Brownian motion and the scale of random change

A standard one-dimensional Brownian motion starts at zero, has continuous paths almost surely, and has independent Gaussian increments. Over a time interval of length h, the increment has mean zero and variance h. Independence refers to increments over disjoint intervals, not to positions at different times. Later positions contain earlier increments, so they are correlated. A Brownian motion relative to a filtration also requires the future increment to be independent of the information available at the earlier endpoint.

For times s≤t, decompose Bₜ as Bₛ plus its independent future increment. This immediately gives covariance s and conditional mean Bₛ. The conditional variance of Bₜ given the past through s is t−s. Marginal normality alone would not imply these properties: a process Xₜ=√t Z built from one standard normal variable has the same one-time distributions but completely different dependence. A process is specified by compatible joint distributions and path behavior, not a histogram at a single time.

If time runs a times faster, increment variances become a times larger. Dividing amplitude by √a restores unit variance per unit time. The rescaled process therefore has the same finite-dimensional distributions and continuous-path law as standard Brownian motion. Equality in distribution does not say that the two constructions produce identical values on each particular realization. This distinction lets us reason about process laws without confusing them with one coupled simulation or one displayed trajectory.

The square-root scale also explains the normalization of symmetric random walks. A sum of n independent centered unit-variance steps has standard deviation √n, so observe it at accelerated time and divide displacement by √n. Under a functional central limit theorem, suitably interpolated paths converge in distribution to Brownian motion. This is a limit theorem with assumptions on the increments and a specified path-space topology. Correlated steps or infinite-variance tails can lead to different limits, so Brownian behavior is not a universal consequence of having many disturbances.

Brownian sample paths are almost surely nowhere differentiable. Over short intervals, increments have size of order √h, which is much larger than h; a putative difference quotient therefore does not settle into an ordinary velocity. More precisely, on uniform partitions of a fixed interval [0,T], the sum of squared increments converges to T in mean square. Along suitable refining sequences one can also obtain almost-sure statements. This nonzero quadratic variation is the source of Itô's second-derivative correction.

A continuous finite-variation path has vanishing sums of squared increments as the partition mesh tends to zero. Brownian quadratic variation therefore rules out finite variation almost surely, on every positive-length interval. A smooth line joining finitely many sampled points hides this distinction: each interpolated path has finite variation, even though the limiting stochastic object does not. A plotted curve can communicate a scale or a realization, but neither visual roughness nor a dense display proves the required limiting property.