Interacting particles, the stochastic Fisher–Kolmogorov–Petrovsky–Piscounov equation, and duality

The stochastic Fisher–Kolmogorov–Petrovsky–Piscunov equation is for 0 U 1 where η(x,t) is a Gaussian white noise process in space and time. Here D, γ and are parameters and the equation is interpreted as the continuum limit of a spatially discretized set of Itô equations. Solutions of this stochastic partial differential equation have an exact connection to the A A+A reaction–diffusion system at appropriate values of the rate coefficients and particles’ diffusion constant. This relationship is called “duality” by the probabilists; it is not via some hydrodynamic description of the interacting particle system. In this paper we present a complete derivation of the duality relationship and use it to deduce some properties of solutions to the stochastic Fisher–Kolmogorov–Petrovsky–Piscunov equation.