[Paper Review] Asymptotic expansion of Markov random evolution

Igor Samoilenko|ArXiv.org|Nov 1, 2009

Stochastic processes and statistical mechanics9 references3 citations

TL;DR

This paper develops a rigorous asymptotic expansion for solutions of singularly perturbed Markov random evolution equations in $\mathbb{R}^d$, decomposing the solution into regular and singular (boundary layer) components via a multiscale expansion in $\varepsilon$. The key contribution is a constructive algorithm that determines the regular part independently of the singular terms, with explicit estimates on the remainder term, improving upon prior methods in convergence analysis and diffusion approximation.

ABSTRACT

Is studied asymptotic expansion for solution of singularly perturbed equation for Markov random evolution in Rd. The views of regular and singular parts of solution are found.

Motivation & Objective

• To analyze the asymptotic behavior of Markov random evolution (MRE) as the small parameter $\varepsilon \to 0$.
• To decompose the solution into regular and singular (boundary layer) components to capture fast and slow dynamics.
• To develop a recursive algorithm for computing the regular part of the expansion without relying on singular terms.
• To derive explicit estimates on the remainder of the asymptotic expansion for $C^2$ test functions.
• To improve the convergence analysis in the hydrodynamic limit and diffusion approximation framework.

Proposed method

• Formal asymptotic expansion in powers of $\varepsilon$ is constructed as $\Phi_t^\varepsilon(u,x) = u^{(0)}(t) + \sum_{k=1}^\infty \varepsilon^k [u^{(k)}(t) + w^{(k)}(t/\varepsilon)]$, separating regular and boundary layer terms.
• The regular part $u^{(k)}(t)$ satisfies a hierarchy of deterministic ODEs derived from the generator $\mathbb{V}(x)\varphi(u) = v(u;x)\varphi'(u)$.
• The singular part $w^{(k)}(\tau)$, $\tau = t/\varepsilon$, is solved via an integral equation involving the resolvent $R_0$ of the Markov generator $Q$, ensuring fast decay in the boundary layer.
• Initial conditions for the regular terms are derived independently using $c^{(k)}(0) = \mathbb{V}\widetilde{w}^{(k-1)}(0)$, avoiding recursive coupling with singular terms.
• The remainder $\tilde{\Phi}^\varepsilon(t)$ is estimated via Gronwall-Bellman inequality applied to the residual equation $\frac{d}{dt}\tilde{\Phi}^\varepsilon - L^\varepsilon\tilde{\Phi}^\varepsilon = \varepsilon\theta^\varepsilon$, using the bounded inverse $ (L^\varepsilon)^{-1} $.
• A refined estimate $||\tilde{\Phi}^\varepsilon(t)|| \leq \varepsilon||\tilde{\Phi}^\varepsilon(0)|| \exp\{\varepsilon L||\theta^\varepsilon||\} $ is established with $L \geq 2||(L^\varepsilon)^{-1}||$.

Experimental results

Research questions

• RQ1 How can the solution of a singularly perturbed Markov random evolution be decomposed into regular and singular components as $\varepsilon \to 0$?
• RQ2 What is the structure of the asymptotic expansion for the evolution semigroup $\Phi_t^\varepsilon(u,x)$, and how are the regular and singular terms determined recursively?
• RQ3 Can the regular part of the expansion be computed independently of the singular (boundary layer) terms?
• RQ4 What is the rate of convergence of the MRE to its averaged limit, and how can the remainder be quantitatively estimated?
• RQ5 How does the proposed algorithm improve upon existing methods in terms of computational decoupling and remainder control?

Key findings

• The regular part of the asymptotic expansion $u^{(k)}(t)$ is determined by a recursive system of ODEs derived from the generator $\mathbb{V}(x)\varphi(u) = v(u;x)\varphi'(u)$, with initial conditions $c^{(k)}(0) = \mathbb{V}\widetilde{w}^{(k-1)}(0)$.
• The singular (boundary layer) terms $w^{(k)}(\tau)$ are solved via an integral equation involving the resolvent $R_0 = \int_0^\infty (P(t) - \pi) dt$, ensuring exponential decay in $\tau = t/\varepsilon$.
• The initial condition for the $k$-th regular term is explicitly determined without requiring knowledge of the singular terms, enabling a separate recursive algorithm for the regular part.
• A remainder estimate is derived: $||\tilde{\Phi}^\varepsilon(t)|| \leq \varepsilon||\tilde{\Phi}^\varepsilon(0)|| \exp\{\varepsilon L||\theta^\varepsilon||\}$, where $L \geq 2||(L^\varepsilon)^{-1}||$, valid for $C^2$ test functions.
• For higher-order expansions, the remainder estimate generalizes to $||\tilde{\Phi}^\varepsilon_{N+1}(t)|| \leq \varepsilon^N||\tilde{\Phi}^\varepsilon(0)|| \exp\{\varepsilon^N L||\theta^\varepsilon_N||\}$, showing improved convergence for larger $N$.
• The method improves on prior algorithms by decoupling the computation of regular terms from singular terms, enhancing computational efficiency and analytical clarity.