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
Ahmed, E., and A. S. Elgazzar. 2007.
“On Fractional Order Differential Equations Model for Nonlocal Epidemics.” Physica A: Statistical Mechanics and Its Applications 379 (2): 607–14.
Bendahmane, Mostafa, Ricardo Ruiz-Baier, and Canrong Tian. 2015.
“Turing Pattern Dynamics and Adaptive Discretization for a Super-Diffusive Lotka-Volterra Model.” Journal of Mathematical Biology 72 (6): 1441–65.
Bucur, Claudia, and Enrico Valdinoci. 2016.
Nonlocal Diffusion and Applications. Vol. 20. Lecture Notes of the Unione Matematica Italiana. New York, NY: Springer International Publishing.
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.
Hurvich, Clifford M. 2002.
“Multistep Forecasting of Long Memory Series Using Fractional Exponential Models.” International Journal of Forecasting, Forecasting Long Memory Processes, 18 (2): 167–79.
Nguyen Tien, Dung. 2013.
“Fractional Stochastic Differential Equations with Applications to Finance.” Journal of Mathematical Analysis and Applications 397 (1): 334–48.
Ortigueira, Manuel D., and J. A. Tenreiro Machado. 2015.
“What Is a Fractional Derivative?” Journal of Computational Physics, Fractional PDEs, 293 (July): 4–13.
Sabzikar, Farzad, Mark M. Meerschaert, and Jinghua Chen. 2015.
“Tempered Fractional Calculus.” Journal of Computational Physics, Fractional PDEs, 293 (July): 14–28.
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.
Williams, Paul. 2007. “Fractional Calculus of Schwartz Distributions.”
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