Brownian motion

A standard Brownian motion starts at zero, has almost surely continuous paths, and has independent Gaussian increments

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 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 $t{B}_{1/t}$.

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.

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.

For every $\theta \in \mathbb{R}$ and standard Brownian motion $B$,is a positive martingale. Applying Itô formula cancels the drift, and its expectation is one by the moment-generating function of the normal distribution.

For Brownian motion started at $x$, the time integral is a Gaussian random variable withThe variance follows by integrating the covariance kernel $\mathrm{cov}(B_r,B_s)=\min (r,s)$ twice.

The displayed continuous centered Gaussian process, with value zero at time zero, has the Brownian covariance kernel. Hence it is a Brownian motion in its own natural filtration. More generally, replacing three by $c$ gives covariance $s+c(c-3)(s/2-s^2/(6t))$ for $0<s\le t$, so three is the only nonzero valid coefficient. Its conditional future increment in the original Brownian filtration is $3(t-s)t^{-1}(s^{-1}{\int }_0^s{W}_{u}du-W_s)$, and is generally nonzero.

The diffusion coefficient sets the linear growth of mean-square displacement. In one dimension the convention $\mathbb{E}[(X_t-X_0{)}^2]=2Dt$ is common.

Solutal diffusivity is the diffusion coefficient governing transport of a dissolved constituent down its concentration gradient. For uniform $D$ and a stationary solvent, the diffusion equation is $C_t=D{\nabla }^{2}C$. It need not equal the thermal diffusivity.

If $B$ is Brownian motion in ${\mathbb{R}}^d$ and $U$ is an orthogonal matrix, then $UB$ is Brownian motion. Its increments remain independent centered Gaussian vectors and have covariance $(t-s)UI{U}^T=(t-s)I$.

Two independent one-dimensional Brownian motions started at $-a$ and $a$ meet at the first time a Brownian motion started at $\sqrt{2}a$ hits zero. The meeting time has densityAn orthogonal sum coordinate is independent of this hitting time and makes the meeting position conditionally $N(0,u/2)$.

Wiener measure is the probability distribution of Brownian motion on a continuous-path space, usually $C_0([0,T])$ or $C_0([0,\infty ))$.

On $C_0([0,T])$, the Cameron-Martin space consists of the absolutely continuous functions $h$ with $h(0)=0$ and $h^{\prime }\in L^2[0,T]$. Its norm is

Translation of Wiener measure by a path $h$ is equivalent to Wiener measure exactly when $h$ belongs to the Cameron-Martin space of Wiener measure. In that case the density isTranslation by any continuous path outside that space produces a singular measure.

For the Cameron-Martin space of Wiener measure path $h(s)=\mu s$, the Cameron-Martin theorem says that translation of Brownian motion by this linear function changes its law through time $t$ by the density

For ${\tau }_{a,b}=\inf \{t\ge 0:B_t+bt=a\}$ with $a,b>0$,The Cameron-Martin likelihood at a driftless path hitting $a$ at time $t$ is $e^{ab-b^{2}t/2}$.

On every compact time interval, a Brownian path is almost surely a Hölder continuous function of every exponent $\alpha <1/2$, and of no exponent $\alpha \ge 1/2$. In particular, $B_t=o(t^{\gamma })$ as $t\downarrow 0$ for every $\gamma <1/2$ after choosing a Hölder exponent strictly between $\gamma$ and $1/2$.

The zero set $\{t\ge 0:B_t=0\}$ of one-dimensional Brownian motion is almost surely closed, uncountable, and has zero Lebesgue measure. The last fact follows from Tonelli theorem because $\mathbb{P}(B_t=0)=0$ for every $t>0$.

If $B$ is standard Brownian motion, then $-B$ is also standard Brownian motion. More generally, applying an orthogonal transformation to a multidimensional Brownian motion preserves its law.

For $a>0$ and $T_a=\inf \{t\ge 0:B_t+t=a\}$,Changing measure with the density $e^{-B_t-t/2}$ turns $B_t+t$ into Brownian motion. At $T_a$ the density equals $e^{-a+T_a/2}$, and Brownian motion hits $a$ almost surely.

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 a three-dimensional Bessel process started at one, $R$ stays strictly positive and satisfies $dR=dW+R^{-1}dt$. The Itô formula makes its reciprocal a local martingale, but $\mathbb{E}[R_t^{-1}]=2\Phi (1/\sqrt{t})-1<1$ for $t>0$. Thus the reciprocal is a strict local martingale. It can deflate the market with constant bank account and stock price $R$ without yielding a true pricing density for the bank account.

For a Bessel process of dimension $\delta$, started at $x>0$, the Itô formula giveson its stochastic interval before hitting zero. Thus the power is a local martingale after localization away from zero. For $\delta \ge 2$ this interval is the whole lifetime; at $\delta =2$ the power is constant. The unqualified assertion fails for a reflecting Bessel process of dimension one, which acquires local time of a semimartingale at zero.

For a Bessel process of dimension $\delta >2$ started at $x>0$, its scale function of a one-dimensional diffusion gives ${\mathbb{P}}_x({\tau }_r<\infty )=(r/x{)}^{\delta -2}$ for $0<r<x$. In particular, ${\inf }_{t\ge 0}{X}_t>0$ almost surely. The minimum is attained because the process tends to infinity; the hitting formula alone already implies positivity of the infimum.

A Bessel process of dimension $d>0$ started away from zero hits zero almost surely exactly when $d<2$. For $d\ne 2$, its scale function of a one-dimensional diffusion is $s(x)=x^{2-d}$; for $d=2$ it is $s(x)=\log x$. The boundary hitting probability from a diffusion scale function then gives the classification by taking the inner boundary to zero and the outer boundary to infinity.

If $X$ is a Bessel process of dimension $d$ started at $x$ and $r>0$, thenis a Bessel process of dimension $d$ started at $rx$. This follows from Brownian scaling in the stochastic differential equation and uniqueness in law.

For $Z_t=e^{B_t+at}$, use the clock $C_t={\int }_0^t{Z}_s^{2}ds$. After the inverse time change, the Dambis-Dubins-Schwarz theorem givesso the time-changed process is a Bessel process of dimension $d=2a+2$. Its lifetime is $C_{\infty }$, which is finite almost surely exactly when $d<2$.

If $X$ is a Bessel process of dimension $d$ and $\alpha >0$, then $X^{\alpha }$, after the clock ${\alpha }^2\int X_s^{2\alpha -2}ds$, is a Bessel process of dimension

One-dimensional reflected Brownian motion on $[0,\infty )$ has the same law as $|B_t|$. The Tanaka formula represents it as a Brownian motion plus a nondecreasing local time of a semimartingale that grows only at zero.

The occupation time of a measurable set $A$ through time $t$ is ${\int }_0^t{1}_{\{B_s\in A\}}ds$. The occupation-times formula expresses such integrals through local time of a semimartingale. Recurrence and the Strong Markov property imply that one-dimensional Brownian motion spends an unbounded total time in every nonempty open interval.

A one-dimensional Brownian motion spends an infinite total time in every nonempty open interval almost surely. To prove this for $(-1,1)$ after first hitting zero, alternate exits from that interval with returns to zero. By recurrence of one-dimensional Brownian motion all these stopping times are finite, and the Strong Markov property makes the successive exit durations independent and identically distributed random variables that are strictly positive. Some fixed positive duration is exceeded with positive probability, so the Borel-Cantelli lemma makes the sum of the durations infinite. Translate and scale for other intervals. In particular, if a nonnegative function is bounded below by a positive constant on such an interval, its infinite-horizon Brownian occupation time integral diverges. Recurrent visits alone, without this duration argument, would not prove divergence of an integral.

For a continuous semimartingale $X$ with local time of a semimartingale $L_t^a(X)$,for every nonnegative measurable function $f$.

Planar Brownian motion is the random vector $(B_t^{(1)},B_t^{(2)})$ formed from two independent standard one-dimensional Brownian motions.

Started at radius one, planar Brownian motion avoids the origin, and its logarithmic radius is a continuous local martingale because the logarithmic potential is harmonic away from zero. It is a strict local martingale: at deterministic $t>0$, the angular average of a logarithmic potential gives $\mathbb{E}\log |B_t|={\int }_1^{\infty }{e}^{-r^2/(2t)}dr/r>0$, whereas the initial value is zero. Absolute integrability follows from integrability of $r|\log r|$ near zero and the Gaussian tail. Its positive expectation does not contradict nonnegative-local-martingale bounds, since this process takes both signs.

For independent coordinate Brownian motions, the stochastic area with the stated orientation is $Z_t={\int }_0^t{Y}_{s}d{X}_s-X_{s}d{Y}_s$. Its bracket is ${\int }_0^t(X_s^2+Y_s^2)ds$. Reversing orientation changes its sign but not its law or quadratic variation.

The radial martingale $H=\int XdX+YdY$ and the planar Brownian stochastic area $Z=\int YdX-XdY$ have common bracket $A=\int (X^2+Y^2)ds$ and zero cross variation, because their integrand vectors $(X,Y)$ and $(Y,-X)$ are orthogonal. The Itô formula also gives $X_t^2+Y_t^2=2H_t+2t$.

Jointly time-change the radial and area martingales by the inverse of $A=\int R^{2}ds$. The bracket matrix becomes $u{I}_2$, so the Lévy characterization of multidimensional Brownian motion gives two independent Brownian coordinates. The clock is strictly increasing; if it were finite at infinity, the radial martingale would converge and $R_t^2=2t+O(1)$ would force its integral to diverge. Thus $R_t^2=2W_{A(t)}+2t$ and $Z_t=B_{A(t)}$ with $W,B$ independent.

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.

For a compact H-hull avoiding zero, let ${\psi }_A$ map its complement to the half-plane, fix zero, and have derivative one at infinity. The Brownian excursion in the upper half-plane avoids $A$ with probability ${\psi }_A^{\prime }(0)$. From an interior starting point $z$, the corresponding transformed process avoids $A$ with probability $\mathrm{Im}{g}_A(z)/\mathrm{Im}z$; taking $z=i\epsilon$ and $\epsilon \downarrow 0$ gives the boundary derivative.

If $B$ is planar Brownian motion in a domain $D$ and $f:D\to D^{\prime }$ is a nonconstant holomorphic function, then $f(B)$, run with the clock ${\int }_0^t|f^{\prime }(B_s){|}^{2}ds$, is planar Brownian motion in $D^{\prime }$ until the relevant exit time. In particular, conformal maps transport harmonic measure between planar domains.

The map $z\mapsto z^{\pi /\alpha }$ takes a wedge of opening $\alpha$ to a half-plane and the circular boundary of radius $r$ to radius $r^{\pi /\alpha }$. Conformal invariance of planar Brownian motion preserves which boundary part is reached first, although it changes the clock. Use the branch defined by the wedge argument; the origin is a polar point for planar Brownian motion, so localization handles the vertex.

A planar Brownian motion starting at $1$ exits the disk of radius $r>1$ before leaving a wedge of opening $\alpha \in (0,2\pi ]$, centered on the positive axis, with probability $(4/\pi )\arctan (r^{-\pi /\alpha })$. The power-map reduction for Brownian exit from a wedge, the reflection identity for Brownian exit from a half-disc and the Möbius calculation of circular Brownian exit give the formula. Its large-radius decay exponent is $\pi /\alpha$.

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 planar Brownian motion exits a conformal image of a bounded planar domain in finite time almost surely. If the image lifetime were infinite, the inverse derivative is bounded away from zero on a small interior disc. Recurrence of planar Brownian motion and the Strong Markov property give infinite Brownian occupation time there, forcing the inverse conformal Brownian clock to diverge. It cannot exceed the finite original exit time. The expected image exit time can nevertheless be infinite.

Applying any fixed orthogonal transformation to planar Brownian motion produces another planar Brownian motion. In particular, rotationally invariant initial data produce rotationally invariant hitting distributions.

Planar Brownian motion almost surely visits every neighborhood of every point. Equivalently, it hits every disc almost surely, although it does not hit any prescribed point with positive probability.

A prescribed point is almost surely never visited by planar Brownian motion started elsewhere. The harmonic function $\log |z-w|$ gives an annular hitting probability that tends to zero as the inner radius decreases to zero. This does not contradict recurrence of planar Brownian motion, which concerns neighbourhoods.

If $f$ is a continuous probability density on ${\mathbb{R}}^2$ and $B$ is planar Brownian motion, thenalmost surely. The density is bounded below on some disc, recurrence supplies infinitely many visits to a smaller concentric disc, and the Strong Markov property gives infinitely many visits of a fixed positive duration.

To couple Brownian motions starting at points exchanged by reflection in a hyperplane, reflect one path until the first path hits that hyperplane and then make the paths agree. Reflection invariance and the strong Markov property give the correct marginal laws.

The harmonic measure of a boundary set $A\subseteq \partial D$ viewed from $x\in D$ is the probability that Brownian motion started at $x$ first exits $D$ through $A$. As a function of $x$, it is harmonic in $D$.

This capacity measures the asymptotic probability that planar Brownian motion hits a compact H-hull before the real axis. It scales linearly with length and is distinct from half-plane capacity. A vertical slit of height $h$ has capacity $2h$; a half-disc of radius $r$ has capacity $4r$. Monotonicity of Brownian hitting probabilities gives $\mathrm{cap}(K)\le 4\mathrm{rad}(K)$, where the radius is that of the least real-centred enclosing half-disc. For finite slit hulls the capacity is the length of the image of the two-sided absorbing boundary under the mapping-out function.

For a connected slit joining $0$ to $x+iy$ with $x^2+y^2=1$, reflect across the vertical line through $x$. The two slits separate the vertical segment below the common tip and the intervening real boundary from infinity. Brownian reflection symmetry bounds the corresponding model hitting probability by twice the original hull-hitting probability. Mapping out the vertical segment gives total boundary-image length $1+y$ for its two banks together with the real interval between $0$ and $x$. The harmonic-measure asymptotic at infinity gives the displayed bound. This argument requires a separating connected barrier; radius gives no positive lower bound for disconnected harmonic hull capacity.

For $c=\sqrt{1-{\epsilon }^2}$, three vertical slits of height $\epsilon$ rooted at $-c,0,c$ form a compact H-hull of enclosing radius one. Subadditivity of harmonic hull capacity gives capacity at most $6\epsilon$, which can tend to zero. Thus a uniform lower bound requires additional geometric hypotheses.

When a finite union is a compact H-hull, hitting that union before the real axis implies hitting at least one member before the real axis. The union bound followed by the defining limit proves subadditivity of harmonic capacity from infinity in the upper half-plane.

For a compact H-hull and a Borel subset of the intrinsic boundary of a simply connected domain, hydrodynamic normalization at infinity and the Poisson kernel for the upper half-plane give this limit as $y\to \infty$ with $x/y\to 0$. After multiplying the kernel by $\pi y$, it tends pointwise to one and is eventually uniformly bounded. Dominated convergence proves the finite-measure case and Fatou's lemma proves the infinite-measure case.

A planar Brownian motion beginning at $a\in (0,1)$ in the unit disc has circular exit law obtained by pulling back uniform angular measure through ${\psi }_a(w)=(w-a)/(1-aw)$. This Möbius transformation sends the starting point to zero. The right semicircle maps to an arc with endpoints at angles $\pm (\pi /2+2\arctan a)$, giving positive-half-plane exit probability $1/2+(2/\pi )\arctan a$.

For planar Brownian motion starting on the positive real axis inside a disk, let $T$ be circular exit and $S$ first contact with the imaginary axis. Reflecting the path after $S$ fixes the disk and swaps positive and negative circular exits. The Strong Markov property makes their probabilities equal on $S<T$. Therefore $\mathbb{P}(T<S)=\mathbb{P}(\mathrm{Re}{B}_T>0)-\mathbb{P}(\mathrm{Re}{B}_T<0)$.

For planar Brownian motion started at $x+iy$ in $0<y<\pi$, this is its unconditional exit density on the bottom boundary. Its mass is $1-y/\pi$; the remaining mass $y/\pi$ lies on the top boundary. The conformal bijection $z\mapsto e^z$, the Poisson kernel for the upper half-plane and the boundary Jacobian determinant derive the formula. Conditional bottom-boundary density requires division by its mass.

For planar Brownian motion started at $(x,y)$ with $y>0$, the probability that its first hit of the real axis lies in $(0,\infty )$ isThis is the bounded harmonic function in the upper half-plane with boundary values zero on the negative half-axis and one on the positive half-axis.

For planar Brownian motion started at $x$ with $0<r<|x|<R$,This is optional stopping applied to the harmonic function $\log |x|$ on the annulus.

For planar Brownian motion started at $iy$, the distribution of its first hit of the closed unit disc converges as $y\to \infty$ to normalized arc length on the unit circle. Inversion maps the exterior disc to the punctured disc and the starting point to $-i/y\to 0$, where rotational invariance makes harmonic measure uniform.

Two Brownian motions in ${\mathbb{R}}^d$ can be coupled by letting corresponding coordinates evolve independently until they meet and then using the same increments in that coordinate. One-dimensional recurrence makes every coordinate meeting time finite almost surely, so the two processes eventually coalesce while each marginal remains Brownian.

For Brownian motion, $B_t/t\to 0$ almost surely as $t\to \infty$. Apply the strong law of large numbers to unit-time increments and control the oscillations on the intervening unit intervals.

Brownian loop measure is a conformally invariant sigma-finite measure on unrooted planar Brownian loops. Taking outer boundaries connects it to conformal-restriction measures on self-avoiding loops.

In the unit disc, the relevant pointed Brownian loop measure is rooted at $0$ and normalized by its conformal deficit: for simply connected $U$ containing $0$, its mass of loops not contained in $U$ is minus the logarithm of the conformal radius. Its outer-boundary pushforward gives the corresponding pointed simple-loop restriction measure. It is not the full unrooted loop measure or a probability law of a fixed-duration Brownian bridge.

A sigma-finite measure on simple closed curves has conformal restriction if restricting to loops in a simply connected domain and then applying a conformal map agrees with first transporting the domain and restricting there. The usual nontrivial measures are finite on bounded macroscopic cutoff windows and are determined up to normalization by their pointed conformal deficits.

For simple loops surrounding a marked point, containment in two simply connected domains is containment in their intersection component containing that point. Finite deficits therefore determine all finite intersections of avoidance events by inclusion-exclusion. Restrict to finite windows $H_r={\mathbf{L}}_D\setminus {\mathbf{L}}_{rD}$ and use the sigma-finite uniqueness theorem for measures; as $r\downarrow 0$ these windows exhaust the pointed loop space. Finiteness of the windows is essential to this argument.

Brownian paths are almost surely nowhere differentiable. An increment over duration $h$ is typically of order $\sqrt{h}$, so its difference quotient is of order $h^{-1/2}$ rather than approaching a finite limit.

A continuous local martingale $M$ with $M_0=0$ is standard Brownian motion exactly when its quadratic variation satisfies $[M{]}_t=t$.

A continuous vector-valued local martingale $M$ starting at zero is standard multidimensional Brownian motion if its quadratic covariations satisfy $[M^i,M^j{]}_t={\delta }_{ij}t$. For each deterministic vector $\xi$, the Itô formula makes $\exp (i\xi \cdot M_t+|\xi {|}^{2}t/2)$ a complex local martingale. Its absolute value is bounded on each finite horizon, so it is a true martingale. Its conditional expectation identity givesThis conditional characteristic function identifies a centered multivariate normal distribution with covariance $(t-s)I$, independent of the past. Thus the coordinates are independent Brownian motions. The converse follows immediately from the standard coordinate quadratic covariations.

For every $c>0$, the process $(c^{-1/2}{B}_{ct}{)}_{t\ge 0}$ is again Brownian motion.

One-dimensional Brownian motion satisfiesalmost surely. Its continuous path therefore visits every point, including zero, infinitely often.

Brownian motion in ${\mathbb{R}}^d$ is transient for $d\ge 3$: its distance from the origin tends to infinity almost surely. In dimension three, the positive local martingale $|B_t{|}^{-1}$ and Brownian scaling give a short proof.

For standard three-dimensional Brownian motion started at $x$ with $r<|x|<R$,The harmonic function $x\mapsto 1/|x|$ gives a bounded stopped martingale in the annulus, so the optional stopping theorem gives the formula. Taking $R\to \infty$ yields ${\mathbb{P}}_x({\tau }_r<\infty )=r/|x|$ for $|x|>r$. This limit uses path continuity on the finite interval before a hit, rather than assuming transience to prove transience.

For a domain $D$, the Brownian exit time is $T_D=\inf \{t\ge 0:B_t\notin D\}$. It is a stopping time, and it has finite expectation when $D$ is bounded.

For one-dimensional Brownian motion started at zero and $T={\tau }_b\wedge {\tau }_{-a}$ with $a,b>0$,Optional stopping of $B_t$ and $B_t^2-t$ proves the two identities.

For standard Brownian motion started at zero and first exit ${\tau }_x$ from $(-x,x)$, the Laplace transform is $1/\cosh (x\sqrt{2\lambda })$ for $\lambda >0$. The stochastic process $e^{-\lambda t}\cosh (\sqrt{2\lambda }{B}_t)$ is the average of two copies of the Exponential martingale for Brownian motion. Its values stopped at $t\wedge {\tau }_x$ are bounded, so bounded-time application of the optional stopping theorem followed by dominated convergence evaluates the transform without assuming uniform integrability of an unstopped exponential martingale on the infinite horizon.

For standard Brownian motion started at zero, let ${\tau }_x$ be its first exit from $(-x,x)$, with $x>0$. Bounded-time stopping of $B_t^2-t$ first proves integrability of the exit time and then gives its mean $x^2$. Stopping the quartic Hermite polynomial martingale gives a uniform bound on $\mathbb{E}(t\wedge {\tau }_x{)}^2$, proving second-moment integrability before passing to the limit. The resulting second moment is $5x^4/3$ and the variance is $2x^4/3$.

The exit time conditioned on leaving through the upper endpoint satisfiesOptional stopping of the cubic martingale $B_t^3-3t{B}_t$, together with the lower moments of the interval exit, gives the formula.

For a suitable twice differentiable function $u$ and an integrable stopping time $T$,