ACTA MATHEMATICA UNIVERSITATIS COMENIANAE

Vol. LXXI, 2(2002)
p. 163

Lyapunov Exponents for the Parabolic Anderson Model
M. Cranston, T. S. Mountford and T. Shiga


Abstract.  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.

AMS subject classification:  60H15, 60H30
Keywords:  Parabolic Anderson model, Feynman-Kac formula, Lyapunov exponents, Borell's inequality, percolation

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