Random dynamical system

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 Γ {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 Γ {displaystyle Gamma } that represents the random choice of map. Motion in a random dynamical system can be informally thought of as a state X ∈ S {displaystyle Xin S} evolving according to a succession of maps randomly chosen according to the distribution Q. 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 Γ {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 Γ {displaystyle Gamma } that represents the random choice of map. Motion in a random dynamical system can be informally thought of as a state X ∈ S {displaystyle Xin 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 f : R d → R d {displaystyle f:mathbb {R} ^{d} o mathbb {R} ^{d}} be a d {displaystyle d} -dimensional vector field, and let ε > 0 {displaystyle varepsilon >0} . Suppose that the solution X ( t , ω ; x 0 ) {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 ω ∈ Ω {displaystyle omega in Omega } , where W : R × Ω → R d {displaystyle W:mathbb {R} imes Omega o mathbb {R} ^{d}} denotes a d {displaystyle d} -dimensional Wiener process (Brownian motion). Implicitly, this statement uses the classical Wiener probability space