Path properties of levy processes

Abstract

This thesis can be split into two main components, the �first of which looks at the fractal dimension, specifically, box-counting dimension, of sets related to subordinators (non-decreasing L�evy processes). It was recently shown in [111] that lim�!0 U(�)N(t; �) = t almost surely, where N(t; �) is the minimal number of boxes of size at most � needed to cover a subordinator's range up to time t, and U(�) is the subordinator's renewal function. The main result in this section is a central limit theorem (CLT) for N(t; �), complementing and refining work in [111]. Box-counting dimension is defined in terms of N(t; �), but for subordinators we prove that it can also be defined using a new process obtained by shortening the original subordinator's jumps of size greater than �. This new process can be manipulated with remarkable ease in comparison to N(t; �), and allows better understanding of the box-counting dimension of a subordinator's range in terms of its L�evy measure, improving upon [111, Corollary 1]. We prove corresponding CLT and almost sure convergence results for the new process. The second main component of this thesis studies Markov processes conditioned so that their local time must grow slower than a prescribed function. Building upon recent work on Brownian motion with constrained local time in [8] and [78], we study whether or not the conditioned process is transient or recurrent, working with a broad class of Markov processes. In order to understand the local time, it is equivalent to study the inverse local time, which is itself a subordinator. The problem at hand is effectively equivalent to determining the distribution of a subordinator (the inverse local time) conditioned to remain above a given function. In conditioning a subordinator to remain above a curve of the form g(t); t � 0, the process is restricted to a time-dependent region, in contrast to previous works in which a process is conditioned to remain in a fixed region (e.g. cones in [43] and [60]). This means that we study boundary crossing probabilities for a family of curves, and must obtain uniform asymptotics for such a family. The main result in this section is a necessary and su�fficient condition for transience or recurrence of the conditioned Markov process. We will explicitly determine the distribution of the inverse local time for the conditioned process, and in the transient case, we explicitly determine the law of the conditioned Markov process. In the recurrent case, we characterise the entropic repulsion envelope via necessary and suffi�cient conditions.