Stochasticitu

Stochasticitu, commonly called stochasticity, is the property of systems or phenomena that involve randomness. It is used across mathematics, physics, biology, economics, and engineering to describe outcomes that cannot be predicted with certainty and are modeled with probability distributions.

In formal terms, a stochastic process is a family of random variables indexed by time or space, representing the evolution of a quantity of interest. Key concepts include independence, stationarity, and the memoryless property (as in Markov processes). Gaussian processes and Poisson processes are common examples; Brownian motion (Wiener process) models continuous-time random movement with normally distributed increments.

Stochasticity is distinct from determinism. Deterministic models yield a single outcome for given initial conditions; stochastic models acknowledge uncertainty and yield distributions over possible outcomes, often described by expectation, variance, and higher moments.

Types of stochasticity include intrinsic (demographic) noise arising from discrete events at the individual level, and extrinsic (environmental) noise from external fluctuations. In genetics and ecology, stochastic effects can influence population dynamics and evolution.

Methods and applications: stochastic differential equations with Ito or Stratonovich calculus describe continuous-time dynamics; Monte Carlo simulation and Gillespie-type algorithms model complex systems. Stochastic models are used in finance (asset prices), physics (particle diffusion), epidemiology (random transmission), and computer science (randomized algorithms).