uniform convergence to equilibrium for granular media we study the long time asymptotics of a nonlinear nonlocal equation used in the modelling of granular media we prove a uniform exponential convergence to equilibrium for degenerately convex and non convex interaction or confinement potentials improving in particular results by j a carrillo r j mccann and c villani the method is based on studying the dissipation of the wasserstein distance between a solution and the steady state

multi to one dimensional optimal transport we consider the monge kantorovich problem of transporting a probability density on m to another on the line so as to optimize a given cost function we introduce a nestedness criterion relating the cost to the densities under which it becomes possible to solve this problem uniquely by constructing an optimal map one level set at a time this map is continuous if the target density has connected support we use level set dynamics to develop and quantify a local regularity theory for this map and the kantorovich potentials solving the dual linear program we identify obstructions to global regularity through examples more specifically fix probability densities f and g on open sets x m and y n with m n 1 consider transporting f onto g so as to minimize the cost s x y we give a nondegeneracy condition on s c1 1 that ensures the set of x paired with g a e y y lie in a codimension n submanifold of x specializing to the case m n 1 we discover a nestedness criterion relating s to f g that allows us to construct a unique optimal solution in the form of a map f x y when s c2 w3 1 and g and f are bounded the kantorovich dual potentials u satisfy cloc1 1 y and the normal velocity v of f 1 y with respect to changes in y is given by v x syy dxsy x f x positivity of v locally implies a lipschitz bound on f moreover c2 if f 1 y intersects x c1 transversally on subsets where this nondegeneracy positivity and transversality can be quantified for each integer r 1 the norms of u cr 1 1 and f cr 1 are controlled by these bounds g log f x cr 1 1 x w2 1 s cr 1 1 and the smallness of f 1 y we give examples showing regularity extends from x to part of x but not from y to y we also show that when s remains nested for all f g the problem in m reduces to a supermodular problem in 2017 wiley periodicals inc

a mass transportation approach to quantitative isoperimetric inequalities a sharp quantitative version of the anisotropic isoperimetric inequality is established corresponding to a stability estimate for the wulff shape of a given surface tension energy this is achieved by exploiting mass transportation theory especially gromov s proof of the isoperimetric inequality and the brenier mccann theorem a sharp quantitative version of the brunn minkowski inequality for convex sets is proved as a corollary

rates of decay to equilibria for p laplacian type equations abstract the long time asymptotics for p laplacian type equations t p m div m p 2 m in r n is studied for p 1 and m n p 1 n p 1 the non negative solutions of the equations are shown to behave asymptotically as t like barenblatt type solutions and the explicit rates of decay are established for the convergence of the relative energy the convergence with respect to the wasserstein distances and the convergence with respect to the l 1 norm the rates are proved to be optimal for p 2 the method used is based on mass transportation inequalities

a splitting method for nonlinear diffusions with nonlocal nonpotential drifts we prove an existence result for nonlinear diffusion equations in the presence of a nonlocal density dependent drift which is not necessarily potential the proof is constructive and based on the helmholtz decomposition of the drift and a splitting scheme the splitting scheme combines transport steps by the divergence free part of the drift and semi implicit minimization steps a la jordan kinderlherer otto to deal with the potential part

lp theory for the multidimensional aggregation equation we consider well posedness of the aggregation equation tu div uv 0 v k u with initial data in input amssym cal p 2 rm bbb r d rm cap l p bbb r d in dimensions 2 and higher we consider radially symmetric kernels where the singularity at the origin is of order x 2 d and prove local well posedness in input amssym cal p 2 bbb r d rm cap l p bbb r d for sufficiently large p ps in the special case of k x x the exponent ps d d 1 is sharp for local well posedness in that solutions can instantaneously concentrate mass for initial data in input amssym cal p 2 bbb r d rm cap l p bbb r d with p ps we also give an osgood condition on the potential k x that guarantees global existence and uniqueness in input amssym cal p 2 bbb r d rm cap l p bbb r d 2010 wiley periodicals inc

displacement convexity of entropy and related inequalities on graphs we introduce the notion of an interpolating path on the set of probability measures on finite graphs using this notion we first prove a displacement convexity property of entropy along such a path and derive prekopa leindler type inequalities a talagrand transport entropy inequality certain hwi type as well as log sobolev type inequalities in discrete settings to illustrate through examples we apply our results to the complete graph and to the hypercube for which our results are optimal by passing to the limit we recover the classical log sobolev inequality for the standard gaussian measure with the optimal constant

existence of solutions for a class of fourth order cross diffusion systems of gradient flow type abstract this article is concerned with the existence and the long time behavior of weak solutions to certain coupled systems of fourth order degenerate parabolic equations of gradient flow type the underlying metric is a wasserstein like transportation distance for vector valued functions with nonlinear mobilities in each component under the hypothesis of flat convexity of the driving free energy functional weak solutions are constructed by means of the variational minimizing movement scheme for metric gradient flows the essential regularity estimates are derived by variational methods

from a large deviations principle to the wasserstein gradient flow a new micro macro passage we study the connection between a system of many independent brownian particles on one hand and the deterministic diffusion equation on the other for a fixed time step h 0 a large deviations rate functional j h characterizes the behaviour of the particle system at t h in terms of the initial distribution at t 0 for the diffusion equation a single step in the time discretized entropy wasserstein gradient flow is characterized by the minimization of a functional k h we establish a new connection between these systems by proving that j h and k h are equal up to second order in h as h 0 this result gives a microscopic explanation of the origin of the entropy wasserstein gradient flow formulation of the diffusion equation simultaneously the limit passage presented here gives a physically natural description of the underlying particle system by describing it as an entropic gradient flow

optimal mass transportation in the heisenberg group in this paper we consider the problem of optimal transportation of absolutely continuous masses in the heisenberg group hn in the case when the cost function is either the square of the carnot caratheodory distance or the square of the koranyi norm in both cases we show existence and uniqueness of an optimal transport map in the former case the proof requires a delicate analysis of minimizing geodesics of the group and of the differentiability properties of the squared distance function in the latter case the proof requires some fine properties of bv functions in the heisenberg group

regularization and interpolation of positive matrices we construct certain matricial analogues of mass transport for positive definite matrices of equal trace the framework aims to devise ways of interpolating positive definite matrices that tradeoff between aligning up their eigenstructure and scaling the corresponding eigenvalues motivation for the work is provided by power spectral analysis of multivariate time series where linear interpolation between matrix valued power spectra generates push pop unrealistic and undesirable artifacts

the reconstruction problem for the euler poisson system in cosmology the motion of a continuum of matter subject to gravitational interaction is classically described by the euler poisson system prescribing the density of matter at initial and final times we are able to obtain weak solutions for this equation by minimizing the action of the lagrangian which is a convex functional through this variational formulation the reconstruction problem becomes very similar to an optimal transportation problem then we see that such minimizing solutions are consistent with smooth solutions of the euler poisson system and enjoy some special regularity properties

entire solutions of nonhomogenous differential equations abstract in this paper entire solutions of nonhomogenous differential equations of second order with polynomial coefficients in complex domain are considered bounds on the type and maximum modulus are obtained conditions for a solution to be entire and univalent are also given

existence and uniqueness of monotone measure preserving maps introduction given a pair of borel probability measures and v on rd it is natural to ask whether v can be obtained from by redistributing its mass in a canonical way in the case of the line d 1 the answer is clear as long as both measures are free from atoms x v x 0mthere is a map y x of the line to itself for which uniquely determined almost everywhere this map may be taken to be nondecreasing by a suitable choice of y x e rw o at the remaining points interpreting and v as the initial and final distribution of a one dimensional fluid the transformation y x gives a rearrangement of fluid particles yielding final state v from the initial state this rearrangement is characterized by the fact that it preserves particle ordering obviating any need for two particles to cross although the generalization of this construction to higher dimensions is the focus of this paper the one dimensional case will be pursued slightly further when the measures are absolutely continuous with respect to lebesgue d x f x dx and dr y g y dymthen formally at least neglecting regularity issues the fundamental theorem of calculus yields

phase transitions and symmetry breaking in singular diffusion the long time asymptotics are determined for fast nonlinear diffusion by linearizing otto s gradient descent model at the barenblatt profile the spectrum of the entropy is explicitly determined the dynamics are found to undergo a phase transition in which rotational symmetry is broken as the strength of the nonlinearity is varied

euler poisson systems as action minimizing paths in the wasserstein space this paper uses a variational approach to establish existence of solutions t vt for the 1 d euler poisson system by minimizing an action we assume that the initial and terminal points 0 t are prescribed in mathcal p 2 mathbb r the set of borel probability measures on the real line of finite second order moments we show existence of a unique minimizer of the action when the time interval 0 t satisfies t these solutions conserve the hamiltonian and they yield a path t t in mathcal p 2 mathbb r when t y t is a dirac mass the euler poisson system reduces to ddot y y 0 the kinetic version of the euler poisson that is the vlasov poisson system was studied in ambrosio and gangbo comm pure appl math to appear as a hamiltonian system

displacement convexity and minimal fronts at phase boundaries we show that certain free energy functionals that are not convex with respect to the usual convex structure on their domain of definition are strictly convex in the sense of displacement convexity under a natural change of variables we use this to show that in certain cases the only critical points of these functionals are minimizers this approach based on displacement convexity permits us to treat multicomponent systems as well as single component systems the developments produce new examples of displacement convex functionals and in the multi component setting jointly displacement convex functionals

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

a proof of a loomis whitney type inequality via optimal transport abstract the paper is devoted to exhibiting a proof of an analytical extension of the well known loomis whitney inequality such a proof is completely independent of the original one and it is based on the technique of optimal transport which leads also to fully characterize the equality case

solution of the monge ampere equation on wiener space for general log concave measures abstract in this work we prove that the unique 1 convex solution of the monge kantorovitch measure transportation problem between the wiener measure and a target measure which has an h log concave density in the sense of feyel and ustunel j funct anal 176 2000 400 428 w r t the wiener measure is also the strong solution of the monge ampere equation in the frame of infinite dimensional frechet spaces we further enhance the polar factorization results of the mappings which transform a spread measure to another one in terms of the measure transportation of monge kantorovitch and clarify the relation between this concept and the ito solutions of the monge ampere equation

on the law of large numbers for the empirical measure process of generalized dyson brownian motion we study the generalized dyson brownian motion gdbm of an interacting n particle system with logarithmic coulomb interaction and general potential v under reasonable condition on v we prove the existence and uniqueness of strong solution to sde for gdbm we then prove that the family of the empirical measures of gdbm is tight on mathcal c 0 t mathscr p mathbb r and all the large n limits satisfy a nonlinear mckean vlasov equation inspired by previous works due to biane and speicher carrillo mccann and villani and blower we prove that the mckean vlasov equation is indeed the gradient flow of the voiculescu free entropy on the wasserstein space of probability measures over mathbb r using the optimal transportation theory we prove that if v ge k for some constant k in mathbb r the mckean vlasov equation has a unique weak solution in the space of probability measures mathscr p mathbb r this establishes the law of large numbers and the propagation of chaos for the empirical measures of gdbm with non quadratic external potentials which are not necessarily convex finally we prove the longtime convergence of the mckean vlasov equation for c 2 convex potentials v

computation of optimal transport on discrete metric measure spaces in this paper we investigate the numerical approximation of an analogue of the wasserstein distance for optimal transport on graphs that is defined via a discrete modification of the benamou brenier formula this approach involves the logarithmic mean of measure densities on adjacent nodes of the graph for this model a variational time discretization of the probability densities on graph nodes and the momenta on graph edges is proposed a robust descent algorithm for the action functional is derived which in particular uses a proximal splitting with an edgewise nonlinear projection on the convex subgraph of the logarithmic mean thereby suitable chosen slack variables avoid a global coupling of probability densities on all graph nodes in the projection step for the time discrete action functional gamma convergence to the time continuous action is established numerical results for a selection of test cases show qualitative and quantitative properties of the optimal transport on graphs finally we use our algorithm to implement a jko scheme for the gradient flow of the entropy in discrete transportation distance which is known to coincide with underlying markov semigroup and test our results against a classical backward euler discretization of this discrete heat flow

the regularity of mappings with a convex potential in this work we apply the techniques developed in cl to the problem of mappings with a convex potential between domains that is given two bounded domains q q2 of rn and two nonnegative real functions fi defined in qi that are bounded away from zero and infinity we want to study the map vf v w for a lipschitz convex v such that v maps ql onto q 2 in the a e sense and in some weak sense 1 f2 vyv det dij v f1 x in recent work y brenier showed existence and uniqueness of such a map provided that jaqil 0 under the obvious compatibility condition

functional inequalities thick tails and asymptotics for the critical mass patlak keller segel model we investigate the long time behavior of the critical mass patlak keller segel equation this equation has a one parameter family of steady state solutions rhohls lambda 0 with thick tails whose second moment is not bounded we show that these steady state solutions are stable and find basins of attraction for them using an entropy functional fcfd coming from the critical fast diffusion equation in r 2 we construct solutions of patlak keller segel equation satisfying an entropy entropy dissipation inequality for fcfd while the entropy dissipation for fcfd is strictly positive it turns out to be a difference of two terms neither of which need to be small when the dissipation is small we introduce a strategy of em controlled concentration to deal with this issue and then use the regularity obtained from the entropy entropy dissipation inequality to prove the existence of basins of attraction for each stationary state composed by certain initial data converging towards rhohls in the present paper we do not provide any estimate of the rate of convergence but we discuss how this would result from a stability result for a certain sharp gagliardo nirenberg sobolev inequality

matrix optimal mass transport a quantum mechanical approach in this paper we describe a possible generalization of the wasserstein 2 metric originally defined on the space of scalar probability densities to the space of hermitian matrices with trace one and to the space of matrix valued probability densities our approach follows a control theoretic optimization formulation of the wasserstein 2 metric having its roots in fluid dynamics and utilizes certain results from the quantum mechanics of open systems in particular the lindblad equation it allows determining the gradient flow for the quantum entropy relative to this matricial wasserstein metric

existence and stability for fokker planck equations with log concave reference measure we study markov processes associated with stochastic differential equations whose non linearities are gradients of convex functionals we prove a general result of existence of such markov processes and a priori estimates on the transition probabilities the main result is the following stability property if the associated invariant measures converge weakly then the markov processes converge in law the proofs are based on the interpretation of a fokker planck equation as the steepest descent flow of the relative entropy functional in the space of probability measures endowed with the wasserstein distance

gradient flows and evolution variational inequalities in metric spaces i structural properties abstract this is the first of a series of papers devoted to a thorough analysis of the class of gradient flows in a metric space x d that can be characterized by evolution variational inequalities evi we present new results concerning the structural properties of solutions to the evi formulation such as contraction regularity asymptotic expansion precise energy identity stability asymptotic behavior and their link with the geodesic convexity of the driving functional under the crucial assumption of the existence of an evi gradient flow we will also prove two main results the equivalence with the de giorgi variational characterization of curves of maximal slope the convergence of the minimizing movement jko scheme to the evi gradient flow with an explicit and uniform error estimate of order 1 2 with respect to the step size independent of any geometric hypothesis as upper or lower curvature bounds on d in order to avoid any compactness assumption we will also introduce a suitable relaxation of the minimizing movement algorithm obtained by the ekeland variational principle and we will prove its uniform convergence as well

dealing with moment measures via entropy and optimal transport a recent paper by cordero erausquin and klartag provides a characterization of the measures mu on r d which can be expressed as the moment measures of suitable convex functions u i e are of the form nabla u e u for u r d to r cup infty and finds the corresponding u by a variational method in the class of convex functions here we propose a purely optimal transport based method to retrieve the same result the variational problem becomes the minimization of an entropy and a transport cost among densities rho and the optimizer rho turns out to be e u this requires to develop some estimates and some semicontinuity results for the corresponding functionals which are natural in optimal transport the notion of displacement convexity plays a crucial role in the characterization and uniqueness of the minimizers

entropic measure and wasserstein diffusion we construct a new random probability measure on the circle and on the unit interval which in both cases has a gibbs structure with the relative entropy functional as hamiltonian it satisfies a quasi invariance formula with respect to the action of smooth diffeomorphism of the sphere and the interval respectively the associated integration by parts formula is used to construct two classes of diffusion processes on probability measures on the sphere or the unit interval by dirichlet form methods the first one is closely related to malliavin s brownian motion on the homeomorphism group the second one is a probability valued stochastic perturbation of the heat flow whose intrinsic metric is the quadratic wasserstein distance it may be regarded as the canonical diffusion process on the wasserstein space

bakry emery curvature dimension condition and riemannian ricci curvature bounds the aim of the present paper is to bridge the gap between the bakry emery and the lott sturm villani approaches to provide synthetic and abstract notions of lower ricci curvature bounds we start from a strongly local dirichlet form mathcal e admitting a carre du champ gamma in a polish measure space x mathfrak m and a canonical distance mathsf d mathcal e that induces the original topology of x we first characterize the distinguished class of riemannian energy measure spaces where mathcal e coincides with the cheeger energy induced by mathsf d mathcal e and where every function f with gamma f le1 admits a continuous representative in such a class we show that if mathcal e satisfies a suitable weak form of the bakry emery curvature dimension condition mathrm be k infty then the metric measure space x mathsf d mathfrak m satisfies the riemannian ricci curvature bound mathrm rcd k infty according to duke math j 163 2014 1405 1490 thus showing the equivalence of the two notions two applications are then proved the tensorization property for riemannian energy spaces satisfying the bakry emery mathrm be k n condition and thus the corresponding one for mathrm rcd k infty spaces without assuming nonbranching and the stability of mathrm be k n with respect to sturm gromov hausdorff convergence

inequalities for quermassintegrals on k convex domains abstract in this paper we study the aleksandrov fenchel inequalities for quermassintegrals on a class of non convex domains our proof uses optimal transport maps as a tool to relate curvature quantities of different orders defined on the boundary of the domain

first order mean field games with density constraints pressure equals price in this paper we study mean field game systems under density constraints as optimality conditions of two optimization problems in duality a weak solution of the system contains an extra term an additional price imposed on the saturated zones we show that this price corresponds to the pressure field from the models of incompressible euler equations a la brenier by this observation we manage to obtain a minimal regularity which allows us to write optimality conditions at the level of single agent trajectories and to define a weak notion of nash equilibrium for our model

optimal transportation between unequal dimensions we establish that solving an optimal transportation problem in which the source and target densities are defined on spaces with different dimensions is equivalent to solving a new nonlocal analog of the monge ampere equation introduced here for the first time under suitable topological conditions we also establish that solutions are smooth if and only if a local variant of the same equation admits a smooth and uniformly elliptic solution we show that this local equation is elliptic and c 2 alpha solutions can therefore be bootstrapped to obtain higher regularity results assuming smoothness of the corresponding differential operator which we prove under simplifying assumptions for one dimensional targets our sufficient criteria for regularity of solutions to the resulting ode are considerably less restrictive than those required by earlier works

weighted ultrafast diffusion equations from well posedness to long time behaviour in this paper we devote our attention to a class of weighted ultrafast diffusion equations arising from the problem of quantisation for probability measures these equations have a natural gradient flow structure in the space of probability measures endowed with the quadratic wasserstein distance exploiting this structure in particular through the so called jko scheme we introduce a notion of weak solutions prove existence uniqueness bv and h1 estimates l1 weighted contractivity harnack inequalities and exponential convergence to a steady state

wasserstein barycenters over riemannian manifolds we study barycenters in the space of probability measures on a riemannian manifold equipped with the wasserstein metric under reasonable assumptions we establish absolute continuity of the barycenter of general measures p p m p p m on wasserstein space extending on one hand results in the euclidean case for barycenters between finitely many measures of agueh and carlier 1 to the riemannian setting and on the other hand results in the riemannian case of cordero erausquin mccann schmuckenschlager 12 for barycenters between two measures to the multi marginal setting our work also extends these results to the case where is not finitely supported as applications we prove versions of jensen s inequality on wasserstein space and a generalized brunn minkowski inequality for a random measurable set on a riemannian manifold

a gradient flow approach to an evolution problem arising in superconductivity we study an evolution equation proposed by chapman rubinstein and schatzman as a mean field model for the evolution of the vortex density in a superconductor we treat the case of a bounded domain where vortices can exit or enter the domain we show that the equation can be derived rigorously as the gradient flow of some specific energy for the riemannian structure induced by the wasserstein distance on probability measures this leads us to some existence and uniqueness results and energy dissipation identities we also exhibit some entropies that decrease through the flow and allow us to get regularity results solutions starting in lp p 1 remain in lp 2007 wiley periodicals inc

ricci flow entropy and optimal transportation let a smooth family of riemannian metrics g tau satisfy the backwards ricer flow equation on a compact oriented n dimensional manifold m suppose two families of normalized n forms omega tau 0 and omega over bar tau 0 satisfy the forwaids in tau heat equation on m generated by the connection laplacian delta g t if these n forms represent two evolving distributions of particles over m the minimum root mean square distance w 2 omega t omega over tilde t t to transport the particles of omega t onto those of omega over tilde t is shown to be non increasing as a function of t without sign conditions on the curvature of mg t moreover this contractivity property is shown to characterize supersolutions to the ricci flow

stochastic control and nonequilibrium thermodynamics fundamental limits we consider damped stochastic systems in a controlled time varying potential and study their transition between specified gibbs equilibria states in finite time by the second law of thermodynamics the minimum amount of work needed for the transition from one equilibrium state to another is the difference between the helmholtz free energy of the two states and can only be achieved by a reversible infinitely slow process the minimal gap between the work needed in a finite time transition and the work during a reversible one turns out to equal the square of the optimal mass transport wasserstein 2 distance between the two end point distributions times the inverse of the duration needed for the transition this result in fact relates nonequilibrium optimal control strategies protocols to gradient flows of entropy functionals via the jordan kinderlehrer otto scheme the purpose of this paper is to introduce ideas and results from the emerging field of stochastic thermodynamics in the setting of the classical regulator theory and to draw connections and derive such fundamental relations from a control perspective in a multivariable setting

optimal entropy transport problems and a new hellinger kantorovich distance between positive measures we develop a full theory for the new class of optimal entropy transport problems between nonnegative and finite radon measures in general topological spaces these problems arise quite naturally by relaxing the marginal constraints typical of optimal transport problems given a pair of finite measures with possibly different total mass one looks for minimizers of the sum of a linear transport functional and two convex entropy functionals which quantify in some way the deviation of the marginals of the transport plan from the assigned measures as a powerful application of this theory we study the particular case of logarithmic entropy transport problems and introduce the new hellinger kantorovich distance between measures in metric spaces the striking connection between these two seemingly far topics allows for a deep analysis of the geometric properties of the new geodesic distance which lies somehow between the well known hellinger kakutani and kantorovich wasserstein distances

moreau yosida approximation and convergence of hamiltonian systems on wasserstein space abstract in this paper we study the stability property of hamiltonian systems on the wasserstein space let h be a given hamiltonian satisfying certain properties we regularize h using the moreau yosida approximation and denote it by h we show that solutions of the hamiltonian system for h converge to a solution of the hamiltonian system for h as converges to zero we provide sufficient conditions on h to carry out this process

the wasserstein gradient flow of the fisher information and the quantum drift diffusion equation we prove the global existence of non negative variational solutions to the drift diffusion evolution equation partial t u div left u mathrm d left 2 frac delta sqrt u sqrt u f right right 0 under variational boundary condition despite the lack of a maximum principle for fourth order equations non negative solutions can be obtained as a limit of a variational approximation scheme by exploiting the particular structure of this equation which is the gradient flow of the perturbed fisher information functional fancyscript f f u frac 12 int left mathrm d log u right 2 u dx int fu dx with respect to the kantorovich rubinstein wasserstein distance between probability measures we also study long time behavior of the solutions proving their exponential decay to the equilibrium state g e v characterized by delta v frac12 left mathrm d v right 2 f quad int rm e v dx int u 0 dx when the potential v is uniformly convex in this case the functional fancyscript f f coincides with the relative fisher information fancyscript f f u frac12 fancyscript i u g int left mathrm d log u g right 2u dx

equilibrium shapes for planar crystals in an external field the equilibrium shape of a two dimensional crystal in a convex background potential g x is analyzed for g 0 the shape of minimum energy may be deduced from surface tension via the wulff construction but if g is not constant little is known beyond the case of a crystal sitting in a uniform field only an unpublished result of okikiolu shows each connected component of the equilibrium crystal to be convex here it will be shown that any such component minimizes energy uniquely among convex sets of its area if the wulff shape and g x are symmetric under x x it follows that the equilibrium crystal is unique convex and connected this last result leads to a new proof that convex crystals away from equilibrium remain convex as they evolve by curvature driven flow subsequent work with felix otto shows without assuming symmetry that no equilibrium crystal has more than two convex components

the optimal partial transport problem given two densities f and g we consider the problem of transporting a fraction m in 0 min f l 1 g l 1 of the mass of f onto g minimizing a transportation cost if the cost per unit of mass is given by x y 2 we will see that uniqueness of solutions holds for m in f wedge g l 1 min f l 1 g l 1 this extends the result of caffarelli and mccann in ann math in print where the authors consider two densities with disjoint supports the free boundaries of the active regions are shown to be n 1 rectifiable provided the supports of f and g have lipschitz boundaries and under some weak regularity assumptions on the geometry of the supports they are also locally semiconvex moreover assuming f and g supported on two bounded strictly convex sets omega lambda subset mathbb r n and bounded away from zero and infinity on their respective supports c 0 alpha rm loc regularity of the optimal transport map and local c1 regularity of the free boundaries away from omega cap lambda are shown finally the optimal transport map extends to a global homeomorphism between the active regions

convexity and regularity properties for entropic interpolations abstract in this paper we prove a convexity property of the relative entropy along entropic interpolations solutions of the schrodinger problem and a regularity property of the entropic cost along the heat flow then we derive a dimensional evi inequality and a contraction property for the entropic cost along the heat flow as a consequence we recover the equivalent results in the wasserstein space proved by erbar kuwada and sturm

a riemannian interpolation inequality a la borell brascamp and lieb a concavity estimate is derived for interpolations between l 1 m mass densities on a riemannian manifold the inequality sheds new light on the theorems of prekopa leindler borell brascamp and lieb that it generalizes from euclidean space due to the curvature of the manifold the new riemannian versions of these theorems incorporate a volume distortion factor which can however be controlled via lower bounds on ricci curvature the method uses optimal mappings from mass transportation theory along the way several new properties are established for optimal mass transport and interpolating maps on a riemannian manifold

the brunn minkowski inequality for the principal eigenvalue of fully nonlinear homogeneous elliptic operators abstract we prove that the principal eigenvalue of any fully nonlinear homogeneous elliptic operator which fulfills a very simple convexity assumption satisfies a brunn minkowski type inequality on the class of open bounded sets in r n satisfying a uniform exterior sphere condition in particular the result applies to the possibly normalized p laplacian and to the minimal pucci operator the proof is inspired by the approach introduced by colesanti for the principal frequency of the laplacian within the class of convex domains and relies on a generalization of the convex envelope method by alvarez lasry lions we also deal with the existence and log concavity of positive viscosity eigenfunctions

the mean field schrodinger problem ergodic behavior entropy estimates and functional inequalities we study the mean field schrodinger problem mfsp that is the problem of finding the most likely evolution of a cloud of interacting brownian particles conditionally on the observation of their initial and final configuration its rigorous formulation is in terms of an optimization problem with marginal constraints whose objective function is the large deviation rate function associated with a system of weakly dependent brownian particles we undertake a fine study of the dynamics of its solutions including quantitative energy dissipation estimates yielding the exponential convergence to equilibrium as the time between observations grows larger and larger as well as a novel class of functional inequalities involving the mean field entropic cost i e the optimal value in mfsp our strategy unveils an interesting connection between forward backward stochastic differential equations and the riemannian calculus on the space of probability measures introduced by otto which is of independent interest

small noise limit and convexity for generalized incompressible flows schrodinger problems and optimal transport this paper is concerned with six variational problems and their mutual connections the quadratic monge kantorovich optimal transport the schrodinger problem brenier s relaxed model for incompressible fluids the so called brodinger problem recently introduced by arnaudon et al an entropic interpolation problem for incompressible viscid fluids arxiv preprint arxiv 1704 02126 2017 the multiphase brenier model and the multiphase brodinger problem all of them involve the minimization of a kinetic action and or a relative entropy of some path measures with respect to the reversible brownian motion as the viscosity parameter nu rightarrow 0 we establish gamma convergence relations between the corresponding problems and prove the convergence of the associated pressures arising from the incompressibility constraints we also present new results on the time convexity of the entropy for some of the dynamical interpolations along the way we extend previous results by lavenant calc var partial differ equ 56 6 170 2017 and benamou et al generalized incompressible flows multi marginal transport and sinkhorn algorithm arxiv preprint arxiv 1710 08234 2017

on the convergence of augmented lagrangian method for optimal transport between nonnegative densities the dynamical formulation of the optimal transport problem introduced by j d benamou and y brenier corresponds to the time space search of a density and a momentum minimizing a transport energy between two densities in order to solve this problem an algorithm has been proposed to estimate a saddle point of a lagrangian we will study the convergence of this algorithm to a saddle point of the lagrangian in the most general conditions particularly in cases where initial and final densities vanish on some areas of the transportation domain the principal difficulty of our study will consist in the proof under these conditions of the existence of a saddle point and especially in the uniqueness of the density momentum component indeed these conditions imply to have to deal with non regular optimal transportation maps for these reasons a detailed study of the properties of the velocity field associated to an optimal transportation map in quadratic space is required

monge kantorovitch measure transportation and monge ampere equation on wiener space let w h be an abstract wiener space assume two i i 1 2 probabilities on w w we give some conditions for the wasserstein distance between 1 and 2 with respect to the cameron martin space to be finite where the infimum is taken on the set of probability measures on w w whose first and second marginals are 1 and 2 in this case we prove the existence of a unique cyclically monotone map t iw with w h such that t maps 1 to 2 moreover if 2 then t is stochastically invertible i e there exists s w w such that s t iw 1 a s and t s iw 2 a s if in addition 1 then there exists a 1 convex function in the gaussian sobolev space such that these results imply that the quasi invariant transformations of the wiener space with finite wasserstein distance from can be written as the composition of a transport map t and a rotation i e a measure preserving map we give also 1 convex sub solutions and ito type solutions of the monge ampere equation on w

asymptotic behaviour of a pile up of infinite walls of edge dislocations we consider a system of parallel straight edge dislocations and we analyse its asymptotic behaviour in the limit of many dislocations the dislocations are represented by points in a plane and they are arranged in vertical walls each wall is free to move in the horizontal direction the system is described by a discrete energy depending on the one dimensional horizontal positions x i 0 of the n walls the energy contains contributions from repulsive pairwise interactions between all walls a global shear stress forcing the walls to the left and a pinned wall at x 0 that prevents the walls from leaving through the left boundary we study the behaviour of the energy as the number of walls n tends to infinity and characterise this behaviour in terms of convergence there are five different cases depending on the asymptotic behaviour of the single dimensionless parameter n corresponding to beta n ll 1 n 1 n ll beta n ll 1 and beta n gg 1 and the two critical regimes n 1 n and n 1 as a consequence we obtain characterisations of the limiting behaviour of stationary states in each of these five regimes the results shed new light on the open problem of upscaling large numbers of dislocations we show how various existing upscaled models arise as special cases of the theorems of this paper the wide variety of behaviour suggests that upscaled models should incorporate more information than just dislocation densities this additional information is encoded in the limit of the dimensionless parameter n

convergence of the mass transport steepest descent scheme for the subcritical patlak keller segel model variational steepest descent approximation schemes for the modified patlak keller segel equation with a logarithmic interaction kernel in any dimension are considered we prove the convergence of the suitably interpolated in time implicit euler scheme defined in terms of the euclidean wasserstein distance associated with this equation for subcritical masses as a consequence we recover the recent result about the global in time existence of weak solutions to the modified patlak keller segel equation for the logarithmic interaction kernel in any dimension in the subcritical case moreover we show how this method performs numerically in dimension one in this particular case this numerical scheme corresponds to a standard implicit euler method for the pseudoinverse of the cumulative distribution function we demonstrate its capabilities to reproduce the blow up of solutions for supercritical masses easily without the need of mesh refinement

a bridge between sobolev and escobar inequalities and beyond abstract the classical sobolev and escobar inequalities are embedded into the same one parameter family of sharp trace sobolev inequalities on half spaces equality cases are characterized for each inequality in this family by tweaking a well known mass transportation argument and lead to a new comparison theorem for trace sobolev inequalities the case p 2 corresponds to a family of variational problems on conformally flat metrics which was previously settled by carlen and loss with their method of competing symmetries in this case minimizers interpolate between conformally flat spherical and hyperbolic geometries passing through the euclidean geometry defined by the fundamental solution of the laplacian

a unifying perspective on the moyal and voros products and their physical meanings the moyal and voros formulations of non commutative quantum field theory have been a point of controversy in the recent past here we address this issue in the context of non commutative non relativistic quantum mechanics in particular we show that the two formulations simply correspond to two different representations associated with two different choices of basis on the quantum hilbert space from a mathematical perspective the two formulations are therefore completely equivalent but we also argue that only the voros formulation admits a consistent physical interpretation these considerations are elucidated by considering the free particle transition amplitude in the two representations

the distance between two random vectors with given dispersion matrices for two p dimensional random vectors x and y with dispersion matrices 11 and 22 respectively we determine that covariance matrix 0 of x and y that minimizes the l2 distance between x and y there is a dual to this problem that is of interest in another context

univalence of a function f and its successive derivatives when f satisfies a differential equation ii abstract solutions of second order differential equations with quadratic polynomials as coefficients are studied by a suitable choice of constants it is shown that there exists a solution which is an entire function of exponential type one and such that this function and all its derivatives are close to convex in the unit disc by another choice of constants we get an entire solution of exponential type 2 with gap power series such that all odd or all even derivatives are close to convex in the unit disc

stability for a gns inequality and the log hls inequality with application to the critical mass keller segel equation starting from the quantitative stability result of bianchi and egnell for the 2 sobolev inequality we deduce several dierent stability results for a gagliardo nirenberg sobolev inequality in the plane then exploiting the connection between this inequality and a fast diusion equation we get stability for the log hls inequality finally using all these estimates we prove a quantitative convergence result for the critical mass keller segel system

on patlak keller segel system for several populations a gradient flow approach abstract we study the global in time existence of solutions to the parabolic elliptic patlak keller segel system of multi species populations we prove that if the initial mass satisfies an appropriate notion of sub criticality then the system has a solution defined for all time we explore the gradient flow structure in the wasserstein space to study the question of existence moreover we show that the obtained solution satisfies energy dissipation inequality

generalized ricci curvature bounds for three dimensional contact subriemannian manifolds measure contraction property is one of the possible generalizations of ricci curvature bound to more general metric measure spaces in this paper we discover necessary and sufficient conditions for a three dimensional contact subriemannian manifold to satisfy this property

a convexity principle for interacting gases a new set of inequalities is introduced based on a novel but natural interpolation between borel probability measures on r d using these estimates in lieu of convexity or rearrangement inequalities the existence and uniqueness problems are solved for a family of attracting gas models in these models the gas interacts with itself through a force which increases with distance and is governed by an equation of state p p relating pressure to density p d 1 d is assumed non decreasing for a d dimensional gas by showing that the internal and potential energies for the system are convex functions of the interpolation parameter an energy minimizing stateunique up to translationis proven to exist the concavity established for t pd q as a function of t 0 1 generalizes the brunnminkowski inequality from sets to measures 1997 academic press

a generalization of caffarelli s contraction theorem via reverse heat flow a theorem of l caffarelli implies the existence of a map pushing forward a source gaussian measure to a target measure which is more log concave than the source one which contracts euclidean distance in fact caffarelli showed that the optimal transport brenier map topt is a contraction in this case we generalize this result to more general source and target measures using a condition on the third derivative of the potential by providing two different proofs the first uses a map t whose inverse is constructed as a flow along an advection field associated to an appropriate heat diffusion process the contraction property is then reduced to showing that log concavity is preserved along the corresponding diffusion semi group by using a maximum principle for parabolic pde in particular caffarelli s original result immediately follows by using the ornstein uhlenbeck process and the prekopa leindler theorem the second uses the map topt by generalizing caffarelli s argument employing in addition further results of caffarelli as applications we obtain new correlation and isoperimetric inequalities

data driven optimal transport the problem of optimal transport between two distributions x and y is extended to situations where the distributions are only known through a finite number of samples xi and yj a weak formulation is proposed based on the dual of the kantorovich formulation with two main modifications replacing the expected values in the objective function by their empirical means over the xi and yj and restricting the dual variables u x and v y to a suitable set of test functions adapted to the local availability of sample points a procedure is proposed and tested for the numerical solution of this problem based on a fluidlike flow in phase space where the sample points play the role of active lagrangian markers 2016 wiley periodicals inc

optimal transportation meshfree approximation schemes for fluid and plastic flows we develop an optimal transportation meshfree otm method for simulating general solid and fluid flows including fluid structure interaction the method combines concepts from optimal transportation theory with material point sampling and max ent meshfree interpolation the proposed otm method generalizes the benamou brenier differential formulation of optimal mass transportation problems to problems including arbitrary geometries and constitutive behavior the otm method enforces mass transport and essential boundary conditions exactly and is free from tension instabilities the otm method exactly conserves linear and angular momentum and its convergence characteristics are verified in standard benchmark problems we illustrate the range and scope of the method by means of two examples of application the bouncing of a gas filled balloon off a rigid wall and the classical taylor anvil benchmark test extended to the hypervelocity range

displacement convexity for the generalized orthogonal ensemble the generalized orthogonal ensemble of n n real symmetric matrices x has probability measure nu n left dx right z n 1 exp left ntracev left x right right where dx is the product of lebesgue measure on the matrix entries and v left x right geqslant left 2 delta right log left x right with 0 the eigenvalue distribution is concentrated on left a mathord left vphantom a 2 right kern nulldelimiterspace 2 a mathord left vphantom a 2 right kern nulldelimiterspace 2 right for some a 0 or if the variance of the trace is o 1 n2 then the empirical distribution of eigenvalues converges weakly almost surely to some non random probability measure on a 2 a 2 as o left sqrt log n mathord left vphantom n n 2 right kern nulldelimiterspace n 2 right these conditions are satisfied for certain polynomial potentials the logarithmic energy is displacement convex as a functional on charge distributions with fixed mean along the real line when the trace distribution satisfies a logarithmic sobolev inequality or equivalently a quadratic transportation inequality the joint eigenvalue distributions and the limiting equilibrium measure likewise satisfy quadratic transportation inequalities in the sense of talagrand 24

barriers of the mckean vlasov energy via a mountain pass theorem in the space of probability measures abstract we show that the empirical process associated with a system of weakly interacting diffusion processes exhibits a form of noise induced metastability the result is based on an analysis of the associated mckean vlasov free energy which for suitable attractive interaction potentials has at least two distinct global minimisers at the critical parameter value c on the torus one of these states is the spatially homogeneous constant state and the other is a clustered state we show that a third critical point exists at this value as a result we obtain that the probability of transition of the empirical process from the constant state scales like exp n with the energy gap at c the proof is based on a version of the mountain pass theorem for lower semicontinuous and geodesically convex functionals on the space of probability measures p 2 m equipped with the 2 wasserstein metric where m is a complete connected and smooth riemannian manifold

on the geometry of metric measure spaces ii we introduce a curvature dimension condition cd k n for metric measure spaces it is more restrictive than the curvature bound underline text curv left m text d m right k introduced in sturm k t 2006 on the geometry of metric measure spaces i acta math 196 65 131 which is recovered as the borderline case cd k the additional real parameter n plays the role of a generalized upper bound for the dimension for riemannian manifolds cd k n is equivalent to text ric m left xi xi right k left xi right 2 and dim m n the curvature dimension condition cd k n is stable under convergence for any triple of real numbers k n l the family of normalized metric measure spaces m d m with cd k n and diameter l is compact condition cd k n implies sharp version of the brunn minkowski inequality of the bishop gromov volume comparison theorem and of the bonnet myers theorem moreover it implies the doubling property and local scale invariant poincare inequalities on balls in particular it allows to construct canonical dirichlet forms with gaussian upper and lower bounds for the corresponding heat kernels

solution of a model boltzmann equation via steepest descent in the 2 wasserstein metric we study a model boltzmann equation closely related to the bgk equation using a steepest descent method in the wasserstein metric and prove global existence of energy and momentum conserving solutions we also show that the solutions converge to the manifold of local maxwellians in the large time limit and obtain other information on the behavior of the solutions we show how the wasserstein metric is natural for this problem because it is adapted to the study of both the free streaming and the collisions

optimal transport for particle image velocimetry particle image velocimetry piv is a method used to measure the velocity field of a fluid flow traditionally cross correlation algorithms are used to retrieve the flow velocity recently a new method for piv based on the l 2 optimal mass transportation problem ot piv was introduced and analyzed in l p saumier b khouider and m agueh comm math sci 13 2015 pp 269 296 for the case of synthetically generated data and for particles of equal mass or brightness here we extend the work of saumier et al to the case of particles with different masses and randomly seeded particles more importantly we compare the ot piv method with a typical cross correlation algorithm for the case of real data using a combination of theory and numerical experiments we demonstrate that in the presence of particles with different weights brightness the ot method is more accurate for the largest brightest particles and it is more faithful when the particles are far enough from each other this makes it more suitable for the so called particle tracking regime of piv i e when the seeding density is low we demonstrate in particular that for low seeding densities the ot method performs better than a typical cross correlation algorithm based on these new results and the previous ones of saumier et al we propose a suite of postprocessing algorithms for the ot piv method it is found that the ot piv with postprocessing performs very well on the synthetic and real data when compared to the cross correlation method

optimal transport and cournot nash equilibria we study a class of games with a continuum of players for which a cournot nash equilibria can be obtained by the minimisation of some cost related to optimal transport this cost is not convex in the usual sense in general but it turns out to have hidden strict convexity properties in many relevant cases this enables us to obtain new uniqueness results and a characterisation of equilibria in terms of some partial differential equations a simple numerical scheme in dimension one as well as an analysis of the inefficiency of equilibria

global in time weak measure solutions and finite time aggregation for nonlocal interaction equations in this paper we provide a well posedness theory for weak measure solutions of the cauchy problem for a family of nonlocal interaction equations these equations are continuum models for interacting particle systems with attractive repulsive pairwise interaction potentials the main phenomenon of interest is that even with smooth initial data the solutions can concentrate mass in finite time we develop an existence theory that enables one to go beyond the blow up time in classical norms and allows for solutions to form atomic parts of the measure in finite time the weak measure solutions are shown to be unique and exist globally in time moreover in the case of sufficiently attractive potentials we show the finite time total collapse of the solution onto a single point for compactly supported initial measures our approach is based on the theory of gradient flows in the space of probability measures endowed with the wasserstein metric in addition to classical tools we exploit the stability of the flow with respect to the transportation distance to greatly simplify many problems by reducing them to questions about particle approximations

equality in borell brascamp lieb inequalities on curved spaces abstract by using optimal mass transportation and a quantitative holder inequality we provide estimates for the borell brascamp lieb deficit on complete riemannian manifolds accordingly equality cases in borell brascamp lieb inequalities including brunn minkowski and prekopa leindler inequalities are characterized in terms of the optimal transport map between suitable marginal probability measures these results provide several qualitative applications both in the flat and non flat frameworks in particular by using caffarelli s regularity result for the monge ampere equation we give a new proof of dubuc s characterization of the equality in borell brascamp lieb inequalities in the euclidean setting when the n dimensional riemannian manifold has ricci curvature ric m n 1 k for some k r it turns out that equality in the borell brascamp lieb inequality is expected only when a particular region of the manifold between the marginal supports has constant sectional curvature k a precise characterization is provided for the equality in the lott sturm villani type distorted brunn minkowski inequality on riemannian manifolds related results for not necessarily reversible finsler manifolds are also presented

an analog of the 2 wasserstein metric in non commutative probability under which the fermionic fokker planck equation is gradient flow for the entropy let mathfrak c denote the clifford algebra over mathbb r n which is the von neumann algebra generated by n self adjoint operators qj j 1 n satisfying the canonical anticommutation relations qiqj qjqi 2 iji and let denote the normalized trace on mathfrak c this algebra arises in quantum mechanics as the algebra of observables generated by n fermionic degrees of freedom let mathfrak p denote the set of all positive operators rho in mathfrak c such that 1 these are the non commutative analogs of probability densities in the non commutative probability space mathfrak c tau the fermionic fokker planck equation is a quantum mechanical analog of the classical fokker planck equation with which it has much in common such as the same optimal hypercontractivity properties in this paper we construct a riemannian metric on mathfrak p that we show to be a natural analog of the classical 2 wasserstein metric and we show that in analogy with the classical case the fermionic fokker planck equation is gradient flow in this metric for the relative entropy with respect to the ground state we derive a number of consequences of this such as a sharp talagrand inequality for this metric and we prove a number of results pertaining to this metric several open problems are raised

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

polar factorization and monotone rearrangement of vector valued functions given a probability space x p and a bounded domain r in rd equipped with the lebesgue measure 1 i normalized so that 10 i i it is shown under additional technical assumptions on x and q that for every vector valued function u e lp x p rd there is a unique polar factorization u v s where is a convex function defined on r and s is a measure preserving mapping from x p into q i i provided that u is nondegenerate in the sense that p u e 0 for each lebesgue negligible subset e of rd through this result the concepts of polar factorization of real matrices helmholtz decomposition of vector fields and nondecreasing rearrangements of real valued functions are unified the monge amgre equation is involved in the polar factorization and the proof relies on the study of an appropriate monge kantorovich problem

the geometry of optimal transportation introduction 113 1 summary of main results 120 2 background on optimal measures 126 part i strictly convex costs 133 3 existence and uniqueness of optimal maps 133 4 characterization of the optimal map 137 part ii costs which are strictly concave as a function of d i s t a n c e 141 5 the role of optimal maps 141 6 uniqueness of optimal solutions 144 part iii appendices 148 a legendre transforms and conjugate costs 148 b examples of c concave potentials 152 c regularity of c concave potentials 154 references 159