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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 $$\mathbf{\Lambda}:=\lim_{| x| \rightarrow\infty}| \log u_{0}(x)| /(\log| x| )^{2/3}$$.
##### MSC:
 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 35R60 PDEs with randomness, stochastic partial differential equations
##### Keywords:
stochastic heat equation
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##### References:
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