  We propose new equations of motion under the theory of the Brownian motion to
connect the states of quantum, diffusion, soliton, and periodic localization.
The new equations are nothing but the classical equations of motion with two
additional terms and the one of them can be regarded as the the quantum
potential. By choosing a parameter space, various important states are
obtained. Further, the equations contain other interesting phenomena such as
general dynamics of diffusion process, collapse of the soliton, the nonlinear
extension of the Schr\"dinger equation, and the dynamics of phase transition.
