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Large Deviation Principle for Multi-Scale Fully Local Monotone Stochastic Dynamical Systems with Multiplicative Noise

Wei Hong, Wei Liu, Luhan Yang

Abstract

This paper is devoted to proving the small noise asymptotic behaviour, particularly large deviation principle, for multi-scale stochastic dynamical systems with fully local monotone coefficients driven by multiplicative noise. The main techniques are based on a combination of the weak convergence approach, the time discretization technique and the theory of pseudo-monotone operator. The main results derived in this paper have broad applicability to various multi-scale models, where the slow component could be such as stochastic porous medium equations, stochastic Cahn-Hilliard equations and stochastic 2D Liquid crystal equations.

Large Deviation Principle for Multi-Scale Fully Local Monotone Stochastic Dynamical Systems with Multiplicative Noise

Abstract

This paper is devoted to proving the small noise asymptotic behaviour, particularly large deviation principle, for multi-scale stochastic dynamical systems with fully local monotone coefficients driven by multiplicative noise. The main techniques are based on a combination of the weak convergence approach, the time discretization technique and the theory of pseudo-monotone operator. The main results derived in this paper have broad applicability to various multi-scale models, where the slow component could be such as stochastic porous medium equations, stochastic Cahn-Hilliard equations and stochastic 2D Liquid crystal equations.
Paper Structure (12 sections, 16 theorems, 231 equations)

This paper contains 12 sections, 16 theorems, 231 equations.

Table of Contents

  1. Introduction
  2. Large deviation principle
  3. Weak convergence analysis
  4. Hypotheses and main results
  5. Stochastic controlled equations
  6. Some energy estimates
  7. A time increment estimate
  8. Proof of main results
  9. Verification of (b)
  10. Verification of (a)
  11. Application to examples
  12. Appendix

Key Result

Lemma 2.1

Assume that $\mathcal{G}^\varepsilon:C([0,T];\mathscr{U}_0)\to \mathscr{E}$ is a family of measurable mappings. If $X^{\varepsilon}=\mathcal{G}^\varepsilon(W_\cdot)$ and there exists a measurable map $\mathcal{G}^0:C([0,T];\mathscr{U}_0)\to \mathscr{E}$ such that the following two conditions hold: ( in distribution as $\varepsilon\to0.$ (b) For any $M<\infty,$ the set is a compact subset in $\mat

Theorems & Definitions (39)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.1
  • Lemma 2.2
  • Definition 2.4
  • Remark 2.1
  • Remark 2.2
  • Theorem 2.1
  • Remark 2.3
  • ...and 29 more