Dynamic Weighted Fractional Entropy for Time-Fractional Diffusion Processes via Moment Formulas

We investigate dynamic weighted fractional information-theoretic measures for linear stochastic differential equations driven by fractional Brownian motion with Hurst parameter H∈(1/2,1). Motivated by recent constructions of fractional Deng entropy and building upon explicit Gaussian solutions and closed-form fractional moments derived in previous work, we establish fully analytical expressions for the Shannon entropy, Rényi entropy, Tsallis entropy, extropy, and a continuous weighted fractional entropy EXtp(−logpXt(Xt)) for p≥0, expressed directly in terms of known fractional moments without density estimation. All derived measures share a universal asymptotic scaling law growing as Hlogt, establishing a precise quantitative link between long-memory effects and information dynamics. The weighted fractional entropy further reveals remarkable structural properties as a function of the weighting order p, exposing a dual role of long memory on the system’s informational content. As a concrete application, we characterize anomalous diffusion in aging soft materials through an explicit critical time linking maximal uncertainty to the memory exponent H and the macroscopic aging rate. All results are validated through extensive Monte-Carlo simulations, demonstrating excellent agreement with the closed-form expressions across a wide range of Hurst exponents H and weighting orders p.