ASYMPTOTIC ANALYSIS AND APPLICATION FOR SOLUTIONS OF MARKOV NOISE SYSTEM BY THE SINGULAR PERTURBATION METHOD

研究结果如下：(1)对于离散时间和连续时间的马氏链，对应于Ehrenfest骨灰盒模型的推广，我们确定了转移概率和平稳分布的结构。(2)对于被标记为多粒子的随机过程，在奇异摄动下，我们推导并辨识了极限过程，并阐明了相互作用力的影响。(3)对于马尔可夫过程，即非线性三维Lorenz型随机微分方程解，通过适当的时空变换和渐近分析，我们得到了一个以随机力为极限过程的随机Duffing振子。(4)我们利用计算机网络和信息处理对上述结果进行了研究，这些结果已发展到图像压缩、人口动力学、经济时间序列、信息安全和随机分形等噪声分析中。(5)我们在日本SIAM会议上介绍了上述结果。(6)将上述结果推广到具有自相似噪声的随机微分方程离散逼近的理论和应用，以及多媒体中分形图像的数据压缩。

The research results are as follows :(1) For Markov chains with discrete-time and continuous-time, which correspond to a generalization of the Ehrenfest urn model, we have determined the structure of transition probabilities and stationary distributions. The results have developed into applications to reliability theory and queueing system in operations research.(2) For stochastic processes with fast-motion and delayed motion, which are tagged to multi-particle and considered as solutions of linear stochastic differential equations with mean-field interaction of the McKean type under singular perturbation, we have derived and identified the limit process and clarified the effect of interacting force.(3) For Markov processed, which are solutions of the nonlinear 3-dimensional stochastic differential equations of the Lorenz type, we have obtained a stochastic Duffing oscillator with random force as a limit process by a suitable space-time transformation and an asymptotic analysis.(4) We have investigated the above-cited results by computer network and information processing, that have developed into applications to noise analyses, such as image compression, population dynamics, economic time-series, information security and random fractal.(5) We have presented the above-cited results at the Japan SIAM meeting, the RIMS(Kyoto Univ.) workshop and the Japan IMA meeting.(6) We are going to extend the above-cited results to the theory and application of discrete approximation for stochastic differential equations with self-similar noises and data compression of fractal images in multi-media.