nLab stochastic process

Idea

A stochastic process is a function from a linearly ordered set $T$ (“time”, typically the integers or positive real axis) into a set of random variables (on fixed probability space), that is a collection $(X_t)_{t\in T}$ of random variables.

By a random process physicists usually mean the same, but mathematicians sometimes allow random processes indexed by more general sets, not usually with meaning of time or equipped with a linear order; maybe “random collection” would be a better term in that case.

The most studied examples include Brownian motion, Ornstein-Uhlenbeck process? and Lévy process?es.

References

• Wikipedia stochastic process

With an eye towards quantum noise;

• Stéphane Attal, Stochastic Processes, Lecture 4 in: Lectures on Quantum Noises [pdf, webpage]