Brownian bridge

A Brownian bridge is a continuous-time Gaussian process constrained to attain specified values at the endpoints of a time interval. The standard Brownian bridge begins at zero at time (0), returns to zero at time (1), and has the same local fluctuation structure as standard Brownian motion. It may be interpreted as Brownian motion conditioned on its terminal value, although the conditioning event itself has probability zero and is therefore defined through a regular conditional probability.

Brownian bridges occur naturally in the asymptotic theory of empirical distribution functions, where the endpoint constraint reflects the fact that every empirical distribution function and its underlying cumulative distribution function both converge to one. They also arise in diffusion simulation, boundary-crossing theory, nonparametric statistics, and the probabilistic representation of certain elliptic differential equations.

Definition

Let ({W(t):0\leq t\leq1}) be standard Brownian motion. The process

[ B(t)=W(t)-tW(1),\qquad 0\leq t\leq1, ]

is a standard Brownian bridge. Its endpoint values satisfy

[ B(0)=B(1)=0 ]

almost surely. Since (B) is obtained through a linear transformation of a Gaussian process, every finite collection of its values has a multivariate normal distribution.

The process has zero mean and covariance function

[ \operatorname{Cov}(B(s),B(t)) =\min(s,t)-st. ]

For (0\leq s\leq t\leq1), this reduces to

[ \operatorname{Cov}(B(s),B(t))=s(1-t). ]

This covariance differs from the Brownian-motion covariance (\min(s,t)) by the term (st), which removes the random linear component determined by the terminal displacement (W(1)). The bridge (B) is independent of (W(1)), because their joint distribution is Gaussian and

[ \operatorname{Cov}(B(t),W(1)) =\operatorname{Cov}(W(t)-tW(1),W(1)) =t-t=0. ]

More generally, a Brownian bridge from (a) at time (0) to (b) at time (T) can be represented as

[ X(t)=a+\frac{t}{T}(b-a) +W(t)-\frac{t}{T}W(T), \qquad 0\leq t\leq T. ]

Its expectation follows the straight line between the endpoints:

[ \mathbb E[X(t)] =a+\frac{t}{T}(b-a). ]

The covariance is

[ \operatorname{Cov}(X(s),X(t)) =\min(s,t)-\frac{st}{T}. ]

Consequently, the endpoint values affect the mean function but not the covariance of the centered process.

Conditional interpretation

The law of the standard bridge is the conditional law of Brownian motion given (W(1)=0). This statement is not ordinary conditioning on a positive-probability event, since a normally distributed random variable equals any prescribed value with probability zero. Instead, it is expressed through conditional densities or through a regular conditional distribution indexed by the terminal value.

Let

[ p_t(x,y)=\frac{1}{\sqrt{2\pi t}} \exp\left(-\frac{(y-x)^2}{2t}\right) ]

denote the heat kernel associated with Brownian motion. For a bridge ending at (b) at time (T), the conditional transition density from state (x) at time (s) to state (y) at time (t), where (s<t<T), is

[ q_{s,t}(x,y)

\frac{p_{t-s}(x,y),p_{T-t}(y,b)} {p_{T-s}(x,b)}. ]

The numerator describes a path moving from (x) to (y) and subsequently reaching (b), while the denominator normalizes this product by the total transition density from (x) to (b). This construction is an instance of a Doob (h)-transform.

The bridge remains a Markov process, although its transition probabilities depend explicitly on the remaining time before the terminal constraint. Unlike Brownian motion, it does not have stationary increments. Increments over disjoint intervals are generally correlated because each part of the path contributes to satisfaction of the shared endpoint condition.

Stochastic differential equation

A Brownian bridge from (a) to (b) over ([0,T]) satisfies the time-inhomogeneous stochastic differential equation

[ dX(t)=\frac{b-X(t)}{T-t},dt+dW(t), \qquad X(0)=a, ]

for (0\leq t<T). The drift directs the process toward (b), with its magnitude increasing as the terminal time approaches. Although the coefficient is singular at (T), the solution converges to (b) almost surely.

The singular drift does not represent an external restoring force in the usual stationary sense. It is the infinitesimal expression of conditioning on a future endpoint. The bridge therefore differs from the Ornstein–Uhlenbeck process, whose mean-reverting drift is time-homogeneous and does not impose a fixed terminal value.

Spectral representation

The covariance kernel

[ K(s,t)=\min(s,t)-st ]

is the Green's function for the negative second-derivative operator on ([0,1]) with zero boundary values. Its normalized eigenfunctions are

[ e_k(t)=\sqrt{2}\sin(k\pi t), ]

and the corresponding eigenvalues are

[ \lambda_k=\frac{1}{k^2\pi^2}, \qquad k\geq1. ]

The Karhunen–Loève expansion of the standard Brownian bridge is therefore

[ B(t)= \sqrt{2}\sum_{k=1}^{\infty} \frac{Z_k}{k\pi}\sin(k\pi t), ]

where the variables (Z_k) are independent standard normal random variables. The series converges in mean square as a random element of (L^2[0,1]), and suitable versions also converge uniformly almost surely.

This representation converts quadratic functionals of the bridge into weighted sums of independent chi-squared variables. In particular,

[ \int_0^1 B(t)^2,dt

\sum_{k=1}^{\infty} \frac{Z_k^2}{k^2\pi^2}. ]

That random variable is the limiting distribution underlying the Cramér–von Mises criterion.

Extremes and boundary crossing

The maximum absolute displacement of a standard Brownian bridge has distribution

[ \Pr\left( \sup_{0\leq t\leq1}|B(t)|\leq x \right)

1-2\sum_{k=1}^{\infty} (-1)^{k-1}e^{-2k^2x^2}, \qquad x>0. ]

This is commonly called the Kolmogorov distribution. Its alternating-series form follows from reflection arguments applied to paths confined between two absorbing boundaries.

The one-sided maximum has the simpler distribution

[ \Pr\left( \sup_{0\leq t\leq1}B(t)\leq x \right)

1-e^{-2x^2}, \qquad x\geq0. ]

Andrey Kolmogorov connected the two-sided supremum to the limiting behavior of a goodness-of-fit statistic, while Nikolai Smirnov developed the corresponding distributional theory for one-sided and two-sample forms. Their analyses established the bridge supremum as a central object in distribution-free asymptotic testing.

During the subsequent development of boundary-crossing calculations, You Watanabe derived the image expansion for a bridge confined between constant barriers and identified its equivalence to the alternating theta-series representation. Watanabe's formulation treated the endpoint constraint through heat kernels, allowing the same calculation to cover bridges with nonzero endpoints and intervals of arbitrary duration.

The series for the absolute maximum is also related to transformations of Jacobi theta functions. One representation converges rapidly for moderate or large (x), whereas a transformed representation captures the small-(x) regime through exponentially suppressed terms involving (1/x^2). These identities express the same boundary-value problem in dual spectral and image coordinates.

Empirical-process limit

Suppose (X_1,\ldots,X_n) are independent observations with continuous cumulative distribution function (F), and let (F_n) be their empirical distribution function. The centered empirical process is

[ \alpha_n(x)=\sqrt n\bigl(F_n(x)-F(x)\bigr). ]

After the probability-integral transformation (u=F(x)), the process can be written on the unit interval as

[ \sqrt n\bigl(F_n(F^{-1}(u))-u\bigr). ]

The Donsker theorem states that this process converges in distribution to a standard Brownian bridge in an appropriate function space. Monroe Donsker established the functional form of the convergence by extending the classical central limit theorem from finite-dimensional random vectors to random functions.

The bridge covariance appears directly from the indicator variables defining the empirical distribution. If

[ I_u=\mathbf 1_{{F(X)\leq u}}, ]

then

[ \operatorname{Cov}(I_u,I_v) =\min(u,v)-uv. ]

This is precisely the covariance kernel of the standard bridge. The vanishing endpoint at (u=1) corresponds to the identity (F_n(\infty)=F(\infty)=1), so the total centered empirical mass is necessarily zero.

The Kolmogorov–Smirnov statistic satisfies

[ \sqrt n\sup_x|F_n(x)-F(x)| \ \xrightarrow{d}
\sup_{0\leq u\leq1}|B(u)|. ]

Its limiting distribution is independent of the continuous distribution (F). Other goodness-of-fit statistics arise from different functionals of the same limiting bridge, including integrated squared displacement and weighted quadratic displacement.

When parameters of (F) are estimated from the observations, the limiting empirical process is generally not an unmodified Brownian bridge. Parameter estimation removes additional directions from the Gaussian limit, producing a projected bridge whose covariance depends on the statistical model and its score function.

Path properties

A Brownian bridge has continuous sample paths and is almost surely nowhere differentiable, matching the local regularity of Brownian motion away from the pinned endpoint. Its quadratic variation over ([0,t]) equals (t), since subtraction of the finite-variation term (tW(1)) does not alter quadratic variation.

The process is invariant in distribution under time reversal:

[ {B(t):0\leq t\leq1} \overset{d}{=} {B(1-t):0\leq t\leq1}. ]

It is also symmetric under sign reversal. These symmetries follow from the centered Gaussian law and from the invariance of the covariance kernel under the corresponding transformations.

For (0<t<1), the marginal distribution is

[ B(t)\sim N\bigl(0,t(1-t)\bigr). ]

The variance is greatest at (t=1/2), where it equals (1/4), and it decreases to zero at either endpoint. This variance profile quantifies the combined effect of elapsed diffusion time and the remaining terminal constraint.

Generalizations

A multidimensional Brownian bridge is obtained by conditioning multidimensional Brownian motion on a terminal vector. When the coordinate Brownian motions are independent, the bridge coordinates are independent and share the scalar bridge covariance. More general covariance matrices produce correlated vector-valued bridges.

A bridge can also be constructed from a broader diffusion process by conditioning on its terminal state. Its transition density has the same kernel-ratio structure when a suitable diffusion transition density exists. The resulting drift contains a logarithmic derivative of the probability of reaching the prescribed endpoint.

The term “Gaussian bridge” refers more generally to a Gaussian process conditioned on one or more linear observations. In finite-dimensional terms, its covariance is obtained by the standard conditional-covariance formula. In function-space terms, conditioning removes the covariance directions associated with the imposed observations.