“Super diffusive” systems, non-Markov processes… Classically, (stochastic or deterministic) ODEs are “memoryless” in the sense that the current state (and not the history) of the system determines the future states/distribution of states.
One way you can destroy this is by using fractional derivatives in the formulation of the equation. (Why this choice, as opposed to putting in explicit integrals over the history of the process, I have no idea. Perhaps it leads to more elegant parameterization or solutions?)
I’ll make this precise later, but want to note some evocative similarities to other branching processes which I usually study in discrete index and/or state space.
Popular in modelling Dengue and pharmacokinetics, whatever that is. Connections to to Lévy flights.
To learn: connection to long memory models. Why not presume a state filter model with hidden states, and learn that? Seems more general and no less tractable.
- SaRC15: (2015) A mathematical model of dengue transmission with memory. Communications in Nonlinear Science and Numerical Simulation, 22(1–3), 511–525. DOI
- GrJo80: (1980) An Introduction to Long-Memory Time Series Models and Fractional Differencing. Journal of Time Series Analysis, 1(1), 15–29. DOI
- Hurv02: (2002) Multistep forecasting of long memory series using fractional exponential models. International Journal of Forecasting, 18(2), 167–179. DOI
- AhEl07: (2007) On fractional order differential equations model for nonlocal epidemics. Physica A: Statistical Mechanics and Its Applications, 379(2), 607–614. DOI
- BeRT15: (2015) Turing pattern dynamics and adaptive discretization for a super-diffusive Lotka-Volterra model. Journal of Mathematical Biology, 72(6), 1441–1465. DOI