regularity of potential functions of the optimal transportation problem the potential function of the optimal transportation problem satisfies a partial differential equation of monge ampere type in this paper we introduce the notion of a generalized solution and prove the existence and uniqueness of generalized solutions of the problem we also prove the solution is smooth under certain structural conditions on the cost function

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

quantitative stability of the entropy power inequality we establish quantitative stability results for the entropy power inequality epi specifically we show that if uniformly log concave densities nearly saturate the epi then they must be close to gaussian densities in the quadratic kantorovich wasserstein distance furthermore if one of the densities is gaussian and the other is log concave or more generally has positive spectral gap then the deficit in the epi can be controlled in terms of the l 1 kantorovich wasserstein distance or relative entropy respectively as a counterpoint an example shows that the epi can be unstable with respect to the quadratic kantorovich wasserstein distance when densities are uniformly log concave on sets of measure arbitrarily close to one our stability results can be extended to non log concave densities provided certain regularity conditions are met the proofs are based on mass transportation

analytic content and the isoperimetric inequality in higher dimensions abstract this paper establishes a conjecture of gustafsson and khavinson which relates the analytic content of a smoothly bounded domain in r n to the classical isoperimetric inequality the proof is based on a novel combination of partial balayage with optimal transport theory

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

a perturbation argument for a monge ampere type equation arising in optimal transportation we prove some interior regularity results for potential functions of optimal transportation problems with power costs the main point is that our problem is equivalent to a new optimal transportation problem whose cost function is a sufficiently small perturbation of the quadratic cost but it does not satisfy the well known condition a 3 guaranteeing regularity the proof consists in a perturbation argument from the standard monge ampere equation in order to obtain first interior c1 1 estimates for the potential and second interior holder estimates for second derivatives in particular we take a close look at the geometry of optimal transportation when the cost function is close to quadratic in order to understand how the equation degenerates near the boundary

towards the smoothness of optimal maps on riemannian submersions and riemannian products of round spheres in particular the variant a3w of ma trudinger and wang s condition for regularity of optimal transportation maps is implied by the non negativity of a pseudo riemannian curvature which we call cross curvature induced by the transportation cost for the riemannian distance squared cost it is shown that 1 cross curvature non negativity is preserved for products of two manifolds 2 both a3w and cross curvature non negativity are inherited by riemannian submersions as is domain convexity for the exponential maps and 3 the n dimensional round sphere satisfies cross curvature non negativity from these results a large new class of riemannian manifolds satisfying cross curvature non negativity thus a3w is obtained including many whose sectional curvature is far from constant all known obstructions to the regularity of optimal maps are absent from these manifolds making them a class for which it is natural to conjecture that regularity holds this conjecture is confirmed for certain riemannian submersions of the sphere such as the complex projective spaces cp n

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

fluctuations and correlations in an individual based model of biological coevolution we extend our study of a simple model of biological coevolution to its statistical properties starting with a complete description in terms of a master equation we provide its relation to the deterministic evolution equations used in previous investigations the stationary states of the mutationless model are generally well approximated by gaussian distributions so that the fluctuations and correlations of the populations can be computed analytically several specific cases are studied by monte carlo simulations and there is an excellent agreement between the data and the theoretical predictions

strict convexity and c1 α regularity of potential functions in optimal transportation under condition a3w abstract in this paper we prove the strict c convexity and the c 1 regularity for potential functions in optimal transportation under condition a3w these results were obtained by caffarelli 1 3 4 for the cost c x y x y 2 by liu 11 loeper 15 trudinger and wang 20 for costs satisfying the condition a3 for costs satisfying the condition a3w the results have also been proved by figalli kim and mccann 6 assuming that the initial and target domains are uniformly c convex see also 21 and by guillen and kitagawa 8 assuming the cost function satisfies a3w in larger domains in this paper we prove the strict c convexity and the c 1 regularity assuming either the support of source density is compactly contained in a larger domain where the cost function satisfies a3w or the dimension 2 n 4

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

free discontinuities in optimal transport we prove a nonsmooth implicit function theorem applicable to the zero set of the difference of convex functions this theorem is explicit and global it gives a formula representing this zero set as a difference of convex functions which holds throughout the entire domain of the original functions as applications we prove results on the stability of singularities of envelopes of semi convex functions and solutions to optimal transport problems under appropriate perturbations along with global structure theorems on certain discontinuities arising in optimal transport maps for the bilinear cost c x bar x langle x bar x rangle for x bar x in bf r n for targets whose components satisfy additional convexity separation multiplicity and affine independence assumptions we show that these discontinuities occur on submanifolds of the appropriate codimension which are parameterized locally as differences of convex functions dc hence c 2 rectifiable and depending on the precise assumptions c 1 alpha smooth under these hypotheses any n 1 affinely independent components of the target measure select at most one point from the source measure where the transport divides between all n 1 specified target components

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

interior w 2 p estimates for solutions of the monge ampere equation in this work we adapt the techniques developed in c1 to prove interior estimates for solutions of perturbations of the monge ampere equation

monotonicity properties of optimal transportation and the fkg and related inequalities optimal transportation between densities f x g y can be interpreted as a joint probability distribution with marginally f x and g y we prove monotonicity and concavity properties of optimal transportation y x under suitable assumptions on f and g as an application we obtain the fortuin kasteleyn ginibre correlation inequalities as well as some generalizations of the brascamp lieb momentum inequalities

submanifolds isoperimetric inequalities and optimal transportation abstract the aim of this paper is to prove isoperimetric inequalities on submanifolds of the euclidean space using mass transportation methods we obtain a sharp weighted isoperimetric inequality and a nonsharp classical inequality similar to the one obtained in michael and simon 1973 19 the proof relies on the description of a solution of the problem of monge when the initial measure is supported in a submanifold and the final one supported in a linear subspace of the same dimension

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

on the regularity of solutions of optimal transportation problems we give a necessary and sufficient condition on the cost function so that the map solution of monge s optimal transportation problem is continuous for arbitrary smooth positive data this condition was first introduced by ma trudinger and wang 24 30 for a priori estimates of the corresponding monge ampere equation it is expressed by a socalled cost sectional curvature being non negative we show that when the cost function is the squared distance of a riemannian manifold the cost sectional curvature yields the sectional curvature as a consequence if the manifold does not have non negative sectional curvature everywhere the optimal transport map cannot be continuous for arbitrary smooth positive data the non negativity of the cost sectional curvature is shown to be equivalent to the connectedness of the contact set between any cost convex function the proper generalization of a convex function and any of its supporting functions when the cost sectional curvature is uniformly positive we obtain that optimal maps are continuous or holder continuous under quite weak assumptions on the data compared to what is needed in the euclidean case this case includes the quadratic cost on the round sphere

quantitative linearization results for the monge ampere equation this paper is about quantitative linearization results for the monge ampere equation with rough data we develop a large scale regularity theory and prove that if a measure is close to the lebesgue measure in wasserstein distance at all scales then the displacement of the macroscopic optimal coupling is quantitatively close at all scales to the gradient of the solution of the corresponding poisson equation the main ingredient we use is a harmonic approximation result for the optimal transport plan between arbitrary measures this is used in a campanato iteration which transfers the information through the scales

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

continuity of optimal transport maps and convexity of injectivity domains on small deformations of 2 given a compact riemannian manifold we study the regularity of the optimal transport map between two probability measures with cost given by the squared riemannian distance our strategy is to define a new form of the so called ma trudinger wang condition and to show that this condition together with the strict convexity on the nonfocal domains implies the continuity of the optimal transport map moreover our new condition again combined with the strict convexity of the nonfocal domains allows us to prove that all injectivity domains are strictly convex too these results apply for instance on any small c4 deformation of the 2 sphere 2009 wiley periodicals inc

regularity of optimal maps on the sphere the quadratic cost and the reflector antenna building on the results of ma et al in arch rational mech anal 177 2 151 183 2005 and of the author loeper in acta math to appear we study two problems of optimal transportation on the sphere the first corresponds to the cost function d2 x y where d is the riemannian distance of the round sphere the second corresponds to the cost function log x y known as the reflector antenna problem we show that in both cases the cost sectional curvature is uniformly positive and establish the geometrical properties so that the results of loeper in acta math to appear and ma et al in arch rational mech anal 177 2 151 183 2005 can apply global smooth solutions exist for arbitrary smooth positive data and optimal maps are holder continuous under weak assumptions on the data

weak existence for the semigeostrophic equations formulated as a coupled monge ampere transport problem hoskins s semigeostrophic equations are reformulated as a coupled monge ampere transport problem b j hoskins quart j royal met soc 97 1971 pp 139 153 existence of global weak solutions is obtained for this formulation

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

locality and orthomodular structure of compound systems a p lattice is defined as a complete orthomodular atomic lattice l which is formed by the set of propositions of a physical system a composition of physical systems in the framework of p lattices is considered and some notions of locality are given it is shown that the following statements about compound systems are equivalent a all atoms of a compound system are reducible to those of its subsystems b all pure states of a compound system are separable into those of its subsystems c a compound system has statistical property independence d at least one of the subsystems is classical e bell type inequalities hold

a localization property of viscosity solutions to the monge ampere equation and their strict convexity the purpose of this note is to show a localization property of convex viscosity solutions to the monge ampere inequality 0 1 2 n are strictly convex

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

continuity equations and ode flows with non smooth velocity in this paper we review many aspects of the well posedness theory for the cauchy problem for the continuity and transport equations and for the ordinary differential equation ode in this framework we deal with velocity fields that are not smooth but enjoy suitable weak differentiability assumptions we first explore the connection between the partial differential equation pde and the ode in a very general non smooth setting then we address the renormalization property for the pde and prove that such a property holds for sobolev velocity fields and for bounded variation velocity fields finally we present an approach to the ode theory based on quantitative estimates

regularity of optimal transport in curved geometry the nonfocal case we explore some geometric and analytic consequences of a curvature condition introduced by ma trudinger and wang in relation to the smoothness of optimal transport in curved geometry we discuss a conjecture according to which a strict version of the ma trudinger wang condition is sufficient to prove regularity of optimal transport on a riemannian manifold we prove this conjecture under a somewhat restrictive additional assumption of nonfocality at the same time we establish the striking geometric property that the tangent cut locus is the boundary of a convex set partial extensions are presented to the case when there is no pure focalization on the tangent cut locus

holder continuity and injectivity of optimal maps consider transportation of one distribution of mass onto another chosen to optimize the total expected cost where cost per unit mass transported from x to y is given by a smooth function c x y if the source density f x is bounded away from zero and infinity in an open region u subset mathbf r n and the target density f y is bounded away from zero and infinity on its support overline v subset mathbf r n which is strongly c convex with respect to u and the transportation cost c satisfies the mathbf a3 rm w condition of trudinger and wang ann sc norm super pisa cl sci 5 8 1 143 174 2009 we deduce the local holder continuity and injectivity of the optimal map inside u so that the associated potential u belongs to c 1 alpha loc u here the exponent 0 depends only on the dimension and the bounds on the densities but not on c our result provides a crucial step in the low interior regularity setting in a sequel figalli et al j eur math soc jems 1131 1166 2013 we use it to establish regularity of optimal maps with respect to the riemannian distance squared on arbitrary products of spheres three key tools are introduced in the present paper namely we first find a transformation that under mathbf a3 rm w makes c convex functions level set convex as was also obtained independently from us by liu calc var partial diff eq 34 435 451 2009 we then derive new alexandrov type estimates for the level set convex c convex functions and a topological lemma showing that optimal maps do not mix the interior with the boundary this topological lemma which does not require mathbf a3 rm w is needed by figalli and loeper calc var partial diff eq 35 537 550 2009 to conclude the continuity of optimal maps in two dimensions in higher dimensions if the densities f are holder continuous our result permits continuous differentiability of the map inside u in fact c 2 alpha loc regularity of the associated potential to be deduced from the work of liu et al comm partial diff eq 35 1 165 184 2010

bayesian inference with optimal maps we present a new approach to bayesian inference that entirely avoids markov chain simulation by constructing a map that pushes forward the prior measure to the posterior measure existence and uniqueness of a suitable measure preserving map is established by formulating the problem in the context of optimal transport theory we discuss various means of explicitly parameterizing the map and computing it efficiently through solution of an optimization problem exploiting gradient information from the forward model when possible the resulting algorithm overcomes many of the computational bottlenecks associated with markov chain monte carlo advantages of a map based representation of the posterior include analytical expressions for posterior moments and the ability to generate arbitrary numbers of independent posterior samples without additional likelihood evaluations or forward solves the optimization approach also provides clear convergence criteria for posterior approximation and facilitates model selection through automatic evaluation of the marginal likelihood we demonstrate the accuracy and efficiency of the approach on nonlinear inverse problems of varying dimension involving the inference of parameters appearing in ordinary and partial differential equations

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

on the reflector shape design in this paper we study the problem of recovering the reflecting surface in a reflector system which consists of a point light source a reflecting surface and an object to be illuminated this problem involves a fully nonlinear partial differential equation of monge

on strict convexity and continuous differentiability of potential functions in optimal transportation this note concerns the relationship between conditions on cost functions and domains the convexity properties of potentials in optimal transportation and the continuity of the associated optimal mappings in particular we prove that if the cost function satisfies the condition a3 introduced in our previous work with xinan ma the densities and their reciprocals are bounded and the target domain is convex with respect to the cost function then the potential is continuously differentiable and its dual potential strictly concave with respect to the cost function our results extend by different and more direct proof similar results of loeper proved by approximation from our earlier work on regularity

best constant in sobolev trace inequalities on the half space using a mass transportation method we study optimal sobolev trace inequalities on the half space and prove a conjecture made by escobar in 1988 about the minimizers

existence and regularity of the reflector surfaces in mathbb r n 1 in this paper we study the problem of constructing reflector surfaces from the near field data the light is transmitted as a collinear beam and the reflected rays illuminate a given domain on the fixed receiver surface we consider two types of weak solutions and prove their equivalence under some convexity assumptions on the target domain the regularity of weak solutions is a very delicate problem and the positive answer depends on a number of conditions characterizing the geometric positioning of the reflector and receiver in fact we show that there is a domain mathcal d in the ambient space such that the weak solution is smooth if and only if its graph lies in mathcal d

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

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

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

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

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