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Quadratic variation and local times of the horizontal component of the Peano curve (square filling curve)

Phumlani L. Zondi, Darlington Hove, Rafał M. Łochowski, Farai J. Mhlanga

TL;DR

This work analyzes the x-coordinate of the deterministic Peano curve through the lens of quadratic variation along Lebesgue partitions. By developing one-step and $k$-step recursions for grids of the form $\left(p/3^{n}\right)\mathbb{Z}+\left(r/3^{n}\right)$, it establishes the existence of a finite, grid-dependent quadratic variation for the horizontal component, culminating in an explicit limit $[x]_t=3^{\lfloor -\log_{3}p\rfloor}p\{1-\tfrac{3}{4}(3^{\lfloor -\log_{3}p\rfloor}p)\}t$ (for irrational $r$). The paper further shows that $x$ possesses a local time with respect to the Lebesgue measure, expressible as a weak limit of normalized numbers of interval crossings, albeit with a non-smooth normalization reflecting a phenomenon of regularisation by noise. Together, these results distinguish the Peano curve’s deterministic path from Brownian trajectories, highlighting grid-dependent regularity phenomena and providing a precise occupation-density framework for a fractal space-filling curve.

Abstract

We show that the horizontal component of the Peano curve has quadratic variation equal the limit of quadratic variations along the Lebesgue partitions for grids of the form $3^{-n}p\mathbb{Z}+3^{-n}r$, $n=1,2,\ldots$, where $p$ is a rational number, while $r$ is irrational number, but the value of such quadratic variation depends on $p$. This also yields that the horizontal component of the Peano curve is an example of a deterministic function possessing local time (density of the occupation measure) with respect to the Lebesgue measure, whose local time can be expressed as the limit of normalized numbers of interval crossings by this function but the normalization is not a smooth function of the width of the intervals. These two features distinct the horizontal component of the Peano curve from the trajectories of the Wiener process, which is widely used in financial models.

Quadratic variation and local times of the horizontal component of the Peano curve (square filling curve)

TL;DR

This work analyzes the x-coordinate of the deterministic Peano curve through the lens of quadratic variation along Lebesgue partitions. By developing one-step and -step recursions for grids of the form , it establishes the existence of a finite, grid-dependent quadratic variation for the horizontal component, culminating in an explicit limit (for irrational ). The paper further shows that possesses a local time with respect to the Lebesgue measure, expressible as a weak limit of normalized numbers of interval crossings, albeit with a non-smooth normalization reflecting a phenomenon of regularisation by noise. Together, these results distinguish the Peano curve’s deterministic path from Brownian trajectories, highlighting grid-dependent regularity phenomena and providing a precise occupation-density framework for a fractal space-filling curve.

Abstract

We show that the horizontal component of the Peano curve has quadratic variation equal the limit of quadratic variations along the Lebesgue partitions for grids of the form , , where is a rational number, while is irrational number, but the value of such quadratic variation depends on . This also yields that the horizontal component of the Peano curve is an example of a deterministic function possessing local time (density of the occupation measure) with respect to the Lebesgue measure, whose local time can be expressed as the limit of normalized numbers of interval crossings by this function but the normalization is not a smooth function of the width of the intervals. These two features distinct the horizontal component of the Peano curve from the trajectories of the Wiener process, which is widely used in financial models.
Paper Structure (10 sections, 2 theorems, 142 equations, 5 figures)

This paper contains 10 sections, 2 theorems, 142 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Quadratic variation of a continuous curve based on the Lebesgue partition -- definitions
  3. Quadratic variation of the Peano curve along the sequence of the Lebesgue partitions for the grids $\left(p/3^{n}\right)\mathbb{Z}+\left(r/3^{n}\right)$, $p\in\mathbb{Q}$, $n\in\mathbb{N}$
  4. One-step recursion
  5. $k$-step recursion
  6. Quadraric variations along the Lebesgue partitions for the grids $\left(p/3^{n}\right)\mathbb{Z}+\left(r/3^{n}\right)$ when $r$ is irrational
  7. Limit of quadratic variations of $x$ along the Lebesgue partitions for the grids $\left(p/3^{n}\right)\mathbb{Z}+\left(r/3^{n}\right)$, $p\in\mathbb{Q}_{+}$, $r\in\mathbb{R}\setminus\mathbb{Q}$
  8. Truncated variation and numbers of interval crossings of the $x$--component of the Peano curve
  9. Local times of the $x$--component of the Peano curve
  10. Local time of the $x$-component of the Peano curve expressed as the weak limit of normalized numbers of interval crossings

Key Result

Theorem 2.3

Let $x:[0,1]\rightarrow\mathbb{R}$ be the horizontal component of the Peano curve, defined by (eq:px). Let $p=p'/q'$ where $p'$ and $q'$ are positive integers not divisible by $3$ and $r$ be such that $r=\theta p$ where $\theta$ is not a multiple of any $3^{m}/p'$, $m\in \mathbb{Z}$. Then for $t \in

Figures (5)

  • Figure 1: Approximating polygon of the Peano curve (first step of iteration procedure). Some lines were altered for better illustration of the parametrization of the curve.
  • Figure 2: Approximating polygon of the Peano curve (second step of iteration procedure). Some lines were altered for better illustration of the iteration procedure.
  • Figure 3: The $x$ component of the first approximating polygon.
  • Figure 4: The $x$ component of the second approximating polygon.
  • Figure 5: The graph of the function $\varphi(c)$ (blue) together with graphs of $3c$ and $4c$.

Theorems & Definitions (5)

  • Remark 2.1
  • Remark 2.2
  • Theorem 2.3
  • Definition 2.4
  • Theorem 3.1