Stochastically

Stochastically is an adverb describing an action or process that involves randomness or is governed by a stochastic process. In mathematics, probability theory and statistics, a stochastic process is a family of random variables indexed by time or another parameter. Saying something happens stochastically means its behavior is described by probability distributions rather than a fixed deterministic rule.

This contrasts with deterministic descriptions, where given the same initial conditions the outcome is fixed. In practice, many natural and engineered systems are modeled stochastically because randomness is inherent or because uncertainty is treated explicitly. Examples include particle diffusion described by Brownian motion, stock prices modeled by geometric Brownian motion, and population dynamics with random birth and death events. Statements like "the system evolves stochastically" typically imply that predictions are probabilistic, expressed through expected values, variances, and distributional properties.

Methods and contexts: Stochastic processes are studied via Markov chains, stochastic differential equations, and stochastic calculus. Computational approaches often rely on Monte Carlo simulation, sampling trajectories stochastically to estimate quantities of interest. In statistics, inference may be carried out on stochastically generated data, and in computer science, randomized algorithms operate stochastically. The term is widely used across physics, finance, biology, and engineering.