DERIVATION OF THE SELECTED PATH-INTEGRAL

To analyze the generalized Brownian motion, i.e. the fractional Browni an motion, we propose a path integral which is governed by the modifie d action along the principal path with fractal natures, i.e. mainly ob servable path in a diffusive phenomena. By modified the definition of the action and summing over fractal paths, the path integral is derive d. We investigate several properties of this integral. The principal p ath has a fractal structure and the path integral represents the trans ition probability of the fractional Brownian motion. The transition pr obability itself has no dependence on the structure of principal paths . The path integral is mainly characterized by two parameters, the Hau sdorff dimension D-H of principal paths representing a microscopic str ucture and the Hurst coefficient H representing a macroscopic structur e, which are independent of each other.