Interacting particle system

In probability theory, an interacting particle system (IPS) is a stochastic process $(X(t))_{{t\in {\mathbb R}^{+}}}$ on some configuration space $\Omega =S^{G}$ given by a site space, a countable-infinite graph $G$ and a local state space, a compact metric space $S$ . 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. Among the main examples are the voter model, the contact process, the asymmetric simple exclusion process (ASEP), the Glauber dynamics and in particular the stochastic Ising model.

IPS are usually defined via their Markov generator giving rise to a unique Markov process using Markov semigroups and the Hille-Yosida theorem. The generator again is given via so-called transition rates $c_{\Lambda }(\eta ,\xi )>0$ where $\Lambda \subset G$ is a finite set of sites and $\eta ,\xi \in \Omega$ with $\eta _{i}=\xi _{i}$ for all $i\notin \Lambda$ . The rates describe exponential waiting times of the process to jump from configuration $\eta$ into configuration $\xi$ . More generally the transition rates are given in form of a finite measure $c_{\Lambda }(\eta ,d\xi )$ on $S^{\Lambda }$ .

The generator $L$ of an IPS has the following form. First, the domain of $L$ is a subset of the space of "observables", that is, the set of real valued continuous functions on the configuration space $\Omega$ . Then for any observable $f$ in the domain of $L$ , one has

$Lf(\eta )=\sum _{\Lambda }\int _{{\xi :\xi _{{\Lambda ^{c}}}=\eta _{{\Lambda ^{c}}}}}c_{\Lambda }(\eta ,d\xi )[f(\xi )-f(\eta )]$ .

For example, for the stochastic Ising model we have $G={\mathbb Z}^{d}$ , $S=\{-1,+1\}$ , $c_{\Lambda }=0$ if $\Lambda \neq \{i\}$ for some $i\in G$ and

c_{i}(\eta ,\eta ^{i})=\exp[-\beta \sum _{{j:|j-i|=1}}\eta _{i}\eta _{j}]

where $\eta ^{i}$ is the configuration equal to $\eta$ except it is flipped at site $i$ . $\beta$ is a new parameter modeling the inverse temperature.

References

Liggett, Thomas M. (1997). "Stochastic Models of Interacting Systems". The Annals of Probability. Institute of Mathematical Statistics. 25 (1): 1–29. doi:10.2307/2959527. ISSN 0091-1798.
Liggett, Thomas M. (1985). Interacting Particle Systems. New York: Springer Verlag. ISBN 0-387-96069-4.

Stochastic processes

Discrete time	Bernoulli process Branching process Chinese restaurant process Galton–Watson process Independent and identically distributed random variables Markov chain Moran process Random walk Loop-erased Self-avoiding

Continuous time	Bessel process Birth–death process Brownian motion Bridge Excursion Fractional Geometric Meander Cauchy process Contact process Continuous-time random walk Cox process Diffusion process Empirical process Feller process Fleming–Viot process Gamma process Hunt process Interacting particle systems Itô diffusion Itô process Jump diffusion Jump process Lévy process Local time Markov additive process McKean–Vlasov process Ornstein–Uhlenbeck process Poisson process Compound Non-homogeneous Point process Schramm–Loewner evolution Semimartingale Sigma-martingale Stable process Superprocess Telegraph process Variance gamma process Wiener process Wiener sausage

Both	Branching process Galves–Löcherbach model Gaussian process Hidden Markov model (HMM) Markov process Martingale Differences Local Sub- Super- Random dynamical system Regenerative process Renewal process Stochastic chains with memory of variable length White noise

Fields and other	Dirichlet process Gaussian random field Gibbs measure Hopfield model Ising model Potts model Boolean network Markov random field Percolation Pitman–Yor process Point process Cox Poisson Random field Random graph

Time series models	Autoregressive conditional heteroskedasticity (ARCH) model Autoregressive integrated moving average (ARIMA) model Autoregressive (AR) model Autoregressive–moving-average (ARMA) model Generalized autoregressive conditional heteroskedasticity (GARCH) model Moving-average (MA) model

Financial models	Black–Derman–Toy Black–Karasinski Black–Scholes Chen Constant elasticity of variance (CEV) Cox–Ingersoll–Ross (CIR) Garman–Kohlhagen Heath–Jarrow–Morton (HJM) Heston Ho–Lee Hull–White LIBOR market Rendleman–Bartter SABR volatility Vašíček Wilkie

Actuarial models	Bühlmann Cramér–Lundberg Risk process Sparre–Anderson

Queueing models	Bulk Fluid Generalized queueing network M/G/1 M/M/1 M/M/c

Properties	Càdlàg paths Continuous Continuous paths Ergodic Exchangeable Feller-continuous Gauss–Markov Markov Mixing Piecewise deterministic Predictable Progressively measurable Self-similar Stationary Time-reversible

Limit theorems	Central limit theorem Donsker's theorem Doob's martingale convergence theorems Ergodic theorem Fisher–Tippett–Gnedenko theorem Large deviation principle Law of large numbers (weak/strong) Law of the iterated logarithm Maximal ergodic theorem Sanov's theorem

Inequalities	Burkholder–Davis–Gundy Doob's martingale Kunita–Watanabe

Tools	Cameron–Martin formula Convergence of random variables Doléans-Dade exponential Doob decomposition theorem Doob–Meyer decomposition theorem Doob's optional stopping theorem Dynkin's formula Feynman–Kac formula Filtration Girsanov theorem Infinitesimal generator Itô integral Itô's lemma Kolmogorov continuity theorem Kolmogorov extension theorem Lévy–Prokhorov metric Malliavin calculus Martingale representation theorem Optional stopping theorem Prokhorov's theorem Quadratic variation Reflection principle Skorokhod integral Skorokhod's representation theorem Skorokhod space Snell envelope Stochastic differential equation Tanaka Stopping time Stratonovich integral Uniform integrability Usual hypotheses Wiener space Classical Abstract

Disciplines	Actuarial mathematics Econometrics Ergodic theory Extreme value theory (EVT) Large deviations theory Mathematical finance Mathematical statistics Probability theory Queueing theory Renewal theory Ruin theory Statistics Stochastic analysis Time series analysis Machine learning

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