Kolmogorov's criterion¶

A cycle-product condition characterizing when a Markov chain is reversible with respect to a positive stationary measure.

Version: v1 · 2026-09-08 · History
Domain-specific #: 5213
Origin domain: markov chain theory
Subdomain: markov chain theory

Core Idea¶

Kolmogorov’s reversibility criterion requires the product of transition rates or probabilities around every finite directed cycle to equal the product around the reversed cycle, subject to support conditions. Pairwise detailed-balance ratios telescope around cycles; conversely path-independent products construct stationary weights when every closed-loop affinity vanishes. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

The load-bearing residual is not the broad topic of markov chain theory. It is the domain-specific identity determined by every admissible closed cycle has equal forward and reverse transition product under the declared discrete- or continuous-time convention.

Scope of Application¶

Kolmogorov's criterion belongs to markov chain theory and is useful where the analyst can specify the typed markov chain theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate every admissible closed cycle has equal forward and reverse transition product under the declared discrete- or continuous-time convention. The scope is broad within that domain but bounded by the need for every admissible closed cycle has equal forward and reverse transition product under the declared discrete- or continuous-time convention. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.

Clarity¶

The abstraction clarifies a crowded vocabulary by making every admissible closed cycle has equal forward and reverse transition product under the declared discrete- or continuous-time convention the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Kolmogorov's criterion can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.

Manages Complexity¶

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Kolmogorov's criterion. Kolmogorov's criterion compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning¶

Identify the carrier. State what the elements, states, objects, or observations are: the typed markov chain theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express every admissible closed cycle has equal forward and reverse transition product under the declared discrete- or continuous-time convention independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of markov chain theory because they reuse the typed markov chain theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, Pairwise detailed-balance ratios telescope around cycles; conversely path-independent products construct stationary weights when every closed-loop affinity vanishes., and type the carrier, state every parameter and convention in the definition, test that every admissible closed cycle has equal forward and reverse transition product under the declared discrete- or continuous-time convention, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Kolmogorov's criterion Domain-specific

Parents (1) — more general patterns this builds on

Kolmogorov's criterion is a kind of Detailed balance Prime

The proposed strict upward parent is prime:detailed_balance.

Hierarchy path (1) — routes to 1 parentless root

Kolmogorov's criterion → Detailed balance → Equilibrium → Fixed Point

Neighborhood in Abstraction Space¶

Kolmogorov's criterion sits in a crowded region of the domain-specific corpus (35^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Stochastic Processes & Markov Dynamics (38 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-09-08