On the duality between $p$-modulus and probability measures

Well-defined Lagrangian flows for absolutely continuous curves of probabilities on the line Mohamed H Amsaad Doctor Of Philosophy in Mathematics throughout my life. Finally, I gratefully acknowledge support from CBIE grant No. 1515 as well as funds from the State of WV and Libyan Ministry of Higher Education. vi Contents Approval Page 2 Acknowledgments v List of Figures ix Notation x Preface It is proved in the book [17] of Ambrosio, Gigli and Savaré that, if 1 < p < +∞, then any such curve admits a unique Borel velocity field v : (0, T) × R d → R d such that (μ, v) satisfies (CE) in the sense of distributions in (0, T) × R d , and v(t, •) ∈ L p (μ(t, •); R d) for a.e. t ∈ (0, T), (0, T) t → v(t, •) L p (μ(t,•);R d) ∈ L q (0, T). This v is called the velocity of minimal norm associated to to the path μ, as it minimizes w(t, •) q L p (μ(t,•);R d) = |μ | q (t) for L 1-a.e. t ∈ (0, T) among all Borel maps w : (0, T) × R d → R d that pair up with μ to satisfy the multidimensional continuity equation above. Moreover, Nguyen and Tudorascu [92] showed that this minimal norm assumption is, in fact, redundant in the case d = 1. They proved that, this "velocity of minimal norm" is unique, in the sense that if v 1 and v 2 satisfy the assumptions above for a.e. t ∈ (0, T), then v 1 (t, •) ≡ v 2 (t, •) μ t-a.e.. They called v a velocity associated to the path μ. Furthermore, within the one-dimensional setting d = 1, the following is true [115], [116]: suppose μ i ∈ P p (R), i = 1, 2 and let M i : I → R be the unique a.e. monotone nondecreasing maps such that M i# χ = μ i , respectively, where χ is the onedimensional Lebesgue measure restricted to I := (0, 1). Then W p (μ 1 , μ 2) = M 1 − M 2 L p (I) , and there is only one Borel velocity v : (0, T) × R → R satisfying (CE), so the minimality of the L p (μ t)-norm as a selection principle is unnecessary here. The most general problem (Flow) discussed here in this thesis assumes only Borel regularity on v and we only study solutions X belonging to some time-Sobolev spaces W 1,q (0, T ; L p (I)) for 1 ≤ q ≤ +∞ and 1 ≤ p < +∞. The reason for this extra-requirement on the object defined in Definition 5.1 will become clear in Chapter 5, where we prove that any map X as in Definition 5.1 which also satisfies X t# χ = μ t for all t ∈ [0, T ] for some μ ∈ AC q (0, T ; P p (R)

Journal of Functional Analysis

It is well known that on arbitrary metric measure spaces, the notion of minimal p-weak upper gradient may depend on p. In this paper we investigate how a first-order condition of the metric-measure structure, that we call Bounded Interpolation Property, grants that in fact such dependence is not present. The kind of independence we obtain is stronger than previously available results in the setting of PI spaces. We also show that the Bounded Interpolation Property is stable for pointed measure Gromov Hausdorff convergence and holds on a large class of spaces satisfying curvature dimension conditions.

Albachiara Cogo, 2019

Handbook of Differential Equations: Evolutionary Equations, 2007

Complex Manifolds

Given a Kähler manifold (Z, J, ω) and a compact real submanifold M ⊂ Z, we study the properties of the gradient map associated with the action of a noncompact real reductive Lie group G on the space of probability measures on M. In particular, we prove convexity results for such map when G is Abelian and we investigate how to extend them to the non-Abelian case.

Proceedings of the American Mathematical Society, 1994

If ft is a finite complex Borel measure and T a Lipschitz graph in the complex plane, then for X > 0 jzer:sup / (C-z)-'^C >4 <c(T)X-l\\fi\U. (e>0 V|C-z|>e J It follows that for any finite Borel measure /i and any rectifiable curve T the finite principal value lim/ (f-z)-'^C exists for almost all (with respect to length) zeT. c,,E(z)= I (t-zy'dpt, Jc\B(z,e) C*(z) = sup \Cn,s(z)\, £>0

Journal of Dynamical and Control Systems, 2007

The Michigan Mathematical Journal, 2000

Let Ω be a bounded Lipschitz regular open subset of R d and let µ, ν be two probablity measures on Ω. It is well known that if µ = f dx is absolutely continuous, then there exists, for every p > 1, a unique transport map T p pushing forward µ on ν and which realizes the Monge-Kantorovich distance W p (µ, ν). In this paper, we establish an L ∞ bound for the displacement map T p x − x which depends only on p, on the shape of Ω and on the essential infimum of the density f .

Theory of Probability & Its Applications, 2006

Given two Borel probability measures µ and ν on R d such that dν/dµ = g, we consider certain mappings of the form T (x) = x + F (x) that transform µ into ν. Our main results give estimates of the form

Calculus of Variations and Partial Differential Equations, 2009

We introduce a new class of distances between nonnegative Radon measures in R d . They are modeled on the dynamical characterization of the Kantorovich-Rubinstein-Wasserstein distances proposed by Benamou and Brenier (Numer Math 84:375-393, 2000) and provide a wide family interpolating between the Wasserstein and the homogeneous W −1, p γ -Sobolev distances. From the point of view of optimal transport theory, these distances minimize a dynamical cost to move a given initial distribution of mass to a final configuration. An important difference with the classical setting in mass transport theory is that the cost not only depends on the velocity of the moving particles but also on the densities of the intermediate configurations with respect to a given reference measure γ . We study the topological and geometric properties of these new distances, comparing them with the notion of weak convergence of measures and the well established Kantorovich-Rubinstein-Wasserstein theory. An example of possible applications to the geometric theory of gradient flows is also given.

Revista Matemática Iberoamericana, 2013

We compare several notions of weak (modulus of) gradients in metric measure spaces and prove their equivalence. Using tools from optimal transportation theory we prove density in energy of Lipschitz maps independently of doubling and Poincaré assumptions on the metric measure space.

this is a paper about math

Hiroshima Mathematical Journal

Let C be a cone in a linear space. Under some weak regularity conditions we show that every subadditive function p: C -> R such that p(rx) < rp(x) for some re(0,1) and all xeC must be positively homogenous. As an application we obtain a new characterization of ZΛnorm. This permits us to prove among other things the following converse of Minkowski's inequality.

Journal of Theoretical Probability

The duality theory of the Monge-Kantorovich transport problem is investigated in an abstract measure theoretic framework. Let (X , F , µ) and (Y, G, ν) be any probability spaces and c : X × Y → R a measurable cost function such that f 1 + g 1 ≤ c ≤ f 2 + g 2 for some f 1 , f 2 ∈ L 1 (µ) and g 1 , g 2 ∈ L 1 (ν). Define α(c) = inf P c dP and α * (c) = sup P c dP , where inf and sup are over the probabilities P on F ⊗ G with marginals µ and ν. Some duality theorems for α(c) and α * (c), not requiring µ or ν to be perfect, are proved. As an example, suppose X and Y are metric spaces and µ is separable. Then, duality holds for α(c) (for α * (c)) provided c is upper-semicontinuous (lower-semicontinuous). Moreover, duality holds for both α(c) and α * (c) if the maps x → c(x, y) and y → c(x, y) are continuous, or if c is bounded and x → c(x, y) is continuous. This improves the existing results in [14] if c satisfies the quoted conditions and the cardinalities of X and Y do not exceed the continuum.