Fluctuation and Complex Stochastic Dynamics : From Micro-system to Macro-system

概要

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We proposed the method of complex diffusion equation. Then, we started with a general stochastic deferential equation, and got to a general complex equation which is similar to wave equation of quantum mechanics. If a change of the floating diffusion coefficient is so small, the complex equation is almost equivalent to Schrodinger Equation. We showed that it is safety to adopt a same complex diffusion equation if there are the common fundamental phenomena as Markov process or stochastic dynamics. The concept of the scaling law is related with a renormalization group. An expectation value and dispersion were defined by the wave function of the complex diffusion equation. The method of the complex diffusion equation (MCDE) is a generalization of functional data analysis. We think that minimum uncertainties have essentially nothing to do with the fuzziness of statistical method. At the next step, we have shown a new application of the system analysis, where probability density function obeys the wave equation or Fokker-Planck equation. We consider the fuzziness of the data by introducing fuzzy derivative, we can estimate the width of the predicted data by the fuzzy integration. This method does not assume the shape of the population of sample data. In this method, every knowledge of the system is accumulated in the real and imaginary potentials, so if we accumulate the knowledge, we can read the trend of the past data or foretell the future events