The entropy functional, the information path functional's essentials and their connections to Kolmogorov's entropy, complexity and physics

Introduction The entropy functional, defined on a Markov diffusion process, plays an important roll in theory of information and statistical physics [1-3], informational macrodynamics and control systems [4]. However in the known references, we did not find the mathematical results related to the entropy functional’s connections with the Kolmogorov’s entropy, complexity, and the Lyapunov’s characteristics [5]. We analyze these connections through the information path functional (IPF) of controllable diffusion process, considering the IPF’s essentials and specifics, including the singularities of the IPF extremal equations and the created invariant relations. The paper is a part of the information path functional approach [6-8] in solving the problems of information evolutionary dynamics. Searching a law, which governs complex dynamics of interacting process, leads us to the process’ statistical dynamics first, and then to finding a variation principle that, according to R. Feynman (The character of physical law), might describe regularities of such dynamics, possibly at a macroscopic level. Sec.1 introduces the entropy functional expressed via an additive functional of controllable diffusion process and the parameters of corresponding stochastic equation, applied to the IPF variation problem. Sec.2. presents an essence of the IPF approach, considering both the problem statement and its formalization for a class of random systems, modeled by the solutions of controlled Ito’s stochastic differential equations; defines a probabilistic measure of the functional’s distance between a current process’ trajectory and some given process; applies the entropy functional, defined on these solutionstrajectories, and presents the dynamic approximation of both this probabilistic measure and the conditional entropy functional by the corresponding path functionals. We illustrate a specific of the formulated variation problem’s solution using both the Kolmogorov (K) eqs. for the functional of a Markov process and the Jacobi-Hamilton (J-H) eq. for the dynamic approximation of this functional as the IPF. Because the fulfillment of both J-H and K equations in the same field’s region of a space is possible at some “punched” discretely selected points (DP), the extremal trajectory, solving the variation problem (VP) is divided on the extremal’s segments at each DP. Between the extremal segments (at DP) exists a “window”, where the random information affects the dynamic process on the extremals, creating it’s piece-wise dependency upon the observed data.