The backward generator of the drift--diffusion is $\mathcal L=\alpha\partial_y+D\partial_y^2$. The <survival probability> satisfies the <Kolmogorov backward equation>
$$
\boxed{\partial_tQ=D\partial_y^2Q+\alpha\partial_yQ,
\qquad 0<y<L,}
$$
with
$$
\boxed{Q(0,t)=0,
\qquad \partial_yQ(L,t)=0,
\qquad Q(y,0)=1.}
$$
The target is absorbing, while reflection gives the Neumann boundary condition at $L$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 356 / 1 / a / Solution

{title2=$Q(x,t)$}

The survival probability is $Q(x,t)=\mathbb P_x(T_A>t)$. When absorption eventually occurs,
$$
\mathbb E_xT_A=\int_0^\infty Q(x,t)\,dt.
$$


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