The thesis considers the Ginzburg-Landau process on the lattice $\Z^d$ 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 $C_g t^{-d/2} + o\left(t^{-d/2}\right)$, for any local function $g$ that is bounded, mean zero, and having finite triple norm; $\triplenorm{g}=\sum_{x \in \Z^d} \norm{\partial_{\eta(x)}g}_\infty.$ The constant $C_g$ 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 $\int_{0}^{t} g(\eta_s) ds$ of Ginzburg-Landau processes, where $g$ is a bounded, mean zero, local function having finite triple norm. A central limit is proven for $a^{-1}(t)\int_{0}^{t} g(\eta_s) ds$ with $a(t)= \sqrt{t}$ in $d \ge 3$, $a(t)=\sqrt{t \log{t}}$ in $d=2$, and $a(t)= t^{3/4}$ 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 $d \ge 3$. Martingale methods together with $L^2$ 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]).
| Identifer | oai:union.ndltd.org:TORONTO/oai:tspace.library.utoronto.ca:1807/30040 |
| Date | 14 November 2011 |
| Creators | Sheriff, John |
| Contributors | Quastel, Jeremy |
| Source Sets | University of Toronto |
| Language | en_ca |
| Detected Language | English |
| Type | Thesis |
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