Short-time statistics of extinction and blowup in reaction kinetics

We study the statistics of extinction and blowup times in well-mixed systems of stochastically reacting particles. We focus on the short-time tail, T→0, of the extinction- or blowup-time distribution Pm(T), where m is the number of particles at t=0. This tail often exhibits an essential singularity at T=0, and we show that the singularity is captured by a time-dependent WKB (Wentzel-Kramers-Brillouin) approximation applied directly to the master equation. This approximation, however, leaves undetermined a large pre-exponential factor. We show how to calculate this factor by applying a leading- and a subleading-order WKB approximation to the Laplace-transformed backward master equation. Accurate asymptotic results can be obtained when this WKB solution can be matched to another approximate solution (the “inner” solution), valid for not too large m. We demonstrate and verify this method on three examples of reactions which are also solvable without approximations.