# Conditional distribution of Ornstein-Uhlenbeck on two fixed points

The conditional distribution of a Ornstein-Uhlenbeck $$X(t)$$ conditional on $$X(0)$$ is given by

$$X(t)|X(0) = X(0)e^{-t} + \mu(1 - e^{-t})$$

This process is usually only defined for $$t>0$$ (future values), but I see no reason why this shouldn't extend backwards as well:

$$X(|t|)|X(0) = X(0)e^{-|t|} + \mu(1 - e^{-|t|})$$

However, I was wondering if this can also be extended between two points as well. I know this is done in kriging, but I was wondering if something similar can be applicable here as well:

$$X(t)|X(a),X(b) = \left\{\begin{matrix} X(a)e^{-(a-t)} + \mu(1 - e^{-(a-t)}) & t < a \\ \color{red}{???} & a < t < b \\ X(b)e^{-(t-b)} + \mu(1 - e^{-(t-b)}) & b < t \end{matrix}\right.$$