In mathematics, 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.

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.

Let ${\displaystyle f:\mathbb {R} ^{d}\to \mathbb {R} ^{d}}$ be a ${\displaystyle d}$-dimensional vector field, and let ${\displaystyle \varepsilon >0}$. Suppose that the solution ${\displaystyle X(t,\omega ;x_{0})}$ to the stochastic differential equation

exists for all positive time and some (small) interval of negative time dependent upon ${\displaystyle \omega \in \Omega }$, where ${\displaystyle W:\mathbb {R} \times \Omega \to \mathbb {R} ^{d}}$ denotes a ${\displaystyle d}$-dimensional Wiener process (Brownian motion). Implicitly, this statement uses the classical Wiener probability space

In this context, the Wiener process is the coordinate process.

Now define a flow map or (solution operator) ${\displaystyle \varphi :\mathbb {R} \times \Omega \times \mathbb {R} ^{d}\to \mathbb {R} ^{d}}$ by

(whenever the right hand side is well-defined). Then ${\displaystyle \varphi }$ (or, more precisely, the pair ${\displaystyle (\mathbb {R} ^{d},\varphi )}$) is a (local, left-sided) random dynamical system. The process of generating a "flow" from the solution to a stochastic differential equation leads us to study suitably defined "flows" on their own. These "flows" are random dynamical systems.

An i.i.d random dynamical system in the discrete space is described by a triplet ${\displaystyle (S,\Gamma ,Q)}$.