Survival probability from an absorbing transition density ($S(t|x_i)={\int }_D{K}_D(x,t|x_i,0)dx$)

A transition density with absorbing boundaries counts the joint event of survival up to time $t$ and a specified final position. Its integrated mass is survival probability, not generally one. Dividing by that mass gives the endpoint density conditional on survival, a different object.

For diffusion starting at the centre of $(-L,L)$ with absorbing endpoints, only even-parity spatial eigenfunctions contribute. Integrating their cosine expansion gives the displayed series. The leading late-time term has prefactor $4/\pi$ and decay rate ${\pi }^{2}D/(4L^2)$; its first relative correction is $-\frac{1}{3}{e}^{-2{\pi }^{2}Dt/L^2}$.