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Fast, slow convergence, and concentration in the house of cards replicator-mutator model

Bertrand Cloez, Pierre Gabriel

Abstract

We propose a fine analysis of the various possible long time behaviours of the solutions of the replicator-mutator equation with so-called Kingman's house of cards mutations. In particular, we give what is to our knowledge the first concentration result for this model.

Fast, slow convergence, and concentration in the house of cards replicator-mutator model

Abstract

We propose a fine analysis of the various possible long time behaviours of the solutions of the replicator-mutator equation with so-called Kingman's house of cards mutations. In particular, we give what is to our knowledge the first concentration result for this model.
Paper Structure (9 sections, 9 theorems, 158 equations)

This paper contains 9 sections, 9 theorems, 158 equations.

Table of Contents

  1. Introduction
  2. A related conservative equation
  3. Geometric convergence
  4. Algebraic convergence
  5. Proof of \ref{['eq:algebraic-decay-inf']}
  6. Proof of \ref{['eq:algebraic-decay-r']}
  7. Weak convergence
  8. The non-conservative linear equation
  9. The nonlinear conservative equation

Key Result

Theorem 1.1

We suppose that (H$\mathcal{X}$)-(H$Q$)-(H$a$) are met. Then the following results hold:

Theorems & Definitions (28)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • proof : Proof of Theorem \ref{['th:main-lin-cons']}-(\ref{['th:main-lin-cons-fast-TV']})
  • Proposition 2.2: $\Phi$-entropy decay
  • proof : Proof of Proposition \ref{['prop:logsob']}
  • proof : Proof of Theorem \ref{['th:main-lin-cons']}-(\ref{['th:main-lin-cons-fast-Lr_1<r<2']})-(\ref{['th:main-lin-cons-fast-Lr_r>2']})-(\ref{['th:main-lin-cons-fast-Linf']})-(\ref{['th:main-lin-cons-fast-KL']})
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • ...and 18 more