| The thesis considers the Ginzburg-Landau process on the lattice Zd whose potential is a bounded perturbation of the Gaussian potential. For such processes the thesis establishes the decay rate to equilibrium in the variance sense is Cgt–d /2 + o (t –d/2), for any local function g that is bounded, mean zero, and having finite triple norm; ||| g||| = x∈Zd ||∂η(x) g||∞. The constant Cg is computed explicitly. This extends the decay to equilibrium result of Janvresse, Landim, Quastel, and Yau [JLQY99] for zero-range process, and the related result of Landim and Yau [LY03] for Ginzburg-Landau processes.;The thesis also considers additive functionals ft0 g(ηs)ds of Ginzburg-Landau processes, where g is a bounded, mean zero, local function having finite triple norm. A central limit is proven for a–1(t) ft0 g(ηs)ds with a(t) = t in d ≥ 3, a(t) = tlogt in d = 2, and a(t) = t3/4 in d = 1 and an explicit form of the asymptotic variance in each case. Corresponding invariance principles are also obtained. Standard arguments of Kipnis and Varadhan [KV86] are employed in the case d ≥ 3. Martingale methods together with L2 decay estimates for the semigroup associated with the process are employed to establish the result in the cases d = 1 and d = 2. This extends similar results for noninteracting random walks (see[CG84]), the symmetric simple exclusion processes (see [Kip87]), and the zero-range process (see [QJS02]). |
