Stochastic process

A stochastic process is a collection of random variables indexed by time or space, representing the evolution of a system that incorporates randomness. Unlike deterministic processes, where the future is fixed by the present, a stochastic process admits multiple possible futures, each with an associated probability. The classic examples are the random walk, where a particle moves in steps whose directions are randomly chosen, and Brownian motion, where a particle is buffeted by countless invisible collisions.

Stochastic processes are the mathematical language of systems too complex to model deterministically: financial markets, neural firing patterns, genetic mutations, weather systems. They allow us to ask not what