We are interested in some biological phenomena which are evolutional ran-
dom complex systems. Simplest cases are expressed in the form of stochastic processes
X(t) that are functionals of white noise. In more general cases those phenomena in
question are viewed as random fields X(C) depending on a manifold C running through
a space-time Euclidean space R^n. We may assume that C is an (n-1)-dimensional
smooth ovaloid so that variational calculus can be applied smoothly.
Our approach starts with the step of reduction of the complex random systems.
This means that we try to find a system of idealized elemental random variables (abb.
i.e.r.v.'s) that has the same information as the given random system, in addition, in
a causal manner. Once the system is expressed as functionals of the i.e.r.v.'s, we
are ready to analyze them by appealing to the white noise analysis which has been
extensively developed in recent years. That is the step of the analysis. Having
established the analysis, namely after the systems are well investigated, we can finally
come to the step of applications.
The system of i.e.r.v.'s. is the so-called the innovation of the random evolutional
system, for either stochastic processes or random fields.
The innovation, if it exists, appears in the stochastic differential equations for
stochastic processes X(t) and in the variational equations for random fields X(C).
We now assume that i.e.r.v. is a (Gaussian) white noise. The variational equation is
a generalization, in a sense, of a stochastic differential equation. In addition to the
usual technique for ordinary stochastic differential equations, we need new method of
calculus for which white noise analysis can be a powerful tool. Generalized white noise
functionals and creation and annihilation operators are efficiently used. Applications
to image processing give us interesting questions, in particular, in medical images.
More examples illustrate our idea.