On the Asymptotics of an Infinite-Dimensional Stochastic Dynamical System

13 Nov 2017 11:43

I am working on an idea where I need to show that, in the long run, a certain infinite-dimensional discrete-time stochastic dynamical system has a stable limiting distribution, and calculate certain properties of that distribution. This notebook is for storing related references.

I'm a little paranoid about being scooped, so I won't say much more, only that there are certain connections with genetic algorithms, and that the finite-dimensional analog is related to the replicator equation of evolutionary biology. In that representation, the fitness function would show a particular kind of frequency-dependence, with random fluctuations which are not necessarily independent over time, though I'd be willing to ignore serial dependence if the alternatives were simply intractable. Now, the replicator equation can be re-written as a linear system (or so Nihat tells me), which may be helpful...

Update: Actually, the project was this paper, and I ended up being able to solve the problem without gong through the more direct route I had in mind here. Which I still think is interesting!

See also: Ergodic Theory; Filtering (since related equations occur in nonlinear filtering); Stochastic Processes