Random Dynamical System
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High Quality Content by WIKIPEDIA articles! In mathematics, a random dynamical system is a measure-theoretic formulation of a dynamical system with an element of "randomness", such as the dynamics of solutions to a stochastic differential equation. It consists of a base flow, the "noise", and a cocycle dynamical system on the "physical" phase space.Let f : mathbb{R}^{d} to mathbb{R}^{d} be a d-dimensional vector field, and let varepsilon 0. Suppose that the solution X(t, ;x0) to the stochastic differential equation left{ begin{matrix} mathrm{d} X = f(X) , mathrm{d} t + varepsilon , mathrm{d} W...
High Quality Content by WIKIPEDIA articles! In mathematics, a random dynamical system is a measure-theoretic formulation of a dynamical system with an element of "randomness", such as the dynamics of solutions to a stochastic differential equation. It consists of a base flow, the "noise", and a cocycle dynamical system on the "physical" phase space.Let f : mathbb{R}^{d} to mathbb{R}^{d} be a d-dimensional vector field, and let varepsilon 0. Suppose that the solution X(t, ;x0) to the stochastic differential equation left{ begin{matrix} mathrm{d} X = f(X) , mathrm{d} t + varepsilon , mathrm{d} W (t); X (0) = x_{0}; end{matrix} right.exists for all positive time and some (small) interval of negative time dependent upon omega in Omega, where W : mathbb{R} times Omega to mathbb{R}^{d} denotes a d-dimensional Wiener process (Brownian motion).
