[Paper Review] Large deviations for slow-fast stochastic partial differential equations

Wei WangA, J. Roberts+1 more|arXiv (Cornell University)|Jan 26, 2010

Stochastic processes and financial applications45 references3 citations

TL;DR

This paper establishes a large deviation principle (LDP) for slow-fast stochastic partial differential equations (SPDEs) with coupled components. It proves that the rate function for the slow component is equivalent to that of an averaged equation plus a small Gaussian fluctuation, confirming the effectiveness of the averaged dynamics with stochastic deviation as an approximation under small noise and fast-scale separation.

ABSTRACT

A large deviation principle is derived for stochastic partial differential equations with slow-fast components. The result shows that the rate function is exactly that of the averaged equation plus the fluctuating deviation which is a stochastic partial differential equation with small Gaussian perturbation. This also confirms the effectiveness of the approximation of the averaged equation plus the fluctuating deviation to the slow-fast stochastic partial differential equations.

Motivation & Objective

• To derive a large deviation principle (LDP) for slow-fast SPDEs with coupled stochastic and deterministic components.
• To confirm the validity of approximating the slow component by an averaged equation plus a small Gaussian fluctuation.
• To analyze the metastability and rare event probabilities in multiscale SPDEs with small noise.
• To extend Freidlin–Wentzell theory to infinite-dimensional, coupled SPDE systems with two time scales.
• To provide rigorous justification for the use of averaged SPDEs with fluctuating deviations in modeling complex stochastic systems.

Proposed method

• Derives the LDP for a class of coupled SPDEs with a slow component $ u^\epsilon $ and a fast component $ v^\epsilon $, where $ \epsilon \to 0 $.
• Uses auxiliary systems and contraction principles to relate the LDP of the original system to that of a perturbed averaged SPDE with small Gaussian noise.
• Applies exponential tightness estimates to handle the infinite-dimensional nature of the SPDEs and ensure convergence of the LDP.
• Employs a martingale-based approximation method to show that the deviation $ u^\epsilon - u $ is asymptotically $ \sqrt{\epsilon} z(t) $, where $ z(t) $ is a Gaussian process.
• Constructs explicit approximations of the slow and fast fields using multiple scales and stochastic averaging, including higher-order corrections.
• Validates the LDP approximation via a detailed example of a stochastic reaction-diffusion SPDE near a pitchfork bifurcation.

Experimental results

Research questions

• RQ1 Does the large deviation principle for slow-fast SPDEs reduce to that of an averaged equation plus a small Gaussian perturbation?
• RQ2 How accurate is the approximation of the slow component by the averaged dynamics plus a stochastic deviation under small noise and fast-scale separation?
• RQ3 Can the LDP for the original system be derived via auxiliary systems and contraction principles despite non-autonomous transformations?
• RQ4 What is the role of the fast component's ergodicity and noise intensity in shaping the rate function of the slow component?
• RQ5 How do higher-order corrections in the amplitude dynamics affect the correspondence between the original and averaged systems?

Key findings

• The rate function for the slow component $ u^\epsilon $ is exactly the same as that of the averaged SPDE with a small Gaussian perturbation, confirming the LDP structure.
• The deviation $ u^\epsilon(t) - u(t) $ is asymptotically approximated by $ \sqrt{\epsilon} z(t) $, where $ z(t) $ is a Gaussian process, validating the fluctuation term.
• The LDP for the original system is derived via exponential tightness and contraction principles applied to auxiliary systems, overcoming issues with non-autonomous fast dynamics.
• The superslow manifold dynamics derived from the LDP-averaged SPDE match those of the original system up to $ \mathcal{O}(\epsilon) $ errors, confirming structural consistency.
• In the example near a stochastic pitchfork bifurcation, the LDP-averaged system reproduces the correct amplitude evolution with $ \mathcal{O}(\epsilon) $-accurate convolutions and noise terms.
• The correspondence between the original and averaged systems is stronger than weak convergence, as evidenced by identical convolution structures in the amplitude equations.