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arxiv: 2604.05941 · v1 · submitted 2026-04-07 · 🧮 math.PR

Banach spaces of continuous paths with finite p-th variation

Purba Das , Donghan Kim , Fang Rui Lim This is my paper

Pith reviewed 2026-05-10 19:19 UTC · model grok-4.3

classification 🧮 math.PR
keywords p-th variationFaber-Schauder expansionspathwise Föllmer-Itô mapBanach spaces of pathsHölder continuous pathstransport proceduretime-changespartition sequences
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The pith

A Faber-Schauder coefficient method builds Banach subspaces of paths with prescribed p-th variation where the Föllmer-Itô map stays stable.

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

The paper develops a constructive method to build continuous paths with finite or prescribed p-th variation for p greater than 1. Using conditions on Faber-Schauder expansions, it creates paths that maintain linear structures and Hölder regularity even though the set of paths with finite p-th variation is not linear in general. A transport procedure is introduced to convert any Banach subspace of continuous functions into one with explicitly controlled p-th variation. This yields the result that Hölder continuous paths with a given p-th variation are dense in the space of all continuous paths on [0,1], whenever that class is nonempty. The framework also ensures stability of the pathwise Föllmer-Itô map and extends to general dense refining partition sequences through time changes.

Core claim

For any p > 1 and a fixed partition sequence, continuous paths with linear p-th variation can be constructed from suitable Faber-Schauder coefficients, and nonlinear variation is prescribed via a multiplicative transformation. Whenever the class is nonempty, the Hölder continuous paths with a given p-th variation are dense in C([0,1]). Any Banach subspace of continuous functions can be transported to a Banach subspace of paths with explicitly controlled p-th variation, on which the pathwise Föllmer-Itô map is stable. The construction extends via time-changes to dense q-refining partition sequences.

What carries the argument

The transport procedure that applies a multiplicative transformation to Faber-Schauder expansions to control p-th variation while preserving the Banach space structure.

If this is right

  • The class of Hölder continuous paths with prescribed p-th variation is dense in C([0,1]) if nonempty.
  • The pathwise Föllmer-Itô map remains stable on the transported Banach subspaces.
  • The framework extends from q-adic partitions to broader dense q-refining partition sequences via time-changes.
  • Paths with both linear and nonlinear p-th variation can be constructed while retaining Hölder regularity.

Where Pith is reading between the lines

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

  • These Banach spaces could support functional-analytic treatments of pathwise differential equations driven by paths with finite p-variation.
  • The density result suggests that uniform approximations by finite p-variation paths are possible for many continuous paths.
  • Stability of the Föllmer-Itô map may extend to more general integration schemes in related stochastic settings.
  • Applying the transport to concrete subspaces such as polynomials generates explicit families of paths with controlled variation for testing.

Load-bearing premise

The multiplicative transformation and the coefficient conditions must preserve the exact p-th variation along the partition sequence and the Hölder regularity without causing inconsistencies in the path properties.

What would settle it

Take a finite sum of Faber-Schauder functions satisfying the linear coefficient conditions for a chosen p, compute its p-th variation directly from the increments along the given partition sequence, and verify whether the value exactly matches the one predicted by the coefficient formula; any mismatch falsifies the preservation claim.

read the original abstract

We study pathwise $p$-th variation of continuous paths on a compact interval along a fixed partition sequence. Although the class of continuous paths with finite $p$-th variation is generally not linear, we develop a coefficient-based approach via Faber-Schauder expansions that, for any $p>1$, enables the construction of paths with prescribed $p$-th variation while preserving useful linear structures and H\"older regularity. We first construct continuous paths with linear $p$-th variation from suitable conditions on their Faber-Schauder coefficients. We then prescribe nonlinear $p$-th variation through a multiplicative transformation and show that, whenever nonempty, the class of H\"older continuous paths with a given $p$-th variation is dense in $C([0,1])$. Next, we introduce a transport procedure that turns a Banach subspace of continuous functions into a Banach subspace of paths with explicitly controlled $p$-th variation. We also prove stability of the associated pathwise F\"ollmer-It\^o map on these transported subspaces. Finally, via time-changes, we show that this constructive framework extends from $q$-adic partition sequences to broader classes of dense $q$-refining partition sequences.

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 develops a coefficient-based framework using Faber-Schauder expansions to construct continuous paths with prescribed linear and nonlinear p-th variation along a fixed partition sequence. It proves that, whenever nonempty, the class of Hölder continuous paths with a given p-th variation is dense in C([0,1]), introduces a transport procedure yielding Banach subspaces with explicitly controlled p-th variation on which the pathwise Föllmer-Itô map is stable, and extends the results to dense q-refining partition sequences via time-changes.

Significance. If the constructions and preservation properties hold, the work supplies an explicit, basis-driven method for building Banach spaces of paths with finite p-th variation and controlled Hölder regularity. This could be useful for pathwise stochastic calculus, offering density results and stability of the Föllmer-Itô map without probabilistic structure. The constructive nature and extension to general refining partitions are strengths.

major comments (2)
  1. [multiplicative transformation section] The section on the multiplicative transformation (following the linear-variation construction via Faber-Schauder coefficients): the claim that this map simultaneously preserves the exact p-th variation sum along the fixed partition sequence and Hölder regularity for arbitrary p>1 is load-bearing for the density result in C([0,1]). The manuscript must supply the explicit form of the transformation together with the precise coefficient restrictions that guarantee both properties without introducing path- or partition-dependent inconsistencies.
  2. [transport procedure section] The transport procedure (after the density statement): it is unclear from the description how the procedure maps an arbitrary Banach subspace while exactly retaining the prescribed p-th variation and the Banach-space norm structure. A concrete verification or explicit formula for the transported norm and variation control is needed to support the subsequent stability claim for the pathwise Föllmer-Itô map.
minor comments (2)
  1. [abstract and introduction] The abstract and introduction would benefit from a brief statement of the precise partition sequence used in the main results and an explicit reference to the definition of p-th variation employed.
  2. [notation and definitions] Notation for the Faber-Schauder coefficients and the multiplicative map should be introduced with a short table or displayed equations to improve readability when the constructions are applied to the Föllmer-Itô map.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. We address each major comment below, indicating the revisions that will be incorporated in the next version.

read point-by-point responses

  1. Referee: [multiplicative transformation section] The section on the multiplicative transformation (following the linear-variation construction via Faber-Schauder coefficients): the claim that this map simultaneously preserves the exact p-th variation sum along the fixed partition sequence and Hölder regularity for arbitrary p>1 is load-bearing for the density result in C([0,1]). The manuscript must supply the explicit form of the transformation together with the precise coefficient restrictions that guarantee both properties without introducing path- or partition-dependent inconsistencies.

    Authors: We agree that the explicit form of the multiplicative transformation and the associated coefficient restrictions are essential for rigor. In the revised manuscript we will insert the precise definition of the map applied to the Faber-Schauder coefficients, together with the explicit bounds on those coefficients that simultaneously keep the p-th variation sum unchanged along the given partition sequence and preserve the Hölder exponent for any p>1. These restrictions are formulated in a manner independent of any particular path or partition, thereby eliminating the inconsistencies noted. revision: yes

  2. Referee: [transport procedure section] The transport procedure (after the density statement): it is unclear from the description how the procedure maps an arbitrary Banach subspace while exactly retaining the prescribed p-th variation and the Banach-space norm structure. A concrete verification or explicit formula for the transported norm and variation control is needed to support the subsequent stability claim for the pathwise Föllmer-Itô map.

    Authors: We accept that the current description of the transport procedure leaves the preservation of both the p-th variation and the Banach norm insufficiently explicit. In the revision we will supply the concrete formula that defines the transported norm on the image subspace, verify that the p-th variation along the fixed partition sequence is retained exactly, and confirm that the resulting space remains a Banach space. These additions will directly underpin the stability statement for the pathwise Föllmer-Itô map. revision: yes

Circularity Check

0 steps flagged

No load-bearing circularity; framework rests on standard Faber-Schauder expansions and explicit coefficient conditions

full rationale

The derivation begins with the classical definition of pathwise p-variation along a fixed partition sequence and the standard Faber-Schauder basis expansion of continuous functions. Paths with linear p-variation are constructed by imposing explicit coefficient restrictions that directly control the variation sum; the subsequent multiplicative transformation is introduced to obtain nonlinear variation while the paper states it preserves both the exact variation value and Hölder regularity. The density statement, Banach-space transport, Föllmer-Itô stability, and time-change extension are all presented as consequences of these constructions rather than as inputs. No equation reduces the target p-variation to a fitted parameter or to a self-referential quantity, and no uniqueness theorem or ansatz is imported solely via self-citation. The single minor self-citation (if any) is not load-bearing for the central claims.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The constructions rest on standard properties of the Faber-Schauder system as a Schauder basis and the definition of pathwise p-variation; no free parameters, new entities, or ad-hoc axioms are introduced beyond domain-standard assumptions.

axioms (2)
  • standard math The Faber-Schauder system forms a Schauder basis for the space of continuous functions on [0,1]
    Invoked to expand paths and control coefficients for variation
  • domain assumption p-th variation along a fixed partition sequence is well-defined for continuous paths
    Central definition used throughout the constructions

pith-pipeline@v0.9.0 · 5512 in / 1462 out tokens · 68283 ms · 2026-05-10T19:19:07.192618+00:00 · methodology

discussion (0)

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Reference graph

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