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Intermittence and nonlinear parabolic stochastic partial differential equations. (English) Zbl 1190.60051
Summary: We consider nonlinear parabolic SPDEs of the form $$\partial_tu={\mathcal L}u+ \sigma(u)\dot w$$, where $$\dot w$$ denotes space-time white noise, $$\sigma:\mathbb R\to\mathbb R$$ is (globally) Lipschitz continuous, and $${\mathcal L}$$ is the $$L^2$$-generator of a Lévy process. We present precise criteria for existence as well as uniqueness of solutions. More significantly, we prove that these solutions grow in time with at most a precise exponential rate. We establish also that when $$\sigma$$ is globally Lipschitz and asymptotically sublinear, the solution to the nonlinear heat equation is “weakly intermittent”, provided that the symmetrization of $${\mathcal L}$$ is recurrent and the initial data is sufficiently large.
Among other things, our results lead to general formulas for the upper second-moment Lyapunov exponent of the parabolic Anderson model for $${\mathcal L}$$ in dimension $$(1+1)$$. When $${\mathcal L}= \kappa\partial_{xx}$$ for $$\kappa>0$$, these formulas agree with the earlier results of statistical physics [M.Kardar, Replica Bethe ansatz studies of two-dimensional interfaces with quenched random impurities, Nuclear Phys. B 290, 582–602 (1987); J. Krug and H. Spohn, Kinetic roughening of growing surfaces, in: C. Godrèche (ed.), Solids Far From Equilibrium: Growth, Morphology, and Defects, Cambridge: Cambridge University Press (1991); E. H. Lieb and W. Liniger, Phys. Rev., II. Ser. 130, 1605–1616 (1963; Zbl 0138.23001), and also probability theory [L. Bertini and N. Cancrini, J. Stat. Phys. 78, No. 5–6, 1377–1401 (1995; Zbl 1080.60508); R. A. Carmona and S. A. Molchanov, Mem. Am. Math. Soc. 518, 125 p. (1994; Zbl 0925.35074)] in the two exactly-solvable cases. That is when $$u_0=\delta_0$$ or $$u_0\equiv 1$$; in those cases the moments of the solution to the SPDE can be computed [Bertini and Cancrini (loc. cit.)].

##### MSC:
 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 60H20 Stochastic integral equations
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