I know from Karatzas & Shreve (1991) that a Brownian Bridge $B(t)$ from $a$ to $b$ on time interval $[0,T]$ satisfies:
$$B(t)=a(1-t/T) + b*t/T + [W(t) - W(T)*t/T]$$
where $W(t)$ is a standard one-dimensional Brownian motion.
By the above equation we can get its distribution.
My question is what's the distribution of the Brownian Bridge $B(t)$ from $a$ to $b$ on time interval $[T_1, T_2]$?
Any idea or reference?