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

TL;DR

This paper introduces a novel two-parameter memory-weighted velocity operator to model systems with evolving, power-law memory, establishing its fundamental properties and demonstrating its consistency with classical derivatives in special cases.

Contribution

It defines and analyzes a new mathematical operator with two independent memory exponents, providing foundational properties and insights into systems with non-stationary memory effects.

Findings

Operator has explicit integral representation and linearity.

Memory exponents' difference influences boundedness and estimates.

Recovers classical derivative as time approaches zero.

Abstract

We introduce and analyze a **memory-weighted velocity operator** \(\mathscr{V}_{\alpha,\beta}\) as a mathematical framework for describing rates of change in systems with time-varying, power-law memory. The operator employs two independent continuous exponents \(\alpha(t)\) and \(\beta(t)\) 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 \(\beta(t)-\alpha(t)\) modulates \(\mathscr{V}_{\alpha,\beta}[x](t)\), leading to conditions under which \(\mathscr{V}_{\alpha,\beta}\) 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 \(\alpha=\beta\equiv1\), we prove that \(\mathscr{V}_{\alpha,\beta}[x](t)\) **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, \(\mathscr{V}_{\alpha,\beta}\) 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.