Spectral theory of large deviations in birth-death systems

Full Text: We suggest a general spectral method for calculating statistics of multi-step birth-death processes and chemical reactions of the type mA→nA (m and n are positive integers) which possess an absorbing state: a state from which there is a zero probability of exiting. The method employs the generating-function formalism in conjunction with the Sturm-Liouville theory of linear differential operators. It yields accurate results for the extinction statistics and for the quasi-stationary probability distribution, including large deviations, of the metastable state. The power of the method will be demonstrated on the example of binary annihilation and triple branching 2A→Φ, A→3A, representative of the rather general class of dissociation-recombination reactions