Multidimensional Brownian motion is simultaneously a vector of independent one-dimensional Brownian motions and an isotropic diffusion. Linear transformations generate correlated Brownian motions; radial functions reveal the dimension-dependent transition from recurrence to transience.
Definition and covariance structure
Definition. A standard \(d\)-dimensional Brownian motion \(B_t\) satisfies \(B_0=0\), has continuous paths and independent increments, and \[ B_t-B_s\sim N_d(0,(t-s)I_d), \qquad0\le s<t. \] Equivalently, its coordinates are independent standard one-dimensional Brownian motions.
Proof of equivalence. Independent coordinate Brownian motions clearly produce the stated vector increment law. Conversely, every increment vector has diagonal covariance and is jointly Gaussian, so its coordinates are independent. Applying this to all finite collections of disjoint time increments gives independent coordinate processes.
Proposition. \[ \mathbb E[B_sB_t^\top]=(s\wedge t)I_d. \] For every \(u\in\mathbb R^d\), \(u^\top B_t\) is a one-dimensional Brownian motion with variance parameter \(\|u\|^2\): \[ \operatorname{Var}(u^\top B_t)=t\|u\|^2. \]
Orthogonal and linear transformations
Theorem. If \(Q\) is orthogonal, then \(QB\) is standard \(d\)-dimensional Brownian motion. If \(A\in\mathbb R^{m\times d}\), then \(X_t=AB_t+\mu t\) has Gaussian independent increments with \[ \mathbb E(X_t-X_s)=\mu(t-s), \qquad \operatorname{Cov}(X_t-X_s) =(t-s)AA^\top. \]
Proof. Linear maps preserve continuity, Gaussianity, and independence of increments. Covariances transform as \[ A\{(t-s)I_d\}A^\top=(t-s)AA^\top. \] For orthogonal \(Q\), this is \((t-s)I_d\), so the law is unchanged.
Radial martingales and Bessel processes
Proposition. \[ \|B_t\|^2-dt \] is a martingale. Away from zero, the radial process \(R_t=\|B_t\|\) satisfies formally \[ \mathrm{d}R_t = \mathrm{d}\beta_t+\frac{d-1}{2R_t}\,\mathrm{d} t, \] where \(\beta\) is a one-dimensional Brownian motion. This is the \(d\)-dimensional Bessel process.
Proof. Summing the coordinate martingales \((B_t^i)^2-t\) gives the first statement. For the second, apply multidimensional Ito to \(r(x)=\|x\|\): \[ \nabla r(x)=\frac x{\|x\|}, \qquad \Delta r(x)=\frac{d-1}{\|x\|}. \] The stochastic term \(\int(B_s/\|B_s\|)^\top\mathrm{d}B_s\) has bracket \(t\) before hitting zero and is therefore Brownian after localization.
Harmonic functions and hitting probabilities
Theorem. Let \(d\ge3\), \(0<r<\|x\|<R\), and let \(\tau_r,\tau_R\) be the first hitting times of the spheres of radii \(r,R\). Then \[ P_x(\tau_r<\tau_R) = \frac{\|x\|^{2-d}-R^{2-d}} {r^{2-d}-R^{2-d}}. \] Letting \(R\to\infty\), \[ P_x(\tau_r<\infty) =\left(\frac r{\|x\|}\right)^{d-2}<1. \]
Proof. The radial function \(h(x)=\|x\|^{2-d}\) satisfies \(\Delta h=0\) away from zero. Therefore \(h(B_{t\wedge\tau_r\wedge\tau_R})\) is a bounded martingale. Optional stopping gives \[ \|x\|^{2-d} =r^{2-d}p+R^{2-d}(1-p), \] where \(p=P_x(\tau_r<\tau_R)\). Solve for \(p\), then let \(R\to\infty\).
Theorem (dimension dichotomy). Brownian motion is recurrent in dimensions \(d=1,2\) and transient in dimensions \(d\ge3\). More precisely, for \(d\ge3\) every bounded ball is eventually left forever almost surely.
Proof outline. The preceding hitting probability is strictly below one in \(d\ge3\), and the strong Markov property plus exits through expanding spheres yields transience. In \(d=2\), replace \(\|x\|^{2-d}\) by the harmonic function \(\log\|x\|\). The annulus formula becomes \[ P_x(\tau_r<\tau_R) =\frac{\log R-\log\|x\|} {\log R-\log r}\to1 \] as \(R\to\infty\), proving repeated returns to every disk. The one-dimensional case follows from level hitting.
Exit from a ball
Proposition. If \(\tau_R=\inf\{t:\|B_t\|=R\}\) and \(\|x\|<R\), then \[ \mathbb E_x\tau_R=\frac{R^2-\|x\|^2}{d}. \]
Proof. Stop the martingale \(\|B_t\|^2-dt\) at \(\tau_R\wedge n\). The stopped position is bounded by \(R\), and monotone convergence handles the times: \[ \mathbb E_x\|B_{\tau_R}\|^2-d\mathbb E_x\tau_R =\|x\|^2. \] Since \(\|B_{\tau_R}\|=R\), solve for the expectation.