multifractional

Multifractional refers to mathematical constructs in which the order of differentiation or the degree of roughness of a process is allowed to vary with time or space. It is most often encountered in multifractional Brownian motion (mBm), a generalization of fractional Brownian motion where the Hurst parameter H, which controls regularity and self-similarity, is replaced by a function H(t).

In fractional Brownian motion, H is constant; in mBm, H(t) can be smooth, piecewise smooth, or even irregular, yielding local behavior that can change over the domain. The result is a Gaussian process with non-stationary increments and local Hölder regularity matching H(t).

The term also appears in multifractional calculus, a generalization of fractional calculus where the order of differentiation α(t) depends on time, leading to variable-order derivatives. This approach has been studied in contexts such as signal processing and anomalous diffusion.

Properties and implications include local regularity determined by H(t) or α(t), and a lack of global self-similarity, providing more flexible modeling of nonstationary data. The mathematical analysis often requires tools from stochastic calculus, functional analysis, and variable-order calculus.

Applications encompass modeling complex time series with changing roughness, such as financial volatility, internet traffic, geophysical signals, and textures in image analysis. Multifractional frameworks offer a way to capture time-varying regularity that cannot be represented by classical fixed-order models.