The stochastic trajectories of molecules in living cells, as well as the dynamics in many other complex systems, often exhibit memory in their path over long periods of time. In addition, these systems can show dynamic heterogeneities due to which the motion changes along the trajectories. Such effects manifest themselves as spatiotemporal correlations. Despite the broad occurrence of heterogeneous complex systems in nature, their analysis is still quite poorly understood and tools to model them are largely missing. We contribute to tackling this problem by employing an integral representation of Mandelbrot’s fractional Brownian motion that is compliant with varying motion parameters while maintaining long memory. Two types of switching fractional Brownian motion are analysed, with transitions arising from a Markovian stochastic process and scalefree intermittent processes. We obtain simple formulas for classical statistics of the processes, namely the mean squared displacement and the power spectral density. Further, a method to identify switching fractional Brownian motion based on the distribution of displacements is described. A validation of the model is given for experimental measurements of the motion of quantum dots in the cytoplasm of live mammalian cells that were obtained by singleparticle tracking.
An inverse random source problem for the time fractional diffusion equation driven by a fractional Brownian motion
This paper is concerned with the mathematical analysis of an inverse random source problem for the time fractional diffusion equation, where the source is driven by a fractional Brownian motion. Given the random source, the direct problem is to study the stochastic time fractional diffusion equation. The inverse problem is to determine the statistical properties of the source from the expectation and variance of the final time data. For the direct problem, we show that it is wellposed and has a unique mild solution under a certain condition. For the inverse problem, the uniqueness is proved and the instability is characterized. The major ingredients of the analysis are based on the properties of the Mittag–Leffler function and the stochastic integrals associated with the fractional Brownian motion.
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 Award ID(s):
 1912704
 NSFPAR ID:
 10182370
 Date Published:
 Journal Name:
 Inverse problems
 Volume:
 36
 ISSN:
 02665611
 Page Range / eLocation ID:
 045008
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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