arxiv: 2601.04900 · v3 · submitted 2026-01-08 · 💱 q-fin.MF · math.PR

Recognition: 2 theorem links

· Lean Theorem

Visible absorbing decompositions and uniqueness of invariant probabilities

Jean-Gabriel Attali

Authors on Pith no claims yet

Pith reviewed 2026-05-16 16:29 UTC · model grok-4.3

classification 💱 q-fin.MF math.PR

keywords Markov kernelinvariant probabilityabsorbing decompositionuniquenessJordan decompositionmeasurable space

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The pith

A Markov kernel has more than one invariant probability exactly when it admits a visible absorbing decomposition.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that uniqueness of invariant probabilities fails for a Markov kernel if and only if there exist two disjoint absorbing sets, each carrying full mass under a distinct invariant probability. This visible absorbing decomposition is the precise measurable obstruction, unlike ordinary absorbing components that may carry no invariant mass at all. The argument proceeds directly from the Jordan decomposition of the difference between any two invariant probabilities, without extra assumptions on the kernel beyond measurability of the sets. A sympathetic reader sees this as a clean if-and-only-if characterization that replaces indirect conditions like global irreducibility with an explicit splitting. The result therefore supplies both a necessary and sufficient test for when multiple stationary distributions must exist.

Core claim

A Markov kernel has more than one invariant probability if and only if it admits a visible absorbing decomposition, namely two disjoint absorbing sets each having full mass for an invariant probability. The proof uses only the Jordan decomposition of the difference of two invariant probabilities.

What carries the argument

Visible absorbing decomposition: two disjoint absorbing sets, each carrying full mass under a distinct invariant probability.

If this is right

Existence of a visible absorbing decomposition forces at least two distinct invariant probabilities.
Absence of any visible absorbing decomposition forces uniqueness of the invariant probability.
Ordinary absorbing decompositions that fail to be visible do not obstruct uniqueness.
The Jordan decomposition alone suffices to detect or rule out the splitting.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Applied modelers could search directly for measurable absorbing sets that each attract positive mass under the kernel to decide uniqueness without computing all invariants.
In contexts with natural partitions of the state space, such as regime-switching models, the criterion reduces to checking whether each regime supports its own stationary distribution.
The same splitting idea may extend to continuous-time processes by replacing the kernel with the resolvent or generator while preserving the Jordan step.
Numerical approximation of the kernel could be used to test for candidate absorbing sets and verify the mass condition empirically.

Load-bearing premise

The state space is a measurable space on which the Jordan decomposition of signed measures applies and the absorbing sets are measurable.

What would settle it

Exhibit a Markov kernel possessing at least two distinct invariant probabilities whose difference cannot be decomposed into two positive measures each supported on a disjoint absorbing set, or conversely a kernel with such a decomposition but only a single invariant probability.

read the original abstract

We identify the measurable absorbing obstruction to uniqueness of invariant probability measures for a Markov kernel. Ordinary absorbing decompositions obstruct global irreducibility and recurrence, but not necessarily uniqueness: an absorbing component may have full mass for no invariant probability. We prove that a Markov kernel has more than one invariant probability if and only if it admits a visible absorbing decomposition, namely two disjoint absorbing sets, each having full mass for an invariant probability. The proof uses only the Jordan decomposition of the difference of two invariant probabilities.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper gives a clean iff characterization of non-uniqueness for invariant probabilities of Markov kernels via visible absorbing decompositions.

read the letter

Hi colleague, The one thing to know is that this paper proves an if and only if statement linking multiple invariant probabilities for a Markov kernel to the existence of a visible absorbing decomposition—specifically two disjoint absorbing measurable sets, each carrying full mass under its own invariant probability. This distinguishes it from standard absorbing decompositions that may not block uniqueness. The work does well in identifying why ordinary absorbing components fall short: they might not support any invariant probability with total mass one. By using only the Jordan decomposition on the difference of two invariants, the proof stays elementary and avoids unnecessary complications. That approach earns credit for being straightforward. A minor soft spot is the potential need to verify that the supports from the decomposition can be made absorbing while keeping the full mass property. The abstract claims the proof uses solely the Jordan step, so presumably the construction works directly, but seeing the explicit argument would confirm there are no gaps in handling the absorption condition. This paper is aimed at researchers in ergodic theory and Markov processes, with relevance to quantitative finance for understanding stationary distributions. Readers looking for a precise measurable condition for non-uniqueness will get something concrete from it. It deserves peer review because the characterization is sharp and grounded in standard tools. Best regards

Referee Report

1 major / 1 minor

Summary. The paper identifies the measurable absorbing obstruction to uniqueness of invariant probability measures for a Markov kernel. It proves that a Markov kernel has more than one invariant probability if and only if it admits a visible absorbing decomposition, namely two disjoint absorbing sets, each having full mass for an invariant probability. The proof of the nontrivial direction uses only the Jordan decomposition of the difference of two invariant probabilities.

Significance. If the result holds, it provides a clean characterization of non-uniqueness of invariant probabilities via visible absorbing decompositions, distinguishing these from ordinary absorbing sets that need not support any invariant probability with full mass. The reliance on the standard Jordan decomposition is a strength, as it keeps the argument within basic measure theory and could aid analysis of Markov processes on general measurable spaces.

major comments (1)

[Proof of the main theorem (nontrivial direction)] Proof of the main theorem (nontrivial direction): The abstract states that the argument uses only the Jordan decomposition of σ = μ − ν. However, the supports H+ and H− of the positive and negative parts are not automatically absorbing under the kernel P. The manuscript must explicitly construct absorbing sets A ⊃ H+ and B ⊃ H− (e.g., via the set of points whose forward orbits remain in H+) such that μ(A) = 1 and ν(B) = 1, and confirm that this step introduces no extra structure beyond the decomposition. This construction is load-bearing for the iff claim.

minor comments (1)

[Abstract] The term 'visible absorbing decomposition' is defined only after the abstract; adding a short parenthetical gloss in the abstract would improve accessibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need to make the absorbing-set construction explicit in the nontrivial direction of the main theorem. We address the comment below and will revise the manuscript to incorporate the requested clarification.

read point-by-point responses

Referee: Proof of the main theorem (nontrivial direction): The abstract states that the argument uses only the Jordan decomposition of σ = μ − ν. However, the supports H+ and H− of the positive and negative parts are not automatically absorbing under the kernel P. The manuscript must explicitly construct absorbing sets A ⊃ H+ and B ⊃ H− (e.g., via the set of points whose forward orbits remain in H+) such that μ(A) = 1 and ν(B) = 1, and confirm that this step introduces no extra structure beyond the decomposition. This construction is load-bearing for the iff claim.

Authors: We agree that the supports H+ and H− arising from the Jordan decomposition of σ = μ − ν are not a priori absorbing, and that the proof requires an explicit construction to complete the argument. In the revised manuscript we will add the following step: define A as the set of points x for which the forward orbit satisfies P^n(x, H+) = 1 for every n ≥ 0, and define B analogously with respect to H−. We will verify that A and B are absorbing (i.e., P(x, A) = 1 for all x ∈ A), that they are disjoint, that μ(A) = 1 and ν(B) = 1, and that the construction relies solely on the invariance of μ and ν together with the definition of the Jordan supports. No additional measurable structure is introduced. This explicit construction will be inserted immediately after the invocation of the Jordan decomposition, thereby making the nontrivial implication fully rigorous while preserving the paper’s reliance on basic measure theory. revision: yes

Circularity Check

0 steps flagged

No circularity: direct application of Jordan decomposition

full rationale

The paper establishes an iff characterization between non-uniqueness of invariant probabilities and the existence of a visible absorbing decomposition. The nontrivial direction is derived by applying the Jordan decomposition directly to the signed measure σ = μ − ν for any two distinct invariant probabilities μ, ν. This yields the positive and negative parts whose supports are then shown to be absorbing sets carrying full mass. No parameters are fitted to data and renamed as predictions, no self-citations are invoked as load-bearing uniqueness theorems, and the argument does not reduce any claimed result to its own inputs by definition. The derivation remains self-contained once the standard Jordan decomposition on the given measurable space is granted.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The result rests on the standard Jordan decomposition theorem for signed measures and the definition of absorbing sets in a measurable space; no free parameters or new postulated entities beyond the defined visible decomposition are introduced.

axioms (1)

standard math Jordan decomposition theorem for signed measures on a measurable space
Invoked to decompose the difference of any two invariant probabilities in the proof.

invented entities (1)

visible absorbing decomposition no independent evidence
purpose: To provide the exact measurable condition equivalent to non-uniqueness of invariant probabilities
Defined as two disjoint absorbing sets each carrying full mass under some invariant probability; no independent evidence outside the paper is supplied.

pith-pipeline@v0.9.0 · 5365 in / 1233 out tokens · 43977 ms · 2026-05-16T16:29:22.892798+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We prove that a Markov kernel has more than one invariant probability if and only if it admits a visible absorbing decomposition, namely two disjoint absorbing sets, each having full mass for an invariant probability. The proof uses only the Jordan decomposition of the difference of two invariant probabilities.
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Definition 1 (Indecomposability). ... two disjoint nonempty measurable sets A, B ... P(x,A)=1 for all x∈A, P(x,B)=1 for all x∈B.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.