The mathematical formulation of the model for molecular movement of single
motor proteins driven by cyclic biochemical reactions in an aqueous environ
ment leads to a drifted Brownian motion characterized by coupled diffusion
equations. In this article, we introduce the basic notion for the continuou
s model and review some asymptotic solutions for the problem. (For the latt
ice model see [17,47].) Stochastic, non-equilibrium thermodynamic interpret
ations of the mathematical equations and their solutions are presented. Som
e relevant mathematics, mainly in the field of stochastic processes, are di
scussed.