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Abstract We introduce the stochastic process of incremental multifractional Brownian motion (IMFBM), which locally behaves like fractional Brownian motion with a given local Hurst exponent and diffusivity. When these parameters change as function of time the process responds to the evolution gradually: only new increments are governed by the new parameters, while still retaining a power-law dependence on the past of the process. We obtain the mean squared displacement and correlations of IMFBM which are given by elementary formulas. We also provide a comparison with simulations and introduce estimation methods for IMFBM. This mathematically simple process is useful in the description of anomalous diffusion dynamics in changing environments, e.g. in viscoelastic systems, or when an actively moving particle changes its degree of persistence or its mobility.
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Path Properties of a Generalized Fractional Brownian Motion
The generalized fractional Brownian motion is a Gaussian self-similar process whose increments are not necessarily stationary. It appears in applications as the scaling limit of a shot noise process with a power-law shape function and non-stationary noises with a power-law variance function. In this paper, we study sample path properties of the generalized fractional Brownian motion, including Hölder continuity, path differentiability/non-differentiability, and functional and local law of the iterated logarithms.
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- Award ID(s):
- 2008427
- PAR ID:
- 10221197
- Date Published:
- Journal Name:
- Journal of Theoretical Probability
- ISSN:
- 0894-9840
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s10959-020-01066-1
