Fluctuations in $P(\phi)_1$ Processes

Abstract

Let $H = -d^2/dx^2 + P(x)$ on $L^2(\mathbb{R}, dx)$ and let $E = \inf \operatorname{spec} (H)$. Let $\Omega$ be a normalized vector with $H\Omega = E\Omega$. Let $q(t)$ be the Markov process with generator $G = \Omega^{-1} (H - E)\Omega$, which is a Brownian motion with drift. We investigate behavior of $q(t)$ as $t \rightarrow \infty$ and in particular prove that if $P(x) = a_{2m} x^{2m} + \cdots + a_0; a_{2m} > 0$, then $$\lim \sup_{t\rightarrow\infty} \int^{t+1}_t q(s) ds/(\ln t)^{1/2m} = (a_{2m})^{-1/2m}$$ with probability one. These represent fluctuations in the sense that the $\lim \inf$ is $-(a_{2m})^{-1/2m}$. We obtain some weaker results for the $P(\phi)_2$ Euclidean field theory.