Brownian bridge

Remarks

If $W$ is a Wiener process, then $X_t = W_t - t W_1$ is a Brownian bridge. Check: $X$ is a linear image of the Gaussian process $W$, hence Gaussian and centred, with continuous paths, and $E[X_s X_t] = \min(s, t) - st - st + st = \min(s, t) - st$. The Wiener process $W$ restricted to $[0, 1]$ satisfies clauses 1 and 3 but fails clause 2, since $E[W_s W_t] = \min(s, t)$; for instance $\operatorname{Var}(W_1) = 1$, while clause 2 forces $X_1 = 0$. Clause 2 gives $X_0 = X_1 = 0$ almost surely and $\operatorname{Var}(X_t) = t(1 - t)$. The law of a Brownian bridge is the conditional law of $(W_t)_{0 \le t \le 1}$ given $W_1 = 0$, in a sense made precise by a limit.