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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 $$\partial_tU(x,t)=D\partial_{xx}U+\gamma U(1-U)+\varepsilon\sqrt{U(1-U)}\eta(x,t)$$ for $0\leqslant U\leqslant 1$ where $\eta(x,t)$ is a Gaussian white noise process in space and time. Here $D$, $\gamma$ and $\varepsilon$ 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 $A\rightleftharpoons A+A$ 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-Piskunov equation.

60H15Stochastic partial differential equations
35K57Reaction-diffusion equations
Full Text: DOI
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