  A recently developed treatment of stochastic processes leads to the
construction of a potential landscape for the dynamical evolution of complex
systems. Since the existence of a potential function in generic settings has
been frequently questioned in literature,herewe study several related
theoretical issues that lie at core of the construction. We showthat the novel
treatment,via a transformation,is closely related to the symplectic structure
that is central in many branches of theoretical physics. Using this insight, we
demonstrate an invariant under the transformation. We further explicitly
demonstrate, in one-dimensional case, the contradistinction among the new
treatment to those of Ito and Stratonovich, as well as others.Our results
strongly suggest that the method from statistical physics can be useful in
studying stochastic, complex systems in general.
