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For solutions with independent Brownian drivers in a common filtration, combine the drivers with weights $\sqrt{Z/(Z+Z^{\prime })}$ and $\sqrt{Z^{\prime }/(Z+Z^{\prime })}$ where the sum is positive. At zero use one original driver. The resulting continuous local martingale has bracket $t$ and is Brownian motion by the Lévy characterization of Brownian motion. Multiplying its differential by $\sqrt{Z+Z^{\prime }}$ gives exactly the sum of the original stochastic differentials.

For independently driven nonnegative squared Bessel processes $X,Y$, put $Z=X+Y$. Where $Z>0$, use Brownian weights $\sqrt{X/Z}$ and $\sqrt{Y/Z}$. Where $Z=0$, fill the weights with $(1,0)$. Their squared norm is always one and their cross-variation is zero, so their stochastic integral is Brownian motion by the Lévy characterization of Brownian motion. Multiplying by $2\sqrt{Z}$ recovers the sum's martingale part even at zero, and the drifts add to $\alpha +\beta$.

A physicist whose work connects special relativity, general relativity, Brownian motion and the photoelectric effect. He worked at the Institute for Advanced Study from 1933 until his death in 1955.^[ref]

In 1905, Albert Einstein used the photoelectric effect to explore quantized light, related Brownian motion to molecular agitation, formulated special relativity, and connected changes in mass and energy. These problems link visible measurements to otherwise inaccessible structure: emitted electron energies, suspended-particle displacements and comparisons of clocks.^[ref]

The Millikan photoelectric experiment follows one of those predictions into the laboratory. Its stopping-voltage measurements test a quantitative relation; the experiment's apparatus, surface preparation and electrical corrections matter to that inference. The Einstein field equations and general relativity lead to a different set of tests involving gravity and spacetime.

The Nobel Prize in Physics in 1921 recognized his theoretical work, particularly the law of the photoelectric effect; the award was presented in 1922.^[ref]

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.

Arthur Ashkin's 1970 first experiment suspended transparent latex spheres in water inside a $120\mu \mathrm{m}$ glass cell. A horizontal $514.5\mathrm{nm}$ argon laser beam, focused to a $6.2\mu \mathrm{m}$ radius, illuminated individual beads viewed through a microscope. An off-centre $2.68\mu \mathrm{m}$ bead moved toward the beam axis and downstream until reaching the cell wall; blocking the beam released it into Brownian motion. Transparent particles and liquid reduced competing thermal effects. This observed sequence separates transverse attraction from downstream radiation pressure: wall-assisted confinement is not free three-dimensional confinement by one beam.^[ref]

The beam-blocking comparison is particularly informative: persistence against a wall while illuminated and release when illumination stops test an optical cause. To infer a force from the downstream velocity, one additionally needs a linear drag model and the bead's distance from the wall; the observed motion alone does not calibrate an unrestricted harmonic optical trap.

For standard Brownian motion started at zero and exit time $T$ from $(-b,a)$, $a,b>0$, stop the two exponential Brownian martingales with parameters $\lambda$ and $-\lambda$. Their stopped values are bounded by $e^{|\lambda |\max (a,b)}$. The optional stopping theorem and dominated convergence theorem give $e^{\lambda a}p+e^{-\lambda b}q=1$ and $e^{-\lambda a}p+e^{\lambda b}q=1$, where $p,q$ are the discounted upper and lower exit probabilities. Solving gives the displayed formula and $q=\sinh (\lambda a)/\sinh (\lambda (a+b))$. ThusAt $\lambda =0$, the first ratio has its removable value $b/(a+b)$, and the second equals one. Finiteness of $T$ follows from a uniformly positive one-unit-time chance to leave any fixed bounded interval.

For Brownian motion, write the later maximum as $B_b+Z+H$, where $Z=B_c-B_b$ has a nondegenerate normal distribution and $H={\sup }_{0\le u\le d-c}(B_{c+u}-B_c)$. The independent increments make $Z$ independent of the earlier history and of $H$. Conditional on those two objects, equality of the two maxima prescribes one value of $Z$, an event of probability zero. This argument uses the positive gap $c-b$ and does not assume independence of the two absolute maxima.

A Bessel process of dimension $d$ is the radial part of $d$-dimensional Brownian motion. Away from zero it solves the stochastic differential equationThis equation also defines noninteger dimensions through its weak solutions.

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 the completed natural Brownian filtration, every event in the germ sigma-field of Brownian motion has probability zero or one. Indeed, an event measurable at every positive time is independent of all increments after each such time. Letting that time decrease to zero and using path continuity makes the event independent of the entire Brownian path sigma-field. Since it is itself measurable in that sigma-field, its probability equals its square.

Under a reference measure where $X$ is Brownian motion started at zero, let $T$ be its exit time from $(-1,1)$. The displayed density is the stochastic exponential with integrand $X{1}_{[0,T]}$, by the Itô formula. It is bounded by $e^{1/2}$, hence defines a uniformly integrable density process with terminal density $Z_T$. Under that measure change, $X_t-{\int }_0^{t\wedge T}{X}_{s}ds$ is Brownian motion. At exit, $Z_T\ge e^{1/2-T}$, implying $\mathbb{P}(T\le t)\ge e^{1/2-t}\tilde{\mathbb{P}}(T\le t)$. This is a terminal density on the whole probability space, not merely an unspecified finite-horizon change of measure.

For standard one-dimensional Brownian motion and a fixed $h>0$, the Brownian reflection principle gives the survival probability $2\Phi (h/\sqrt{t})-1$. The standard normal distribution function has derivative $1/\sqrt{2\pi }$ at zero, giving the displayed asymptotic. Thus the persistence exponent is one half, with leading constant $h\sqrt{2/\pi }$.

For standard Brownian motion, the Brownian bridge $U_t=B_t-t{B}_1$, $0\le t\le 1$, is independent of $B_1$. Every finite vector of bridge coordinates is jointly multivariate normal with $B_1$, and its cross-covariances with $B_1$ vanish. The independence of uncorrelated jointly normal variables and independence extended from generating pi-systems prove independence of the whole stochastic process.

For $0<x<M$, a three-dimensional Bessel process stopped at $M$ has the law of Brownian motion conditioned to hit $M$ before zero and stopped at $M$. On $(\epsilon ,M)$, the Girsanov theorem and the Itô formula give terminal density $X_{\tau }/x$. Its value is $M/x$ at the upper exit and $\epsilon /x$ at the lower exit. Letting $\epsilon \downarrow 0$ in bounded stopped-path functionals gives density $(M/x)1_{\{T_M<T_0\}}$ relative to the corresponding stopped Wiener measure. The exit probability is $x/M$ by the optional stopping theorem, so the density is normalized. On the unstopped full path space the same density defines the conditioned Brownian law; pushing it forward by the stopping map gives the stopped Bessel law. The already-stopped law is not absolutely continuous relative to unstopped Wiener measure, since eventually constant paths have Wiener probability zero.

The covariance kernel of standard Brownian motion on $[0,1]$ is $c(s,t)=\min (s,t)$. Its integral operator has eigenfunctions $\sqrt{2}\sin ((k-\frac{1}{2})\pi t)$ and eigenvalues $((k-\frac{1}{2})\pi {)}^{-2}$.

The excursion from $0$ to infinity is the Doob h-transform of killed planar Brownian motion with $h(z)=\mathrm{Im}z$, using the entrance law at zero. Equivalently, its real part is a standard Brownian motion and its independent imaginary part is a Bessel process of dimension three started at zero. This planar, infinite-duration excursion differs from the one-dimensional excursion on a finite time interval.

Planar Brownian motion started at $a+ia$, $a>0$, exits the positive quadrant through either positive coordinate axis with probability one-half. The density along each axis is $q_a(r)$ for $r>0$, measured with respect to distance along that axis. Squaring maps the quadrant to the upper half-plane and sends the starting point to $2i{a}^2$. The boundary square therefore has the centered Cauchy distribution of scale $2a^2$; pulling it back by $r\mapsto r^2$ gives the displayed density. The radius has distribution function $2\arctan (r^2/(2a^2))/\pi$, independently of which axis is chosen. There is no atom at the origin.

For independent coordinates of standard Brownian motion and $x_i>0$, the Brownian reflection principle gives ${\mathbb{P}}_x(T>t)={\prod }_i[2\Phi (x_i/\sqrt{t})-1]$. This is asymptotic to $(2/\pi {)}^{n/2}{\prod }_i{x}_i{t}^{-n/2}$. The tail integral formula for expectation therefore proves the stated criterion, with logarithmic divergence in dimension two.

Successive exits from intervals of radius $a$ around the current position embed a simple symmetric random walk into Brownian motion. The Strong Markov property makes the durations independent with common law $a^2{\sigma }_1$; the displacements divided by $a$ are independent symmetric signs. The displacement sequence is independent across exits; no independence between a given displacement and its duration is needed for these conclusions.

A filtration $({\mathcal{F}}_t)$ is a Brownian filtration for $B$ when $B$ is an adapted process and each future increment $B_t-B_s$ is independent of ${\mathcal{F}}_s$. The natural filtration of a Brownian motion has this property. Arbitrary enlargement by future information need not preserve it.