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Exponential mixing of constrained random dynamical systems via controllability conditions

Laurent Mertz, Vahagn Nersesyan, Manuel Rissel

TL;DR

The paper develops a general framework in which deterministic controllability conditions imply exponential mixing for randomly forced constrained dynamics, including non-smooth elasto-plastic models described by differential inclusions. It introduces four core conditions (Lyapunov stability, approximate controllability to an interior point, solid controllability near that point, and decomposability of noise) and proves a main theorem giving a unique stationary measure with exponential convergence in total variation. The authors verify the conditions for low-dimensional elasto-plastic models driven by white or decomposable noise, using a combination of measure transformation, coupling, and recurrence arguments, and they provide a concrete deterministic controllability analysis of the elasto-plastic system. The results offer a robust ergodicity toolkit for non-smooth constrained systems with unbounded state spaces, with potential impact on risk analysis and long-time statistics of plastic deformations under random forcing.

Abstract

We provide deterministic controllability conditions that imply exponential mixing properties for randomly forced constrained dynamical systems with possibly unbounded state space. As an application, new ergodicity results are obtained for non-smooth models in elasto-plasticity driven by various types of noise, including white noise. It is thereby illustrated how tools from control theory can be utilized to tackle regularity issues that commonly arise in the qualitative study of constrained systems.

Exponential mixing of constrained random dynamical systems via controllability conditions

TL;DR

The paper develops a general framework in which deterministic controllability conditions imply exponential mixing for randomly forced constrained dynamics, including non-smooth elasto-plastic models described by differential inclusions. It introduces four core conditions (Lyapunov stability, approximate controllability to an interior point, solid controllability near that point, and decomposability of noise) and proves a main theorem giving a unique stationary measure with exponential convergence in total variation. The authors verify the conditions for low-dimensional elasto-plastic models driven by white or decomposable noise, using a combination of measure transformation, coupling, and recurrence arguments, and they provide a concrete deterministic controllability analysis of the elasto-plastic system. The results offer a robust ergodicity toolkit for non-smooth constrained systems with unbounded state spaces, with potential impact on risk analysis and long-time statistics of plastic deformations under random forcing.

Abstract

We provide deterministic controllability conditions that imply exponential mixing properties for randomly forced constrained dynamical systems with possibly unbounded state space. As an application, new ergodicity results are obtained for non-smooth models in elasto-plasticity driven by various types of noise, including white noise. It is thereby illustrated how tools from control theory can be utilized to tackle regularity issues that commonly arise in the qualitative study of constrained systems.
Paper Structure (36 sections, 12 theorems, 109 equations, 1 figure)

This paper contains 36 sections, 12 theorems, 109 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Review on stochastic elasto-plastic models
  3. Review on controllability methods
  4. Overview of the manuscript
  5. Exponential mixing via controllability
  6. Abstract framework
  7. Conditions
  8. Main result
  9. Proof of the main result
  10. Transition function near $p$
  11. Step 1. Fixing a projection.
  12. Step 2. Utilizing solid controllability.
  13. Step 3. Measure transformation.
  14. Coupling construction
  15. The auxiliary process.
  16. ...and 21 more sections

Key Result

Theorem 1

If $f$ is smooth (infinitely differentiable) in a neighborhood of some interior point $p \in \mathbb{R} \times (-1,1)$, then the Markov process associated with the problem equation:Application1_Model0.1, equation:Application1_Model_initialcondition0.1 has a unique stationary measure that is exponent

Figures (1)

  • Figure 1: A schematic sketch of a controlled trajectory $x = (y, z)$. An initial state $x_0$ which is, for instance, situated in the second quadrant of the $yz$-plane, is connected with a prescribed target state $x_T$ lying northwest of $x_0$ (cf.\ref{['example:elasto-plastic_controlled']}). The reachable set in the plastic phase is indicated by the shaded parts of the lines $z=\pm1$. Red arrows emphasize the orientation of $t \mapsto (y, z)(t)$.

Theorems & Definitions (29)

  • Theorem
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Example 2.7
  • Theorem 2.8
  • Lemma 2.9
  • ...and 19 more