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Interacting particles, the stochastic Fisher-Kolmogorov-Petrovsky-Piscounov equation, and duality. (English) Zbl 1025.60027

Summary: The stochastic Fisher-Kolmogorov-Petrovsky-Piskunov equation is

t U(x,t)=D xx U+γU(1-U)+εU(1-U)η(x,t)

for 0U1 where η(x,t) is a Gaussian white noise process in space and time. Here D, γ and ε 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 AA+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-Piskunov equation.

60H15Stochastic partial differential equations
35K57Reaction-diffusion equations