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

2003; Elsevier BV; Volume: 325; Issue: 1-2 Linguagem: Inglês

10.1016/s0378-4371(03)00203-6

1873-2119

Charles R. Doering, Carl Mueller, Peter Smereka,

Stochastic processes and statistical mechanics

The stochastic Fisher–Kolmogorov–Petrovsky–Piscunov equation is∂tU(x,t)=D∂xxU+γU(1−U)+εU(1−U)η(x,t)for 0⩽U⩽1 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 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.