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 relate to fractional Brownian motions?