Brownian paths are continuous but live exactly at the boundary where ordinary first-order calculus fails. Their oscillations have order \(\sqrt h\), refined by a logarithmic factor; total variation is infinite; deterministic-partition quadratic variation is time. These facts explain both the correction term in Ito's formula and the geometry of diffusion paths.
H?lder continuity below one half
Definition. A function \(f\) is H?lder continuous of order \(\alpha\in(0,1]\) on \([0,T]\) if \[ |f(t)-f(s)|\le C|t-s|^\alpha \] for all \(s,t\), for some finite \(C\). Local H?lder continuity means this holds on every compact time interval.
Theorem. For every \(\alpha<1/2\), Brownian paths are locally H?lder continuous of order \(\alpha\) almost surely.
Proof. For \(p>2\), \[ \mathbb E|B_t-B_s|^p =C_p|t-s|^{p/2} =C_p|t-s|^{1+(p/2-1)}. \] Kolmogorov-Chentsov gives H?lder continuity of every order \[ \gamma<\frac{p/2-1}{p}=\frac12-\frac1p. \] Given \(\alpha<1/2\), choose \(p>(1/2-\alpha)^{-1}\). Intersecting the resulting probability-one events over rational \(\alpha<1/2\) proves the simultaneous claim.
Sharp modulus of continuity
Theorem (Levy modulus). On every fixed interval \([0,T]\), \[ \limsup_{h\downarrow0} \frac{ \sup_{\substack{0\le s<t\le T\\t-s\le h}} |B_t-B_s|} {\sqrt{2h\log(1/h)}}=1 \quad\text{almost surely}. \]
Proof outline. For the upper bound, cover \([0,T]\) by dyadic intervals of length \(2^{-n}\). Reflection and Gaussian tail bounds show that the probability any interval oscillates more than \((1+\varepsilon)\sqrt{2\cdot2^{-n}n\log2}\) is summable after a slightly coarser blocking argument. Borel-Cantelli and chaining control all sufficiently small increments. For the lower bound, use a disjoint collection of increments of length \(h\); their normalized values are independent standard Gaussians, whose maximum is asymptotic to \(\sqrt{2\log(1/h)}\). A second Borel-Cantelli argument completes the proof.
Corollary. Brownian paths are almost surely not H?lder continuous of order \(1/2\) on any interval.
Proof. The ratio of the modulus to \(\sqrt h\) grows like \(\sqrt{2\log(1/h)}\to\infty\), contradicting any finite \(1/2\)-H?lder constant.
Law of the iterated logarithm
Theorem. Almost surely, \[ \limsup_{t\to\infty} \frac{B_t}{\sqrt{2t\log\log t}}=1, \qquad \liminf_{t\to\infty} \frac{B_t}{\sqrt{2t\log\log t}}=-1. \] By time inversion, the analogous local statement is \[ \limsup_{t\downarrow0} \frac{B_t}{\sqrt{2t\log\log(1/t)}}=1 \] with liminf \(-1\).
Proof outline. On a geometric time sequence \(t_n=c^n\), Gaussian tails and Borel-Cantelli provide the upper bound. For the lower bound, use independent increments over well-separated geometric blocks and choose \(c\) large enough that the preceding position is negligible relative to the new block. Let \(c\downarrow1\) and control times between grid points by the reflection principle. Time inversion turns the large-time result into the local result.
Nowhere differentiability
Theorem. With probability one, a Brownian path has no finite derivative at any \(t\ge0\).
Proof outline. If a continuous function is differentiable at some \(t\) with derivative bounded by \(M\), then on all sufficiently small neighboring intervals its increments are bounded by a constant times \(M\) times the interval length. On a dyadic mesh, Brownian increments have size \(2^{-n/2}\), whereas this differentiability bound has size \(2^{-n}\). Independence and Gaussian small-ball estimates show that the probability of three consecutive dyadic increments all satisfying such a linear bound is summable over locations and sufficiently sparse scales. Borel-Cantelli rules out a point with a finite derivative. Taking a countable union over integer \(M\) covers every finite derivative.
Quadratic variation
Theorem. Let \(\pi_n=\{0=t_{n,0}<\cdots<t_{n,k_n}=T\}\) be deterministic partitions with mesh \(|\pi_n|\to0\). Then \[ Q_n(T)=\sum_{j=1}^{k_n} (B_{t_{n,j}}-B_{t_{n,j-1}})^2 \longrightarrow T \quad\text{in }L^2. \] If additionally \(\sum_n|\pi_n|<\infty\), convergence holds almost surely.
Proof. Independent increments give \[ \mathbb EQ_n(T)=\sum_j(t_{n,j}-t_{n,j-1})=T. \] If \(Z\sim N(0,h)\), then \(\operatorname{Var}(Z^2)=2h^2\). Thus \[ \operatorname{Var}(Q_n(T)) =2\sum_j(t_{n,j}-t_{n,j-1})^2 \le2T|\pi_n|\to0. \] This proves \(L^2\)-convergence. Chebyshev gives \[ P(|Q_n(T)-T|>\varepsilon) \le\frac{2T|\pi_n|}{\varepsilon^2}, \] and summability plus Borel-Cantelli gives the almost-sure assertion.
Corollary. Brownian paths have infinite total variation on every nontrivial interval almost surely.
Proof. For a continuous function of finite total variation \(V\), along any partitions with mesh tending to zero, \[ \sum_j(\Delta f_j)^2 \le\left(\max_j|\Delta f_j|\right) \sum_j|\Delta f_j| \le\left(\max_j|\Delta f_j|\right)V\to0. \] Brownian quadratic variation along dyadic partitions converges instead to \(T>0\), a contradiction.
\(p\)-variation threshold
Proposition. Almost surely, Brownian paths have finite \(p\)-variation on compact intervals for every \(p>2\), and infinite \(p\)-variation for \(1\le p\le2\).
Proof outline. For \(p>2\), choose \(\alpha\in(1/p,1/2)\). H?lder continuity gives \[ \sum_j|B_{t_j}-B_{t_{j-1}}|^p \le C^p\sum_j|t_j-t_{j-1}|^{\alpha p} \le C^pT^{\alpha p}, \] uniformly over partitions. At \(p=2\), adaptive refinements together with Brownian oscillations make the supremum over all partitions infinite even though deterministic refining partitions have quadratic-variation limit \(T\). For \(p<2\), the divergence follows from the \(p=2\) behavior and small-increment comparison.
Quadratic variation is therefore not ordinary \(2\)-variation as a supremum over partitions. It is a probabilistic limit along refining partitions, and it is exactly the second-order quantity retained by stochastic calculus.