The Impulse Interactive Cuts of Entropy Functional Measure on Trajectories of Markov Diffusion Process, Integrating in Information Path Functional, Encoding and Applications

Abstract

Integrating discrete information, composed of information Bits extracted from observing random process, solves the impulse cutting off entropy functional (EF) measure on trajectories Markov diffusion process whose information integrates path functional (IPF). Each cut brings memory of the entropy being cut, which provides both reduction of the process entropy and discrete unit of the cutting entropy -a Bit. Consequently, information is memorized entropy cutting in random observations which process interactions. The origin of information associates with an "anatomy of creation of impulse" enables both cut and stipulate random process while generating information under the cut. Memory stands as the impulse' cut time interval. Defining the EF via the process additive functional with functions drift and diffusion allows reducing this functional on trajectories to a regular integral functional. Compared to conventional Shannon entropy measure of a random state, cutting the process on separated states decreases quantity of process information by the amount, concealed in correlation connections between these states, which hold hidden process information. The EF-IPF measure integrates information covered in both continuous process and discrete impulses which generate information and transmit it between the impulses.The n -dimensional process cutoff generates a finite information measure, integrated in the IPF whose information approaches the EF measure at n → ∞ , restricting maximal information of the Markov diffusion process. Studied impulse delta-function cutoff and the discrete impulse deliver equivalent information at each cutoff. The constructed finite restriction limits the impulses' discrete stepwise actions applied for cutting the regular integral on the functional increments between the cutoffs. Finite impulse step-up action transfers EF increment to following impulse whose step-down action cuts off information and step-up action starts imaginary (virtual) impulse carrying entropy increment to next real cut.Step-down cut generates maximal information while the step-up action delivers minimal information from impulse cut to next impulse step-down action. A virtual impulse transfers conjugated entropy increments during a microprocess ending with adjoining increment within actual step-down action at cutoff. Extracting maximum of minimal impulse information and transferring minimal entropy between impulses implement maxmin-minimax principle of converting process entropy to information. Each cutoff sequentially and automatically converts entropy to information, holding information Bit from random process, which connects the Bits sequences in the IPF and predicts next cut. The built information macroprocess, as the EF minimax extremal, integrates both imaginary entropy of microprocess and cutoff information of real impulses in IPF information physical process. Each IPF dimensional cut measures Feller kernel information. Cutting entropy memorizes the cutting time interval which freezes the probability of events with re...

Natural encoding of information through interacting impulses

“…But the EP preserves invariant impulse entropy measure independently of size scale and curvature during the dynamic model of the observing natural interactive movements. The impulse correlation" curving cut is orthogonal [46]. Within the observing process, the growing Bayes a posteriori probability along neighbor impulses may merge, generating interactive jump on a such impulse border.…”

“…Each process" dimensional cut measures the IPF finite Feller kernel" [19] information [46], which, at infinite dimensions, approaches the EF measure restricting maximal information of the Markov multidimensional diffusion process.…”

“…But the EP preserves invariant impulse entropy measure independently of size scale and curvature during the dynamic model of the observing natural interactive movements. The impulse correlation" curving cut is orthogonal [46].…”