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Foundations and Fundamental Properties of a Two-Parameter Memory-Weighted Velocity Operator

Jiahao Jiang

TL;DR

The paper introduces a memory-weighted velocity operator $\mathscr{V}_{\alpha,\beta}$ that incorporates time-varying memory via two exponents $\alpha(t)$ and $\beta(t)$, yielding a nonlocal rate of change that decouples state-increment memory from elapsed-time memory. It provides a rigorous operator framework: explicit integral representation, linearity, and continuous dependence on the memory laws, together with weighted estimates that reveal a compensation mechanism between the two memory weightings. A key result is that $\mathscr{V}_{\alpha_n,\beta_n}[x]$ converges locally uniformly to $\mathscr{V}_{\alpha,\beta}[x]$ when $(\alpha_n,\beta_n)\to(\alpha,\beta)$ uniformly; in the uniform-memory case $\alpha=\beta=1$, the operator recovers the classical derivative $\dot{x}(0)$ as $t\to0^{+}$. The framework provides a self-contained, analytically robust tool for modeling evolution equations with non-stationary memory, with potential applications in viscoelasticity and anomalous transport where memory characteristics adapt over time.

Abstract

We introduce and analyze a **memory-weighted velocity operator** \(\mathscr{V}_{α,β}\) as a mathematical framework for describing rates of change in systems with time-varying, power-law memory. The operator employs two independent continuous exponents \(α(t)\) and \(β(t)\) that separately weight past state increments and elapsed time scaling, motivated by physical systems where these memory aspects may evolve differently -- such as viscoelastic materials with stress-dependent relaxation or anomalous transport with history-dependent characteristics. We establish the operator's foundational properties: an explicit integral representation, linearity, and **continuous dependence** on the memory exponents with respect to uniform convergence. Central to the analysis are **weighted pointwise estimates** revealing how the exponent difference \(β(t)-α(t)\) modulates \(\mathscr{V}_{α,β}[x](t)\), leading to conditions under which \(\mathscr{V}_{α,β}\) defines a bounded linear operator between standard function spaces. These estimates exhibit a natural compensation mechanism between the two memory weightings. For the uniform-memory case \(α=β\equiv1\), we prove that \(\mathscr{V}_{α,β}[x](t)\) **asymptotically recovers** the classical derivative \(\dot{x}(0)\) as \(t\to 0^{+}\), ensuring consistency with local calculus. The mathematical framework is supported by self-contained technical appendices. By decoupling the memory weighting of state increments from that of elapsed time, \(\mathscr{V}_{α,β}\) provides a structured approach to modeling systems with independently evolving memory characteristics, offering potential utility in formulating evolution equations for complex physical processes with non-stationary memory.

Foundations and Fundamental Properties of a Two-Parameter Memory-Weighted Velocity Operator

TL;DR

The paper introduces a memory-weighted velocity operator that incorporates time-varying memory via two exponents and , yielding a nonlocal rate of change that decouples state-increment memory from elapsed-time memory. It provides a rigorous operator framework: explicit integral representation, linearity, and continuous dependence on the memory laws, together with weighted estimates that reveal a compensation mechanism between the two memory weightings. A key result is that converges locally uniformly to when uniformly; in the uniform-memory case , the operator recovers the classical derivative as . The framework provides a self-contained, analytically robust tool for modeling evolution equations with non-stationary memory, with potential applications in viscoelasticity and anomalous transport where memory characteristics adapt over time.

Abstract

We introduce and analyze a **memory-weighted velocity operator** as a mathematical framework for describing rates of change in systems with time-varying, power-law memory. The operator employs two independent continuous exponents \(α(t)\) and \(β(t)\) that separately weight past state increments and elapsed time scaling, motivated by physical systems where these memory aspects may evolve differently -- such as viscoelastic materials with stress-dependent relaxation or anomalous transport with history-dependent characteristics. We establish the operator's foundational properties: an explicit integral representation, linearity, and **continuous dependence** on the memory exponents with respect to uniform convergence. Central to the analysis are **weighted pointwise estimates** revealing how the exponent difference \(β(t)-α(t)\) modulates \(\mathscr{V}_{α,β}[x](t)\), leading to conditions under which defines a bounded linear operator between standard function spaces. These estimates exhibit a natural compensation mechanism between the two memory weightings. For the uniform-memory case , we prove that \(\mathscr{V}_{α,β}[x](t)\) **asymptotically recovers** the classical derivative \(\dot{x}(0)\) as , ensuring consistency with local calculus. The mathematical framework is supported by self-contained technical appendices. By decoupling the memory weighting of state increments from that of elapsed time, provides a structured approach to modeling systems with independently evolving memory characteristics, offering potential utility in formulating evolution equations for complex physical processes with non-stationary memory.
Paper Structure (16 sections, 18 theorems, 38 equations)

This paper contains 16 sections, 18 theorems, 38 equations.

Table of Contents

  1. INTRODUCTION
  2. MATHEMATICAL FOUNDATIONS AND BASIC PROPERTIES OF THE MEMORY-WEIGHTED VELOCITY OPERATOR
  3. Function Spaces and Notational Conventions
  4. Time-Varying Power-Law Memory Weight Functions
  5. Definition of the Memory-Weighted Velocity Operator
  6. Continuous dependence on the memory exponents
  7. The Uniform-Memory Regime: Asymptotic Recovery of the Classical Derivative
  8. Weighted Estimates and Mapping Properties: Boundedness Analysis of $\mathscr{V}_{\alpha,\beta}$
  9. Technical Lemmas on Logarithmic and Power Function Inequalities
  10. Auxiliary convergence results
  11. Systematic Construction of Error-Control Functions
  12. Basic Setup and Notation
  13. Well‑Posedness and Fundamental Properties
  14. Limit Behavior at the Origin
  15. Monotonicity and the Uniform Control Parameter $\epsilon_{t}$
  16. ...and 1 more sections

Key Result

Lemma 2.1

Let $\varrho \in \mathcal{C}(I)$ be a time-varying memory exponent and let $K_{\varrho}$ be the associated kernel defined in eq:memory_kernel. For any $t \in (0,T]$, Consequently, for each fixed $t$, the kernel $K_{\varrho}(t,\cdot)$ is integrable on $[0,t]$ and its total mass is a continuous function of $t$.

Theorems & Definitions (35)

  • Definition 2.1: Time-varying memory exponent
  • Definition 2.2: Time-varying power-law memory kernel
  • Remark 2.1: Causal structure and relation to existing work
  • Lemma 2.1: Integral formula for the time-varying memory kernel
  • Remark 2.2: Special choices of the memory exponent
  • Definition 2.3: Memory-weighted velocity operator
  • Remark 2.3: Interpretation and well-definedness
  • Corollary 2.1: Simplified expression
  • Proposition 2.1: Linearity of the memory-weighted velocity operator
  • Remark 2.4: Origin of the linearity
  • ...and 25 more