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Large deviations for slow-fast processes on connected complete Riemannian manifolds

Yanyan Hu, Richard C. Kraaij, Fubao Xi

Abstract

We consider a class of slow-fast processes on a connected complete Riemannian manifold $M$.The limiting dynamics as the scale separation goes to $\infty$ is governed by the averaging principle. Around this limit, we prove large deviation principles with an action-integral rate function for the slow process by nonlinear semigroup methods together with the Hamilton-Jacobi-Bellman equation techniques. The innovation is solving a comparison principle for viscosity solutions on $M$ and the existence of a viscosity solution via a control problem for a non-smooth Hamiltonian.

Large deviations for slow-fast processes on connected complete Riemannian manifolds

Abstract

We consider a class of slow-fast processes on a connected complete Riemannian manifold .The limiting dynamics as the scale separation goes to is governed by the averaging principle. Around this limit, we prove large deviation principles with an action-integral rate function for the slow process by nonlinear semigroup methods together with the Hamilton-Jacobi-Bellman equation techniques. The innovation is solving a comparison principle for viscosity solutions on and the existence of a viscosity solution via a control problem for a non-smooth Hamiltonian.
Paper Structure (20 sections, 30 theorems, 147 equations, 3 figures)

This paper contains 20 sections, 30 theorems, 147 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Constructing a diffusion process with fast switching on Riemannian manifold
  4. The main results
  5. The strategy of proof \ref{['thm_LDPmanifold']}
  6. The proof of \ref{['thm_LDPmanifold']}
  7. Identification of a single valued Hamiltonian
  8. Multivalued convergent Hamiltonian
  9. Single valued Hamiltonian
  10. Exponential tightness on Riemannian manifold
  11. Comparison principle on Riemannian manifold
  12. The smooth distance function to prove comparison principle
  13. The necessary operators for proving comparison principle
  14. Main propositions: comparison principle
  15. Proof of \ref{['pro_compa_prin']}
  16. ...and 5 more sections

Key Result

Lemma 3.1

Let $f\in C^{\infty}(M)$, and $\Tilde{f}=f\circ \mathbf{p}$ its lift to $O(M)$. Then for any $u\in O(M)$, where $x=\mathbf{p}u$.

Figures (3)

  • Figure 1: The strategy of proof \ref{['thm_LDPmanifold']}
  • Figure 2: An arrow connecting an operator $A$ with operator $B$ with subscript 'sub’ means that viscosity subsolutions of $f-\lambda Af=h$ are also viscosity subsolutions of $f-\lambda Bf=h$. Similarly, we get the description for arrows with a subscript 'super'. The middle gray box around the operators $H_\dagger$ and $H_\ddagger$ indicates that the comparison principle holds for subsolutions of $f-\lambda H_\dagger f=h$ and supersolutions of $f-\lambda H_\ddagger f=h$. The left blue box indicates that $H$ is an implicit and multivalued operator. The right blue box indicates $\mathbf{H}$ is an explicit single valued operator.
  • Figure 3: Add lemmas in Figure \ref{['3']}

Theorems & Definitions (64)

  • Definition 2.1
  • Definition 2.2: Absolutely continuous curves
  • Lemma 3.1: Proposition 3.1.2 of H2002MR1882015
  • Lemma 3.2: Mass conservation BGL2014MR3155209
  • Theorem 3.9
  • proof
  • Theorem 3.10: Large deviation principle
  • Definition 4.1: Operator convergence
  • Proposition 4.2: Adaptation of Theorem 5.15, Theorem 8.27 and Corollary 8.28 in FK2006 to our context
  • Theorem 4.3: existence of viscosity solutions
  • ...and 54 more