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Lyapunov exponents for the parabolic Anderson model. (English) Zbl 1046.60057
An asymptotic almost sure behaviour of the solution of the equation \[ u(t,x)= u_0(x) + \kappa \int _0^t\Delta u(s,x)\,ds + \int _0^t u(s,x)\,\partial B_x(s) \] is studied, where \((B_x(t), x\in \mathbb Z^d)\) is a field of independent Brownian motions, \(\Delta \) denotes the discrete Laplacian and \(\partial \) stands for the Stratonovich integral. It is shown that if \(u_0\) is a bounded nonnegative, not identically \(0\) function, there exists a (positive Lyapunov exponent) \(\lambda (\kappa )\) independent of \(u_0\) such that \[ \lim _{t\to \infty } \frac {1}{t}\log u(t,x)= \lambda (\kappa )\text{ a.s.}, \quad\text{and} \quad \lim _{\kappa \to 0+} \lambda (\kappa )\log \frac {1}{\kappa } = c, \] where \(c\) is identified by means of a subadditive ergodic theory argument.

MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60G15 Gaussian processes
60K35 Interacting random processes; statistical mechanics type models; percolation theory
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