Stochastic Process

Stochastic process is a mathematical model of a system that evolves over time in a manner that is not fully determined by its initial conditions, but involves randomness at each step. Formally, it is a collection of random variables indexed by time — a probability distribution over trajectories rather than a single deterministic path. The random walk is the simplest example; more complex instances include Brownian motion, Poisson processes, and Markov chains.

The significance of stochastic processes extends beyond their use as noise models. They are the mathematical language for systems where microscopic unpredictability produces macroscopic structure that cannot be captured by deterministic averages. In physics, they describe diffusion and fluctuation phenomena. In biology, they model genetic drift and mutation accumulation. In finance, they underlie the pricing of derivatives and risk management.

The theory connects to ergodic theory — the study of whether time averages equal ensemble averages — and to the foundations of statistical mechanics. A stochastic process is ergodic if a single sufficiently long realization reveals the statistical properties of the entire process. Not all processes are ergodic, and the distinction matters: non-ergodic systems cannot be understood by observing one trajectory, no matter how long.