Multidimensional Brownian Motion from Independent Coordinates

Building Brownian Motion Coordinate by Coordinate

A particle moving in several spatial dimensions has a position vector rather than a single real-valued position. The simplest model makes each coordinate an independent one-dimensional Brownian motion. Independence is what gives the vector process the standard isotropic covariance: over an interval of length \(h\), every coordinate has variance \(h\), and distinct coordinates have covariance zero. This construction lets us calculate vector increment laws and covariance matrices directly from the one-dimensional process.

Work on a probability space \((\Omega,\mathcal{F},\mathbb{P})\), and let \(d\ge1\) be an integer. A one-dimensional standard Brownian motion \(W\) starts at zero, has almost surely continuous paths, has independent increments, and satisfies \(W_t-W_s\sim N(0,t-s)\) for \(0\le s<t\). For the vector process, use the natural filtration \(\mathcal{F}_t=\sigma(W^r_u:0\le u\le t,\ 1\le r\le d)\). No conditioning is needed below, but this filtration records the information available up to time \(t\).

For a random vector \(Y\) with finite second moments, its covariance matrix is \(\operatorname{Cov}(Y)=\mathbb{E}[(Y-\mathbb{E}Y)(Y-\mathbb{E}Y)^T]\). Write \(I_d\) for the \(d\) by \(d\) identity matrix. A Gaussian vector with mean \(m\) and covariance matrix \(\Sigma\) is denoted \(N_d(m,\Sigma)\). When \(h>0\), the matrix \(hI_d\) is positive definite; when \(h=0\), the corresponding zero increment is degenerate at the zero vector.

Proof. Each coordinate starts at zero and has continuous paths with probability one. There are finitely many coordinates, so the intersection of their probability-one continuity events still has probability one; on this event the vector path is continuous. For disjoint time intervals, the increments in any one coordinate are independent by the one-dimensional Brownian property. The coordinate processes are independent of one another, so the collections of increments from different coordinates are also independent. Together these facts give independence of the vector increments over disjoint intervals.

Fix \(s<t\) and let \(h=t-s>0\). For each \(r\), the coordinate increment \(W^r_t-W^r_s\) has law \(N(0,h)\). These \(d\) increments are independent, so their joint density at \(x=(x_1,\ldots,x_d)^T\) is the product of the one-dimensional densities: \((2\pi h)^{-d/2}\exp(-\sum_{r=1}^d x_r^2/(2h))\). This is the density of \(N_d(0,hI_d)\). Its mean is zero and its covariance matrix has diagonal entries \(h\) and off-diagonal entries zero, as also follows from independence of the coordinates. If \(s=t\), the increment equals zero, so the stated distribution is the degenerate Gaussian with covariance \(0I_d\). \(\square\)

The theorem gives the vector law as well as the coordinate laws. The coordinates of an increment are independent normal variables, but a general linear combination of them need not have independent components. To compute such combinations, the covariance matrix provides a compact and reliable method.

Covariances Across Coordinates and Times

Proof. If \(r\ne q\), the two coordinate processes are independent, so \(W^r_s\) and \(W^q_t\) are independent and their covariance is zero. If \(r=q\), first suppose \(s\le t\). Write \(W^r_t=W^r_s+(W^r_t-W^r_s)\). The increment \(W^r_t-W^r_s\) is independent of \(W^r_s\), and both variables have mean zero. Therefore \(\operatorname{Cov}(W^r_s,W^r_t)=\operatorname{Var}(W^r_s)+\operatorname{Cov}(W^r_s,W^r_t-W^r_s)=s+0=s\). If \(t<s\), interchange the two times in the covariance, which is symmetric, to obtain \(t\). Thus the covariance is \(\min(s,t)\) when \(r=q\), and zero otherwise. These are exactly the entries of \(\min(s,t)I_d\). \(\square\)

For a single interval, the proposition also gives the increment covariance by expanding the covariance of a difference. The independent-increment theorem supplies the full Gaussian law; the covariance calculation is a useful separate check. When several times are stacked into a longer vector, the same formula fills in every matrix entry. The ordering of the stacked variables must be stated, because it determines where those entries appear.

Linear Combinations of Vector Increments

If \(\Delta B=B_t-B_s\) and \(A\) is a fixed real \(k\) by \(d\) matrix, then \(A\Delta B\) is a Gaussian vector with mean zero and covariance \(A(hI_d)A^T=hAA^T\), where \(h=t-s\). This follows from the affine transformation rule for Gaussian vectors. It is often more efficient to use this matrix identity than to calculate each covariance from scratch. In particular, two linear combinations that share a coordinate can be correlated, even though the original Brownian coordinates are independent.

Why Independence Is Essential

Having standard Brownian motion in each coordinate is not by itself enough to produce standard Brownian motion in \(\mathbb{R}^d\). The coordinates must also be independent. For example, take \(d=2\) and define \(B_t=(W_t,W_t)^T\) using one standard one-dimensional Brownian motion \(W\). Each coordinate separately is a standard Brownian motion, and the vector path is continuous. But for \(h=t-s>0\), the increment is \((Z,Z)^T\), where \(Z\sim N(0,h)\). Its covariance matrix is \(\begin{bmatrix}h&h\\h&h\end{bmatrix}\), not \(hI_2\). The increment lies on the diagonal of \(\mathbb{R}^2\), so it cannot have the standard two-dimensional Brownian increment law.

Common mistake. A tempting but wrong calculation is: “Both coordinates have increment variance \(h\), so the covariance matrix is \(hI_2\).” This fills in the diagonal entries correctly but assumes, without justification, that the off-diagonal entry is zero. In the example \(B_t=(W_t,W_t)^T\), the off-diagonal covariance is actually \(\operatorname{Cov}(Z,Z)=\operatorname{Var}(Z)=h\). The correction is to check coordinate independence (or calculate cross-covariances) before writing down a diagonal covariance matrix. For independent Brownian coordinates those cross-covariances are zero; for duplicated coordinates they are not.

The coordinate construction therefore gives a precise route from one-dimensional Brownian motion to its standard multidimensional counterpart. It also supplies a practical calculation rule: identify the time interval, use \(h=t-s\), assign covariance \(h\) within each coordinate and zero across independent coordinates, and then apply matrix multiplication for any linear combinations.