= Mean first-passage time
{title2=$\tau(x)=\mathbb E_xT_A$}
The <Mean first-passage time> to a target $A$ is $u(x)=\mathbb E_xT_A$, which can be infinite. For a <diffusion process>[diffusion] with <diffusion generator>[backward generator] $L$, a finite sufficiently regular <mean> solves the <Poisson equation>
$$
Lu=-1
$$
outside $A$, with $u=0$ on the absorbing target. Reflecting parts of the boundary require the reflecting condition of the generator, rather than an absorbing value. These conditions express different physical rules and cannot be exchanged.
For <Brownian motion>[Brownian diffusion] with generator $D\,d^2/dx^2$ on $[0,\ell]$, $D>0$, absorption at both endpoints gives
$$
Du''=-1,\quad u(0)=u(\ell)=0,\qquad
u(x)=\frac{x(\ell-x)}{2D}.
$$
If zero instead reflects while $\ell$ absorbs, the <boundary condition>[boundary conditions] are $u'(0)=0$, $u(\ell)=0$ and the solution is $u(x)=(\ell^2-x^2)/(2D)$. In particular the <mean> time from the reflecting endpoint is $\ell^2/(2D)$, displaying the squared-distance <diffusion process>[diffusion] scale.
<Dynkin formula>[Dynkin's formula] explains the equation: before absorption, $u(X_{t\wedge T_A})+(t\wedge T_A)$ has constant <expected value>. For these bounded-interval nonnegative bounded solutions, this identity first bounds $\mathbb E_x(t\wedge T_A)$ by $u(x)$, proving <integrability> of $T_A$; passing to the limit then proves $u(x)=\mathbb E_xT_A$. Boundary equations without a finiteness or minimality condition can have extraneous solutions on unbounded domains.
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