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Parabolic Anderson problem and intermittency. (English) Zbl 0925.35074
Mem. Am. Math. Soc. 518, 125 p. (1994).
We consider the stochastic partial differential equation \[ \frac {\partial u}{\partial t}= \kappa \Delta u+\xi_t(x)u. \] The potential \(\xi_i(x)\) is assumed to be a mean zero homogeneous Gaussian field. We pay special attention to the white noise case. In order to minimize the technical difficulties we consider only the case the discrete Laplacian \(\Delta\) on the lattice \(\mathbb{Z}^d\). We prove existence and uniqueness (for almost every realization of the random potential) for nonnegative initial conditions. These results are proved by means of the Feynman-Kac representation of the minimal solutions. Infinite dimensional Ito and Stratonovich equations are needed to study the white noise case. We then prove that the solutions have moments of all orders. In the case of a white noise potential we derive a family of closed equations for these moments.
We then prove the existence of the moment Lyapunov exponents and we study their dependence upon the diffusion constant \(\kappa\). As a consequence, we show that there is full intermittency of the solution when the dimension \(d\) is not greater than 2 while the same intermittency only holds for large values of the diffusion constant in higher dimensions. The fundamental equation can be viewed as a parabolic Anderson model and this phase transition is natural from the point of view of localization theory.
Finally, the last chapter is devoted to the study of the almost sure Lyapunov exponent. We prove its existence as we derive their asymptotic behavior for small \(\kappa\).

35K10 Second-order parabolic equations
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35B40 Asymptotic behavior of solutions to PDEs
60H05 Stochastic integrals
60H25 Random operators and equations (aspects of stochastic analysis)
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