Random dynamical system

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In the mathematical field of dynamical systems, a random dynamical system is a dynamical system in which the equations of motion have an element of randomness to them. Random dynamical systems are characterized by a state space S, a set of maps ${\displaystyle \Gamma }$ from S into itself that can be thought of as the set of all possible equations of motion, and a probability distribution Q on the set ${\displaystyle \Gamma }$ that represents the random choice of map. Motion in a random dynamical system can be informally thought of as a state ${\displaystyle X\in S}$ evolving according to a succession of maps randomly chosen according to the distribution Q.^[1]

An example of a random dynamical system is a stochastic differential equation; in this case the distribution Q is typically determined by noise terms. It consists of a base flow, the "noise", and a cocycle dynamical system on the "physical" phase space. Another example is discrete state random dynamical system; some elementary contradistinctions between Markov chain and random dynamical system descriptions of a stochastic dynamics are discussed.^[2]