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Uniform large deviations and metastability of random dynamical systems

Jifa Jiang, Jian Wang, Jianliang Zhai, Tusheng Zhang

Abstract

In this paper, we first provide a criterion on uniform large deviation principles (ULDP) of stochastic differential equations under Lyapunov conditions on the coefficients, which can be applied to stochastic systems with coefficients of polynomial growth and possible degenerate driving noises. In the second part, using the ULDP criterion we preclude the concentration of limiting measures of invariant measures of stochastic dynamical systems on repellers and acyclic saddle chains and extend Freidlin and Wentzell's asymptotics theorem to stochastic systems with unbounded coefficients. Of particular interest, we determine the limiting measures of the invariant measures of the famous stochastic van der Pol equation and van der Pol Duffing equation whose noises are naturally degenerate. We also construct two examples to match the global phase portraits of Freidlin and Wentzell's unperturbed systems and to explicitly compute their transition difficulty matrices. Other applications include stochastic May-Leonard system and random systems with infinitely many equivalent classes.

Uniform large deviations and metastability of random dynamical systems

Abstract

In this paper, we first provide a criterion on uniform large deviation principles (ULDP) of stochastic differential equations under Lyapunov conditions on the coefficients, which can be applied to stochastic systems with coefficients of polynomial growth and possible degenerate driving noises. In the second part, using the ULDP criterion we preclude the concentration of limiting measures of invariant measures of stochastic dynamical systems on repellers and acyclic saddle chains and extend Freidlin and Wentzell's asymptotics theorem to stochastic systems with unbounded coefficients. Of particular interest, we determine the limiting measures of the invariant measures of the famous stochastic van der Pol equation and van der Pol Duffing equation whose noises are naturally degenerate. We also construct two examples to match the global phase portraits of Freidlin and Wentzell's unperturbed systems and to explicitly compute their transition difficulty matrices. Other applications include stochastic May-Leonard system and random systems with infinitely many equivalent classes.
Paper Structure (11 sections, 18 theorems, 234 equations, 9 figures)

This paper contains 11 sections, 18 theorems, 234 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Uniform large deviation principles
  3. The main result on ULDP
  4. A Sufficient Condition for ULDP
  5. Proof of Theorem \ref{['thmULDP']}
  6. The concentration of limiting measures and metastability
  7. Applications
  8. The realization of Freidlin's two phase portraits of dynamical systems
  9. Stochastic van der Pol equation and single diode circuit
  10. Random perturbation of May-Leonard model
  11. An unbounded example with infinite equivalent classes

Key Result

Proposition 2.1

For any $0<{\varepsilon}<1$, under Assumptions ass1 and ass2, there exists a unique solution to Eq. 12 defined on $[0, +\infty)$.

Figures (9)

  • Figure 1: The phase portrait of system (\ref{['DS']}).
  • Figure 2: The phase portrait of system (\ref{['EE']}).
  • Figure 3: The phase portrait of system (\ref{['DS2']}).
  • Figure 4: The unique limit cycle.
  • Figure 5: Single Diode Circuit.
  • ...and 4 more figures

Theorems & Definitions (30)

  • Definition 2.1
  • Definition 2.2: Freidlin-Wentzell ULDP
  • Definition 2.3
  • Proposition 2.1
  • Proposition 2.2
  • Theorem 2.1
  • Remark 2.1
  • Proposition 2.3
  • Theorem 2.2: A Criteria of Budhiraja-Dupuis
  • Theorem 2.3
  • ...and 20 more