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，我们确定了其转移概率和平稳分布的结构，这是Escherifest模型的推广。其结果已在可靠性理论和运筹学中的决策系统中得到应用。(2)对于具有快运动和延迟运动的随机过程，我们将其标记为多粒子，并将其视为具有平均场相互作用的McKean型线性随机微分方程在奇异摄动下的解，导出并识别了极限过程，阐明了相互作用力的影响. (3)对于三维Lorenz型非线性随机微分方程的Markov过程解，通过适当的时空变换和渐近分析，得到了一个具有随机力的极限过程的随机Duffing振子. (4)我们利用计算机网络和信息处理技术对上述结果进行了研究，并将其应用于图像压缩、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)我们已经在日本SIAM会议、RIMS（京都大学）研讨会和日本IMA会议。(6)我们将把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和多媒体分形图像数据压缩的理论和应用中。

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.