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Ergodic properties of Kantorovich operators

Nassif Ghoussoub, Malcolm Bowles

TL;DR

The paper develops a unified ergodic-thermodynamic framework for general Kantorovich operators, nonlinear counterparts of Markov operators, connecting optimal transport, weak KAM theory, and ergodic optimization. It introduces the Mather constant $c(T)$ and Mañé-type notions, proves existence of backward weak KAM solutions $h$ with $T^- h + c(T)=h$, and constructs idempotent weak KAM operators $T_\infty$ that commute with $T$, along with the Peierls barrier $\mathcal{T}_\infty$. The authors extend the theory to continuous time via semigroups, establishing a continuous-time Mather constant and weak KAM theory for Hamilton–Jacobi–Bellman equations and Lagrangian dynamics, including stochastic versions. They provide a breadth of examples—ranging from convex-energy and balayage transfers to optimal transport and symbolic dynamics—and apply the framework to ergodic optimization in deterministic and stochastic holonomic settings, thereby unifying diverse approaches under a common Kantorovich-operator perspective.

Abstract

Kantorovich operators are non-linear extensions of Markov operators and are omnipresent in several branches of mathematical analysis. The asymptotic behaviour of their iterates plays an important role even in classical ergodic, potential and probability theories, which are normally concerned with linear Markovian operators, semi-groups, and resolvents. The Kantorovich operators that appear implicitly in these cases, though non-linear, are all positively 1-homogenous. General Kantorovich operators amount to assigning "a cost" to most operations on measures and functions normally conducted "for free" in these classical settings. Motivated by extensions of the Monge-Kantorovich duality in mass transport, the stochastic counterpart of Aubry-Mather theory for Lagrangian systems, weak KAM theory à la Fathi-Mather, and ergodic optimization of dynamical systems, we study the asymptotic properties of general Kantorovich operators.

Ergodic properties of Kantorovich operators

TL;DR

The paper develops a unified ergodic-thermodynamic framework for general Kantorovich operators, nonlinear counterparts of Markov operators, connecting optimal transport, weak KAM theory, and ergodic optimization. It introduces the Mather constant and Mañé-type notions, proves existence of backward weak KAM solutions with , and constructs idempotent weak KAM operators that commute with , along with the Peierls barrier . The authors extend the theory to continuous time via semigroups, establishing a continuous-time Mather constant and weak KAM theory for Hamilton–Jacobi–Bellman equations and Lagrangian dynamics, including stochastic versions. They provide a breadth of examples—ranging from convex-energy and balayage transfers to optimal transport and symbolic dynamics—and apply the framework to ergodic optimization in deterministic and stochastic holonomic settings, thereby unifying diverse approaches under a common Kantorovich-operator perspective.

Abstract

Kantorovich operators are non-linear extensions of Markov operators and are omnipresent in several branches of mathematical analysis. The asymptotic behaviour of their iterates plays an important role even in classical ergodic, potential and probability theories, which are normally concerned with linear Markovian operators, semi-groups, and resolvents. The Kantorovich operators that appear implicitly in these cases, though non-linear, are all positively 1-homogenous. General Kantorovich operators amount to assigning "a cost" to most operations on measures and functions normally conducted "for free" in these classical settings. Motivated by extensions of the Monge-Kantorovich duality in mass transport, the stochastic counterpart of Aubry-Mather theory for Lagrangian systems, weak KAM theory à la Fathi-Mather, and ergodic optimization of dynamical systems, we study the asymptotic properties of general Kantorovich operators.
Paper Structure (7 sections, 46 theorems, 269 equations)

This paper contains 7 sections, 46 theorems, 269 equations.

Table of Contents

  1. Introduction
  2. Basic properties of Kantorovich operators
  3. The Mather constant and weak KAM solutions for a Kantorovich operator
  4. Idempotent Kantorovich operators
  5. Weak KAM operators associated to Kantorovich operators
  6. Semi-groups of Kantorovich operators and weak KAM theories
  7. Linear Transfers and Ergodic Optimization

Key Result

Theorem 1.2

G Let $T$ be a map from $C(Y)$ to $USC(X)$, where $X$ and $Y$ are two compact metric spaces. The following are then equivalent: The gambling house is then, where $T_\#\mu (g)=\int_XTg \, d\mu$ for every $g\in C(Y)$.

Theorems & Definitions (99)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Theorem 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Definition 2.5
  • Theorem 2.6
  • Proposition 2.7
  • ...and 89 more