Let $X$ be a real-valued Markov process, $ \mathsf P_x\{X_1\in A\} = K(x,A) $ is its transition kernel. Let $\tau = \inf\{n\geq 0:X_n\geq a\}$ be the first hitting time of the level $a$. I am interested in distribution $ L_k(x;A) = \mathsf P_x\{X_\tau\in A|\tau = k\} $
In the latter event we cannot just put $\mathsf P_x\{X_\tau\in A|\tau = k\} = \mathsf P_x\{X_k\in A\}$ since it doesn't hold e.g. for $A = [a,\infty)$. Any hints are appreciated.