random variables and dynamical systems Dynamical systems have two parts: the state (usually a vector), and the rule (usually a matrix) such that the vector and matrix are compatible. Often enough I have seen how dynamical systems are represented as deterministic equations. 
What I haven't seen dynamical systems expressed in terms of random variables. I would like to know if any of you have worked with these. I am interested to know how properties like eigenvalues behave. (I'm sure they will also have distributions, just like the random variables). Stability of the matrix would be interesting to study, too. 
Any insights would be great. 
 A: Random variables that are functions of time are called stochastic processes. A stochastic process can be, but doesn't have to be described as dynamical (i.e. the future arising from the past via a rule), and they are not limited to time as the independent variable (see random field).
For discrete states and discrete time, a typical form to describe a dynamical stochastic process is a Markov process, whose dynamics are described by transition probabilities between states from one time point to the next. For continuous states and discrete time, a typical form to describe linear dynamics are autoregressive processes, where the next state is a linear combination of previous states plus noise. For continuous states and continuous time, which is probably closest to what you are familiar with in deterministic dynamics, a stochastic process may be described by a stochastic differential equation or Langevin equation.
A dynamics that is stochastic with respect to the system states can alternatively be described as a dynamics that is deterministic with respect to the probability distribution over states. For the Langevin equation this representation leads to the Fokker–Planck equation, for discrete states and continuous time to the Master equation.
Stochastic and deterministic dynamics are not completely separate. One the one side, deterministic dynamics can be understood as the limiting case of stochastic dynamics for vanishing noise amplitude; on the other side, it may prove useful to characterize deterministic (maybe chaotic) dynamics in terms of the evolution of distributions over state space (see e.g. Probabilistic Properties of Deterministic Systems). Moreover, "noise" can be considered as the result of coupling to a very large (very high-dimensional) deterministic dynamical system whose state evolution cannot be tracked in detail.
Be aware that this answer can only be a tiny overview over a vast field.
