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.
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…
How do these related to fractional Brownian motions?
References
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https://doi.org/10.1007/s00285-015-0917-9.
Bucur, Claudia, and Enrico Valdinoci. 2016.
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Camrud, Evan. 2017.
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https://doi.org/10.18576/pfda/040402.
Granger, C. W. J., and Roselyne Joyeux. 1980.
“An Introduction to Long-Memory Time Series Models and Fractional Differencing.” Journal of Time Series Analysis 1 (1): 15–29.
https://doi.org/10.1111/j.1467-9892.1980.tb00297.x.
Hurvich, Clifford M. 2002.
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https://doi.org/10.1016/S0169-2070(01)00151-0.
Nguyen Tien, Dung. 2013.
“Fractional Stochastic Differential Equations with Applications to Finance.” Journal of Mathematical Analysis and Applications 397 (1): 334–48.
https://doi.org/10.1016/j.jmaa.2012.07.062.
Ortigueira, Manuel D., and J. A. Tenreiro Machado. 2015.
“What Is a Fractional Derivative?” Journal of Computational Physics, Fractional PDEs, 293 (July): 4–13.
https://doi.org/10.1016/j.jcp.2014.07.019.
Sabzikar, Farzad, Mark M. Meerschaert, and Jinghua Chen. 2015.
“Tempered Fractional Calculus.” Journal of Computational Physics, Fractional PDEs, 293 (July): 14–28.
https://doi.org/10.1016/j.jcp.2014.04.024.
Sardar, Tridip, Sourav Rana, and Joydev Chattopadhyay. 2015.
“A Mathematical Model of Dengue Transmission with Memory.” Communications in Nonlinear Science and Numerical Simulation 22 (1–3): 511–25.
https://doi.org/10.1016/j.cnsns.2014.08.009.
Williams, Paul. 2007. “Fractional Calculus of Schwartz Distributions.”
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