Fractional brownian motion run with a nonlinear clock.

Area of Science:

• * Stochastic Processes and Statistical Physics
• * Nonlinear Dynamics and Complex Systems

Background:

• * Traditional stochastic processes often assume stationary or independent increments, limiting their applicability to complex natural phenomena.
• * Anomalous diffusion describes systems where particle spread deviates from standard Brownian motion, often seen in disordered media.
• * Existing models struggle to independently control key characteristics like fractal dimension and mean-square displacement.

Purpose of the Study:

• * To construct a novel family of stochastic processes with nonstationary, correlated increments.
• * To enable independent selection of fractal dimension and mean-square displacement.
• * To explore the characteristics and implications of this new process for anomalous diffusion.

Main Methods:

• * Construction of a generalized fractional Brownian motion (fBm) framework.
• * Introduction of a nonlinear clock mechanism to modify the temporal scaling of fBm.
• * Mathematical analysis of the resulting fractional Brownian motion with a nonlinear clock (fBm-nlc) process.

Main Results:

• * The fractal dimension of the fBm-nlc process is identical to that of the underlying fBm process.
• * The process allows for independent tuning of fractal dimension and mean-square displacement.
• * Analysis of p-variation highlights limitations in differentiating certain diffusive processes.

Conclusions:

• * The fBm-nlc process offers a flexible tool for modeling complex transport phenomena.
• * It demonstrates that the spectrum of anomalous diffusive processes is broader than previously recognized.
• * Further research into nonlinear clock mechanisms can yield new insights into anomalous diffusion.