Fluctuation and Relaxation in Stochastic Systems

A new definition of a stochastic integral is proposed, and the relation among the It6 type, Stratonovich type and new type of stochastic integrals is discussed. A stochastic linear response is studied in a general process. This is used to derive the Fokker-Planck equation from stochastic differential equations. Applications of Kubo's stochastic Liouville equations are presented. With the use of these formulations, relaxation and fluctuation are studied in stochastic systems, particularly near the instability point. The essence of the scaling theory is also presented. The stochastic time variable is briefly mentioned.