“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 branching processes
which I usually study in discrete index and/or state space.
Popular in modelling Dengue and pharmacokinetics, whatever that is.
Connections to Lévy flights.
Why not presume a state filter model
with hidden states, and learn that? Seems more general and no less tractable.
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