Spectral analysis of weakly coupled stochastic lattice Ginzburg-Landau models

We consider the relaxation to equilibrium of solutions phi (t, (x) over bar), t > 0, (x) over bar is an element of Z(d), of stochastic dynamical Langevin equations with white noise and weakly coupled Cinzburg-Landau interactions. Using a Feynman-Kac formula, which relates stochastic expectations to correlation functions of a spatially non-local imaginary time quantum field theory, we obtain results on the joint spectrum of H, (P) over right arrow, where H is the selfadjoint, positive, generator of the semi-group associated with the dynamics, and P-j, j = 1,..., d are the self-adjoint generators of the group of lattice spatial translations. We show that the low-lying energy-momentum spectrum consists of an isolated one-particle dispersion curve and, for the mass spectrum (energy-momentum at zero-momentum), besides this isolated one-particle mass, we show, using a Bethe-Salpeter equation, the existence of an isolated two-particle bound state if the coefficient of the quartic term in the polynomial of the Ginzburg-Landau interaction is negative and d = 1, 2; otherwise, there is no two-particle bound state. Asymptotic values for the masses are obtained.