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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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