02.15.01 · analysis / stochastic-analysis

Brownian motion and the Wiener process

shipped3 tiersLean: none

Anchor (Master): Revuz-Yor, Continuous Martingales and Brownian Motion (Springer Grundlehren 293, 3rd ed. 1999), Ch. I-IV; Karatzas-Shreve §§2.6-3.3 (quadratic variation, Lévy's characterisation, strong Markov, reflection, Wiener measure)

Intuition Beginner

Imagine a speck of pollen floating on still water, jittering this way and that without end. Each tiny push from an unseen water molecule nudges it a little, and the nudges keep coming from every direction. The speck never settles; it wanders. Brownian motion is the mathematical idealisation of that wandering — a path that is continuous, so it never jumps, yet so jagged that it has no sensible direction of travel at any instant.

The wandering has two defining habits. First, where it goes next does not depend on where it has been: the future push is fresh, unrelated to the past. Second, the size of a wiggle depends only on how much time passes, not on the clock reading — a one-second step looks the same whether taken now or an hour from now. These two habits, together with the demand that the path stay unbroken, pin down the object almost completely.

A surprising consequence is scale. Over a span of time of length $t$ , the typical distance the speck has drifted grows like the square root of $t$ , not like $t$ itself. Doubling the time does not double the spread; it multiplies it by about $1.4$ . This square-root law is the fingerprint of diffusion, and it recurs everywhere a quantity is buffeted by many small independent shocks.

The path is also self-similar: zoom in on any short stretch, rescale, and it looks like the whole again. There is no smallest wiggle to find.

Visual Beginner

A single jagged curve runs left to right across the frame, plotting position against time. It rises and falls erratically, never flat for long, never with a clean slope you could read off. Around it, a pale shaded band spreads outward like a sideways trumpet: narrow at the left where time is zero and the position is known, flaring as time grows. The band's edges trace two curves shaped like the square root of time, marking where the path typically stays.

The picture carries two lessons at once. The jaggedness of the single path is genuine: no matter how far you zoom in, the curve keeps wiggling, with no straight piece anywhere. The widening band is a statement about many paths at once — start the speck again and again, and the cloud of where it ends up after time $t$ is a bell curve whose width grows like the square root of $t$ . One picture, one path, but it stands in for a whole spreading family.

Worked example Beginner

Suppose our wanderer starts at position $0$ at time $0$ , and we ask about its position at time $4$ . Call that position $B_{4}$ . The rules say $B_{4}$ is drawn from a bell-shaped (normal) law centred at $0$ , with a spread measured by the square root of the elapsed time. Here the elapsed time is $4$ , so the spread is $4 = 2$ .

A bell curve centred at $0$ with spread $2$ puts about two-thirds of its weight within one spread of the centre. So roughly two times out of three, the wanderer at time $4$ sits somewhere between $- 2$ and $+ 2$ . It strays beyond $\pm 4$ (two spreads) only about one time in twenty.

Now compare two stretches: from time $0$ to time $1$ , and from time $1$ to time $3$ . The first lasts one unit, the second lasts two units. The displacement over the first has spread $1 = 1$ ; the displacement over the second has spread $2 \approx 1.41$ . The longer stretch produces a wider wiggle, but only by the square-root factor, not by doubling. And because the two stretches do not overlap in time, their displacements are independent: knowing the first tells you nothing about the second.

If we wanted the total displacement from time $0$ to time $3$ , we would not add the spreads. We add the elapsed times — $1 + 2 = 3$ — and take the square root: spread $3 \approx 1.73$ . Variances add along disjoint time intervals; spreads do not.

Check your understanding Beginner

Exercise. A Brownian wanderer starts at $0$. Over the time interval from $0$ to $9$, what is the typical spread (one standard deviation) of its final position? If instead you watched two separate, non-overlapping stretches each lasting $4.5$ units, how does the spread of each compare to the spread over the full $9$?

Solution

Over $0$ to $9$ the elapsed time is $9$ , so the spread is $9 = 3$ .

Each half-stretch lasts $4.5$ units, so each has spread $4.5 \approx 2.12$ . The two displacements are independent because the stretches do not overlap, and their variances add: $4.5 + 4.5 = 9$ , recovering the total variance $9$ and total spread $3$ . The lesson: variances (squared spreads) add along disjoint intervals, while the spreads themselves combine by the square-root rule. This additivity of variance over disjoint time is exactly the property that will become the identity $⟨ B ⟩_{t} = t$ at the higher tiers.

Formal definition Intermediate+

Fix a probability space $(Ω, F, P)$ . A filtration is an increasing family $(F_{t})_{t \geq 0}$ of sub- $σ$ -algebras of $F$ , meaning $F_{s} \subseteq F_{t}$ whenever $s \leq t$ . We read $F_{t}$ as the information available by time $t$ . The filtration satisfies the usual conditions when it is right-continuous, $F_{t} = ⋂_{u > t} F_{u}$ , and $F_{0}$ contains every $P$ -null set.

For an integrable random variable $X \in L^{1} (Ω, F, P)$ and a sub- $σ$ -algebra $G \subseteq F$ , the conditional expectation $E [X ∣ G]$ is the unique (up to null sets) $G$ -measurable integrable random variable $Y$ with $$ \int_G Y , d\mathbb{P} ;=; \int_G X , d\mathbb{P} \qquad \text{for every } G \in \mathcal{G}. $$ Existence and uniqueness follow from the Radon-Nikodym theorem 02.07.08 applied to the signed measure $G \mapsto \int_{G} X d P$ , which is absolutely continuous with respect to the restriction of $P$ to $G$ ; the density is $Y$ .

An adapted, integrable process $(M_{t})_{t \geq 0}$ — adapted meaning each $M_{t}$ is $F_{t}$ -measurable — is a martingale if $E [M_{t} ∣ F_{s}] = M_{s}$ for all $s \leq t$ . It is a submartingale if $\geq$ holds and a supermartingale if $\leq$ holds.

A real-valued process $(B_{t})_{t \geq 0}$ on $(Ω, F, P)$ is a standard Brownian motion (or Wiener process) if

$B_{0} = 0$ almost surely;
for $0 \leq s \leq t$ the increment $B_{t} - B_{s}$ has the centred Gaussian law $N (0, t - s)$ ;
for $0 \leq t_{0} < t_{1} < \dots < t_{n}$ the increments $B_{t_{1}} - B_{t_{0}}, \dots, B_{t_{n}} - B_{t_{n - 1}}$ are independent;
the map $t \mapsto B_{t} (ω)$ is continuous for almost every $ω$ .

Condition (2) makes the increments stationary (their law depends only on $t - s$ ) and Gaussian; (3) makes them independent. Relative to its own filtration $F_{t}^{B} = σ (B_{s} : s \leq t)$ , a Brownian motion is a martingale, since $E [B_{t} - B_{s} ∣ F_{s}^{B}] = E [B_{t} - B_{s}] = 0$ by independence of the increment from the past.

Key theorem with proof Intermediate+

We record the optional-sampling theorem and the existence of Brownian motion, the two structural pillars of everything that follows.

Theorem (Optional sampling, bounded case). Let $(M_{t})$ be a right-continuous martingale and let $σ \leq τ$ be bounded stopping times for the filtration $(F_{t})$ , that is, ${τ \leq t} \in F_{t}$ for all $t$ and $τ \leq K$ for some constant $K$ . Then $M_{σ}, M_{τ} \in L^{1}$ and $$ \mathbb{E}[M_\tau \mid \mathcal{F}\sigma] = M\sigma, \qquad \text{in particular } \mathbb{E}[M_\tau] = \mathbb{E}[M_0]. $$

Proof. First take dyadic discretisations $τ_{n} = 2^{- n} ⌈ 2^{n} τ ⌉ \land K$ and likewise $σ_{n}$ , decreasing to $τ, σ$ . For the discrete martingale obtained by sampling $M$ on the grid ${k 2^{- n}}$ , Doob's discrete optional-sampling theorem gives $E [M_{τ_{n}} ∣ F_{σ_{n}}] = M_{σ_{n}}$ ; this is proved by summing the defining martingale identity over the finitely many grid points and using that ${σ_{n} = k 2^{- n}} \in F_{k 2^{- n}}$ . The family ${M_{τ_{n}}}_{n}$ is uniformly integrable because each $M_{τ_{n}} = E [M_{K} ∣ F_{τ_{n}}]$ is a conditional expectation of the single integrable variable $M_{K}$ , and such a family is uniformly integrable. Right-continuity of $M$ gives $M_{τ_{n}} \to M_{τ}$ and $M_{σ_{n}} \to M_{σ}$ almost surely; uniform integrability upgrades this to $L^{1}$ convergence, and the conditioning identity passes to the limit. $■$

Theorem (Existence; Lévy-Ciesielski). A standard Brownian motion exists on a suitable probability space.

Proof sketch. On $[0, 1]$ , let $(h_{k})$ be the Haar system and $(s_{k})$ the Schauder functions obtained by integrating them, $s_{k} (t) = \int_{0}^{t} h_{k}$ . Draw independent standard Gaussians $(ξ_{k})$ and set $B_{t} = \sum_{k} ξ_{k} s_{k} (t)$ . Each Schauder function is a tent supported on a dyadic subinterval; at generation $n$ the tents have height of order $2^{- n /2}$ , and a Borel-Cantelli estimate on $max_{k} ∣ ξ_{k} ∣$ over the $2^{n}$ indices of generation $n$ shows the series converges uniformly almost surely. The uniform limit of continuous functions is continuous, giving property (4). Because the Schauder functions are obtained from an orthonormal basis of $L^{2} [0, 1]$ , the resulting Gaussian process has covariance $E [B_{s} B_{t}] = \sum_{k} s_{k} (s) s_{k} (t) = ⟨ 1_{[0, s]}, 1_{[0, t]} ⟩_{L^{2}} = s \land t$ , which is exactly the covariance forcing properties (1)-(3). Concatenating independent copies extends the construction from $[0, 1]$ to $[0, \infty)$ ; alternatively, the Kolmogorov extension theorem builds the finite-dimensional Gaussian laws and the Kolmogorov continuity theorem supplies the continuous modification. $■$

Bridge. These two theorems are the foundational reason the rest of stochastic analysis is possible. The optional-sampling identity is exactly the statement that a fair game stays fair when you stop it at a strategy that cannot see the future, and it builds toward the martingale machinery that defines the Itô integral 02.15.02. The Lévy-Ciesielski series generalises the finite-dimensional Gaussian vectors of 26.02.01 to an infinite-dimensional object living on the path space, and the covariance $s \land t$ is dual to the variance-additivity the Beginner tier saw as the square-root law. Putting these together, the central insight is that Brownian motion is simultaneously a Gaussian process and a martingale, and its existence as an a.s.-continuous object is what makes the quadratic-variation identity below meaningful; this same continuous-path object appears again in Lévy's characterisation, the strong Markov property, and the Wiener measure.

Exercises Intermediate+

Exercise 3. Let $\tau = \inf\{t : B_t = a\}$ for $a>0$ be the first hitting time of level $a$. Using optional sampling on the martingale $B_t$ (with a truncation argument) and the continuity of paths, argue heuristically that $\mathbb{E}[B_{\tau\wedge n}] = 0$, and discuss why $\mathbb{E}[B_\tau]=a \ne 0$ does not contradict the martingale property.

Solution

For each fixed $n$ , $τ \land n$ is a bounded stopping time, so optional sampling gives $E [B_{τ \land n}] = E [B_{0}] = 0$ . As $n \to \infty$ , $B_{τ \land n} \to B_{τ} = a$ almost surely on ${τ < \infty}$ (and one shows $τ < \infty$ a.s.), yet the expectations do not pass to the limit. The family ${B_{τ \land n}}$ is not uniformly integrable: before hitting $a$ , the path can dip far negative, so $E ∣ B_{τ \land n} ∣$ does not stay bounded in the way uniform integrability requires. The hypothesis $τ \leq K$ in the optional-sampling theorem cannot be dropped for free; unbounded stopping needs a uniform-integrability or $L^{2}$ -boundedness side condition.

Advanced results Master

We now state the deep sample-path and structural theorems. Throughout, $(B_{t})$ is a standard Brownian motion on a probability space carrying its augmented natural filtration.

Quadratic variation. For a partition $Π = {0 = t_{0} < \dots < t_{n} = t}$ with mesh $∥Π∥ = max_{i} (t_{i} - t_{i - 1})$ , set $V_{Π} = \sum_{i} (B_{t_{i}} - B_{t_{i - 1}})^{2}$ . Then $V_{Π} \to t$ in $L^{2}$ as $∥Π∥ \to 0$ , and along a refining sequence of partitions the convergence is almost sure. One writes $⟨ B ⟩_{t} = [B]_{t} = t$ . The computation is short: $E [V_{Π}] = \sum_{i} (t_{i} - t_{i - 1}) = t$ , and the variance of $V_{Π}$ is $\sum_{i} Var ((B_{t_{i}} - B_{t_{i - 1}})^{2}) = 2 \sum_{i} (t_{i} - t_{i - 1})^{2} \leq 2∥Π∥ t \to 0$ , using that a $N (0, σ^{2})$ variable has fourth moment $3 σ^{4}$ .

Nowhere differentiability. Almost surely the path $t \mapsto B_{t}$ is nowhere differentiable; in fact it is nowhere even locally Hölder of any exponent $α > 1/2$ , while it is locally Hölder of every exponent $α < 1/2$ . The non-differentiability is the path-space shadow of the quadratic variation being a strictly increasing function of $t$ rather than zero, which is the situation for any function of bounded variation.

Law of the iterated logarithm. Almost surely, $$ \limsup_{t\downarrow 0} \frac{B_t}{\sqrt{2t\log\log(1/t)}} = 1, \qquad \liminf_{t\downarrow 0}\frac{B_t}{\sqrt{2t\log\log(1/t)}} = -1, $$ with the analogous statement at $t \to \infty$ obtained by time inversion $t \mapsto t B_{1/ t}$ . This sharpens the square-root scaling: the true envelope of the path is $2 t lo g lo g (1/ t)$ , fractionally larger than $t$ .

Lévy's characterisation. A continuous local martingale $(X_{t})$ with $X_{0} = 0$ and quadratic variation $⟨ X ⟩_{t} = t$ is a standard Brownian motion. This converts a list of distributional axioms into two purely martingale-theoretic conditions.

Strong Markov property and reflection. If $τ$ is a finite stopping time, then $B_{s} := B_{τ + s} - B_{τ}$ is a Brownian motion independent of $F_{τ}$ . Applying this at the hitting time of a level $a$ yields the reflection principle: for $a > 0$ , $$ \mathbb{P}\Big(\max_{0\le s\le t} B_s \ge a\Big) = 2,\mathbb{P}(B_t \ge a) = \mathbb{P}(|B_t| \ge a). $$

Synthesis. Putting these together, the central insight is that all five results are facets of one object whose defining tension is to be simultaneously Gaussian, continuous, and a martingale. The quadratic-variation identity $⟨ B ⟩_{t} = t$ is the foundational reason the path cannot be differentiable: a differentiable path would have zero quadratic variation, so the very roughness measured by $[B]_{t} = t$ is exactly what the nowhere-differentiability and the law of the iterated logarithm describe from the path side. Lévy's characterisation is dual to the construction: existence builds Brownian motion from Gaussian axioms, while Lévy's theorem recovers those axioms from the two intrinsic facts that $X$ is a continuous martingale and $⟨ X ⟩_{t} = t$ — this is exactly the rigidity that makes the Itô integral well posed. The strong Markov property generalises the plain Markov property to random times and is the bridge to the reflection principle, whose hitting-time identity appears again in barrier-option pricing and in the arcsine laws. This pattern recurs throughout the subject: a distributional fact, a martingale fact, and a path fact are three readings of the same structure.

Full proof set Master

Proposition (Lévy's characterisation). Let $(X_{t})_{t \geq 0}$ be a continuous local martingale on a filtered probability space satisfying the usual conditions, with $X_{0} = 0$ and quadratic variation $⟨ X ⟩_{t} = t$ for all $t$ . Then $(X_{t})$ is a standard Brownian motion.

Proof. We use the complex exponential martingale and the (already available) Itô formula for continuous local martingales; the argument fixes the finite-dimensional distributions, which together with the assumed continuity yield Brownian motion. Fix $ξ \in R$ and define $$ Z_t = \exp!\Big(i\xi X_t + \tfrac{1}{2}\xi^2 \langle X\rangle_t\Big) = \exp!\Big(i\xi X_t + \tfrac{1}{2}\xi^2 t\Big). $$ Applying Itô's formula to the function $f (x, a) = exp (i ξ x + \frac{1}{2} ξ^{2} a)$ along $X$ and its quadratic variation $⟨ X ⟩_{t} = t$ , the bounded-variation terms cancel exactly: the $\partial_{a} f d t$ term is $\frac{1}{2} ξ^{2} Z_{t} d t$ , while the second-order term $\frac{1}{2} \partial_{xx} f d ⟨ X ⟩_{t}$ is $- \frac{1}{2} ξ^{2} Z_{t} d t$ , leaving only the local-martingale differential $i ξ Z_{t} d X_{t}$ . Hence $Z_{t}$ is a continuous local martingale; since $∣ Z_{t} ∣ = exp (\frac{1}{2} ξ^{2} t)$ is bounded on each $[0, T]$ , it is a genuine martingale there.

The martingale property of $Z$ gives, for $s \leq t$ , $$ \mathbb{E}\big[e^{i\xi(X_t - X_s)} \mid \mathcal{F}_s\big] = e^{-\frac12\xi^2 (t-s)}. $$ The right-hand side is non-random. Taking expectations shows the conditional characteristic function of $X_{t} - X_{s}$ equals that of a $N (0, t - s)$ variable, so $X_{t} - X_{s} \sim N (0, t - s)$ . That the conditional characteristic function is non-random and $F_{s}$ -free means $X_{t} - X_{s}$ is independent of $F_{s}$ ; iterating over a finite time grid $t_{0} < \dots < t_{n}$ gives independence of the successive increments and identifies each increment's law as $N (0, t_{i} - t_{i - 1})$ . Thus $X$ has independent, stationary, centred Gaussian increments with the correct variances, and $X_{0} = 0$ . The path continuity is a hypothesis. All four defining properties of standard Brownian motion hold, so $X$ is a standard Brownian motion. $■$

Proposition (Reflection principle). For a standard Brownian motion $(B_{t})$ and $a > 0$ , with $M_{t} = max_{0 \leq s \leq t} B_{s}$ , $$ \mathbb{P}(M_t \ge a) = 2,\mathbb{P}(B_t \ge a). $$

Proof. Let $τ = in f {s : B_{s} = a}$ , a stopping time, finite almost surely because $lim sup_{s \to \infty} B_{s} = + \infty$ . On the event ${M_{t} \geq a}$ we have $τ \leq t$ , and by continuity $B_{τ} = a$ . By the strong Markov property the post- $τ$ process $B_{s} = B_{τ + s} - B_{τ}$ is a Brownian motion independent of $F_{τ}$ ; in particular $B_{t - τ}$ is symmetric, so conditionally on $τ \leq t$ the endpoint $B_{t} = a + B_{t - τ}$ is equally likely to lie above or below $a$ . Therefore $$ \mathbb{P}(B_t \ge a) = \mathbb{P}(B_t \ge a,, \tau\le t) = \mathbb{P}(M_t\ge a)\cdot \mathbb{P}(\widetilde B_{t-\tau}\ge 0 \mid \tau \le t) = \tfrac12,\mathbb{P}(M_t \ge a), $$ where the first equality holds because $B_{t} \geq a$ forces $τ \leq t$ . Rearranging gives $P (M_{t} \geq a) = 2 P (B_{t} \geq a)$ , and since $B_{t}$ is symmetric this also equals $P (∣ B_{t} ∣ \geq a)$ . $■$

Connections Master

The construction here builds the Wiener measure $W$ on the path space $C ([0, \infty), R)$ : it is the law of $(B_{t})$ , the pushforward of $P$ under $ω \mapsto B_{\cdot} (ω)$ , a Borel probability measure on the Polish space of continuous paths. This is the rigorous mathematical counterpart of the physics treatment of the Wiener measure and the path integral in 08.14.01; that unit reaches the same measure on path space through the heat kernel and the formal Feynman sum, where the present unit reaches it through Gaussian increments and the Kolmogorov continuity theorem.
The martingale, quadratic-variation, and Lévy-characterisation results assembled here are the exact foundation for the Itô integral and Itô calculus developed in 02.15.02 (co-produced with this unit). The identity $⟨ B ⟩_{t} = t$ is what makes $\int_{0}^{t} H_{s} d B_{s}$ an $L^{2}$ -isometry, and Lévy's theorem is what lets one recognise time-changed continuous martingales as Brownian motions inside that calculus.
The whole apparatus rests on measure-theoretic prerequisites: filtrations are increasing families of the $σ$ -algebras introduced in 02.07.01; the conditional expectation defining martingales is the Radon-Nikodym derivative of 02.07.08; and the $L^{2}$ convergence of quadratic-variation sums and of martingale limits lives in the $L^{p}$ spaces of 02.07.06. The Gaussian increment law and the additivity of variance over disjoint intervals are the continuous-time face of the distributions and independence catalogued in 26.02.01.

Historical & philosophical context Master

The physical phenomenon was recorded by the botanist Robert Brown in 1827, watching pollen grains jitter in water, and given a kinetic explanation by Einstein in 1905 and Smoluchowski in 1906, who derived the linear-in-time growth of the mean-squared displacement that the square-root law encodes. The mathematical object, however, was not secured until Norbert Wiener's 1923 memoir Differential space ^{[Wiener 1923]} constructed a genuine countably additive measure on the space of continuous functions — the first rigorous proof that the intuitive "random continuous path" exists as a well-defined probabilistic object rather than a heuristic. Wiener's achievement was to tame an infinite-dimensional integral, and his measure now bears his name.

The structural theory matured with Paul Lévy, whose Théorie de l'addition des variables aléatoires ^{[Lévy 1937]} established the law of the iterated logarithm and pointed toward the martingale characterisation later sharpened by Doob, Itô, and Watanabe. Philosophically, Brownian motion sits at a fault line: it is the canonical example of an object that is everywhere continuous yet nowhere differentiable, dramatising that the smoothness of classical analysis is a strong and special assumption, not a default of nature. Wiener measure also gave the first concrete model for a genuinely infinite-dimensional measure on a function space, a template that Feynman would later imitate (without rigour) for quantum amplitudes, and that the mathematics of 08.14.01 still works to justify.

Bibliography Master

@book{KaratzasShreve1991,
  author    = {Karatzas, Ioannis and Shreve, Steven E.},
  title     = {Brownian Motion and Stochastic Calculus},
  series    = {Graduate Texts in Mathematics},
  number    = {113},
  edition   = {2nd},
  publisher = {Springer-Verlag},
  year      = {1991}
}

@book{RevuzYor1999,
  author    = {Revuz, Daniel and Yor, Marc},
  title     = {Continuous Martingales and Brownian Motion},
  series    = {Grundlehren der mathematischen Wissenschaften},
  number    = {293},
  edition   = {3rd},
  publisher = {Springer-Verlag},
  year      = {1999}
}

@book{MortersPeres2010,
  author    = {M\"{o}rters, Peter and Peres, Yuval},
  title     = {Brownian Motion},
  series    = {Cambridge Series in Statistical and Probabilistic Mathematics},
  publisher = {Cambridge University Press},
  year      = {2010}
}

@article{Wiener1923,
  author    = {Wiener, Norbert},
  title     = {Differential space},
  journal   = {Journal of Mathematics and Physics},
  volume    = {2},
  pages     = {131--174},
  year      = {1923}
}

@book{Levy1937,
  author    = {L\'{e}vy, Paul},
  title     = {Th\'{e}orie de l'addition des variables al\'{e}atoires},
  publisher = {Gauthier-Villars},
  year      = {1937}
}

Prerequisites

02.07.01
02.07.06
02.07.08
26.02.01

Tier anchors

beginner: Mörters-Peres, Brownian Motion (CUP, 2010), Ch. 1 (informal definition, scaling picture); Klebaner, Introduction to Stochastic Calculus, Ch. 3
intermediate: Karatzas-Shreve, Brownian Motion and Stochastic Calculus (Springer GTM 113, 1991), §§2.1-2.5 (filtrations, martingales, optional sampling, BM definition and Lévy-Ciesielski construction)
master: Revuz-Yor, Continuous Martingales and Brownian Motion (Springer Grundlehren 293, 3rd ed. 1999), Ch. I-IV; Karatzas-Shreve §§2.6-3.3 (quadratic variation, Lévy's characterisation, strong Markov, reflection, Wiener measure)

References

Karatzas, Shreve — Brownian Motion and Stochastic Calculus (Springer GTM 113, 2nd ed. 1991) · §2.2 (filtrations, martingales, optional sampling Thm 2.2.13), §2.3 (Lévy-Ciesielski construction Thm 2.3.1), §2.9-§3.3 (quadratic variation, Lévy's Thm 3.3.16, strong Markov, reflection)
Revuz, Yor — Continuous Martingales and Brownian Motion (Springer Grundlehren 293, 3rd ed. 1999) · Ch. I §1 (filtrations, conditional expectation), Ch. I §2 (BM, Wiener measure), Ch. II (martingales, optional stopping), Ch. IV §1 (quadratic variation), Thm IV.3.6 (Lévy characterisation)
Mörters, Peres — Brownian Motion (Cambridge Series in Statistical and Probabilistic Mathematics, 2010) · Ch. 1 (definition and construction), Thm 1.30 (nowhere differentiability, Paley-Wiener-Zygmund), §5.1 (law of the iterated logarithm), Ch. 2 (strong Markov, reflection principle)
Wiener — Differential space (1923) · Journal of Mathematics and Physics 2, pp. 131--174; the first rigorous construction of a measure on the space of continuous paths
Lévy — Théorie de l'addition des variables aléatoires (Gauthier-Villars, 1937; 2nd ed. 1954) · Ch. VI-VII; the law of the iterated logarithm and the martingale characterisation of the Wiener process

Estimated time

beginner: 18m
intermediate: 45m
master: 85m