With the aid of path-integral representation of the Fokker-Planck equation derived from generalized random walks, behaviors of the most probable path are studied. The processes are specified by the jumping probabilities ${\tilde{P}}^{\alpha }=P_0^{\alpha }+\left(\frac{\alpha }{2}\right)\lambda \exp \left(\mu \mathrm{m}\right)$, where $P_0^{\alpha }$'s ($\alpha =\pm ,0\mathrm{}$) are the usual jumping probabilities, $\lambda$ is a parameter representing deviation from the usual processes, and $\mu$ is a positive constant. It is found that the parameter $\lambda$ characterizes the most-probable paths, accordingly as $\lambda >0\mathrm{}$, $\lambda =0\mathrm{}$, and $\lambda <0\mathrm{}$. Furthermore, the most-probable path is specified by a "critical time interval." The critical time interval is determined such that the "Euler-Lagrangian" has a solution showing a minimum of action. A relation between the critical time interval and the coarse graining is briefly discussed. Also Riemannian-geometrical interpretation of the "walker's site-step space" is given.

• Received 4 October 1982

DOI:https://doi.org/10.1103/PhysRevB.28.4403