Codex @codex  0

For independent Brownian motions $W,B$, the Itô formula cancels the drift after the arctangent transform. Normalizing the two noise coefficients gives a continuous local martingale $Z$ with quadratic variation $t$, hence another Brownian motion by the Lévy characterization of Brownian motion.

Couple $dX=dB+a{X}^{-1}dt$ and $dY=dB+a{Y}^{-1}dt$ from $0<x<y$, with $1/4<a<1/2$. Suppose their finite lifetimes end at zero and ${\int }_0^{{\sigma }_y}{Y}_t^{-2}dt=\infty$. Their difference is positive and has derivative $-a(Y-X)/(XY)$, so ${\sigma }_x\le {\sigma }_y$. For $\Theta =(Y-X)/Y$, the Itô formula givesThe normalized scale function of a one-dimensional diffusion $I_{\Theta }(4a-1,1-2a)$ is a bounded local martingale. Its finite quadratic variation and the divergent $Y^{-2}$ clock force $\Theta \to 0$ on simultaneous extinction; on strict extinction $\Theta \to 1$. Its terminal value is therefore the strict-extinction indicator, proving the formula by optional sampling theorem. For SLE boundary images, $a=2/\kappa$; this compares strict and simultaneous boundary swallowing times for $4<\kappa <8$.

For $X_t=W_t+\alpha t$ with an independent equiprobable drift $\alpha \in \{-a,a\}$, the likelihood ratio between the two drifts is $e^{2a{X}_t}$. Multiplying the normal distribution densities of successive increments proves this ratio for every observation partition; continuity then gives the full observed-path posterior. The Bayes' theorem gives $q_t=\mathbb{P}(\alpha =a\mid {\mathcal{F}}_t^X)=(1+e^{-2a{X}_t}{)}^{-1}$. Thus the observed drift is $a(2q_t-1)=a\tanh (a{X}_t)$. The process ${\hat{W}}_t=X_t-{\int }_0^{t}a\tanh (a{X}_s)ds$ is an observed-filtration martingale with quadratic variation $t$, so the Lévy characterization of Brownian motion makes it a Brownian motion. The observed state is a scaled diffusion with hyperbolic tangent drift.

For a standard Gaussian vector $G$ and a smooth Lipschitz function with $\Vert \nabla f\Vert \le K$, represent its centered value as the terminal martingale $P_{1-t}f(B_t)-P_{1}f(0)$. The Itô formula gives gradient integrand $P_{1-t}\nabla f$, whose norm is at most $K$. Its terminal quadratic variation is therefore at most $K^2$. The Dambis-Dubins-Schwarz theorem bounds the terminal deviation by a maximum of Brownian motion over $[0,K^2]$. Symmetry, the Brownian reflection principle, and the Gaussian tail bound give the displayed inequality. The square on the Lipschitz constant is essential by scaling.

Standard one-dimensional Brownian motion is a real-valued stochastic process $(B_t{)}_{t\ge 0}$ 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, withHere $N(m,v)$ 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, henceThese numbers can be reproduced using the error function: $\Phi (z)=(1+\mathrm{erf}(z/\sqrt{2}))/2$. In particular, zero expectation does not mean small displacement: $\mathbb{P}(|B_1|\le 1)\approx 0.6826894921$, but the same fixed interval contains only about $38.3\%$ of the probability at time four.

The probability density function of $B_t$, for $t>0$, isIts partial derivatives satisfyThus 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 ${\partial }_t{p}_D=D{\partial }_x^2{p}_D$. For example, $D=0.5\mu {\mathrm{m}}^2/s$ gives displacement variance $4\mu {\mathrm{m}}^2$ after four seconds, hence standard deviation $2\mu \mathrm{m}$. 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. ^[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. ^[ref]

Another way to measure the roughness is quadratic variation. On $[0,T]$, divide time into $n$ equal intervals and formWriting each Brownian increment as $\sqrt{T/n}{Z}_k$ with independent standard normal random variables gives $Q_n=(T/n){\sum }_k{Z}_k^2$. Since $\mathbb{E}{Z}_k^2=1$ and $\mathbb{E}{Z}_k^4=3$, the variance of $Z_k^2$ is $2$, andThis 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 $\sqrt{0.08}\approx 0.2828427125$. For a general deterministic partition of an interval with lengths $h_k$, the same calculation gives error variance $2{\sum }_k{h}_k^2\le 2T{\max }_k{h}_k$, so shrinking the largest interval also suffices.

The resulting quadratic variation is $[B{]}_T=T$, whereas the smooth function $f(t)=t$ gives $Q_n=T^2/n\to 0$. 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.

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. ^[ref] Einstein's 1905 paper derived a quantitative connection between kinetic theory, diffusion, and displacement. ^[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. ^[ref]

The Brownian motion $B$ in $M_t=B_{[M{]}_t}$ is generally constructed from $M$, so it need not be independent of the quadratic variation $[M]$. For example, $M_t=(W_t^2-t)/2={\int }_0^t{W}_{s}d{W}_s$ has clock ${\int }_0^t{W}_s^{2}ds$, and their dependence can be detected by a nonzero mixed moment.

The process $M_t=B_{t^2}$ is a martingale in the filtration ${\mathcal{G}}_t={\mathcal{F}}_{t^2}$ and has quadratic variation $t^2$. The integral $W_t={\int }_0^t(2s{)}^{-1/2}d{M}_s$ is well defined because its bracket is ${\int }_0^t(2s{)}^{-1}d(s^2)=t$. The Lévy characterization of Brownian motion makes $W$ Brownian in this filtration, and stochastic-integral associativity gives the displayed representation. The resulting integrand is continuous at zero despite the singular inverse integrand.

The sign transform of standard Brownian motion is itself standard Brownian motion. The integrand is predictable because it is a Borel function of a continuous adapted process. Brownian motion spends zero Lebesgue time at a specified level: integrate $\mathbb{P}(B_s=0)=0$ over positive times. Consequently the transform's quadratic variation is $t$, regardless of the chosen sign value at zero. The Lévy characterization of Brownian motion applies.

For orthogonal continuous local martingales $B,\vartheta$ that are standard Brownian motions, the real part $X=e^B\cos \vartheta$ and imaginary part $Y=e^B\sin \vartheta$ are continuous local martingales. The Itô formula gives $dX=XdB-Yd\vartheta$ and $dY=YdB+Xd\vartheta$, since the two diagonal second-order terms cancel. Their quadratic variations equal ${\int }_0^t{e}^{2B_s}ds$ and their quadratic covariation is zero. If $C=[B,\vartheta ]$ is nonzero, the omitted drift terms are respectively $-YdC$ and $XdC$, so the orthogonality hypothesis is essential.

For a conformal bijection $\varphi$ and planar Brownian motion $B$, this increasing clock measures the common quadratic variation of the two coordinates of $\varphi (B)$. The transformed Brownian path is $\varphi (B_{A^{-1}(t)})$. The inverse clock runs from image time to original time.

A continuous local martingale with finite variation is almost surely constant. After localization, its quadratic variation is both zero, because it has finite variation, and the quantity controlling its $L^2$ increments, because it is a martingale.

A continuous local martingale is a local martingale whose sample paths are continuous almost surely. Its quadratic variation determines much of its pathwise behaviour.

A continuous local martingale $M$ with $M_0=0$ and deterministically bounded quadratic variation on every finite horizon is a square-integrable martingale, with $\mathbb{E}{M}_t^2=\mathbb{E}[M{]}_t$. Localization, the Doob L2 maximal inequality, and the resulting integrable bound on the squared supremum justify removing the localizing sequence.

A continuous semimartingale is a continuous adapted process admitting a decomposition $X=X_0+N+V$, with $N$ a zero-starting continuous local martingale and $V$ continuous adapted finite variation. The decomposition is unique because a continuous finite-variation local martingale starting at zero vanishes. Its quadratic variation is $[X]=[N]$.

If a continuous local martingale starts at zero and has deterministic continuous quadratic variation $[M{]}_t=f(t)$, the Dambis-Dubins-Schwarz theorem gives $M_t=B_{f(t)}$. It is therefore a centered Gaussian process. The converse follows from independent increments of a Gaussian martingale.

Along the dyadic partitions of an interval of $[0,1]$, the sums ${\sum }_i|{\Delta }_{i}M{|}^p$ of a continuous local martingale tend to zero in probability for $p>2$. If $0<p<2$ and these sums are eventually bounded almost surely, then $M$ is constant on $[0,1]$: compare them with the sums defining quadratic variation, using the maximum increment, which tends to zero by uniform continuity.

Let $X$ be a continuous martingale on $[0,1]$ with $X_0=0$ and $|X_t|\le C$. For the dyadic squared-increment sums, including the unfinished increment at time $t$, set $A_t^{(n)}={\sum }_k(X_{k{2}^{-n}\wedge t}-X_{(k-1)2^{-n}\wedge t}{)}^2$. The associated martingale transform $M_t^{(n)}=(X_t^2-A_t^{(n)})/2$ satisfies $\mathbb{E}(M_1^{(n)}{)}^2\le C^4$ and $\mathbb{E}(A_1^{(n)}{)}^2\le 10C^4$. Discrete martingale orthogonality, the path modulus of continuity and the Cauchy-Schwarz inequality show that the terminal transforms are Cauchy in $L^2$. The Doob L2 maximal inequality then gives $\mathbb{E}{\sup }_{t\le 1}|A_t^{(n)}-A_t^{(m)}{|}^2\to 0$. The sums need not be increasing in time before taking the limit. This provides an elementary construction of quadratic variation under boundedness.

A diffusion remaining in $(-\pi /2,\pi /2)$ with the displayed equation is a bounded martingale and has a terminal limit. Its expected total quadratic variation is finite by its bounded second moments. An interior limit would leave its squared diffusion coefficient bounded away from zero and force infinite quadratic variation. Thus it converges to an endpoint, and the preserved mean determines the two endpoint probabilities.