On the behaviour of stochastic heat equations on bounded domains

2014/12/07 by Foondun, Mohammud, Nualart, Eulalia
#60H15 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1412.2343

Abstract

Consider the following equation ∂_t u_t(x)=(1)/(2)∂ _xxu_t(x)+λσ(u_t(x))W(t, x) on an interval. Under Dirichlet boundary condition, we show that in the long run, the second moment of the solution grows exponentially fast if λ is large enough. But if λ is small, then the second moment eventually decays exponentially. If we replace the Dirichlet boundary condition by the Neumann one, then the second moment grows exponentially fast no matter what λ is. We also provide various extensions.

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