Brownian motion as a Gaussian process and a martingale

Brownian motion can be looked at through two complementary lenses, and each unlocks different tools. Through the first lens it is a Gaussian process: a random function whose value at every collection of times is jointly Normal. Through the second it is a martingale: a perfectly fair game, where your best forecast of the future value is simply the present value. The first lens hands you all the algebra of the Normal distribution; the second hands you the powerful theory of fair games (optional stopping, maximal inequalities, convergence). Brownian motion is special precisely because it is BOTH at once.

As a Gaussian process, B is fully described by its mean function and its covariance function — nothing else is needed, because a joint Normal is determined by means and covariances. The mean is zero everywhere, E[B(t)] = 0, and the covariance has a memorable form: Cov(B(s), B(t)) = min(s, t), the smaller of the two times. Why the minimum? Write B(t) = B(s) + (B(t) - B(s)) for s < t; the increment B(t) - B(s) is independent of (and uncorrelated with) the past value B(s), so Cov(B(s), B(t)) = Var(B(s)) = s = min(s, t). That single formula encodes the whole correlation structure: nearby times are strongly correlated, far-apart times only weakly. As a martingale, B satisfies E[B(t) given the history up to time s] = B(s): given everything observed through time s, the expected future value is exactly the current value — no predictable drift up or down, the hallmark of a fair game.

These twin identities are why Brownian motion is computationally tractable despite its monstrous paths. The Gaussian view lets you compute any finite-dimensional probability with Normal integrals and gives clean formulas for things like the covariance. The martingale view lets you import optional stopping (to get hitting probabilities), Doob's maximal inequality (to bound the running maximum), and forms the backdrop for the Ito integral, which is built precisely to preserve the martingale property. The caveat to keep straight: being a martingale is about conditional MEANS (no predictable trend); being Gaussian is about the SHAPE of the joint distribution. They are independent facts that happen to coincide here, and most processes are neither.