First-passage time ($T_A$)

The first-passage time to a target set $A$ is $T_A=\inf \{t\ge 0:X_t\in A\}$. It is also called a hitting time.

For a diffusion with backward generator $L$, the mean first-passage time satisfies $L\tau =-1$ in the interior, with the backward forms of the process's boundary conditions.

The survival probability is $Q(x,t)={\mathbb{P}}_x(T_A>t)$. When absorption eventually occurs,