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Random Walk (Term)

Random Walk — Sun, 11 Jul 2010 06:41:24 -0500

A random walk, sometimes denoted RW, is a mathematical formalisation of a trajectory that consists of taking successive random steps. The results of random walk analysis have been applied to computer science, physics, ecology, economics, psychology and a number of other fields as a fundamental Statistical model for Stochastic process in time. For example, the path traced by a molecule as it travels in a liquid or a gas, the search path of a foraging animal, the price of a fluctuating random walk hypothesis and the financial status of a gambler can all be modeled as random walks. The term random walk was first introduced by Karl Pearson in 1905.

Various different types of random walks are of interest. Often, random walks are assumed to be Markov chains or Markov processes, but other, more complicated walks are also of interest. Some random walks are on graph theory, others on the line, in the plane, or in higher dimensions, while some random walks are on group theory. Random walks also vary with regard to the time parameter. Often, the walk is in discrete time, and indexed by the natural numbers, as in X_0,X_1,X_2,\dots. However, some walks take their steps at random times, and in that case the position X_t is defined for the continuum of times t\ge 0. Specific cases or limits of random walks include the drunkard's walk and Lévy flight. Random walks are related to the diffusion models and are a fundamental topic in discussions of Markov processes. Several properties of random walks, including dispersal distributions, first-passage times and encounter rates, have been extensively studied.
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Lyapunov Function (Term)

Lyapunov Function — Thu, 08 Jul 2010 11:08:05 -0500

In mathematics, Lyapunov functions are functions which can be used to prove the stability of a certain fixed point in a dynamical system or autonomous differential equation. Named after the Russian mathematician Aleksandr Mikhailovich Lyapunov, Lyapunov functions are important to stability theory and control theory. A similar concept appears in the theory of general state space Markov Chains, usually under the name Lyapunov-Foster functions.Functions which might prove the stability of some equilibrium are called Lyapunov-candidate-functions. There is no general method to construct or find a Lyapunov-candidate-function which proves the stability of an equilibrium, and the inability to find a Lyapunov function is inconclusive with respect to stability, which means, that not finding a Lyapunov function doesn't mean that the system is unstable. For dynamical systems (e.g. physical systems), conservation laws can often be used to construct a Lyapunov-candidate-function.
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Probability (Term)

Probability — Tue, 06 Jul 2010 10:12:46 -0500

Probability is a way of expressing knowledge or belief that an Event (probability theory) will occur or has occurred. In mathematics the concept has been given an exact meaning in probability theory, that is used extensively in such areas of study as mathematics, statistics, finance, gambling, science, and philosophy to draw conclusions about the likelihood of potential events and the underlying mechanics of complex systems.
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