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

The stochastic Fisher-Kolmogorov-Petrovsky-Piscunov equation is partial derivative(t)U(x, t) = Dpartial derivative(xx)U + gammaU(1 - U) + epsilonrootU(1 - U)eta(x, t) for 0 less than or equal to U less than or equal to 1 where eta(x, t) is a Gaussian white noise process in space and time. Here D, gamma and epsilon are parameters and the equation is interpreted as the continuum limit of a spatially discretized set of Ito equations. Solutions of this stochastic partial differential equation have an exact connection to the A reversible arrow 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. (C) 2003 Elsevier Science B.V. All rights reserved.