Imaginary noise and parity conservation in the reaction A+A⇌0

The master equation for the reversible reaction A+A⇌0 is considered in Poisson representation, where it is equivalent to a Langevin equation with imaginary noise for a complex stochastic variable φ. Such Langevin equations appear quite generally in field-theoretic treatments of reaction–diffusion problems. For this example we study the probability flow in the complex φ plane both analytically and by simulation. We show that this flow has various curious features that must be expected to occur similarly in other Langevin equations associated with reaction–diffusion problems.