We investigate the temporal evolution and spatial propagation of branching
annihilating random walks in one dimension. Depending on the branching and
annihilation rates, a few-particle initial state can evolve to a propagating
finite density wave, or extinction may occur, in which the number of particles
vanishes in the long-time limit. The number parity conserving case where
2-offspring are produced in each branching event can be solved exactly for unit
reaction probability, from which qualitative features of the transition between
propagation and extinction, as well as intriguing parity-specific effects are
elucidated. An approximate analysis is developed to treat this transition for
general BAW processes. A scaling description suggests that the critical
exponents which describe the vanishing of the particle density at the
transition are unrelated to those of conventional models, such as Reggeon Field
Theory. P. A. C. S. Numbers: 02.50.+s, 05.40.+j, 82.20.-wComment: 12 pages, plain Te