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  "body": "\n## Summary\n\nA random dynamical system is a [[dynamical system|BBZP1R872N]] whose rule of evolution is chosen at random at each time.\nThe randomness comes from a noise that is itself a measure-preserving dynamical system.\nThe evolution is a cocycle over the noise.\nThe state after time $s + t$ is obtained by first evolving for time $s$ and then evolving for time $t$ with the shifted noise.\nStochastic differential equations and random iterations of maps give random dynamical systems.\nThe framework allows notions of deterministic dynamics, such as attractors, invariant measures and [[Lyapunov exponents|NJ1FJSNK6A]], to be used for systems with noise.\n\n## Explanations\n\n### L1 human\n\nImagine moving a point by one of two maps, and choosing which map by tossing a coin each time.\nEach sequence of coin tosses gives one sequence of maps, and the point follows the composition of these maps.\nA random dynamical system keeps track of all coin sequences at once, together with their probabilities.\nThe rule that ties it together is simple.\nRunning for $s$ steps and then $t$ steps is the same as running for $s + t$ steps, as long as the coin tosses are read in the right order.\n",
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