= Brownian motion
{c}
{title2=$B_t$, $W_t$}
{wiki}
Standard one-dimensional <Brownian motion> is a real-valued <stochastic process> $(B_t)_{t\geq0}$ on a <probability space> with three defining properties: $B_0=0$ <almost surely>; its <sample paths> $t\mapsto B_t$ are continuous <almost surely>; and its <Brownian increments> over any finite collection of disjoint time intervals are <independent>, with
$$
B_t-B_s\sim N(0,t-s)
\qquad(0\leq s<t).
$$
Here $N(m,v)$ denotes the <normal distribution> with <expectation> $m$ and <variance> $v$. The alternative symbol $W_t$ denotes the same kind of <stochastic process>. Continuity is a simultaneous statement about an entire <sample path>, rather than a separate assertion at each fixed time. Relative to its <natural Brownian filtration>, a future <Brownian increment> is <independent> of all observations up to its starting time.
The normalization gives $B_t\sim N(0,t)$, so the <standard deviation> of a <Brownian increment> over duration $h$ is $\sqrt h$. For $h=1$, $0.01$, and $0.0001$, the corresponding <standard deviations> are $1$, $0.1$, and $0.01$. Time becoming one hundred times shorter makes typical displacement ten times smaller. <Brownian scaling> expresses this rule for the whole <stochastic process>, while the <Brownian covariance kernel> describes how positions at different times remain related despite <independent increments>.
For a concrete <probability> calculation, let $\Phi$ be the <cumulative distribution function> of $N(0,1)$. At time four, $B_4/2$ has that <normal distribution>, hence
$$
\mathbb P(|B_4|\leq1)=\Phi(1/2)-\Phi(-1/2)=2\Phi(1/2)-1\approx0.3829249225,
\qquad
\mathbb P(B_4>2)=1-\Phi(1)\approx0.1586552539.
$$
These numbers can be reproduced using the <error function>: $\Phi(z)=(1+\operatorname{erf}(z/\sqrt2))/2$. In particular, zero <expectation> does not mean small displacement: $\mathbb P(|B_1|\leq1)\approx0.6826894921$, but the same fixed interval contains only about $38.3\%$ of the <probability> at time four.
The <probability density function> of $B_t$, for $t>0$, is
$$
p(t,x)=\frac{1}{\sqrt{2\pi t}}e^{-x^2/(2t)}.
$$
Its <partial derivatives> satisfy
$$
\partial_t p=\left(-\frac1{2t}+\frac{x^2}{2t^2}\right)p,
\qquad
\partial_x^2p=\left(-\frac1t+\frac{x^2}{t^2}\right)p,
\qquad
\partial_t p=\tfrac12\partial_x^2p.
$$
Thus the <normal distribution> spreads according to the <heat equation>. Starting at a fixed position $x_0$ replaces $x$ by $x-x_0$. This is a useful bridge between individual random trajectories and the deterministic evolution of their <probability density function>: the same <Brownian motion> provides both descriptions.
Physical <diffusion> uses its own scale. For constant <diffusion coefficient> $D>0$, the position model $X_t=x_0+\sqrt{2D}B_t$ has <variance> $2Dt$, and its <probability density function> satisfies $\partial_t p_D=D\partial_x^2p_D$. For example, $D=0.5\,\mathrm{\mu m^2/s}$ gives displacement <variance> $4\,\mathrm{\mu m^2}$ after four seconds, hence <standard deviation> $2\,\mathrm{\mu m}$. Standard <Brownian motion> corresponds numerically to $D=1/2$ in chosen units; omitting the factor two changes the physical convention. Einstein's 1905 paper relates <diffusion> to the mean squared displacement in precisely this convention. https://djalil.chafai.net/docs/M2/history-brownian-motion/Einstein%20-%201905.pdf[Einstein, 1905 paper, English translation, section 4]{ref}
Continuity does not supply a smooth physical <velocity>. The difference quotient $(B_{t+h}-B_t)/h$ has <normal distribution> $N(0,1/h)$, so its <standard deviation> grows as $h^{-1/2}$: for the three durations above it is $1$, $10$, and $100$. This calculation motivates the <nowhere differentiability of Brownian motion>, but it alone is not a proof of the simultaneous assertion at every time. The theorem says that, with <probability> one, no finite <derivative> exists anywhere along the <sample path>. A drawn polygon only displays a finite grid; its straight segments cannot establish this theorem. See Theorem 7.1.6 in <Probability: Theory and Examples (fifth edition) by Rick Durrett>.
For physical particles, <Brownian motion> is a model of position on suitable observation scales. It does not claim that microscopic <velocity> is a <derivative> of this idealized path. Einstein explicitly restricts his assumption of <independent increments> to intervals that are not too short. https://djalil.chafai.net/docs/M2/history-brownian-motion/Einstein%20-%201905.pdf[Einstein, 1905 paper, English translation, section 4]{ref}
Another way to measure the roughness is <quadratic variation>. On $[0,T]$, divide time into $n$ equal intervals and form
$$
Q_n=\sum_{k=1}^n(B_{kT/n}-B_{(k-1)T/n})^2.
$$
Writing each <Brownian increment> as $\sqrt{T/n}Z_k$ with <independent> standard <normal random variables> gives $Q_n=(T/n)\sum_kZ_k^2$. Since $\mathbb EZ_k^2=1$ and $\mathbb EZ_k^4=3$, the <variance> of $Z_k^2$ is $2$, and
$$
\mathbb E Q_n=T,
\qquad
\operatorname{Var}(Q_n)=\frac{2T^2}{n},
\qquad
\mathbb E[(Q_n-T)^2]=\frac{2T^2}{n}\longrightarrow0.
$$
This proves <mean-square convergence> to $T$ along uniform partitions. For $T=2$ and $n=100$, the <expectation> is $2$, the <variance> is $0.08$, and the <root mean square> of the error is $\sqrt{0.08}\approx0.2828427125$. For a general deterministic <partition of an interval> with lengths $h_k$, the same calculation gives error <variance> $2\sum_kh_k^2\leq2T\max_kh_k$, so shrinking the largest interval also suffices.
The resulting <quadratic variation> is $[B]_T=T$, whereas the <smooth function> $f(t)=t$ gives $Q_n=T^2/n\to0$. This difference explains why the ordinary rules of <calculus> need the second-order correction in the <Itô formula>. In the figure, coarser sums reuse the same finest-grid <Brownian increments>; their individual values can oscillate with refinement even though the error tends to zero in <mean-square convergence>. Refining a fixed interpolating polygon forever would instead give zero <quadratic variation>, so the finite-grid illustration must not be confused with the limiting <Brownian motion>.
\Image[/brownian-quadratic-variation.png]
{title=Finite Brownian quadratic variation sums and their exact root mean squared error}
Historically, Brown's 1828 report describes microscopic observations made in 1827, starting with particles released from pollen and extending to inorganic materials; this challenged an explanation based solely on living matter. His observations and the later mathematical model are distinct contributions. https://en.wikisource.org/wiki/The_miscellaneous_botanical_works_of_Robert_Brown/Volume_1/A_brief_account_of_microscopical_observations_made_on_the_particles_contained_in_the_pollen_of_plants[Brown, 1828 report, reprinted in his collected works]{ref} Einstein's 1905 paper derived a quantitative connection between <kinetic theory>, <diffusion>, and displacement. https://djalil.chafai.net/docs/M2/history-brownian-motion/Einstein%20-%201905.pdf[Einstein, 1905 paper, English translation]{ref} Wiener's 1923 Differential-Space developed integration on spaces of paths, a foundation for <Wiener measure>; it also acknowledges earlier work and his own previous papers. These dates mark documented contributions rather than an assertion that each subject had a single discoverer. https://djalil.chafai.net/docs/M2/history-brownian-motion/Wiener%20-%201923.pdf[Wiener, Differential-Space, 1923, sections 1 and 6]{ref}
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