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Surnniary . For a system of diffusions in a domain of Rd with Long-range weak interaction tl:e behavior of the associated empirical process is studied . Under mild growth and smoothness assuniptions 011 the drift and diffusion coefficients ~u c h as coercivity and monotonicity conditions the law of large nnmbers.and, the propagation of. chaos are proved . Existence and uniqueness of tlie \veak solution to tile MCI

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We consider the parabolic Anderson problem ∂tu = ∆u + ξ(x)u on R+ × Z d with localized initial condition u(0, x) = δ0(x) and random i.i.d. potential ξ. Under the assumption that the distribution of ξ(0) has a double-exponential, or slightly heavier, tail, we prove the following geometric characterization of intermittency: with probability one, as t → ∞, the overwhelming contribution to the total mass x u(t, x) comes from a slowly increasing number of "islands" which are located far from each other. These "islands" are local regions of those high exceedances of the field ξ in a box of side length 2t log 2 t for which the (local) principal Dirichlet eigenvalue of the random operator ∆ + ξ is close to the top of the spectrum in the box. We also prove that the shape of ξ in these regions is nonrandom and that u(t, ·) is close to the corresponding positive eigenfunction. This is the geometric picture suggested by localization theory for the Anderson Hamiltonian. Contents