Non-Markovian Stochastic Processes on Fractal Domains

Most stochastic models (e.g., Li et al.'s MDPs or Zhang et al.'s EV networks) assume Markovian dynamics. However, real systems like contaminant transport in clay (Muniruzzaman & Rolle, 2023) exhibit memory due to heterogeneous diffusion. This research develops non-Markovian stochastic processes on fractal curves (extending Golmankhaneh et al., 2023) by incorporating fractional derivatives and long-range correlations. For instance, we could model DOM molecular dynamics (She et al., 2023) as fractal processes with "memory kernels" capturing delayed electrostatic effects. Unlike Golmankhaneh et al.'s mean-square calculus, we introduce path-dependent integrals using Volterra equations. The novelty lies in unifying fractal geometry with non-Markovianity, potentially revolutionizing fields like hydrology (Mantilla et al., 2020) or finance (Hambly et al., 2021).

References:

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2. Relevance of charge interactions for contaminant transport in heterogeneous formations: a stochastic analysis. M. Muniruzzaman, M. Rolle (2023). Stochastic environmental research and risk assessment (Print).
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