arXiv subjects

Malcolm Bowles

Publications and source records attributed to Malcolm Bowles.

3 recordsLinked to original sources

arXiv2023

Ergodic properties of Kantorovich operators

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 \`a la Fathi-Mather, and ergodic optimization of dynamical systems, we study the asymptotic properties of general Kantorovich operators.

math.AP↗

arXiv2019

Mather Measures and Ergodic Properties of Kantorovich Operators associated to General Mass Transfers

We introduce and study the class of linear transfers between probability distributions and the dual class of Kantorovich operators between function spaces. Linear transfers can be seen as an extension of convex lower semi-continuous energies on Wasserstein space, of cost minimizing mass transports, as well as many other couplings between probability measures to which Monge-Kantorovich theory does not readily apply. Basic examples include balayage of measures, martingale transports, optimal Skorokhod embeddings, and the weak mass transports of Talagrand, Marton, Gozlan and others. The class also includes various stochastic mass transports such as the Schrödinger bridge associated to a reversible Markov process, and the Arnold-Brenier variational principle for the incompressible Euler equations. We associate to most linear transfers, a critical constant, a corresponding effective linear transfer and additive eigenfunctions to their dual Kantorovich operators, that extend Mané's critical value, Aubry-Mather invariant tori, and Fathi's weak KAM solutions for Hamiltonian systems. This amounts to studying the asymptotic properties of the nonlinear Kantorovich operators as opposed to classical ergodic theory, which deals with linear Markov operators. This allows for the extension of Mather theory to other settings such as its stochastic counterpart. We also introduce the class of convex transfers, which includes $p$-powers ($ p \geq 1$) of linear transfers, the logarithmic entropy, the Donsker-Varadhan information, optimal mean field plans, and certain free energies as functions of two probability measures, i.e., where the reference measure is also a variable. Duality formulae for general transfer inequalities follow in a very natural way. This paper is an expanded version of a previously posted but not published work by the authors.

math.AP↗

arXiv2018

A Theory of Transfers: Duality and convolution

We introduce and study the permanence properties of the class of linear transfers between probability measures. This class contains all cost minimizing mass transports, but also martingale mass transports, the Schrodinger bridge associated to a reversible Markov process, and the weak mass transports of Tala- grand, Marton, Gozlan and others. The class also includes various stochastic mass transports to which Monge-Kantorovich theory does not apply. We also introduce the cone of convex transfers, which include any p-power (p > 1) of a linear transfer, but also the logarithmic entropy, the Donsker-Varadhan infor- mation and certain free energy functionals. This first paper is mostly about exhibiting examples that point to the pervasiveness of the concept in the important work on correlating probability distributions. Duality formulae for general transfer inequalities follow in a very natural way. We also study the infinite self-convolution of a linear transfer in order to establish the existence of generalized weak KAM solutions that could be applied to the stochastic counterpart of Fathi-Mather theory.

math.AP↗