Classically, (stochastic or deterministic) ODEs are βmemorylessβ in the sense that the current state (and not the history) of the system determines the future states/distribution of states. In the stochastic case, they are Markov.

One way you can destroy this locality/memorylessness is by using *fractional derivatives* in the
formulation of the equation.
These use the Laplace-transform representation to do something like differentiating to a non-integer order.

This is not the only way we could introduce memory; for example we could put some explicit integrals over the history of the process into the defining equations; but that is a different notebook, or a hidden state. But this option fits into certain ODEs elegantly, which is an attraction.

Note some evocative similarities to branching processes which I usually study in discrete index and/or state space, and a connection I have been told exists but do not understand, to fractals.

Popular in modelling Dengue and pharmacokinetics, whatever that is.
Keywords that pop up in the vicinity: *Super diffusive systems*β¦

How do these related to fractional Brownian motions?

## References

*Physica A: Statistical Mechanics and Its Applications*379 (2): 607β14.

*Journal of Mathematical Biology*72 (6): 1441β65.

*Nonlocal Diffusion and Applications*. Vol. 20. Lecture Notes of the Unione Matematica Italiana. New York, NY: Springer International Publishing.

*arXiv:1708.06605 [Math]*, August.

*Journal of Time Series Analysis*1 (1): 15β29.

*International Journal of Forecasting*, Forecasting Long Memory Processes, 18 (2): 167β79.

*Journal of Mathematical Analysis and Applications*397 (1): 334β48.

*Journal of Computational Physics*, Fractional PDEs, 293 (July): 4β13.

*Journal of Computational Physics*, Fractional PDEs, 293 (July): 14β28.

*Communications in Nonlinear Science and Numerical Simulation*22 (1β3): 511β25.

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