Duality-based calculations for transition probabilities in birth-death processes

Transition probabilities in birth-death processes are fomulated via the corresponding dual birth-death processes. In order to obtain the corresponding dual processes, the Doi-Peliti formalism is employed. Conventional numerical evaluation enables us to obtain the transition probabilities from a fixed initial state; on the other hand, the duality relation gives us a useful method to calculate the transition probabilities to a fixed final state. Furthermore, it is clarified that the transition probabilities for various rate constants can be evaluated from only one numerical evaluation of extended dual stochastic processes.