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arxiv: 2410.04051 · v2 · submitted 2024-10-05 · 🧮 math.PR

Non-Markovianity of 2K-B and a degeneration

Yang Chu , Lingfu Zhang This is my paper

Pith reviewed 2026-05-23 20:24 UTC · model grok-4.3

classification 🧮 math.PR
keywords Brownian motionconcave majorantBessel processnon-MarkovianPitman theoremmixturefiltration
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The pith

The process 2K-B formed from Brownian motion and its concave majorant is not Markovian and thus not a BES(5) process.

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

The paper examines 2K-B, where K is the concave majorant of a standard Brownian motion B. A conjecture had proposed that this process follows the law of the BES(5) process because the two share Brownian scaling, quadratic variation, one-point distributions, and the same infinitesimal generator. The authors disprove the conjecture by showing that 2K-B is not Markovian. They then introduce a degeneration of 2K-B and prove that this degeneration equals a mixture of BES(3) processes, from which they derive its multiple-point distributions, generator, and path decomposition at future infimum. They also describe the Markovian structures of 2K-B, B, and K under their respective filtrations.

Core claim

The authors prove that 2K-B is not Markovian under the natural filtration, refuting the conjecture that it has the law of BES(5). For a degeneration of 2K-B, they establish that it is a mixture of BES(3) processes and obtain its multi-point distributions, infinitesimal generator, and path decomposition at the future infimum. They also examine the Markovian structures and filtrations of 2K-B, B, and K.

What carries the argument

The concave majorant K of Brownian motion B, together with the process 2K-B, whose failure to be Markovian is established by showing that shared scaling, variation, and generator properties do not force the Markov property.

If this is right

  • Shared scaling, quadratic variation, one-point law, and generator are insufficient to force 2K-B to be Markovian or equal in law to BES(5).
  • The degeneration of 2K-B equals a mixture of BES(3) processes and admits explicit multi-point distributions.
  • Path decomposition at the future infimum holds for the degeneration.
  • The processes 2K-B, B, and K possess distinct Markovian structures under their natural filtrations.

Where Pith is reading between the lines

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

  • Non-Markovianity of 2K-B implies that its future evolution depends on the global shape of the majorant in ways invisible from the current state alone.
  • Other constructions built from concave majorants of Brownian paths may fail to be Markovian even when they match one-dimensional marginals and generators of classical processes.
  • The filtration generated by K alone may contain additional structure that becomes visible only after the degeneration is taken.

Load-bearing premise

The specific construction of the concave majorant K and the filtration under which the Markov property is tested admit no overlooked Markovian version of 2K-B.

What would settle it

An explicit path history where the conditional law of the future increments of 2K-B given its current value differs from the conditional law under BES(5).

Figures

Figures reproduced from arXiv: 2410.04051 by Lingfu Zhang, Yang Chu.

Figure 1
Figure 1. Figure 1: A simulation of a Brownian motion B and its concave majorant K, and the reflection 2K − B. 1 Introduction As one of the most fundamental random processes, Brownian motion (or Werner process) has been intensively studied in the past century, with most of its properties now well-understood. Among ∗Department of Statistics, University of California, Berkeley. e-mail: yang.chu@berkeley.edu †Department of Stati… view at source ↗
Figure 2
Figure 2. Figure 2: 100 samples of 2K − B and BES(5). procedure. Fix h > 0, and consider (2K(s + t) − B(s + t))t≥0 | 2K(s) − B(s) = z, (1.1) and send s → ∞ next. The degeneration we study is this limit, denoted by by Z = (Z(t))t≥0. There are various motivations of studying the process. First, as will be explained through Lemma 4.1, Z can be regarded as a approximation of 2K − B, by ‘removing the non-Markovian part’. Second, Z… view at source ↗
Figure 3
Figure 3. Figure 3: KS test for λ = (1, 0.1) (left) and λ = (1, 0.5) (right) In this final section we include some numerical simulations comparing X = 2K −B and BES(5). It’s difficult to directly simulate the concave majorant on [0, ∞) numerically, as it includes information about the Brownian motion until time infinity. Instead, we use the formulation in Theorem 2.1 (and also [6]). As proven by [15], the one dimensional dist… view at source ↗
Figure 4
Figure 4. Figure 4: KS test for λ = (1, 1) (left) and λ = (1, 2) (right) [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: KS test for λ = (1, 5) (left) and λ = (1, 10) (right) More precisely, we consider one dimensional projections of the two dimensional random variables. (Note that any probability distribution on R 2 is determined by all of its one dimensional projections, via Fourier transform.) Three different projections are taken, as plotted in Figures 3 to 5, with the KS statistics computed. It can be seen that the KS t… view at source ↗
read the original abstract

We study the process of $2K-B$, where $B$ is a standard one-dimensional Brownian motion and $K$ is its concave majorant. In light of Pitman's $2M-B$ theorem, it was recently conjectured by Ouaki and Pitman \cite{OP} that $2K-B$ has the law of the BES(5) process. The two processes share properties such as Brownian scaling, time inversion and quadratic variation, and the same one point distribution and infinitesimal generator, among many other evidences; and it remains to prove that $2K-B$ is Markovian. However, we show that this conjecture is false. To better understand the similarity between these two processes, we study a degeneration of $2K-B$. We show it is a mixture of BES(3), and get other properties including multiple points distribution, infinitesimal generator, and path decomposition at future infimum. We also further investigate the Markovian structure and the filtrations of $2K-B, B$ and $K$.

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.

Referee Report

2 major / 2 minor

Summary. The paper claims to disprove the Ouaki-Pitman conjecture that 2K-B (K the concave majorant of Brownian motion B) has the law of BES(5), despite sharing Brownian scaling, time inversion, quadratic variation, one-point law, and infinitesimal generator. It shows non-Markovianity of 2K-B under the relevant filtration. It further studies a degeneration of 2K-B, proving it is a mixture of BES(3) processes, and derives multiple-point distributions, the generator, path decomposition at future infimum, and the Markovian structures/filtrations of 2K-B, B, and K.

Significance. If the disproof holds, the result is significant because it separates processes that agree on scaling, quadratic variation, marginals, and generator yet fail to be Markovian due to filtration dependence; this refines understanding of Pitman-type transforms. The degeneration result supplies an explicit mixture representation together with path decomposition and generator, which are concrete additions to the literature on Bessel processes.

major comments (2)
  1. [Abstract and main disproof section] The non-Markovianity claim (abstract and the section establishing the counterexample) rests on the precise construction of the concave majorant K and the filtration generated by (K,B) or by 2K-B. No explicit two-time conditional law is exhibited that visibly depends on an auxiliary variable outside the current state, so the failure of the Markov property cannot be verified from the supplied details.
  2. [Section on degeneration] The degeneration result (section deriving the mixture of BES(3)) asserts that the shared generator is computed under the same filtration as the original process; without the explicit computation of the generator on the degenerated process and confirmation that the filtration is identical, the mixture representation remains formally unanchored.
minor comments (2)
  1. [Introduction] The reference [OP] to Ouaki-Pitman should be expanded to a full bibliographic entry.
  2. [Throughout] Notation for the process 2K-B and the concave majorant K should be introduced once with a clear definition before being used in statements about filtrations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of the significance of the results. We address the two major comments below. Both point to places where additional explicit calculations would improve clarity and verifiability; we agree and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract and main disproof section] The non-Markovianity claim (abstract and the section establishing the counterexample) rests on the precise construction of the concave majorant K and the filtration generated by (K,B) or by 2K-B. No explicit two-time conditional law is exhibited that visibly depends on an auxiliary variable outside the current state, so the failure of the Markov property cannot be verified from the supplied details.

    Authors: We agree that an explicit two-time conditional law would make the non-Markovianity argument more immediately verifiable. The current proof proceeds by exhibiting a specific path configuration in which the future evolution of 2K-B, conditional on the present value, still depends on the position of the concave majorant K at an earlier time that is not recoverable from the current state alone. To address the referee's concern we will add, in the revised version, an explicit computation of P(2K_{t+s} ∈ dy | 2K_t = x, filtration up to t) for a concrete choice of times and states that visibly retains dependence on an auxiliary variable (the location of a prior vertex of K). revision: yes

  2. Referee: [Section on degeneration] The degeneration result (section deriving the mixture of BES(3)) asserts that the shared generator is computed under the same filtration as the original process; without the explicit computation of the generator on the degenerated process and confirmation that the filtration is identical, the mixture representation remains formally unanchored.

    Authors: We accept the observation. While the degeneration is constructed so that the underlying filtration remains the same as that of the original 2K-B process, the manuscript does not recompute the generator explicitly on the degenerated object. In the revision we will supply the direct calculation of the infinitesimal generator for the mixture-of-BES(3) process under the common filtration and include a short paragraph confirming that the filtration generated by the degenerated process coincides with the one used for the original generator computation. revision: yes

Circularity Check

0 steps flagged

No circularity: disproof and degeneration analysis are independent of inputs

full rationale

The paper disproves the Ouaki-Pitman conjecture by establishing non-Markovianity of 2K-B under a specified filtration and studies a degeneration shown to be a mixture of BES(3) processes, deriving multiple-point distributions, generators, and path decompositions. These steps rely on direct analysis of Brownian motion properties, concave majorants, and filtrations rather than any self-definitional equivalence, fitted parameters renamed as predictions, or load-bearing self-citations. The cited conjecture source is external, and no equation or claim reduces by construction to the paper's own inputs. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard properties of Brownian motion (continuous paths, quadratic variation) and the definition of the concave majorant; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • standard math Brownian motion has continuous paths and quadratic variation t.
    Invoked implicitly when defining K and the process 2K-B.
  • domain assumption The concave majorant K is well-defined and measurable with respect to the filtration of B.
    Required for the process 2K-B to be adapted and for the filtration comparisons.

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Reference graph

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