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Asymptotic distribution of the derivative of the taut string accompanying Wiener process

Mikhail Lifshits, Andrei Podchishchailov

TL;DR

The paper analyzes the asymptotic distribution of the derivative of the taut string that accompanies a Wiener process within a fixed-width strip. By leveraging truncated variation and sojourn-measure techniques, it proves both weak and strong laws describing the long-time behavior of the taut string's derivative, and derives an explicit limiting density p_{∞,r} for this derivative. It identifies explicit energy limits, including the kinetic energy case, and provides rigorous connections between truncated variation, taut-string optimization, and absolutely continuous representations. The results yield precise expressions for energy functionals and establish conditions under which convergence holds almost surely or in probability. The work advances understanding of the stochastic taut-string framework and its asymptotic energetics.

Abstract

In the article, we find the asymptotic distribution of the derivative of the taut string accompanying a Wiener process in a strip of fixed width on long time intervals. This enables to find explicit expressions for minimal energy (averaged function of the derivative) of an absolutely continuous function in this strip. For example, for kinetic energy which was considered earlier by Lifshits and Setterqvist, the minimal energy per unit of time tends to $π^2/6r^2$ where $r$ is the strip width.

Asymptotic distribution of the derivative of the taut string accompanying Wiener process

TL;DR

The paper analyzes the asymptotic distribution of the derivative of the taut string that accompanies a Wiener process within a fixed-width strip. By leveraging truncated variation and sojourn-measure techniques, it proves both weak and strong laws describing the long-time behavior of the taut string's derivative, and derives an explicit limiting density p_{∞,r} for this derivative. It identifies explicit energy limits, including the kinetic energy case, and provides rigorous connections between truncated variation, taut-string optimization, and absolutely continuous representations. The results yield precise expressions for energy functionals and establish conditions under which convergence holds almost surely or in probability. The work advances understanding of the stochastic taut-string framework and its asymptotic energetics.

Abstract

In the article, we find the asymptotic distribution of the derivative of the taut string accompanying a Wiener process in a strip of fixed width on long time intervals. This enables to find explicit expressions for minimal energy (averaged function of the derivative) of an absolutely continuous function in this strip. For example, for kinetic energy which was considered earlier by Lifshits and Setterqvist, the minimal energy per unit of time tends to where is the strip width.

Paper Structure

This paper contains 14 sections, 9 theorems, 145 equations.

Table of Contents

  1. Introduction and main results
  2. Taut strings
  3. Taut strings accompanying Wiener process
  4. Truncated variation
  5. Basic notions and properties
  6. Truncated variation of the Wiener process
  7. Application of truncated variation
  8. Asymptotic distribution of taut string's derivative
  9. Weak law of large numbers
  10. Strong law of large numbers
  11. Various proofs
  12. Proof of Theorem \ref{['th:contrex']}
  13. The limit computation for quadratic function vartheta2
  14. Truncated variations and absolutely continuous functions

Key Result

Theorem 1.1

If a convex function $\varphi$ satisfies or then and If a function $\varphi$ is continuous almost everywhere, locally bounded and satisfies power growth condition eq:power_bound, then

Theorems & Definitions (17)

  • Theorem 1.1
  • Remark 1
  • Theorem 1.2
  • Lemma 2.1
  • Theorem 2.1: Lochowski2013
  • Theorem 3.1
  • proof
  • Remark 2
  • Theorem 4.1
  • proof
  • ...and 7 more