Geodesic
Lyapunov Exponents for the Parabolic Anderson Model
Acta mathematica Universitatis Comenianae, Tome 71 (2002) no. 2 
M. Cranston; T. S. Mountford; T. Shiga. Lyapunov Exponents for the Parabolic Anderson Model. Acta mathematica Universitatis Comenianae, Tome 71 (2002) no. 2. http://geodesic.mathdoc.fr/item/AMUC_2002_71_2_a4/
@article{AMUC_2002_71_2_a4,
     author = {M. Cranston and T. S. Mountford and T. Shiga},
     title = {Lyapunov {Exponents} for the {Parabolic} {Anderson} {Model}},
     journal = {Acta mathematica Universitatis Comenianae},
     year = {2002},
     volume = {71},
     number = {2},
     url = {http://geodesic.mathdoc.fr/item/AMUC_2002_71_2_a4/}
}
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Voir la notice de l'article provenant de la source Comenius University

We consider the asymptotic almost sure behavior of the solution of the equation \begin{eqnarray*} 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)\\ &&u(0,x)=u_0(x) \end{eqnarray*} where $\{B_x:x \in \mathbf Z^d\}$ is a field of independent Brownian motions.