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A boundedness trichotomy for the stochastic heat equation. (English. French summary) Zbl 1387.60096
Summary: We consider the stochastic heat equation with a multiplicative white noise forcing term under standard “intermitency conditions.” The main finding of this paper is that, under mild regularity hypotheses, the a.s.-boundedness of the solution \(x\mapsto u(t,x)\) can be characterized generically by the decay rate, at \(\pm\infty\), of the initial function \(u_{0}\). More specifically, we prove that there are 3 generic boundedness regimes, depending on the numerical value of \(\mathbf{\Lambda}:=\lim_{| x| \rightarrow\infty}| \log u_{0}(x)| /(\log| x| )^{2/3}\).
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations
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