A Note on Duality Theorems in Mass Transportation
- PDF / 294,725 Bytes
- 14 Pages / 439.37 x 666.142 pts Page_size
- 86 Downloads / 166 Views
A Note on Duality Theorems in Mass Transportation Pietro Rigo1 Received: 4 June 2019 / Revised: 13 July 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract 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 +g1 ≤ c ≤ f 2 +g2 for some f 1 , f 2 ∈ L 1 (μ) and g1 , g2 ∈ L 1 (ν). Define α(c) = inf P c d P and α ∗ (c) = sup P c d P, 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 Ramachandran and Ruschendorf (Probab Theory Relat Fields 101:311–319, 1995) if c satisfies the quoted conditions and the cardinalities of X and Y do not exceed the continuum. Keywords Duality theorem · Mass transportation · Perfect probability measure · Probability measure with given marginals · Separable probability measure Mathematics Subject Classification (2010) 60A10 · 60E05 · 28A35
1 Introduction Throughout, (X , F, μ) and (Y, G, ν) are probability spaces and H=F ⊗G is the product σ -field on X × Y. Further, (μ, ν) is the collection of probability measures P on H with marginals μ and ν, namely
B 1
Pietro Rigo [email protected] Dipartimento di Matematica “F. Casorati”, Universita’ di Pavia, Via Ferrata 1, 27100 Pavia, Italy
123
Journal of Theoretical Probability
P(A × Y) = μ(A) and P(X × B) = ν(B) for all A ∈ F and B ∈ G. For any probability space (, A, Q), we write L 1 (Q) to denote the class of Ameasurable and Q-integrable functions φ : → R (without identifying maps which agree Q-a.s.). We also write Q(φ) = φ d Q for φ ∈ L 1 (Q). With a slight abuse of notation, for any maps f : X → R and g : Y → R, we still denote by f and g the functions on X ×Y given by (x, y) → f (x) and (x, y) → g(y). Thus, f + g is the map on X × Y defined as ( f + g)(x, y) = f (x) + g(y) for all (x, y) ∈ X × Y. In this notation, we let L = { f + g : f ∈ L 1 (μ), g ∈ L 1 (ν)}. Let c : X × Y → R be an H-measurable function satisfying f 1 + g1 ≤ c ≤ f 2 + g2
for some f 1 + g1 ∈ L and f 2 + g2 ∈ L.
(1)
For such a c, we define α(c) = inf P(c) : P ∈ (μ, ν) , α ∗ (c) = sup P(c) : P ∈ (μ, ν) , β(c) = sup μ( f ) + ν(g) : f + g ∈ L, f + g ≤ c , β ∗ (c) = inf μ( f ) + ν(g) : f + g ∈ L, f + g ≥ c . It is not hard to see that β(c) ≤ α(c) ≤ α ∗ (c) ≤ β ∗ (c). A duality theorem (for both α(c) and α ∗ (c)) is the assertion that α(c) = β(c) and α ∗ (c) = β ∗ (c).
(2)
Indeed, duality theorems arise in a plenty of frameworks. The main one is poss
Data Loading...
