Brownian Motion

Brownian motion is the continuous-time limit of random walk, the canonical continuous-path process, and the first place where the existence questions of extension-theorem.md become unavoidable rather than pedantic. Its finite-dimensional distributions are prescribed in one line; that a process with continuous paths realizes them is a theorem, and that those paths are nowhere differentiable is another.

This page is about existence and path structure. Stochastic calculus — the Itô integral, SDEs, Feynman–Kac — is stated in §5 and deliberately taken no further (README §Scope).

References: Mörters–Peres, Brownian Motion; Karatzas–Shreve, Brownian Motion and Stochastic Calculus, ch. 1–2; Durrett, Probability, ch. 7.

1. Definition

Definition. A standard Brownian motion (Wiener process) is a process $(B_t{)}_{t\ge 0}$ with

1. $B_0=0$ a.s.;
2. independent increments: for $t_0<\cdots <t_n$ the increments $B_{t_i}-B_{t_{i-1}}$ are independent;
3. stationary Gaussian increments: $B_t-B_s\sim N(0,t-s)$ for $s<t$;
4. continuous paths: $t\mapsto B_t(\omega )$ is continuous for a.e. $\omega$.

Conditions (1)–(3) fix the finite-dimensional distributions: $(B_{t_1},\dots ,B_{t_n})$ is a centred Gaussian vector with covariance

$$\mathrm{Cov}(B_s,B_t)=\min (s,t),$$

and these are consistent in the sense of extension-theorem.md §1 (marginalizing a Gaussian gives a Gaussian with the restricted covariance). Condition (4) is of a different kind: as §4 of that page shows, it is not expressible in the product $\sigma$-algebra that Kolmogorov's theorem supplies. It must be obtained by a separate argument.

The Gaussian shape is not an arbitrary choice: it is forced by requiring stationary independent increments with finite variance and continuous paths, since $B_t$ is then a sum of $n$ i.i.d. increments for every $n$ and the CLT applies. Brownian motion is the CLT made into a process.

2. Existence

Three standard constructions, each illuminating a different aspect.

(a) Kolmogorov extension + continuous version. Apply extension-theorem.md §3 to the Gaussian fdds, then repair the paths with the Kolmogorov–Chentsov criterion: since $B_t-B_s\sim N(0,t-s)$,

$$E[|B_t-B_s{|}^4]=3|t-s{|}^2,$$

so with $\alpha =4$, $\beta =1$ there is a version with locally Hölder paths of every exponent $\gamma <1/4$. Iterating with higher moments ($\alpha =2k$) pushes this to every $\gamma <1/2$.

(b) Lévy–Ciesielski. Expand in the Haar/Schauder basis: with $({\xi }_k)$ i.i.d. $N(0,1)$ and $({\Delta }_k)$ the Schauder functions, $B_t={\sum }_k{\xi }_k{\Delta }_k(t)$ converges uniformly a.s. on $[0,1]$ (Borel–Cantelli plus a Gaussian tail bound), giving continuity by construction. This is the most explicit route and makes the process a random element of $C[0,1]$ from the start.

(c) Donsker's invariance principle. Let $(X_i)$ be i.i.d. with mean $0$, variance $1$, $S_n={\sum }_{i\le n}{X}_i$, and linearly interpolate the rescaled walk $W_t^{(n)}=S_{\lfloor nt\rfloor }/\sqrt{n}$. Then $W^{(n)}\Rightarrow B$ weakly in $C[0,1]$. Existence follows from tightness plus Prokhorov (convergence-modes.md §4), and the theorem simultaneously explains the universality: like the CLT, the limit does not depend on the step distribution.

Wiener measure. Whichever construction is used, the object produced is a probability measure $\mathbb{W}$ on $(C[0,\infty ),\mathcal{B})$ — the Borel $\sigma$-algebra of the topology of uniform convergence on compacts, which coincides with the $\sigma$-algebra generated by the coordinate maps. On this space "the path is continuous" is not an event of probability one; it is the whole space. Choosing the carrier space well makes the §1(4) problem disappear.

3. Path Properties

Almost every Brownian path is far stranger than the smooth pictures suggest.

Theorem (nowhere differentiability, Paley–Wiener–Zygmund). Almost surely, $t\mapsto B_t$ is differentiable at no point.

Idea. If $B$ were differentiable at some $t$, the increments near $t$ would be $O(h)$; a union bound over a dyadic grid, using the Gaussian scaling $B_{t+h}-B_t\sim \sqrt{h}\cdot N(0,1)$, makes the probability of that happening anywhere zero. $\square$

The exact modulus of continuity is $\sqrt{2h\log (1/h)}$ (Lévy), so paths are Hölder of every exponent $<1/2$ and of none $\ge 1/2$ — the exponent $1/2$ is the scaling $B_{ct}\stackrel{d}{=}\sqrt{c}{B}_t$ read off the covariance. Further consequences:

• Unbounded variation on every interval, so path integrals $\int fdB$ cannot be defined pathwise as Stieltjes integrals. This is the reason stochastic calculus needs a new construction (§5).
• Quadratic variation ${\sum }_i(B_{t_{i+1}}-B_{t_i}{)}^2\to t$ in probability as the mesh $\to 0$. The heuristic $(dB{)}^2=dt$ is this theorem, and it is the source of the extra term in Itô's formula.
• Recurrence and level sets. $B$ hits every level a.s.; its zero set is a closed Lebesgue-null set of Hausdorff dimension $1/2$, homeomorphic to the Cantor set. The graph has dimension $3/2$.
• Martingale structure. $B_t$, $B_t^2-t$, and $\exp (\theta B_t-{\theta }^{2}t/2)$ are martingales (martingales.md); optional stopping applied to them computes exit probabilities and expected exit times in a line or two.
• Markov and strong Markov. $(B_{t+s}-B_t{)}_{s\ge 0}$ is a Brownian motion independent of ${\mathcal{F}}_t$, and this remains true at stopping times — the strong Markov property, which is what makes the reflection principle $P({\sup }_{s\le t}{B}_s\ge a)=2P(B_t\ge a)$ work.

Two independent smallness notions again. The zero set is null yet uncountable and perfect; the path is continuous yet nowhere differentiable. Both are instances of the dissociation between measure and topology first met with the Cantor set in measure-theory.md §6.

4. The Gaussian Family

Brownian motion is the simplest member of a family defined by covariance alone: $(X_t)$ is a Gaussian process if every finite-dimensional vector is Gaussian, and it is then determined by its mean and covariance functions. Kolmogorov's theorem applies to any positive-semidefinite covariance kernel, and Kolmogorov–Chentsov decides continuity. Brownian bridge ($\min (s,t)-st$), fractional Brownian motion, and the Ornstein–Uhlenbeck process are the standard variants; in QFT the free-field Euclidean measure is a Gaussian process of this kind, with covariance the propagator (QFT path integral).

5. Stochastic Calculus (Stated Only)

Because paths have unbounded variation, ${\int }_0^t{H}_{s}d{B}_s$ cannot be a pathwise Stieltjes integral. Itô's construction defines it as an $L^2$ limit of sums ${\sum }_i{H}_{t_i}(B_{t_{i+1}}-B_{t_i})$ over predictable integrands, evaluating $H$ at the left endpoint so that the result is a martingale. The choice of endpoint is not a convention with no consequences: the midpoint rule gives the Stratonovich integral, which obeys the ordinary chain rule but is not a martingale. The Itô rule

$$df(B_t)=f^{\prime }(B_t)d{B}_t+\frac{1}{2}{f}^{\prime \prime }(B_t)dt$$

carries the correction term $(dB{)}^2=dt$ from §3.

This is as far as this folder goes. SDEs, the Feynman–Kac formula, and the connection to the QM path integral via Wick rotation are a separate subject.

6. Where This Is Used

• Non-standard probability — the hyperfinite random walk gives Brownian motion directly via Loeb measure, replacing §2's limit arguments with an infinitesimal-step walk.
• The Kolmogorov extension theorem — §1(4) is the motivating example of what fdds cannot determine.
• Martingales — the continuous-time theory.
• The physics bridge — diffusion, the heat equation, and Euclidean field theory.