Generation and Motion of Interfaces in One-Dimensional Stochastic Allen–Cahn Equation

Kai Lee ()
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Kai Lee: University of Tokyo

Journal of Theoretical Probability, 2018, vol. 31, issue 1, 268-293

Abstract: Abstract In this paper we study a sharp interface limit for a stochastic reaction–diffusion equation which is parameterized by a sufficiently small parameter $$\varepsilon >0$$ ε > 0 . We consider the case that the noise is a space–time white noise multiplied by $$\varepsilon ^\gamma a(x)$$ ε γ a ( x ) where the function a(x) is a smooth function which has compact support. First, we show a generation of interfaces for a one-dimensional stochastic Allen–Cahn equation with general initial values. We prove that interfaces are generated in time of order $$O(\varepsilon |\log \varepsilon |)$$ O ( ε | log ε | ) . After the generation of interfaces, we connect it to the motion of interfaces which was investigated by Funaki (Probab Theory Relat Fields 102(2):221–288, 1995) for special initial values. Funaki (Probab Theory Relat Fields 102(2):221–288, 1995) proved that the interface moved in a proper time scale obeying a certain stochastic differential equation (SDE) if the interface formed at the initial time. We take the time scale of order $$O(\varepsilon ^{-2\gamma - \frac{1}{2}})$$ O ( ε - 2 γ - 1 2 ) . This time scale is the same as that of Funaki (Probab Theory Relat Fields 102(2):221–288, 1995) and interface moves in this time scale obeying some SDE with high probability.

Keywords: Stochastic partial differential equation; Stochastic reaction–diffusion equation; Allen–Cahn equation; Sharp interface limit; 60H15 (search for similar items in EconPapers)
Date: 2018
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DOI: 10.1007/s10959-016-0717-1

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