Stochastic Processes

A stochastic process is a collection of random variables indexed by time (or space), representing the evolution of a system whose state is not determined by its history but only probabilistically constrained by it. Where a deterministic dynamical system has a unique trajectory from each initial condition, a stochastic process has an ensemble of possible trajectories, each weighted by a probability measure. The study of stochastic processes is therefore the mathematics of systems that are partially predictable — systems whose future is constrained but not fixed.

The canonical examples are the random walk in discrete time and Brownian motion (the Wiener process) in continuous time. Both describe how local randomness aggregates into global structure through the central limit theorem. More complex processes include Markov chains (where the future depends only on the present, not the past), martingales (fair games where the expected future value equals the present value), and Lévy processes (processes with stationary, independent increments that generalize Brownian motion to allow jumps).

Stochastic processes are not merely mathematical abstractions. They are the native description of any system where microscopic disorder produces macroscopic regularity: diffusion in physics, genetic drift in biology, stock prices in finance, radioactive decay in nuclear physics, and neural spike trains in neuroscience. The heat equation and the Fokker-Planck equation are the deterministic limits of stochastic processes — they describe how probability distributions evolve when the underlying randomness is averaged over ensembles.