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?
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