Definition 5.3.1 (Brownian Bridge).label Let $(\Omega, \cf, \bp)$ be a probability space, $a \in \real^{d}$, and $X: \Omega \to C([0, 1]; \real^{d})$ be a Gaussian process, then $X$ is a Brownian bridge from $0$ to $a$ such that:

1. (1)

   For each $t \in [0, 1]$, $\ev(X_{t}) = at$.

2. (2)

   For each $s, t \in [0, 1]$, $\text{Cov}(X_{s}, X_{t}) = (s \wedge t - st)I$.