  Momentum-space representation renders an interesting perspective to theory of
large fluctuations in populations undergoing Markovian stochastic gain-loss
processes. This representation is obtained when the master equation for the
probability distribution of the population size is transformed into an
evolution equation for the probability generating function. Spectral
decomposition then brings about an eigenvalue problem for a non-Hermitian
linear differential operator. The ground-state eigenmode encodes the stationary
distribution of the population size. For long-lived metastable populations
which exhibit extinction or escape to another metastable state, the
quasi-stationary distribution and the mean time to extinction or escape are
encoded by the eigenmode and eigenvalue of the lowest excited state. If the
average population size in the stationary or quasi-stationary state is large,
the corresponding eigenvalue problem can be solved via WKB approximation
amended by other asymptotic methods. We illustrate these ideas in several model
examples.

Statistical Mechanics;Populations and Evolution