Planar Brownian motion

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

A planar Brownian motion range has zero two-dimensional Lebesgue measure almost surely. Assuming finite expected area, Brownian scaling gives half that expected area for each half-time range. Inclusion-exclusion then makes their expected overlap zero. Translate their ranges by the midpoint position; the independent Brownian arms at a deterministic time make the expected overlap the integral of the squared point-hitting probabilities for a half-time range. Those probabilities vanish almost everywhere, so the half-time expected area, and hence the full-time expected area, vanish. This argument does not assert zero hitting probability for every individual point solely from an almost-everywhere integral conclusion.

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 planar Brownian motion started at $1$, continuous winding angles sampled at the displayed radii have independent stationary increments. At $T_r$, apply the Strong Markov property, rotate by the current angle, and rescale space by $e^r$ and time by $e^{-2r}$. The future radial-level winding increment has the original law and is independent of the past. The logarithmic radius and winding angle are independent Brownian coordinates on the common clock $\int |B_s{|}^{-2}ds$. At radial passage they are therefore $U_{{\eta }_r}=-r$ and $V_{{\eta }_r}$, where ${\eta }_r$ is a Brownian first-passage time. Its Laplace transform gives the displayed characteristic function. The level process is a symmetric Cauchy process; replacing the passage clock by its right-continuous version supplies a càdlàg modification without changing any fixed-level law.

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.

For a compact H-hull $A$ whose closure avoids a real boundary point $x$, the Brownian excursion in the upper half-plane from $x$ to infinity avoids $A$ with the displayed probability. From an interior point $z$, the corresponding Doob h-transform avoidance probability is $\mathrm{Im}{g}_A(z)/\mathrm{Im}z$. Letting $z=x+i\epsilon$ approach the boundary and using the Schwarz reflection principle gives the derivative. In particular $0<g_A^{\prime }(x)\le 1$ at every regular real boundary point away from the hull.

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.

For planar Brownian motion started at radius $\epsilon$ and stopped on first hitting radius one, the net winding count divided by $\log \epsilon$ converges to a standard Cauchy random variable divided by $2\pi$. The complex exponential maps Brownian coordinates in the left half-plane to the punctured unit disk after the conformal Brownian clock. The unwrapped argument at exit is the vertical half-plane exit coordinate, with the Cauchy exit law from a Brownian half-plane. Counting integer turns introduces only a uniformly bounded error in winding units.

Planar Brownian motion started at $(-x,0)$ exits the left half-plane with ordinate distributed as $x$ times a standard Cauchy random variable. Conditional on the independent horizontal hitting time, the ordinate is normal with variance equal to that time. The inverse-square Gaussian law of Brownian first passage turns this into a ratio of independent standard normals. Its exit density is $x/(\pi (x^2+y^2))$.

For independent one-dimensional Brownian motions $\beta ,\theta$, the real and imaginary parts of $Z$ are continuous local martingales with equal bracket $A$ and zero cross bracket. The Divergence of a driftless Brownian exponential clock makes $A_{\infty }=\infty$, and the common-clock time change of orthogonal local martingales makes $Z_{A^{-1}(r)}$ planar Brownian motion started at $1$. The exponential never vanishes, proving that this Brownian motion never hits zero. Recurrent visits of $\beta$ to arbitrarily negative levels prove visits arbitrarily near zero at arbitrarily large Brownian times.

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 $D$.

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.

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.