Towards a Mathematical Theory of Adaptive Memory: From Time-Varying to Responsive Fractional Brownian Motion

TL;DR

This paper introduces a rigorous mathematical framework for adaptive stochastic processes with state-dependent memory, including time-varying and responsive fractional Brownian motions, with applications to attention mechanisms.

Contribution

It develops a novel class of processes called Responsive Fractional Brownian Motion with state-dependent Hurst exponents and analyzes their properties and implications for adaptive memory modeling.

Findings

Established properties of TV-fBm including variance scaling and local asymptotics.

Proved well-posedness and regularity of RfBm with state-dependent Hurst exponent.

Derived bounds and stability measures for the induced attention weights.

Abstract

This work develops a comprehensive mathematical theory for a class of stochastic processes whose local regularity adapts dynamically in response to their own state. We first introduce and rigorously analyze a time-varying fractional Brownian motion (TV-fBm) with a deterministic, H\"older-continuous Hurst exponent function. Key properties are established, including its exact variance scaling law, precise local increment asymptotics, local non-determinism, large deviation asymptotics for its increments, and a covariance structure that admits a closed-form hypergeometric representation. We then define a novel class of processes termed Responsive Fractional Brownian Motion (RfBm). Here,the Hurst exponent is governed by a Lipschitz-H\"older response function depending on the process state itself, creating an intrinsic feedback mechanism between state and memory. We establish the well-posedness of this definition, prove pathwise H\"older regularity of the induced instantaneous scaling exponent, and analyze associated cumulative memory processes along with their asymptotic convergence. The mathematical structure of RfBm naturally gives rise to a continuous-time, pathwise attention mechanism. We show that its kernel induces a well-defined attention weight distribution, derive fundamental bounds for these weights, and quantify the stability of attentional allocation through residence measures and volatility functionals. This work develops a stochastic-process-theoretic framework for concepts central to adaptive memory and content-sensitive information processing, offering a mathematically grounded perspective that may complement existing empirical approaches.