So the Brownian bridge is a Wiener process conditioned on fixed endpoints, i.e. $$B_t = \{W_t | W_0 =0, W_T = a\}$$
Is the analysis of the brownian bridge equivalent to the diffusion process with standard noise and a drift so that $\mathbb{E}[X_T] = a$? I.e. $$dX_t = \frac{a}{T} dt + dW_t$$
I read that $W(t) + at$ is equivalent in law to the Brownian bridge that starts at $0$ and ends at $a$, but I'm not sure what this implies.