary systems subjected to a white noise force field

existence of periodic probability solutions to fokker planck equations with applications abstract in the present paper we consider a fokker planck equation associated to periodic stochastic differential equations with irregular coefficients we define periodic probability solutions to be periodic analogs of stationary measures for stationary fokker planck equations and study their existence in both non degenerate and degenerate cases in the non degenerate case a lyapunov condition is imposed to ensure the existence of periodic probability solutions to the fokker planck equation with sobolev coefficients in the degenerate case with slightly more regular coefficients the existence is established under the same lyapunov condition as applications of our results we construct periodic probability solutions to fokker planck equations associated to stochastic damping hamiltonian systems and stochastic differential inclusions

new results on discrete copulas and quasi copulas abstract in this paper we prove that the set of all irreducible discrete quasi copulas coincides with the set of all discrete quasi copulas with minimal range we also provide answers throughout some counterexamples to two questions posed in aguilo et al 2 regarding both discrete quasi copulas with minimal range and irreducible discrete quasi copulas some additional results concerning discrete quasi copulas are also given

probabilistic representation for solutions of an irregular porous media type equation we consider a porous media type equation over all of d d 1 with monotone discontinuous coefficient with linear growth and prove a probabilistic representation of its solution in terms of an associated microscopic diffusion the interest in such singular porous media equations is due to the fact that they can model systems exhibiting the phenomenon of self organized criticality one of the main analytic ingredients of the proof is a new result on uniqueness of distributional solutions of a linear pde on 1 with not necessarily continuous coefficients

uniqueness of continuous solutions for bv vector fields we consider a vector field whose coefficients are functions of bounded variation with a bounded divergence we prove the uniqueness of continuous solutions for the cauchy problem

zero linear copulas abstract we introduce a new method for constructing copulas c a special type of aggregation functions based on the idea that c u v 0 if u g v for any continuous and decreasing function g 0 1 0 1 such that g 1 0 and c is linear if u g v 1 we also study the relationship between these copulas and other known families of copulas we provide some salient properties and measure the dependence and asymmetry in this class of copulas

propagation of chaos for the landau equation with moderately soft potentials we consider the 3d landau equation for moderately soft potentials 2 0 2 0 with the usual notation as well as a stochastic system of nn particles approximating it we first establish some strong weak stability estimates for the landau equation which are fully satisfactory only when 1 0 1 0 we next prove under some appropriate conditions on the initial data the so called propagation of molecular chaos that is that the empirical measure of the particle system converges to the unique solution of the landau equation the main difficulty is the presence of a singularity in the equation when 1 0 1 0 the strong weak uniqueness estimate allows us to use a coupling argument and to obtain a rate of convergence when 2 1 2 1 we use the classical martingale method introduced by mckean to control the singularity we have to take advantage of the regularity provided by the entropy dissipation unfortunately this dissipation is too weak for some very rare aligned configurations we thus introduce a perturbed system with an additional noise show the propagation of chaos for this system and finally prove that the additional noise is almost never used in the limit n n

copulas diagonals and tail dependence we present some known and novel aspects about bivariate copulas with prescribed diagonal section by highlighting their use in the description of the tail dependence moreover we present the tail concentration function which depends on the diagonal section of a copula as a tool to give a description of tail dependence at finite scale the tail concentration function is hence used to introduce a graphical tool that can help to distinguish different families of copulas in the copula test space moreover it serves as a basis to determine the grouping structure of different financial time series by taking into account their pairwise tail behavior

universal direct cascade in two dimensional turbulence we consider the correlation functions of vorticity ensuremath omega in the region of the direct cascade in a steady two dimensional turbulence the nonlocality of the cascade in k space provides for logarithmic corrections to the expressions obtained by dimension estimates and the main problem is to take those logarithms into account our procedure starts directly with the euler equation rewritten in the comoving reference frame we express the correlation functions of the vorticity via the correlation functions of the pumping force and renormalized strain it enables us to establish a set of integrodifferential equations which gives a logarithmic renormalization of the vorticity correlation functions in the inertial interval we find the indices characterizing the logarithmic behavior of different correlation functions for example the two point simultaneous functions are as follows mathrm ensuremath omega mathit n mathbf r 1 mathrm ensuremath omega mathit n mathbf r 2 ensuremath sim mathit p 2 ln l mathbf r 1 mathbf r 2 ensuremath vert 2 mathit n 3 where l is the pumping scale we demonstrate that the form of those correlation functions is universal i e independent of the pumping the only pumping related value which enters the expressions is the enstrophy production rate mathit p 2 the contributions related to pumping rates mathit p mathit n of the higher order integrals of motion are demonstrated to be small in comparison with the ones induced by mathit p 2 we establish also the time dependence of the correlation functions the correlation time ensuremath tau in the comoving reference frame is the same for the vorticity and strain and is scale dependent ensuremath tau ensuremath propto mathrm ln 2 3 l r we reformulate our procedure in the diagrammatic language to reinforce the conclusions

zero sets of copulas abstract we study conditions on sets in order to be zero sets of semi copulas quasi copulas and with special attention of copulas we find necessary and sufficient conditions for characterizing the zero sets of absolutely continuous copulas and copulas whose support coincides with the closure of the complementary of the zero set moreover we study several topological properties and the lattice theoretic structure and characterize the zero sets of the class of archimedean copulas

best possible bounds on the set of copulas with a given value of spearman s footrule abstract in this paper we find pointwise best possible bounds on the set of copulas with a given value of the spearman s footrule coefficient we show that the lower bound is always a copula but unlike the bounds on sets of copulas with a given value of other measures such as kendall s tau spearman s rho and blonqvist s beta the upper bound can be a copula or a proper quasi copula we characterised both of these cases

the distribution of the probability mass of biconic copulas we study the probability mass distribution of biconic copulas provide a counterexample to a result given in jwaid et al 2012 29 concerning a characterization of singular biconic copulas and provide a new one

ode sde and pde this note consisting of two parts gives a survey of new developments on the topic concerning ordinary differential equations stochastic differential equations and partial differential equations in the first one our aim is to prove the existence of a flow of measurable maps associated to a vector field belonging to a sobolev space to this end we discuss the link with transport equations and continuity equations the second part deals with stochastic differential equations we construct explicitly strong solutions beyond lipschitz conditions the existence of a stochastic flow of homeomorphisms under global lipschitz conditions is fully proved and the relation with fokker planck equations is discussed and finally we emphasize the role of ellipticity to make weak solutions to strong one

well posedness of the transport equation by stochastic perturbation we consider the linear transport equation with a globally holder continuous and bounded vector field with an integrability condition on the divergence while uniqueness may fail for the deterministic pde we prove that a multiplicative stochastic perturbation of brownian type is enough to render the equation well posed this seems to be the first explicit example of a pde of fluid dynamics that becomes well posed under the influence of a multiplicative noise the key tool is a differentiable stochastic flow constructed and analyzed by means of a special transformation of the drift of ito tanaka type

a special class of stationary flows for two dimensional euler equations a statistical mechanics description we consider the canonical gibbs measure associated to an vortex system in a bounded domain at inverse temperature widetilde beta and prove that in the limitn widetilde beta n n 1 where 8 here denotes the vorticity intensity of each vortex the one particle distribution function n nx x converges to a superposition of solutions of the following mean field equation left begin array 20 c varrho beta x frac e beta psi mathop smallint limits lambda e beta psi delta psi varrho beta in lambda psi partial lambda 0 end array right

finite time lyapunov exponents for products of random transformations i show how continuous products of random transformations constrained by a generic group structure can be studied by using iwasawa s decomposition into angular diagonal and shear degrees of freedom in the case of a gaussian process a set of variables adapted to the iwasawa decomposition and still having a gaussian distribution is introduced and used to compute the statistics of the finite time lyapunov spectrum of the process the variables also allow to show the exponential freezing of the shear degrees of freedom which contain information about the lyapunov eigenvectors

a note on stochastic semilinear equations and their associated fokker planck equations abstract the main purpose of this paper is to prove existence and uniqueness of probabilistically weak and strong solutions to stochastic differential equations sde on hilbert spaces under a new approximation condition on the drift recently proposed in 6 to solve fokker planck equations fpe extended in this paper to a considerably larger class of drifts as a consequence we prove existence of martingale solutions to the sde whose time marginals then solve the corresponding fpe applications include stochastic semilinear partial differential equations with white noise and a non linear drift part which is the sum of a burgers type part and a reaction diffusion part the main novelty is that the latter is no longer assumed to be of at most linear but of at most polynomial growth this case so far had not been covered by the existing literature we also give a direct and more analytic proof for existence of solutions to the corresponding fpe extending the technique from 6 to our more general framework which in turn requires to work on a suitable gelfand triple rather than just the hilbert state space

some results on homeomorphisms between fractal supports of copulas abstract we consider parametric classes t r r 0 1 2 of so called transformation matrices and their induced families a r r 0 1 2 and r r 0 1 2 of two dimensional copulas and doubly stochastic measures with fractal support respectively by using tools from symbolic dynamics we show that for each pair r r 0 1 2 with r r there exists a homeomorphism h r r between the supports of r and r mapping a borel set of r measure one to a set of r measure zero differentiability properties of these homeomorphisms are studied and hausdorff dimensions of related sets are calculated several examples and graphics illustrate the main results

the distribution of the probability mass of conic copulas in this note we study the probability mass distribution of conic copulas provide counterexamples to a result given in jwaid et al 21 concerning a characterization of singular conic copulas and provide a new one

lp solutions of fokker planck equations abstract we obtain the existence and uniqueness of l p solutions 1 p for fokker planck equations with irregular and degenerate coefficients moreover as an application we also derive the existence and uniqueness of l 1 solutions for the fokker planck boltzmann equation

extensions of subcopulas abstract in view of sklar s theorem the probability distribution function of every not necessarily continuous random vector can be uniquely decomposed in terms of the marginal distributions of its components and a suitable subcopula the study of such latter functions is therefore of interest for understanding the dependence information of non continuous variables here we investigate some analytical properties of the class of subcopulas including compactness with respect to a novel metric approximations and baire category results moreover under a suitable assumption we describe all possible extensions from a subcopula to a copula in any dimension

inverse cascade and intermittency of passive scalar in one dimensional smooth flow random advection of a lagrangian tracer scalar field ensuremath theta t x by a one dimensional spatially smooth and short correlated in time velocity field is considered scalar fluctuations are maintained by a source concentrated at the integral scale l the statistical properties of both scalar differences and the dissipation field are analytically determined exploiting the dynamical formulation of the model the gaussian statistics known to be present at small scales for incompressible velocity fields emerges here at large scales x ensuremath gg l these scales are shown to be excited by an inverse cascade of ensuremath theta 2 and the probability distribution function pdf of the corresponding scalar differences to approach the gaussian form as larger and larger scales are considered small scale x ensuremath ll l statistics is shown to be strongly non gaussian a collapse of scaling exponents for scalar structure functions takes place moments of order pg 1 all scale linearly independently of the order p smooth scaling x p is found for ensuremath 1lpl1 tails of the scalar difference pdf are exponential while at the center a cusped shape tends to develop when smaller and smaller ratios x l are considered the same tendency is present for the scalar gradient pdf with respect to the inverse of the p eclet number the pumping to diffusion scale ratio the tails of the latter pdf are however much more extended decaying as a stretched exponential of exponent 2 3 smaller than unity this slower decay is physically associated with the strong fluctuations of the dynamical dissipative scale

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

kac s program in kinetic theory this paper is devoted to the study of propagation of chaos and mean field limits for systems of indistinguishable particles undergoing collision processes the prime examples we will consider are the many particle jump processes of kac and mckean kac in proceedings of the third berkeley symposium on mathematical statistics and probability 1954 1955 vol iii pp 171 197 1956 mckean in j comb theory 2 358 382 1967 giving rise to the boltzmann equation we solve the conjecture raised by kac proceedings of the third berkeley symposium on mathematical statistics and probability 1954 1955 vol iii pp 171 197 1956 motivating his program on the rigorous connection between the long time behavior of a collisional many particle system and the one of its mean field limit for bounded as well as unbounded collision rates

the homogeneous boltzmann hierarchy and statistical solutions to the homogeneous boltzmann equation an existence and uniqueness result for the homogeneous boltzmann hierarchy is proven by exploiting the statistical solutions to the homogeneous boltzmann equation

on kac s chaos and related problems this paper is devoted to establish quantitative and qualitative estimates related to the notion of chaos as firstly formulated by m kac cite kac1956 in his study of mean field limit for systems of n undistinguishable particles as n to infty first we quantitatively liken three usual measures of it kac s chaos some involving the all n variables other involving a finite fixed number of variables the cornerstone of the proof is a new representation of the monge kantorovich wasserstein mkw distance for symmetric n particle probability measures in terms of the distance between the law of the associated empirical measures on the one hand and a new estimate on some mkw distance on probability measures spaces endowed with a suitable hilbert norm taking advantage of the associated good algebraic structure next we define the notion of it entropy chaos and it fisher information chaos in a similar way as defined by carlen et al cite ccllv we show that it fisher information chaos is stronger than it entropy chaos which in turn is stronger than it kac s chaos more importantly with the help of the hwi inequality of otto villani we establish a quantitative estimate between these quantities which in particular asserts that it kac s chaos plus it fisher information bound implies it entropy chaos we then extend the above quantitative and qualitative results about chaos in the framework of probability measures with support on the it kac s spheres revisiting cite ccllv and giving a possible answer to cite open problem 11 ccllv additionally to the above mentioned tool we use and prove an optimal rate local clt in l infty norm for distributions with finite 6 th moment and finite l p norm for some p 1 last we investigate how our techniques can be used without assuming chaos in the context of probability measures mixtures introduced by de finetti hewitt and savage in particular we define the level 3 fisher information for mixtures and prove that it is l s c and affine as that was done in cite rr for the level 3 boltzmann s entropy

a note on an idempotent transformation of absolutely continuous archimedean copulas abstract in de baets et al 6 the authors define for a copula c an idempotent transformation d c which in some cases generates new copulas in this note we give a negative answer to a question posed in 6 showing that d c is not necessarily a copula when c is an absolutely continuous archimedean copula

on the fokker planck boltzmann equation we consider the boltzmann equation perturbed by fokker planck type operator to overcome the lack of strong a priori estimates and to define a meaningful collision operator we introduce a notion of renormalized solution which enables us to establish stability results for sequences of solutions and global existence for the cauchy problem with large data the proof of stability and existence combines renormalization with an analysis of a defect measure

collective behavior models with vision geometrical constraints truncated noises and propagation of chaos abstract we consider large systems of stochastic interacting particles through discontinuous kernels which has vision geometrical constrains we rigorously derive a vlasov fokker planck type of kinetic mean field equation from the corresponding stochastic integral inclusion system more specifically we construct a global in time weak solution to the stochastic integral inclusion system and derive the kinetic equation with the discontinuous kernels and the inhomogeneous noise strength by employing the 1 wasserstein distance

ordinary differential equations transport theory and sobolev spaces we obtain some new existence uniqueness and stability results for ordinary differential equations with coefficients in sobolev spaces these results are deduced from corresponding results on linear transport equations which are analyzed by the method of renormalized solutions

existence and uniqueness of martingale solutions for sdes with rough or degenerate coefficients abstract in this paper we extend recent results on the existence and uniqueness of solutions of odes with non smooth vector fields to the case of martingale solutions in the stroock varadhan sense of sdes with non smooth coefficients in the first part we develop a general theory which roughly speaking allows to deduce existence uniqueness and stability of martingale solutions for l d almost every initial condition x whenever existence and uniqueness is known at the pde level in the l setting and conversely if existence and uniqueness of martingale solutions is known for l d a e initial condition then existence and uniqueness for the pde holds in the second part of the paper we consider situations where on the one hand no pointwise uniqueness result for the martingale problem is known and on the other hand well posedness for the fokker planck equation can be proved thus the theory developed in the first part of the paper is applicable in particular we will study the fokker planck equation in two somehow extreme situations in the first one assuming uniform ellipticity of the diffusion coefficients and lipschitz regularity in time we are able to prove existence and uniqueness in the l 2 setting in the second one we consider an additive noise and assuming the drift b to have bv regularity and allowing the diffusion matrix a to be degenerate also identically 0 we prove existence and uniqueness in the l setting therefore in these two situations our theory yields existence uniqueness and stability results for martingale solutions

on α β homogeneous copulas we introduce and characterize the family of a b homogeneous copulas and study some of its salient properties such as the greatest and the lowest copulas measures of association and the relationships with the a migrativity and invariant copulas

stochastic particle approximations for generalized boltzmann models and convergence estimates fied boltzmann equation with general motion between collisions and nonlinear bounded jump collision operator and give the nonlinear martingale problem it solves we consider various linear interacting particle systems in order to approximate this nonlinear process we prove propagation of chaos in variation norm on path space with a precise rate of convergence using coupling and interaction graph techniques and a representation of the nonlinear process on a boltzmann tree no regularity nor uniqueness assumption is needed we then consider a nonlinear equation with both vlasov and boltzmann terms and give a weak pathwise propagation of chaos result using a compactness uniqueness method which necessitates some regularity these results imply functional laws of large numbers and extend to multitype models we give algorithms simulating or approximating the particle systems 1 framework and main results the boltzmann equation describes the evolution of the limit density f t x v of molecules of a rarefied gas at time t position x and speed v f is positive and f f t x v dx dv 1 it is given by

a strengthened central limit theorem for smooth densities abstract we consider the central limit theorem for the probability density function n x of the standardized sum of independent and identically distributed random variables with finite variance and regular probability density function by showing boundedness of different convex functionals along the sequence fn we prove convergence in various norms to the normal density

weak solutions to fokker planck equations and mean field games we deal with systems of pdes arising in mean field games theory where viscous hamilton jacobi and fokker planck equations are coupled in a forward backward structure we consider the case of local coupling when the running cost depends on the pointwise value of the distribution density of the agents in which case the smoothness of solutions is mostly unknown we develop a complete weak theory proving that those systems are well posed in the class of weak solutions for monotone couplings under general growth conditions and for superlinear convex hamiltonians as a key tool we prove new results for fokker planck equations under minimal assumptions on the drift through a characterization of weak and renormalized solutions the results obtained give new perspectives even for the case of uncoupled equations as far as the uniqueness of weak solutions is concerned

characterization of copulas with given diagonal and opposite diagonal sections in recent years special attention has been devoted to the problem of finding a copula the diagonal section and opposite diagonal section of which are known for given diagonal function and opposite diagonal functions we provide necessary and sufficient conditions for the existence of a copula to have these functions as diagonal and opposite diagonal sections we make use of techniques related to interpolation between the diagonals the construction of checkerboard copulas and linear programming this result allows us to solve two open problems to characterize the class of copulas where the knowledge of diagonal and opposite diagonal sections determines the copula in a unique way and to formulate necessary and sufficient conditions for each pair of such functions to be the diagonal and opposite diagonal sections of a unique copula

lagrangian flows driven by bv fields in wiener spaces we establish the renormalization property for essentially bounded solutions of the continuity equation associated to bounded variation bv fields in wiener spaces with values in the associated cameron martin space thus obtaining by standard arguments new uniqueness and stability results for correspondent lagrangian flows

stochastic differential equations with coefficients in sobolev spaces abstract we consider the ito stochastic differential equation d x t j 1 m a j x t d w t j a 0 x t d t on r d the diffusion coefficients a 1 a m are supposed to be in the sobolev space w loc 1 p r d with p d and to have linear growth for the drift coefficient a 0 we distinguish two cases i a 0 is a continuous vector field whose distributional divergence a 0 with respect to the gaussian measure d exists ii a 0 has sobolev regularity w loc 1 p for some p 1 assume r d exp 0 a 0 j 1 m a j 2 a j 2 d d for some 0 0 in case i if the pathwise uniqueness of solutions holds then the push forward x t d admits a density with respect to d in particular if the coefficients are bounded lipschitz continuous then x t leaves the lebesgue measure leb d quasi invariant in case ii we develop a method used by g crippa and c de lellis for ode and implemented by x zhang for sde to establish existence and uniqueness of stochastic flow of maps

solution to two open problems on perturbations of the product copula abstract in this note we provide an answer to some open problems concerning perturbations of the product copula in terms of convex strong negations we also approach the case in higher dimensions

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

statistics of temperature increments in fully developed turbulence part ii experiments abstract a partial differential equation for the probability density function pdf of temperature increments in fully developed turbulence has been derived in a previous theoretical analysis s vaienti m ould rouis f auselmet and p le gal physica d 73 1994 99 this equation is a function of two conditional expectations q 1 and q 2 of respectively the increment of velocity and the temperature dissipation it was shown that these two quantities are the only parameters that govern the evolution of the pdfs for the temperature increments with the help of closure hypotheses on q 1 and q 2 the calculation of the asymptotic forms of this pdf is then tractable and displays a notable deviation from the gaussian behavior for large temperature increment fluctuations when the spatial separation r lies within the inertial and dissipative ranges the objective of the present paper is to complete our theoretical approach and to investigate experimentally from simultaneous measurements of velocity and temperature fluctuations obtained in a laboratory wind tunnel the evolution of the marginal and joint pdfs of the velocity and temperature increments attention is particularly paid to the conditional expectations q 1 and q 2

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

orthogonal grid constructions of copulas we establish a general framework for constructing copulas that can be regarded as a patchwork like assembly of arbitrary copulas with nonoverlapping rectangles as patches we derive a family of construction methods that require the choice of a single background copula when this background copula is the greatest copula we retrieve the well known ordinal sum construction while in the case of the smallest copula we obtain a construction method that is dual to the ordinal sum construction also nonsingular background copulas lead to suitable construction methods

on the existence of a trivariate copula with given values of a trivariate quasi copula at several points the existence of a multivariate copula with a given value of a multivariate quasi copula at a single point is known in the bivariate case the existence of a copula with given values of a quasi copula at two or three arbitrary points is also known in this paper we give an alternative proof of the existence of a trivariate copula with a given value of a trivariate quasi copula at a single point this proof relies on the reformulation of the existence problem as a linear programming minimization problem and its solution by means of the simplex algorithm the same method is then used to prove the existence of a trivariate copula with given values of a trivariate quasi copula at two arbitrary points in the unit cube we furthermore establish a counter example showing that the existence for given values at three points is not guaranteed this completes the analysis of the trivariate case

copulas with continuous strictly increasing singular conditional distribution functions abstract using iterated function systems induced by so called modifiable transformation matrices t and tools from symbolic dynamical systems we first construct mutually singular copulas a t with identical possibly fractal or full support that are at the same time singular with respect to the lebesgue measure 2 on 0 1 2 afterwards the established results are utilized for a simple proof of the existence of singular copulas a t with full support for which all conditional distribution functions y f x a t y are continuous strictly increasing and have derivative zero almost everywhere this result underlines the fact that conditional distribution functions of copulas may exhibit surprisingly irregular analytic behavior finally we extend the notion of empirical copula to the case of non i i d data and prove uniform convergence of the empirical copula e n corresponding to almost all orbits of a markov process usually referred to as chaos game to the singular copula a t several examples and graphics illustrate both the chosen approach and the main results

sklar s theorem obtained via regularization techniques abstract sklar s theorem establishes the connection between a joint d dimensional distribution function and its univariate marginals its proof is straightforward when all the marginals are continuous the hard part is the extension to the case where at least one of the marginals has a discrete component we present a new proof of this extension based on some analytical regularization techniques i e mollifiers and on the compactness with respect to the l norm of the class of copulas

flipping of multivariate aggregation functions abstract we consider flipping transformations of multivariate aggregation functions and we investigate the closure of these transformations with respect to the class of aggregation functions with annihilator element equal to 0 moreover the consecutive application of flipping transformations is also discussed the results present interesting connections both with quasi copulas and with suitable modifications of the n increasing property of a multivariate probability distribution function

local existence lower mass bounds and a new continuation criterion for the landau equation abstract we consider the spatially inhomogeneous landau equation with soft potentials first we establish the short time existence of solutions assuming the initial data has sufficient decay in the velocity variable and regularity no decay assumptions are made in the spatial variable next we show that the evolution instantaneously spreads mass throughout the domain the resulting lower bounds are sub gaussian which we show is optimal the proof of mass spreading is based on a stochastic process and makes essential use of nonlocality by combining this theorem with prior results we derive two important applications c smoothing even for initial data with vacuum regions and a continuation criterion the solution can be extended as long as the mass and energy densities stay bounded from above this is the weakest condition known to prevent blow up in particular it does not require a lower bound on the mass density or an upper bound on the entropy density

uniqueness in law for a class of degenerate diffusions with continuous covariance we study the martingale problem associated with the operator lu s x partial su s x frac 1 2 sum i j 1 d 0 a ij s x partial ij u s x sum i j 1 d b ij x j partial iu s x where d0 d we show that the martingale problem is well posed when the function a is continuous and strictly positive definite on mathbb r d 0 and the matrix b takes a particular lower diagonal block form we then localize this result to show that the martingale problem remains well posed when b is replaced by a sufficiently smooth vector field whose jacobian matrix satisfies a nondegeneracy condition

a λ convexity based proof for the propagation of chaos for weakly interacting stochastic particles abstract in this work we give a proof of the mean field limit for convex potentials using a purely variational viewpoint our approach is based on the observation that all evolution equations that we study can be written as gradient flows of functionals at different levels in the set of probability measures in the set of symmetric probability measures on n variables and in the set of probability measures on probability measures this basic fact allows us to rely on convergence tools for gradient flows to complete the proof by identifying the limits of the different terms in the evolutionary variational inequalities evis associated to each gradient flow the convexity of the confining and interaction potentials is crucial for the unique identification of the limits and for deriving the evis at each description level of the interacting particle system

ricci curvature for metric measure spaces via optimal transport we dene a notion of a measured length space x having nonnegative n ricci curvature for n 2 1 1 or having1 ricci curvature bounded below byk fork2 r the denitions are in terms of the displacement convexity of certain functions on the associated wasserstein metric space p2 x of probability measures we show that these properties are preserved under measured gromov hausdor limits we give geometric and analytic consequences this paper has dual goals one goal is to extend results about optimal transport from the setting of smooth riemannian manifolds to the setting of length spaces a second goal is to use optimal transport to give a notion for a measured length space to have ricci curvature bounded below we refer to 11 and 44 for background material on length spaces and optimal transport respectively further bibliographic notes on optimal transport are in appendix f in the present introduction we motivate the questions that we address and we state the main results to start on the geometric side there are various reasons to try to extend notions of curvature from smooth riemannian manifolds to more general spaces a fairly general setting is that of length spaces meaning metric spaces x d in which the distance between two points equals the inmum of the lengths of curves joining the points in the rest of this introduction we assume that x is a compact length space alexandrov gave a good notion of a length space having curvature bounded below by k with k a real number in terms of the geodesic triangles in x in the case of a riemannian manifold m with the induced length structure one recovers the riemannian notion of having sectional curvature bounded below by k length spaces with alexandrov curvature bounded below by k behave nicely with respect to the gromovhausdor topology on compact metric spaces modulo isometries they form