Satunnaiskulku

Satunnaiskulku is a Finnish term that translates to "random walk" in English. It describes a mathematical object that follows a path consisting of a succession of random steps. In its simplest form, a one-dimensional random walk might involve a particle moving left or right with equal probability at each step.

The concept of a random walk has applications in various fields. In physics, it is used to model phenomena such as Brownian motion, the erratic movement of particles suspended in a fluid. In finance, random walks can be used to model the behavior of stock prices, assuming that price changes are unpredictable. Computer science also utilizes random walks for algorithms such as Markov chain Monte Carlo methods.

The mathematical properties of random walks are extensively studied. Key questions include the probability of returning to the origin, the expected distance from the origin after a certain number of steps, and the limiting distributions. These properties can differ significantly depending on the dimension of the space in which the walk occurs and the probabilities associated with each step. For instance, a random walk in one or two dimensions is recurrent, meaning it is guaranteed to return to its starting point infinitely often, while in three or more dimensions, it is transient, with a finite expected number of returns to the origin.