Stochastic dynamical systems are dynamical systems subjected to the effect of noise. The randomness brought by the noise takes into account the variability observed in real-world phenomena. For example, the evolution of a share price typically exhibits long-term behaviors along with faster, smaller-amplitude oscillations, reflecting day-to-day or hour-to-hour variations.

Applications of stochastic systems to data science include methods for statistical inference (such as Markov chain Monte Carlo) and stochastic modeling for time series or geospatial data.

Stochastic discrete-time systems include discrete-time Markov chains. The Markov property means that the state of a system at time n+1 only depends on its state at time n. Stochastic cellular automata, which are stochastic extensions of cellular automata, are particular Markov chains.

As far as continuous-time systems are concerned, Ordinary Differential Equations with noise yield Stochastic Differential Equations (SDEs). Partial...