Duality Theorems in Ergodic Transport

• Published in: Journal of statistical physics Vol. 149; no. 5; pp. 921 - 942
• Main Authors: Lopes, Artur O., Mengue, Jairo K.
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01-11-2012
• Subjects: Mathematical and Computational Physics; Physical Chemistry; Physics; Physics and Astronomy; Quantum Physics; Statistical Physics and Dynamical Systems; Theoretical; Invariant probability; Periodic orbits; Admissible pairs; Kantorovich Duality; Transport Theory; Ergodic Theory; Holonomic probabilities; Optimal plan
• ISSN: 0022-4715; 1572-9613
• DOI: 10.1007/s10955-012-0626-3

We analyze several problems of Optimal Transport Theory in the setting of Ergodic Theory. In a certain class of problems we consider questions in Ergodic Transport which are generalizations of the ones in Ergodic Optimization.
Another class of problems is the following: suppose
σ
is the shift acting on Bernoulli space
X
={1,2,…,
d
}
ℕ
, and, consider a fixed continuous cost function
c
:
X
×
X
→ℝ. Denote by
Π
the set of all Borel probabilities
π
on
X
×
X
, such that, both its
x
and
y
marginals are
σ
-invariant probabilities. We are interested in the optimal plan
π
which minimizes ∫
c

dπ
among the probabilities in
Π
.
We show, among other things, the analogous Kantorovich Duality Theorem. We also analyze uniqueness of the optimal plan under generic assumptions on
c
. We investigate the existence of a dual pair of Lipschitz functions which realizes the present dual Kantorovich problem under the assumption that the cost is Lipschitz continuous. For continuous costs
c
the corresponding results in the Classical Transport Theory and in Ergodic Transport Theory can be, eventually, different.
We also consider the problem of approximating the optimal plan
π
by convex combinations of plans such that the support projects in periodic orbits.