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A Lévy flight, named for French mathematician Paul Lévy, is a random walk in which the step-lengths have a probability distribution that is heavy-tailed. When defined as a walk in a space of dimension greater than one, the steps made are in isotropic random directions. The term "Lévy flight" was coined by Benoît Mandelbrot,〔Mandelbrot (1982, p.289)〕 who used this for one specific definition of the distribution of step sizes. He used the term Cauchy flight for the case where the distribution of step sizes is a Cauchy distribution,〔Mandelbrot (1982, p.290)〕 and Rayleigh flight for when the distribution is a normal distribution〔Mandelbrot (1982, p.288)〕 (which is not an example of a heavy-tailed probability distribution). Later researchers have extended the use of the term "Lévy flight" to include cases where the random walk takes place on a discrete grid rather than on a continuous space.〔〔 A Lévy flight is a random walk in which the steps are defined in terms of the step-lengths, which have a certain probability distribution, with the directions of the steps being isotropic and random. The particular case for which Mandelbrot used the term "Lévy flight"〔 is defined by the survivor function (commonly known as the survival function) of the distribution of step-sizes, ''U'', being〔Mandelbrot (1982, p. 294)〕 : Here ''D'' is a parameter related to the fractal dimension and the distribution is a particular case of the Pareto distribution. Later researchers allow the distribution of step sizes to be any distribution for which the survival function has a power-like tail : for some ''k'' satisfying 1 < ''k'' < 3. (Here the notation ''O'' is the Big O notation.) Such distributions have an infinite variance. Typical examples are the symmetric stable distributions. ==Properties== Lévy flights are, by construction, Markov processes. For general distributions of the step-size, satisfying the power-like condition, the distance from the origin of the random walk tends, after a large number of steps, to a stable distribution due to the generalized Central Limit Theorem first proved by Kolmogorov. Due to this property many processes can be modeled using Lévy flights. The probability densities for particles undergoing a Levy flight can be modeled using a generalized version of the Fokker- Planck equation, which is usually used to model Brownian motion. The equation requires the use of fractional derivatives. For jump lengths which have a symmetric probability distribution, the equation takes a simple form in terms of the Riesz fractional derivative. In one dimenson, the equation reads as : where γ is a constant akin to the diffusion constant, α is the stability parameter and f(x,t) is the potential. The Reisz derivative can be understood in terms of its Fourier Transform. : This can be easily extended to multiple dimensions. Another important property of the Lévy is that of diverging variances in all cases except that of α=2, i.e. Brownian motion. In general, the θ fractional moment of the distribution diverges if α < θ Also, : if θ ≤ α The exponential scaling of the step lengths gives Lévy flights a scale invariant property, and they are used to model data that exhibits clustering. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Lévy flight」の詳細全文を読む スポンサード リンク
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