This book was first published in 2006. Written by two of the foremost researchers in the field, this book studies the local times of Markov processes by employing isomorphism theorems that relate them to certain associated Gaussian processes. It builds to this material through self-contained but harmonized 'mini-courses' on the relevant ingredients, which assume only knowledge of measure-theoretic probability. The streamlined selection of topics creates an easy entrance for students and experts in related fields. The book starts by developing the fundamentals of Markov process theory and then of Gaussian process theory, including sample path properties. It then proceeds to more advanced results, bringing the reader to the heart of contemporary research. It presents the remarkable isomorphism theorems of Dynkin and Eisenbaum and then shows how they can be applied to obtain new properties of Markov processes by using well-established techniques in Gaussian process theory. This original, readable book will appeal to both researchers and advanced graduate students.

Let T (x, ε) denote the first hitting time of the disc of radius ε centered at x for Brownian motion on the two dimensional torus T 2 . We prove that sup x∈T 2 T (x, ε)/| log ε| 2 → 2/π as ε → 0. The same applies to Brownian motion on any smooth, compact connected, two-dimensional, Riemannian manifold with unit area and no boundary. As a consequence, we prove a conjecture, due to Aldous (1989), that the number of steps it takes a simple random walk to cover all points of the lattice torus Z 2 n is asymptotic to 4n 2 (log n) 2 /π. Determining these asymptotics is an essential step toward analyzing the fractal structure of the set of uncovered sites before coverage is complete; so far, this structure was only studied nonrigorously in the physics literature. We also establish a conjecture, due to Kesten and Révész, that describes the asymptotics for the number of steps needed by simple random walk in Z 2 to cover the disc of radius n.

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Let $\mathcal{T}_n(x)$ denote the time of first visit of a point $x$ on the lattice torus $\mathbb {Z}_n^2=\mathbb{Z}^2/n\mathbb{Z}^2$ by the simple random walk. The size of the set of $\alpha$, $n$-late points $\mathcal{L}_n(\alpha )=\{x\in \mathbb {Z}_n^2:\mathcal{T}_n(x)\geq \alpha \frac{4}{\pi}(n\log n)^2\}$ is approximately $n^{2(1-\alpha)}$, for $\alpha \in (0,1)$ [$\mathcal{L}_n(\alpha)$ is empty if $\alpha >1$ and $n$ is large enough]. These sets have interesting clustering and fractal properties: we show that for $\beta \in (0,1)$, a disc of radius $n^{\beta}$ centered at nonrandom $x$ typically contains about $n^{2\beta (1-\alpha /\beta ^2)}$ points from $\mathcal{L}_n(\alpha)$ (and is empty if $\beta <\sqrt{\alpha} $), whereas choosing the center $x$ of the disc uniformly in $\mathcal{L}_n(\alpha)$ boosts the typical number of $\alpha, n$-late points in it to $n^{2\beta (1-\alpha)}$. We also estimate the typical number of pairs of $\alpha$, $n$-late points within distance $n^{\beta}$ of each other; this typical number can be significantly smaller than the expected number of such pairs, calculated by Brummelhuis and Hilhorst [Phys. A 176 (1991) 387--408]. On the other hand, our results show that the number of ordered pairs of late points within distance $n^{\beta}$ of each other is larger than what one might predict by multiplying the total number of late points, by the number of late points in a disc of radius $n^{\beta}$ centered at a typical late point.Comment: Published at http://dx.doi.org/10.1214/009117905000000387 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

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