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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
##### Keywords:
parabolic Anderson model; Lyapunov exponents; percolation
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