 # Random walk Example of eight random walks in one dimension starting at 0. The plot shows the current position on the line (vertical axis) versus the time steps (horizontal axis). A random walk is a mathematical formalization of a path that consists of a succession of random steps. 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 stock and the financial status of a gambler can all be modeled as random walks, although they may not be truly random in reality. The term random walk was first introduced by Karl Pearson in 1905. Random walks have been used in many fields: ecology, economics, psychology, computer science, physics, chemistry, and biology. Random walks explain the observed behaviors of processes in these fields, and thus serve as a fundamental model for the recorded stochastic activity. Various different types of random walks are of interest. . . Lattice random walk .

Markov renewal process In probability and statistics a Markov renewal process is a random process that generalizes the notion of Markov jump processes. Other random processes like Markov chain, Poisson process, and renewal process can be derived as a special case of an MRP (Markov renewal process). Definition Consider a state space Consider a set of random variables , where are the jump times and are the associated states in the Markov chain (see Figure). . Relation to other stochastic processes If we define a new stochastic process for , then the process is called a semi-Markov process. See also References and Further Reading

Markov model Introduction The most common Markov models and their relationships are summarized in the following table: Markov chain The simplest Markov model is the Markov chain. It models the state of a system with a random variable that changes through time. Hidden Markov model A hidden Markov model is a Markov chain for which the state is only partially observable. Markov decision process A Markov decision process is a Markov chain in which state transitions depend on the current state and an action vector that is applied to the system. Partially observable Markov decision process A partially observable Markov decision process (POMDP) is a Markov decision process in which the state of the system is only partially observed. Markov random field A Markov random field (also called a Markov network) may be considered to be a generalization of a Markov chain in multiple dimensions. See also References

Interacting particle system In probability theory, an interacting particle system (IPS) is a stochastic process on some configuration space and a local state space, a compact metric space where is a finite set of sites and with for all . into configuration . on . of an IPS has the following form: Let be an observable in the domain of which is a subset of the real valued continuous function on the configuration space, then For example for the stochastic Ising model we have if for some and is the configuration equal to except it is flipped at site is a new parameter modeling the inverse temperature. Liggett, Thomas M. (1997).

Examples of Markov chains This page contains examples of Markov chains in action. Board games played with dice A game of snakes and ladders or any other game whose moves are determined entirely by dice is a Markov chain, indeed, an absorbing Markov chain. This is in contrast to card games such as blackjack, where the cards represent a 'memory' of the past moves. A center-biased random walk Consider a random walk on the number line where, at each step, the position (call it x) may change by +1 (to the right) or −1 (to the left) with probabilities: (where c is a constant greater than 0) For example if the constant, c, equals 1, the probabilities of a move to the left at positions x = −2,−1,0,1,2 are given by respectively. Since the probabilities depend only on the current position (value of x) and not on any prior positions, this biased random walk satisfies the definition of a Markov chain. A very simple weather model The above matrix as a graph. Predicting the weather or So Citation ranking

Wiener process A single realization of a one-dimensional Wiener process A single realization of a three-dimensional Wiener process The Wiener process has applications throughout the mathematical sciences. In physics it is used to study Brownian motion, the diffusion of minute particles suspended in fluid, and other types of diffusion via the Fokker–Planck and Langevin equations. It also forms the basis for the rigorous path integral formulation of quantum mechanics (by the Feynman–Kac formula, a solution to the Schrödinger equation can be represented in terms of the Wiener process) and the study of eternal inflation in physical cosmology. Characterizations of the Wiener process The Wiener process Wt is characterized by three properties: W0 = 0The function t → Wt is almost surely everywhere continuousWt has independent increments with Wt−Ws ~ N(0, t−s) (for 0 ≤ s < t), where N(μ, σ2) denotes the normal distribution with expected value μ and variance σ2. Basic properties Substituting Thus and .

Dynamics of Markovian particles Two particular features of DMP might be noticed: (1) an ergodic-like relation between the motion of particle and the corresponding steady state, and (2) the classic notion of geometric volume appears nowhere (e.g. a concept such as flow of "substance" is not expressed as liters per time unit but as number of particles per time unit). Bergner—DMP, a kinetics of macroscopic particles in open heterogeneous systems Markov process Markov process example Introduction A Markov process is a stochastic model that has the Markov property. Note that there is no definitive agreement in literature on the use of some of the terms that signify special cases of Markov processes. Markov processes arise in probability and statistics in one of two ways. Markov property The general case Let , for some (totally ordered) index set ; and let be a measurable space. adapted to the filtration is said to possess the Markov property with respect to the if, for each and each with s < t, A Markov process is a stochastic process which satisfies the Markov property with respect to its natural filtration. For discrete-time Markov chains In the case where is a discrete set with the discrete sigma algebra and , this can be reformulated as follows: Examples Gambling Suppose that you start with \$10, and you wager \$1 on an unending, fair, coin toss indefinitely, or until you lose all of your money. , then the sequence .

Kolmogorov backward equations (diffusion) The Kolmogorov backward equation (KBE) (diffusion) and its adjoint sometimes known as the Kolmogorov forward equation (diffusion) are partial differential equations (PDE) that arise in the theory of continuous-time continuous-state Markov processes. Both were published by Andrey Kolmogorov in 1931. Later it was realized that the forward equation was already known to physicists under the name Fokker–Planck equation; the KBE on the other hand was new. Informally, the Kolmogorov forward equation addresses the following problem. ); we want to know the probability distribution of the state at a later time . serves as the initial condition and the PDE is integrated forward in time. is a Dirac delta function centered on the known initial state). The Kolmogorov backward equation on the other hand is useful when we are interested at time t in whether at a future time s the system will be in a given subset of states B, sometimes called the target set. , the indicator function for the set B. for

Continuous-time Markov chain In probability theory, a continuous-time Markov chain (CTMC or continuous-time Markov process) is a mathematical model which takes values in some finite or countable set and for which the time spent in each state takes non-negative real values and has an exponential distribution. It is a continuous-time stochastic process with the Markov property which means that future behaviour of the model (both remaining time in current state and next state) depends only on the current state of the model and not on historical behaviour. The model is a continuous-time version of the Markov chain model, named because the output from such a process is a sequence (or chain) of states. Definitions A continuous-time Markov chain (Xt)t ≥ 0 is defined by a finite or countable state space S, a transition rate matrix Q with dimensions equal to that of the state space and initial probability distribution defined on the state space. There are three equivalent definitions of the process. is used. .

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