隨機過程的縮併理論分析

Abstract

縮併分析是最重要的模型降階方法之一。借由縮併變換，我們可將模型的幾個狀態合併，使模型的維度降低。傳統的縮併理論的主要著墨於找出適當的縮併變換，使得縮併後的低維模型在動力行為上接近於原高維模型以達到模型降階的目的；而該理論主要適用於由確定性方程所支配的模型。 在本論文中，我們利用縮併變換建構出狀態變量的平均行為在動力學或熱力學上彼此等效的網絡，並探討當系統存在內噪聲或受外噪聲影響下，這些網絡的狀態濃度在漲落上的關聯。對於內噪聲的情形，我們將縮併變換從作用至確定性的化學速率方程拓展到作用於隨機性的化學主方程式(chemical master equation)以及化學朗之萬方程(chemical Langevin equation)，並證明動力學等效的網絡能給出自洽的狀態變量漲落分布；而外噪聲的情形，我們則將縮併變換推廣至隨機微分方程上，給出了狀態漲落在動力學等效網絡之間的大小關係，並探討了在近平衡下由不同形式的自由能微擾所產生的狀態漲落的情形。此外，對於不可化約的動力網絡系統，我們證明了在趨向於穩態時，各狀態將趨向於彼此縮併，由此可和非平衡熱力學上的熵產生概念相連繫。

Lumping analysis is one of the most important methods for model reduction, in which states of higher dimensional kinetic networks are merged into fewer number of new states in other lower dimensional kinetic networks. Prior studies of lumping analysis were concentrated on finding appropriate lumping transformations and lumped networks to reduce the dimension of their rate equations. In this dissertation, we construct hierarchical kinetically or thermodynamically equivalent networks in mean kinetics by using lumping transformations and study the state variable fluctuations in these networks under intrinsic or extrinsic noises. For intrinsic noises, we extend the use of lumping transformations from deterministic rate equations to chemical master equations and chemical Langevin equations, which describe stochastic dynamics, and prove that kinetically equivalent networks possess indistinguishable fluctuations in probability distribution. For extrinsic noises, lumping stochastic differential equation gives general ordering relations of state variance among hierarchical networks. As an example, the free energy variations around equilibrium are investigated. Finally, for irreducible networks, we prove that any lumping process can generate a thermodynamically equivalent network. The increasing lumpability between the probabilities of different thermodynamically equivalent networks may be related to the entropy production in non-equilibrium thermodynamics.