Correlation structure of intermittency in the parabolic Anderson model

Consider the Cauchy problem partial derivative u(x, t)/partial derivative t = Hu(x, t) (x is an element of Z(d), t greater than or equal to 0) with initial condition u(x, 0) = 1 and with H the Anderson Hamiltonian H = kappa Delta + xi. Here a is the discrete Laplacian, kappa is an element of (0, infinity) is a diffusion constant, and xi = {xi(x): x is an element of Z(d)} is an i.i.d. random field taking values in IR. Gartner and Molchanov (1990) have shown that if the law of xi(0) is nondegenerate, then the solution u is asymptotically intermittent. In the present paper we study the structure of the intermittent peaks for the special case where the law of xi (0) is (in the vicinity of) the double exponential Prob(xi(0) > s) = exp[-e(s/theta)] (s is an element of R). Here theta is an element of (0, infinity) is a parameter that can be thought of as measuring the degree of disorder in the xi-field. Our main result is that, for fixed x, y is an element of Z(d) and t --> infinity, the correlation coefficient of u(x, t) and u(y, t) converges to parallel to w(rho)parallel to(l2)(-2)Sigma(z is an element of Z)d w(rho) (x + z)w(rho)(y + z) In this expression, rho = theta/kappa while w(rho):Z(d) --> R+ is given by w(rho) = (v(rho))(xd) with v rho:Z --> R+ the unique centered ground state (i.e., the solution in l(2)(Z) with minimal l(2)-norm) of the 1-dimensional nonlinear equation Delta v + 2 rho v log v = 0. The uniqueness of the ground state is actually proved only for large rho, but is conjectured to hold for any rho is an element of (0, infinity). It turns out that if the right tail of the law of xi(0) is thicker (or thinner) than the double exponential, then the correlation coefficient of u (x, t) and u(y, t) converges to delta(x,y) (resp. the constant function 1). Thus, the double exponential family is the critical class exhibiting a nondegenerate correlation structure.