Large deviations and related LIL's for Brownian motions on nested fractals
The paper looks at the Brownian motion \(M=(X_t, P_x)\) on a bounded nested fractal set \( \widetilde E\) in \(R^d\). It is assumed to have the origin as one of its boundary points, to have diameter one, and to be decided by \(N\) \(\alpha\)-similitudes. The paper develops analogs to the Donsker-Varadhan results for stable processes [\textit{M. D. Donsker} and \textit{S. R. S. Varadhan}, Commun. Pure Appl. Math. 30, 707-753 (1977; Zbl 0356.60029)] where the stable index \(\alpha\) is replaced by (for a suitable \(c\)) \[ d_\omega=\frac{\log N - \log(1-c)}{\log\alpha}, \] the walk dimension of \(M\). The paper is self-contained and provides material useful to the non-expert. Section 2 describes Dirichlet forms and Brownian motions on unbounded nested fractals \(E\) and culminates in a large deviation result for occupation time distributions. Section 3 studies limit points of scale-changed occupation time distributions. It establishes that for \(P_x\)-a.e. \(\omega\) and \(x\) in \(E\) \[ \bigcap_N\overline{\bigcup_{m\geq N}\widehat{L}_{t_m}(\omega,\cdot)}=C. \] Here \[ \widehat{L}_{t_m}(\omega,\cdot)=\frac{1}{t_m}\int_0^{t_m}I(\alpha^{-m}X_s) ds, \] and \(t_m\) is a sequence of times. Theorem 3.2 presents bounds for upper and lower semicontinuous functions at \(\widehat{L}_{t_m}\). Section 4 establishes that \(P_x\)-a.e. \(\omega\) and \(x\) in \(E\) \[ \lim\inf_{m\rightarrow\infty} \alpha^{-m} \sup_{0\leq s\leq t_m}|X_s|=a_0 \] for suitable \(a_0\). Consequently they obtain the ``other'' LIL \[ \lim\inf_{t\rightarrow\infty} \left(\frac{\log\log t}{t}\right)^\gamma \sup_{0\leq s\leq t}|X_s|=a_{00}. \] Section 5 establishes a LIL for local times, and Section 6 examines a more general context of symmetric Markov processes with smooth transition functions.
scientific article; zbMATH DE number 1445730