For a large system of independent diffusing particles, each of which is killed at a certain space-time dependent rate, the conditional distribution of surviving trajectories in a bounded time interval is computed, given the approximate form of the initial and final empirical distribution of surviving particles. This generalizes a result for the Brownian case without killing, which was first obtained by Schrödinger [Sitzungsber. Preuss. Akad. Wiss. Phys. Math. Kl. 1931, 144].

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Journal Journal of Mathematical Physics
Dawson, D.A, Gorostiza, L., & Wakolbinger, A. (1990). Schrödinger processes and large deviations. Journal of Mathematical Physics, 31(10), 2385–2388.