Abstract
The author studies the dynamic theory of stochastic processes. The dynamic theory concerns an evolution
equation which contains a potential function , and the diffusion matrix and drift vector. The case with no
potential term can be treated in the framework of the conventional theory of Markov processes of Kolmogorov
and Itˆo, which is a kinematic theory. The kinematic equation determines Markov (diffusion) processes, i.e.,the
movement of systems. By contrast, the author considers the equation of motion in the mechanics part of the
dynamic theory. The equation of motion contains the potential function of external forces. External forces
influence the movement of systems, but not in a direct way. The kinematic equation finally describes sample
paths of the movement of observing systems. In this paper the author clarifies the mathematical structures
which connect three notions, external force, induced drift vector and sample paths of the movement.
Nagasawa, M