$(X_t, Y_t)$ is a two-dimensional Markov stochastic process that runs on time interval $[0, t_f]$. Given its transition function $a(x, y | x', y')$, I would like to condition the process on $\inf_{s \in [0, t_f]} X_s \ge k$ and find the new transition function.

Can the problem be solved at this level of generality? Or must we dig into the specifics of $a$ to find a solution on a case-by-case basis?