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STATE 04 · DYNAMICS

Brownian motion and the Wiener process

A continuous-time path built from Gaussian increments.

The Wiener process

Brownian motion, also called the Wiener process, is a continuous-time process with three defining features. It starts at zero. It has independent increments, so that the movements over disjoint intervals are independent of one another. And the increment over an interval of length t is Gaussian with mean zero and variance equal to t. These three features fix the law of the process completely.

The increments are stationary as well as independent: the law of an increment depends only on the length of the interval, not on when the interval begins. The process therefore carries no preferred origin in time, though it is not stationary in the strict sense, because its variance grows with the elapsed time from the start.

Gaussian increments

Because every increment is Gaussian, every finite collection of values of the process has a joint Gaussian law. The covariance between the values at two times equals the smaller of the two times. This compact covariance structure is the analytic heart of the process, and it lets every question about finite-dimensional behaviour be answered by the machinery of the bivariate and multivariate Gaussian laws.

The increment over a short interval has variance proportional to the interval length. This square root scaling of the typical displacement is the signature of diffusion. Doubling the elapsed time does not double the distance travelled but multiplies it by the square root of two.

time position
Figure 1. A continuous but nowhere smooth path, with the square-root scale of its spread shown as the dashed envelope.
Two wave forms shown together on an analogue display
Figure 2. Two wave forms shown together; the comparison of shapes is the beginning of spectral thinking.

Properties of the paths

The paths of the Wiener process are continuous: there are no jumps. Yet they are nowhere differentiable, so the path has no well-defined instantaneous speed at any point. It wiggles at every scale, and a magnified portion looks statistically like the whole. This self-similarity is exact, and it is the reason the process serves as the prototype of a random continuous curve.

The path also has infinite variation over any interval, even though it has finite quadratic variation. The distinction between these two kinds of variation is technical, but it is the reason the ordinary rules of integration do not apply to functions of the process without modification.

Why the process matters

The Wiener process is the continuous-time limit of the random walk, obtained when the step size shrinks and the step count grows so that the accumulated variance stays finite. It is the natural model for the position of a particle buffeted by countless small shocks, and it is the starting point for a large family of more elaborate diffusions.

Because its increments are Gaussian, the process sits at the meeting point of two themes of this reference: the Gaussian law of State 02 and the dynamics of State 04. A reader who has followed the earlier pages will recognise the covariance structure as a two-by-two Gaussian table in disguise, and the independence of increments as the Markov property in continuous time. With the dynamics in place, the reference turns to the estimation of unknown quantities in estimation theory.