[Paper Review] The impulse cutoff an entropy functional measure on trajectories of Markov diffusion process integrating in information path functional

Vladimir S. Lerner|arXiv (Cornell University)|Apr 24, 2012

Fault Detection and Control Systems29 references3 citations

TL;DR

This paper introduces a novel framework for measuring information in Markov diffusion processes by modeling entropy functional reduction through discrete impulse cutoffs, which convert stochastic trajectory entropy into quantized information bits. The key contribution is the Information Path Functional (IPF), which integrates these entropy cuts into a finite, non-additive information measure that captures hidden process correlations and enables optimal entropy-to-information conversion via a maxmin-minimax principle.

ABSTRACT

The impulses, cutting entropy functional (EF) measure on trajectories Markov diffusion process, integrate information path functional (IPF) composing discrete information Bits extracted from observing random process. Each cut brings memory of the cutting entropy, 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 anatomy creation of impulse enables both cut entropy and stipulate random process generating information under the cut. Memory of the impulse cutting time interval freezes the observing events dynamics in information processes. Diffusion process additive functional defines EF reducing it to a regular integral functional. Compared to Shannon entropy measure of random state, cutting process on separated states decreases quantity information concealed in the states correlation holding hidden process information. Infinite dimensional process cutoffs integrate finite information in IPF whose information approaches EF restricting process maximal information. Within the impulse reversible microprocess, conjugated entropy increments are entangling up to the cutoff converting entropy in irreversible information. Extracting maximum of minimal impulse information and transferring minimal entropy between impulses implement maxmin-minimax principle of optimal conversion process entropy to information. Macroprocess extremals integrate entropy of microprocess and cutoff information of impulses in the IPF information physical process. IPF measures Feller kernel information. Estimation extracting information confirms nonadditivity of EF measured process increments.

Motivation & Objective

• To formalize the transformation of continuous entropy in diffusion processes into discrete, measurable information units through impulse-based cutoffs.
• To develop a non-additive information measure that captures hidden correlations in stochastic trajectories beyond standard Shannon entropy.
• To establish a physical and mathematical framework linking entropy reduction via impulses to irreversible information formation.
• To derive an optimal conversion principle between entropy and information using maxmin-minimax strategies in macroscopic processes.
• To integrate microprocess conjugate entropy increments into macroprocess extremals via the Information Path Functional (IPF).

Proposed method

• Models Markov diffusion processes using additive functionals to reduce entropy functional (EF) to a regular integral form.
• Introduces 'impulse cutoffs' that discretize entropy reduction, each corresponding to one information bit and freezing dynamic memory of the process.
• Defines the Information Path Functional (IPF) as the cumulative measure of entropy cuts, integrating finite information from infinite-dimensional process trajectories.
• Applies the maxmin-minimax principle to optimize entropy-to-information conversion across impulse intervals.
• Uses Feller kernel estimation to measure IPF information content and validate nonadditivity of EF increments.
• Analyzes conjugated entropy increments in reversible microprocesses that become irreversible upon cutoff, forming information.

Experimental results

Research questions

• RQ1 How can entropy in continuous diffusion processes be systematically converted into discrete, measurable information units?
• RQ2 What mathematical structure enables the integration of multiple entropy cutoffs into a finite, non-additive information measure?
• RQ3 How does the IPF capture hidden correlations in stochastic trajectories that standard entropy measures miss?
• RQ4 What principle governs optimal conversion of entropy to information in such systems?
• RQ5 How do impulse cutoffs induce irreversibility and memory in stochastic processes?

Key findings

• Impulse cutoffs transform continuous entropy into discrete information bits, with each cut preserving memory of the entropy reduction.
• The Information Path Functional (IPF) integrates finite information from infinite-dimensional diffusion trajectories, effectively restricting maximal process information.
• Nonadditivity of the entropy functional (EF) is confirmed through estimation of process increments, indicating information is not simply additive across intervals.
• Conjugated entropy increments in reversible microprocesses become irreversible upon cutoff, forming the basis of information creation.
• The maxmin-minimax principle enables optimal conversion of entropy to information by extracting maximum minimal impulse information and minimizing entropy transfer between impulses.
• Macroprocess extremals emerge as integrals of microprocess entropy and impulse cutoff information, forming a physical information process governed by the IPF.