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

Abstract

This work attempts to develop a systematic mathematical description for a class of stochastic processes whose local regularity adjusts dynamically in response to their own state. The investigation proceeds in three stages. First, we consider a Time-Varying Fractional Brownian Motion (TV-fBm) with a deterministic, H\"older-continuous Hurst exponent function, and conduct a further theoretical analysis of its properties and structural features. For this process, we establish a range of fundamental properties, including an exact variance scaling law, local increment asymptotic estimates, local non-determinism, large deviation asymptotics for local increments, and a covariance structure that admits a closed-form hypergeometric representation. These results provide useful insight into the local and global dependence structure of the process. Second, as a central component of this work, we define a new class of processes termed Responsive Fractional Brownian Motion (RfBm). In this construction, the Hurst exponent is governed by a Lipschitz-H\"older response function that depends on the process state itself, thereby introducing an intrinsic feedback mechanism between state and memory. We establish the well-posedness of this definition through local existence and uniqueness results, prove that the induced instantaneous scaling exponent possesses almost sure H\"older regularity, and analyze the associated cumulative memory processes along with their asymptotic convergence properties. The framework offers a reference mathematical foundation for exploring state-dependent memory mechanisms.