Interacting particle systemIn probability theory, an interacting particle system (IPS) is a stochastic process on some configuration space given by a site space, a countably-infinite-order graph and a local state space, a compact metric space . More precisely IPS are continuous-time Markov jump processes describing the collective behavior of stochastically interacting components. IPS are the continuous-time analogue of stochastic cellular automata.
Théorème de DonskerEn théorie des probabilités, le théorème de Donsker établit la convergence en loi d'une marche aléatoire vers un processus stochastique gaussien. Il est parfois appelé le théorème central limite fonctionnel. Ce théorème est une référence pour la convergence en loi de marches aléatoires renormalisées vers un processus à temps continus. De nombreux théorèmes sont alors dits de « type Donsker ». Soient une suite iid de variables aléatoires centrées, de carré intégrable et de variance .
Stochastic cellular automatonStochastic cellular automata or probabilistic cellular automata (PCA) or random cellular automata or locally interacting Markov chains are an important extension of cellular automaton. Cellular automata are a discrete-time dynamical system of interacting entities, whose state is discrete. The state of the collection of entities is updated at each discrete time according to some simple homogeneous rule. All entities' states are updated in parallel or synchronously.
Ruine du joueurIn statistics, 'gambler's ruin' is the fact that a gambler playing a game with negative expected value will eventually go broke, regardless of their betting system. The concept was initially stated: A persistent gambler who raises his or her bet to a fixed fraction of the gambler's bankroll after a win, but does not reduce it after a loss, will eventually and inevitably go broke, even if each bet has a positive expected value.
Deterministic systemIn mathematics, computer science and physics, a deterministic system is a system in which no randomness is involved in the development of future states of the system. A deterministic model will thus always produce the same output from a given starting condition or initial state. Physical laws that are described by differential equations represent deterministic systems, even though the state of the system at a given point in time may be difficult to describe explicitly.
Random fieldIn physics and mathematics, a random field is a random function over an arbitrary domain (usually a multi-dimensional space such as ). That is, it is a function that takes on a random value at each point (or some other domain). It is also sometimes thought of as a synonym for a stochastic process with some restriction on its index set. That is, by modern definitions, a random field is a generalization of a stochastic process where the underlying parameter need no longer be real or integer valued "time" but can instead take values that are multidimensional vectors or points on some manifold.
State space (computer science)In computer science, a state space is a discrete space representing the set of all possible configurations of a "system". It is a useful abstraction for reasoning about the behavior of a given system and is widely used in the fields of artificial intelligence and game theory. For instance, the toy problem Vacuum World has a discrete finite state space in which there are a limited set of configurations that the vacuum and dirt can be in. A "counter" system, where states are the natural numbers starting at 1 and are incremented over time has an infinite discrete state space.
Équation de diffusionLéquation de diffusion est une équation aux dérivées partielles. En physique, elle décrit le comportement du déplacement collectif de particules (molécules, atomes, photons. neutrons, etc.) ou de quasi-particules comme les phonons dans un milieu causé par le mouvement aléatoire de chaque particule lorsque les échelles de temps et d'espace macroscopiques sont grandes devant leurs homologues microscopiques. Dans le cas contraire le problème est décrit par l'équation de Boltzmann.
Décalage de Bernoulli (langage formel)Un décalage de Bernoulli (en anglais Bernoulli shift) est une transformation opérant sur des mots de longueur infinie, étudiée en dynamique symbolique. Étant donné un alphabet Λ, c'est-à-dire un ensemble fini. Un mot infini est une suite à valeurs dans l'alphabet Λ. Le décalage de Bernoulli est l'application qui décale un mot d'un cran vers la gauche : On peut définir de même les décalages de Bernoulli pour des mots infinis indexés sur et les résultats et propriétés énoncés sont similaires.
SemimartingaleIn probability theory, a real valued stochastic process X is called a semimartingale if it can be decomposed as the sum of a local martingale and a càdlàg adapted finite-variation process. Semimartingales are "good integrators", forming the largest class of processes with respect to which the Itô integral and the Stratonovich integral can be defined. The class of semimartingales is quite large (including, for example, all continuously differentiable processes, Brownian motion and Poisson processes).
