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Five eight-step random walks from a central point. Some paths appear shorter than eight steps where the route has doubled back on itself. (animated version)In mathematics, a random walk, sometimes known as a drunkard's walk, is a stochastic process that describes a path that consists of a succession of random steps on some mathematical space.
The actual random walk obeys a stochastic equation of motion, but its probability density function (PDF) obeys a deterministic equation. PDFs of random walks can be formulated in terms of the (discrete in space) master equation [1] [12] [13] and the generalized master equation [3] or the (continuous in space and time) Fokker Planck equation [37] and its generalizations. [10]
It can be considered as a continuous version of the simple random walk. [ 49 ] [ 105 ] The process arises as the mathematical limit of other stochastic processes such as certain random walks rescaled, [ 107 ] [ 108 ] which is the subject of Donsker's theorem or invariance principle, also known as the functional central limit theorem.
where is the number of element at the state , are i.i.d. random variables (usually with mean 0) and where is the indicator function. For the simple case of coin flips this models the different levels of a Galton board.
For random walks on a hyperbolic group, under the finite entropy assumption on the step distribution which always hold for a simple walk (a more general condition is that the first moment be finite) the Poisson boundary is always equal to the Gromov boundary when equipped with the hitting probability measure.
Donsker's invariance principle for simple random walk on .. In probability theory, Donsker's theorem (also known as Donsker's invariance principle, or the functional central limit theorem), named after Monroe D. Donsker, is a functional extension of the central limit theorem for empirical distribution functions.
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A loop-erased random walk in 2D for steps. In mathematics, loop-erased random walk is a model for a random simple path with important applications in combinatorics, physics and quantum field theory. It is intimately connected to the uniform spanning tree, a model for a random tree.