Levy's modulus of continuity

How rough, exactly, is a Brownian path? It is continuous but nowhere differentiable, so it has no local slopes; the right way to measure its roughness is its modulus of continuity — how much |B_(t+h) - B_t| can be, uniformly over t, as the gap h shrinks. Levy's theorem gives the EXACT uniform modulus, the precise gauge function against which Brownian roughness is calibrated, sharpening the cruder statement that paths are Holder continuous of every order below 1/2.

Levy's modulus of continuity says: almost surely, limsup_(h -> 0+) [ sup_(0 <= t <= 1 - h) |B_(t+h) - B_t| ] / sqrt(2 h log(1/h)) = 1. So the maximal increment over a window of length h, taken over all positions t in [0,1], is asymptotically sqrt(2 h log(1/h)). The factor log(1/h) (not log log, as in the LIL) appears because we take a supremum over MANY starting points t — there are about 1/h disjoint windows, and the maximum of that many near-independent Gaussian increments inflates the bound. This is the GLOBAL modulus; the LIL gives the LOCAL modulus at a single fixed point, sqrt(2 h log log(1/h)), which is smaller. The gap between the two — there exist (random) exceptional times where increments are larger than the local LIL rate, namely the fast points — is itself a rich topic in path geometry.

Why it matters: the modulus quantifies exactly why Ito calculus needs a second-order correction (increments of size sqrt(h) accumulate quadratic variation), why Brownian paths are not Holder-1/2 but are Holder-alpha for every alpha < 1/2, and why no pathwise ordinary-calculus differentiation is possible. The honest subtlety is the difference between global and local: the worst increment ANYWHERE (Levy, with log) is strictly larger than the worst increment at a PRE-CHOSEN point (LIL, with log log). Confusing the two — or quoting Holder-1/2 as if it held — is a classic error: at the exact exponent 1/2 the path just fails to be Holder, by the logarithmic factor.