Brownian motion ($B_t$, $W_t$)

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]

A planar self-propelled particle with translational diffusion coefficient $D$, propulsion speed $v$, rotational diffusion $D_r$ and constant angular drift $\omega$ has orientation correlation $e^{-D_r|t|}\cos (\omega t)$. Integrating this correlation twice gives the displayed mean-square displacement. For $D_r>0$, its long-time diffusion coefficient is $D+v^2{D}_r/[2(D_r^2+{\omega }^2)]$. The full position-angle process has the Markov property, whereas position alone generally retains orientational memory. The angular second moment refers to an unwrapped angle.

The Lévy construction builds Brownian motion on $[0,1]$ by successive independent midpoint displacements on dyadic intervals. Start with $B_0=0$ and $B_1=Z_0$, where $Z_0$ has the standard normal distribution. On each dyadic interval $[u,v]$, choose a fresh independent standard normal random variable $Z$ and use the displayed midpoint rule. Conditional on the endpoints, this is the correct Brownian bridge midpoint law.

To verify the increment law, if $D=B_v-B_u\sim N(0,\ell )$ with $\ell =v-u$, the new increments are $D/2\pm \sqrt{\ell }Z/2$. They are Jointly Gaussian, each has variance $\ell /2$, and their covariance is zero, hence they are independent. Induction gives all dyadic Brownian increments. The change between consecutive linear interpolants has uniform norm $2^{-n/2-1}{\max }_{k<2^n}|Z_{n,k}|$. The Gaussian tail bound and union bound bound the chance that this maximum exceeds $n$ by $2^{n+1}{e}^{-n^2/2}$. The First Borel-Cantelli lemma and ${\sum }_{n}n{2}^{-n/2}<\infty$ prove uniform convergence almost surely. The continuous limit has the required Brownian increments at arbitrary times by dyadic approximation.

Let $H_{n,k}(s)=2^{n/2}(1_{(0,1/2)}-1_{(1/2,1)})(2^{n}s-k)$ for $n\ge 0$ and $0\le k<2^n$. Integrating these Haar wavelets gives tents of height $2^{-n/2-1}$ supported on the corresponding dyadic intervals. For independent standard normal random variables $Z_0,Z_{n,k}$, the displayed series, summed level by level, is exactly the succession of midpoint displacements in the Lévy construction of Brownian motion. Its almost sure convergence is uniform, and its limit is standard Brownian motion. This gives a continuous function from Gaussian coefficients even though the unintegrated unit-variance Haar wavelet series diverges in L2 space.

With thermal initial velocities and relaxation time $\tau =m/\zeta$, the one-dimensional displacement variance is $2D[t-\tau (1-e^{-t/\tau })]$. It approaches $(k_{B}T/m)t^2$ at short times and $2Dt$ at long times. Initial velocity preparation changes the short-time result.

For ordinary long-time diffusion, the signed mean displacement vanishes while one-coordinate mean-square displacement is $2Dt$ and the three-dimensional mean-square radial displacement is $6Dt$. An rms distance is neither mean signed displacement nor cumulative path length.

For isotropic three-dimensional diffusion with coefficient $D$, the vector mean displacement is zero, the radial RMS is $\sqrt{6Dt}$ and the mean radial magnitude is $4\sqrt{Dt/\pi }$. The radial distribution is the Maxwell speed distribution arising from three independent Gaussian coordinate increments. State the statistic before calling a quantity an average distance; it is not the cumulative path length.

A colloidal grain in a liquid is large enough to see with a microscope, yet small enough for thermal motion to compete with its tendency to settle. In the Perrin sedimentation experiment, the equilibrium number density falls with height: a visible miniature atmosphere. Counting grains at different heights measures an energy scale; tracking their Brownian motion measures a transport rate. Connecting both to the same Avogadro constant makes the experiment a quantitative test of kinetic theory. The Wiener measure describes an idealized path law; these measurements instead test a physical model and its material parameters.

Jean Baptiste Perrin's 1909 memoir traces his sedimentation experiments to May 1908. It reports relative counts $100$, $47$, $22.6$, $12$ at heights $5$, $35$, $65$, $95\mu \mathrm{m}$, and compares radius estimates from Stokes settling velocity, counted grain mass, and rows of touching grains.^[ref]

The 1926 lecture describes resin precipitated from alcohol into water, washed and fractionated by centrifugation to obtain nearly monodisperse gamboge or mastic grains. Diameter and density were measured separately. A sealed slide-and-cover-glass chamber prevented evaporation; a vertical microscope resolved roughly $2\mu \mathrm{m}$ layers through a roughly $0.1\mathrm{mm}$ depth. Refocusing and photographing compared layer populations after equilibrium; changing temperature reversibly changed the distribution.^[ref]

In Les Atomes (1913), Jean Baptiste Perrin restricts the eyepiece view to fewer than five or six grains and alternates observations between heights. Focus displacement requires a refractive index correction to recover actual layer separation. Horizontal positions were marked with a camera lucida, usually every $30\mathrm{s}$, and distances calibrated with a micrometer; the polygon joins sampled positions rather than recording the intervening trajectory.^[ref][ref]

Take identical rigid spheres of radius $a$, particle density ${\rho }_p$ and displaced liquid density ${\rho }_f$, at uniform absolute temperature $T$. Let $z$ increase upwards from a reference layer and write $\Delta \rho ={\rho }_p-{\rho }_f>0$. The buoyancy correction gives the effective gravitational massThe actual grain mass is ${\rho }_{p}V$; the displaced liquid subtracts ${\rho }_{f}V$ from the effective gravitational mass. In a dilute, noninteracting particle suspension, the Boltzmann distribution therefore givesHere $n$ is number density and $k_B$ is the Boltzmann constant. Equivalently, the dilute osmotic pressure $\Pi =n{k}_{B}T$ supports the grains above it: $d\Pi /dz=-n{m}_{b}g$. Dividing by $n$ and integrating yields the same exponential function. No dynamic viscosity appears in this equilibrium argument. If ${\rho }_p<{\rho }_f$, the population instead increases upwards; at equal densities this gravitational model predicts uniform $n$.

A microscope counts a finite optical slab, not an infinitesimal plane. For equal viewed area $A$ and slab thickness $\delta$, an ideal uniform detection window centred at $z$ givesThus equal optical volumes preserve the exponential function count ratio exactly, even for finite $\delta$, provided the windows lie wholly inside the accessible chamber and have the same detection response. Unequal depth response or overlapping a boundary can corrupt this inference. Repeated observations estimate the mean count; closely spaced visits need not be independent because grains remain in view.

For an original chosen model, set $a=0.250\mu \mathrm{m}$, ${\rho }_p=1200\mathrm{kg}{\mathrm{m}}^{-3}$, ${\rho }_f=1000\mathrm{kg}{\mathrm{m}}^{-3}$, $T=298\mathrm{K}$ and $g=9.81\mathrm{m}{\mathrm{s}}^{-2}$. These are illustrative inputs, not Jean Baptiste Perrin's measurements. Using $k_B=1.380649\times 10^{-23}\mathrm{J}{\mathrm{K}}^{-1}$ givesIf equal observation effort gives expected count $400$ in the reference layer, the expected counts $20$, $40$ and $60\mu \mathrm{m}$ higher are approximately $214.3$, $114.8$ and $61.49$. Fractions describe expected values over observations, not fractional grains. Using the actual grain mass instead of its buoyancy-corrected value would make the inferred Avogadro constant six times too small for these chosen densities.

For two observed layers $z_2>z_1$, define the positive logarithmic count slopeSince $k_B=R/N_A$, where $R$ is the molar gas constant, the sedimentation scale height givesSeveral heights check whether one slope fits the whole profile. A mixture of grain radii generally gives a sum of exponential functions, with the larger grains depleted faster aloft; one apparent radius need not describe every height.

An independent readout uses horizontal displacements at lag $\tau$, avoiding the vertical gravitational drift. Define the one-coordinate mean squared displacement $M_x(\tau )=⟨[x(t+\tau )-x(t){]}^2⟩$. In a stationary liquid without horizontal advection, translational diffusion predictsA camera sees two horizontal coordinates, so its squared radial displacement has twice the one-coordinate mean. Neither observable is path length or instantaneous velocity. The bulk Stokes–Einstein relation, with dynamic viscosity $\eta$, then gives

The 1926 lecture compares displacement-based and equilibrium determinations of the Avogadro constant.^[ref]

Rotational diffusion supplies a further observable when an identifiable grain orientation can be followed; centre positions alone cannot measure it.

For the chosen model above and $\eta =1.00\times 10^{-3}\mathrm{Pa}\mathrm{s}$, the bulk prediction is $D\simeq 0.8731\mu {\mathrm{m}}^2{\mathrm{s}}^{-1}$. At $\tau =10\mathrm{s}$, $M_x\simeq 17.46\mu {\mathrm{m}}^2$ and the planar value is $34.92\mu {\mathrm{m}}^2$. Reconstructing $N_A$ from these stipulated model outputs returns $6.02214076\times 10^{23}\mathrm{mo}{\mathrm{l}}^{-1}$ by construction; this arithmetic consistency check is not experimental confirmation. Historical force comes from comparing separately measured profiles and displacements using independently checked material parameters.

The diffusive flux and gravitational particle flux provide another useful connection. With constant drag coefficient $\zeta$, zero vertical net flux readswhere $v_s$ is the downward settling velocity magnitude. With $\zeta =6\pi \eta a$, the chosen model gives $v_s\simeq 0.02725\mu \mathrm{m}{\mathrm{s}}^{-1}$. The equilibrium profile forgets how rapidly it was established, while displacement measurements retain that rate information.

At fixed measured count slope, $N_A^{\mathrm{sed}}\propto a^{-3}$, whereas at fixed measured diffusion coefficient, $N_A^{\mathrm{diff}}\propto a^{-1}$. First-order propagation of uncertainty follows by taking logarithmic derivatives:For independent small errors, the radius contributes $9({\sigma }_a/a{)}^2$ to the relative variance of $N_A^{\mathrm{sed}}$, compared with $({\sigma }_a/a{)}^2$ for $N_A^{\mathrm{diff}}$. Shared radius and temperature calibrations make the two estimates correlated; their agreement is therefore not two wholly independent determinations. As a finite-error example, overestimating $a$ by $5\%$ multiplies the respective answers by $1.05^{-3}\simeq 0.8638$ and $1.05^{-1}\simeq 0.9524$.

This explains the importance of the radius cross-checks in the 1909 memoir: testing the Stokes drag law through an independent radius estimate helps prevent circular inference.^[ref] Convection must also be excluded: a residual horizontal speed $u$ adds $u^2{\tau }^2$ to an uncentred mean squared displacement, making a linear fit misleading.

Buoyancy changes the effective gravitational mass; a nearby no-slip wall changes the hydrodynamic resistance matrix and therefore the diffusion coefficient. These are distinct corrections. Write the horizontal resistance as ${\zeta }_{\parallel }=6\pi \eta a/f$, with $0<f<1$ representing wall hindrance at a specified height. Applying the Stokes–Einstein relation with that resistance givesif one mistakenly uses the bulk resistance in the inversion. A chosen $f=0.8$ would overestimate $N_A$ by $25\%$; this is a sensitivity example, not a correction measured in Jean Baptiste Perrin's apparatus. Normal and parallel coefficients of the hydrodynamic resistance matrix need not coincide. Stokes settling velocity near a wall can likewise bias a radius inferred using an unbounded-fluid formula.

For a passive dilute particle suspension at uniform temperature, a change of mobility alone does not change the Boltzmann distribution. In the absence of additional wall forces the correct local vertical flux is $j_z=-D_z(z)[dn/dz+m_{b}gn/(k_{B}T)]$, so zero flux still gives the same sedimentation scale height. Walls can nevertheless invalidate that simple inference through adsorption, additional interaction energies, or sampling of inaccessible grain-centre positions. Optical position errors and exposure averaging also affect displacement readout; varying distance from the wall, observation lag and optical depth tests different possible errors.

For $t>0$, let $M_t={\sup }_{s\le t}{B}_s$ and $A_t={\sup }_{s\le t}|B_s|$. The Brownian reflection principle gives $M_t\stackrel{d}{=}|B_t|$, hence $\Vert M_t{\Vert }_2=\sqrt{t}$. Also $A_t\ge M_t$, with strict inequality on the positive-probability event $\{B_t<-\lambda ,M_t<\lambda \}$: reflection gives its probability as $\mathbb{P}(B_t>\lambda )-\mathbb{P}(B_t>3\lambda )>0$. The Doob L2 maximal inequality bounds $\Vert A_t{\Vert }_2\le 2\sqrt{t}$, so $\sqrt{t}<\Vert A_t{\Vert }_2\le 2\sqrt{t}$. At $t=0$ all quantities vanish and the lower inequality is equality.

// Target: probability-theory.bigb

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.

On a real Hilbert space $H$, this centered jointly Gaussian process is linear in its test vector $h$ and has the displayed covariance, with independent increments in time. Equivalently choose an orthonormal basis and independent real Brownian motions $W^j$, and define stochastic integration by $\int ⟨v_s,d{W}_s⟩={\sum }_j\int ⟨v_s,e_j⟩d{W}_s^j$ whenever $\int \Vert v_s{\Vert }^{2}ds<\infty$. In infinite dimension it need not be an $H$-valued random variable; the integral is the meaningful object.

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.

For standard Brownian motion started at zero in ${\mathbb{R}}^n$, every fixed orthogonal transformation preserves the process law. Its transformed increments remain independent centred Gaussian random vectors with covariance $(t-s)I$, and its paths remain continuous. Consequently its exit law from any centred sphere is normalized surface measure. Translations give the same assertion for spheres centred at its starting point.

For standard one-dimensional Brownian motion, the Itô formula for $B^k$ gives this recursion. Stop at exits from $[-n,n]$ to make the stochastic integral a true martingale. Gaussian tail integrability and the Doob Lp maximal inequality dominate all stopped powers on a fixed horizon, justifying the limit. Odd moments vanish and $\mathbb{E}{B}_t^{2k}=(2k)!t^k/(2^{k}k!)$.

// Target: probability-and-statistics.bigb

The process $B_t^k-\mathbb{E}{B}_t^k$ is a martingale only for $k=1,2$. For higher powers its finite-variation drift is $k(k-1)(B_t^{k-2}-\mathbb{E}{B}_t^{k-2})/2$, which is not identically zero. Subtracting the random compensator $k(k-1){\int }_0^t{B}_s^{k-2}ds/2$ instead produces a true martingale on each finite horizon.

// Target: probability-and-statistics.bigb

Planar Brownian motion started away from zero does not hit zero. Stopping the harmonic function $\log |x|$ on an annulus gives probability $(\log R-\log |x|)/(\log R-\log \epsilon )$ of reaching its inner circle before its outer circle. This tends to zero as $\epsilon \downarrow 0$. Exhausting the plane by outer circles proves point avoidance at finite times.

// Target: probability-and-statistics.bigb

Lévy area is the antisymmetric second-level integral of multidimensional Brownian motion. For distinct independent components the integrals are also Itô integrals. Combined with the symmetric tensor $\frac{1}{2}{B}_{s,t}^{\otimes 2}$, it gives the second level of enhanced Brownian motion. It is not continuously determined by the uniform norm of the Brownian path.

A Brownian motion starting outside a fixed affine subspace of codimension at least two almost surely never hits it. Choose a two-dimensional orthogonal projection annihilating the subspace's direction and with nonzero projected initial displacement. The projected process is planar Brownian motion starting away from zero, and a hit of the subspace would hit zero in that projection. The polar point for planar Brownian motion result excludes this. In three dimensions this shows avoidance of every fixed line when started outside it.

A Wiener sausage is the region swept out by a ball of fixed radius along a Brownian motion path. A deterministic continuous displacement $f(s)$ gives the variant ${\bigcup }_{s\le t}\mathcal{B}(B_s+f(s),r)$. On each bounded time interval it has finite expected volume: the region lies inside a ball whose radius is the path's maximum displacement plus $r$, and the Brownian maximum has Gaussian tail bounds and finite moments of every order.

Start traps at a unit-intensity Poisson random measure in Euclidean space and attach mutually independent Brownian motions, independent of the initial measure. For a deterministic target path $f$, independently thin initial positions according to whether their marked path comes within radius one of the target by time $t$. The retention probability $p_t(x)$ integrates, by the Tonelli theorem, to the expected volume of the sausage along $f-B$. Symmetry of Brownian motion makes this the expected volume of the sausage along $f+B$. The Poisson zero-count probability proves the displayed survival formula. This derives the formula directly from independent thinning, without assuming a separate marking or displacement theorem.

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.

An increment of a Brownian motion between deterministic times $s<t$ is $W_t-W_s$. It has normal distribution $N(0,t-s)$ and is independent of the filtration up to time $s$. Increments on disjoint time intervals are independent. Their covariance is the length of overlap of the corresponding intervals; this also follows from the Brownian covariance kernel.

At integer times, $B_n/\sqrt{n}$ has the same standard normal distribution for every $n$. The event that its limit superior exceeds a fixed finite level is a tail event of the independent unit-time Brownian increments, since changing any finite initial segment contributes only a term tending to zero. The Kolmogorov zero-one law applies. Its probability is positive because each fixed-time exceedance has the same positive probability, giving a positive probability of infinitely many exceedances by the decreasing union-of-tail-events argument. Thus its probability is one. Intersecting over integer levels proves the stated divergence, without the lower iterated-logarithm bound.

The ordinary time integral of standard Brownian motion is a centered Gaussian process with continuously differentiable paths and derivative $B_t$. For $0\le s\le t$, its covariance is $\mathbb{E}(I_s{I}_t)=s^2(3t-s)/6$, obtained by integrating the Brownian covariance $\min (u,v)$. Consequently $I_t$ has normal distribution $N(0,t^3/3)$. This time integral is distinct from the Itô integral with respect to Brownian motion.

The quadratic functional $Q={\int }_0^1{W}_t^{2}dt$ is finite almost surely. The Brownian half-integer sine expansion writes it as a sum of independent scaled squared standard normals. Its mean is $1/2$, and its Laplace transform is determined by a hyperbolic cosine product.

The independent squared-normal series for the integrated square of Brownian motion gives the product ${\prod }_n(1+2\lambda /((n-1/2{)}^2{\pi }^2){)}^{-1/2}$. The half-integer cosine product reduces this to $(\cosh \sqrt{2\lambda }{)}^{-1/2}$. Expanding at zero gives mean $1/2$ and variance $1/3$.

For standard Brownian motion and $\alpha ,t\ge 0$, $\beta >0$, the title gives the joint Laplace transform of the terminal square and the integrated square of Brownian motion. At $\beta =0$ the answer is $(1+2\alpha t{)}^{-1/2}$. To prove the formula, use the stochastic exponentialOn every finite interval $[0,T]$, $0<Z_t\le e^{\beta T/2}$, so the bounded local martingale criterion makes it a martingale. The Girsanov theorem changes $B$ into the zero-started Ornstein-Uhlenbeck process $dB=dW-\beta Bdt$ under the terminal density $Z_t$. Its terminal variance is $v=(1-e^{-2\beta t})/(2\beta )$. Multiplying $Z_t$ by $\exp [(\beta /2-\alpha )B_t^2-\beta t/2]$ recovers the integrand on the left of the title. The Gaussian quadratic exponential moment then gives $e^{-\beta t/2}[1-(\beta -2\alpha )v{]}^{-1/2}$, which simplifies to the title formula. At $\alpha =0$ this reduces to the Laplace transform of the integrated square of Brownian motion after Brownian scaling.

For standard Brownian motion and deterministic $T>0$, the process $R_s=B_T-B_{T-s}$, $0\le s\le T$, is standard Brownian motion on that interval in its own natural filtration. Disjoint reversed time intervals give independent centered normal increments with the correct variances. This is different from an arbitrary random-time shift, which can depend on future data. It is also different from the time inversion of Brownian motion transformation $t{B}_{1/t}$.

For a deterministic $h>0$, the processes $B_{h-t}-B_h$, $0\le t\le h$, and $B_{h+t}-B_h$, $t\ge 0$, are independent standard Brownian motions. The first uses reversed, sign-changed past increments; the second uses future increments independent of the entire past. The first process is not generally independent of $B_h$, but translating a range by $B_h$ preserves its Lebesgue measure pathwise.

A longitudinal Brownian field ${\mathcal{B}}_x(z)$ is a centered Gaussian random field withwhere $B$ is a valid transverse covariance kernel. For each fixed $z$ it is a scaled Brownian motion, and increments over disjoint longitudinal intervals are independent. Such a field drives a Markov model of wave propagation in a random medium. Its formal longitudinal derivative is Gaussian white noise in $x$, with transverse correlations described by $B$; it is not in general a Brownian sheet.

The times of local maxima of a Brownian motion path are dense in the half-line almost surely, and every such local maximum is a strict local maximum. Density follows from nowhere monotonicity of Brownian motion and the extreme value theorem. Strictness follows by checking that maxima over separated rational compact time intervals are unequal; the increment across their gap supplies an independent nondegenerate normal distribution.

Almost surely, no restriction of a Brownian motion path to an interval of positive length is a monotone function. On a fixed interval, monotonicity forces the independent centered normal random variables given by every dyadic subdivision's increments to have the same sign. The probability is at most $2^{1-2^m}$ for a subdivision into $2^m$ pieces. A countable union over rational intervals gives the simultaneous pathwise assertion.

For three-dimensional Brownian motion, every radius $r>0$ and every initial point satisfyIf the initial point is on one of the spheres, it is already a hit at time zero. If it is inside a lattice-centered open ball, its eventual exit hits that sphere. Otherwise choose at each integer time a nearest lattice center. The current displacement from it has norm at most $\sqrt{3}/2$, so the multivariate normal distribution of the next unit increment gives a uniform positive chance of entering its radius-$r/2$ open ball. The geometric tail bound from a uniform escape probability makes eventual entry certain, and continuity forces a boundary crossing. This periodic-family recurrence does not contradict the Brownian sphere-hitting probability in dimension three for one fixed sphere.

For every fixed mesh $1/k$, independent increments with the normal distribution imply increments of absolute value greater than one occur infinitely often almost surely, by the second Borel-Cantelli lemma. A countable intersection over $k$ contradicts uniform continuity on $[0,\infty )$.

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.

The natural Brownian filtration is the natural filtration generated by a Brownian motion. Its usual augmentation adds null events and makes it right-continuous; the Brownian martingale representation theorem holds for this augmented natural filtration. A larger Brownian filtration can preserve independence of future increments while containing additional randomness, so it need not have martingale representation with respect to the specified Brownian motion.

A martingale representation theorem states conditions under which every martingale in a filtration can be represented as a stochastic integral with respect to one or more fundamental martingales.

The conditional expectation of $\sin B_T$ at time $t$ is $e^{-(T-t)/2}\sin B_t$, by independent Brownian increment. Itô's formula cancels its time derivative against its second spatial derivative and leaves the displayed stochastic integral. The integrand is bounded, so Itô isometry verifies its square integrability.

Every square-integrable random variable measurable with respect to a Brownian filtration can be written as its expectation plus an Itô integral against that Brownian motion. Equivalently, every square-integrable martingale in that filtration has the formfor a predictable square-integrable process $H$.

In the completed natural Brownian filtration, a bounded continuously differentiable $\varphi$ with bounded derivative satisfiesThe integrand is unique up to $d\mathbb{P}dt$-almost everywhere equality. Gaussian integration by parts first gives the duality between the payoff and every square-integrable Itô integral. The Brownian martingale representation theorem then identifies the integrand as the projection of ${\varphi }^{\prime }(W_T)1_{\{t\le T\}}$ onto the predictable processes in the product $L^2$ space. This explains the conditional expectation in the formula, rather than a nonadapted terminal derivative.