[Paper Review] Asymptotic behavior of multiscale stochastic partial differential equations

Michael Röckner, Longjie Xie+1 more|arXiv (Cornell University)|Oct 28, 2020

Advanced Mathematical Modeling in Engineering34 references6 citations

TL;DR

This paper establishes the averaging principle and normal deviations for a slow-fast stochastic partial differential equation with singular coefficients in Hilbert spaces. Using the Poisson equation in Hilbert space, it proves strong convergence of the slow component to the averaged equation and weak convergence of normalized fluctuations to an Ornstein-Uhlenbeck process, with convergence rates independent of fast component regularity.

ABSTRACT

In this paper, we study the asymptotic behavior of a semi-linear slow-fast stochastic partial differential equation with singular coefficients. Using the Poisson equation in Hilbert space, we first establish the strong convergence in the averaging principe, which can be viewed as a functional law of large numbers. Then we study the stochastic fluctuations between the original system and its averaged equation. We show that the normalized difference converges weakly to an Ornstein-Uhlenbeck type process, which can be viewed as a functional central limit theorem. Furthermore, rates of convergence both for the strong convergence and the normal deviation are obtained, and these convergence are shown not to depend on the regularity of the coefficients in the equation for the fast variable, which coincides with the intuition, since in the limit systems the fast component has been totally averaged or homogenized out.

Motivation & Objective

• To analyze the asymptotic behavior of a semi-linear slow-fast SPDE with singular coefficients in Hilbert spaces.
• To establish the strong convergence of the slow component to its averaged limit, viewed as a functional law of large numbers.
• To characterize the stochastic fluctuations around the averaged equation as a functional central limit theorem.
• To derive explicit convergence rates for both strong convergence and normal deviations, independent of the regularity of the fast component's coefficients.

Proposed method

• Formulates a slow-fast SPDE in Hilbert spaces $H_1 \times H_2$ with a small parameter $\varepsilon$ separating time scales.
• Applies the Poisson equation in Hilbert space to construct a corrector function for the averaging principle.
• Uses Galerkin approximation to handle the infinite-dimensional SPDE and establish moment estimates.
• Employs Kolmogorov equations and Itô's formula to analyze the normalized deviation process.
• Derives moment estimates for the slow and fast components using fractional Sobolev norms and stochastic integration bounds.
• Applies Burkholder-Davis-Gundy inequality and regularity estimates for semigroups to control the noise and drift terms.

Experimental results

Research questions

• RQ1 Does the slow component of the multiscale SPDE converge strongly to the averaged equation as $\varepsilon \to 0$?
• RQ2 What is the rate of convergence for the strong averaging principle in the presence of singular coefficients?
• RQ3 Do the normalized fluctuations between the original and averaged systems converge weakly to a stochastic process?
• RQ4 Can the convergence rates be bounded independently of the regularity of the fast component's coefficients?
• RQ5 What is the limiting behavior of the fluctuation process, and how is it characterized?

Key findings

• The slow component $X_t^\varepsilon$ converges strongly to the averaged solution $\bar{X}_t$ in the $L^q(\Omega)$-sense as $\varepsilon \to 0$, with convergence rate $O(\varepsilon^{\gamma})$ for any $\gamma \in (0,1/2)$.
• The normalized difference $\varepsilon^{-1/2}(X_t^\varepsilon - \bar{X}_t)$ converges weakly to an Ornstein-Uhlenbeck process, establishing a functional central limit theorem.
• The convergence rate for the strong averaging principle is $O(\varepsilon^{\gamma})$, independent of the regularity of the coefficients in the fast equation.
• Moment estimates for $X_t^\varepsilon$ and $\bar{X}_t$ are derived, showing $\mathbb{E}\|AX_t^\varepsilon\|_1^q \leq C_{\theta,\gamma,q,T}(t^{\theta-1} + \varepsilon^{-\gamma})(1 + \|x\|_{(-A)^\theta}^2 + \|y\|_2^{2p})$.
• The averaged equation admits a unique mild solution in $H_1$, and its regularity is controlled via fractional powers of the generator $A$.
• The fluctuation process satisfies a moment bound of order $O(\varepsilon^{\gamma/2})$, consistent with the central limit scaling.