Asymptotic Series for Singularly Perturbed Kolmogorov-Fokker-Planck Equations
Asymptotic Series for Singularly Perturbed Kolmogorov-Fokker-Planck Equations
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DOI：
10.1137/s0036139994270085
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发表时间：
1996-12
期刊：
影响因子：
--
通讯作者：
R. Khasminskii;G. Yin
中科院分区：
文献类型：
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作者：
R. Khasminskii;G. Yin
我们得到了扩散过程的转移密度的极限定理，并发展了一类奇摄动Kolmogorov-Fokker-Planck方程解的渐近展开式。所考虑的模型可以被看作是一个马尔可夫过程有两个时间尺度。其中一个是快速变化的规模，另一个是缓慢变化的规模。这项研究的动机是广泛的应用，涉及奇摄动马尔可夫过程在制造系统，可靠性分析，网络，统计物理，人口生物学，金融经济学，和许多其他相关领域。在这项工作中，渐近展开构造明确。结果表明，在膨胀的初始层项以指数速率衰减。余项的误差界也得到了。扩展的有效性是严格合理的。
We derive limit theorems for the transition densities of diffusion processes and develop asymptotic expansions for solutions of a class of singularly perturbed Kolmogorov–Fokker–Planck equations. The model under consideration can be viewed as a Markov process having two time scales. One of them is a rapidly changing scale, and the other is a slowly varying one. The study is motivated by a wide range of applications involving singularly perturbed Markov processes in manufacturing systems, reliability analysis, queueing networks, statistical physics, population biology, financial economics, and many other related fields. In this work, the asymptotic expansion is constructed explicitly. It is shown that the initial layer terms in the expansion decay at an exponential rate. Error bounds on the remainder terms also are obtained. The validity of the expansion is rigorously justified.