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The chemical birth-death process with Gillespie noise

A nontrivial technical issue has long plagued the literature on stochastic path integrals: it is not clear which definition is correct in the case of multiplicative/state-dependent noise. One reason for this is the unavailability of exactly solvable toy problems with state-dependent noise, that could in principle be used to compare the correctness of different approaches. In this paper, we provide an exact path integral calculation of the transition probability corresponding to a one-dimensional system with state-dependent noise. In particular, we solve the chemical birth-death process with Gillespie noise (the canonical continuous approximation to the discrete birth-death process often used as a toy model in chemical kinetics) using a Martin-Siggia-Rose-Janssen-De Dominicis (MSRJD) path integral. We verify that our result is correct by solving the Fokker-Planck equation via eigenfunction expansion.