Derivation of the Selected Path Integral

Abstract

To analyze the generalized Brownian motion, i.e. the{fractional} Brownian motion, we propose a path integral which isgoverned by the modified action along the principal path with fractalnatures, i.e. mainly observable path in a diffusive phenomena. Bymodified the definition of the action and summing over fractal paths, the path integral is derived. We investigate several properties ofthis integral. The principal path has a fractal structure and the pathintegral represents the transition probability of the fractionalBrownian motion. The transition probability itself has no dependenceon the structure of principal paths. The path integral is mainlycharacterized by two parameters, the Hausdorff dimension D_H ofprincipal paths representing a microscopic structure and the Hurstcoefficient H representing a macroscopic structure, which areindependent of each other.