No abstract

The derivative of self-intersection local time (DSLT) for Brownian motion was introduced by Rosen [Ros05] and subsequently used by others to study the L 2 and L 3 moduli of continuity of Brownian local time. A version of the DSLT for fractional Brownian motion (fBm) was introduced in [YYL08]; however, the definition given there presents difficulties, since it is motivated by an incorrect application of Itô's formula. To rectify this, we introduce a modified DSLT for fBm and prove existence using an explicit Wiener chaos expansion. We will then argue that our modification is the natural version of the DSLT by rigorously proving the corresponding Tanaka formula. This formula corrects a formal identity given in both [Ros05] and [YYL08]. In the course of this endeavor we prove a Fubini theorem for integrals with respect to fBm. The Fubini theorem may be of independent interest, as it generalizes (to Hida distributions) similar results previously seen in the literature. As a further byproduct of our investigation, we also provide a correction to an important technical secondmoment bound for fBm given in [Hu01].

We prove joint Hölder continuity and an occupation-time formula for the self-intersection local time of fractional Brownian motion. Motivated by an occupation-time formula, we also introduce a new version of the derivative of self-intersection local time for fractional Brownian motion and prove Hölder conditions for this process. This process is related to a different version of the derivative of self-intersection local time studied by the authors in a previous work.

This paper presents some results on the expected exit time of Brownian motion from simply connected domains in C. We indicate a way in which Brownian motion sees the identity function and the Koebe function as the smallest and largest analytic functions, respectively, in the Schlicht class. We also give a sharpening of a result of McConnell's concerning the moments of exit times of Schlicht domains. We then show how a simple formula for expected exit time can be applied in a series of examples. Included in the examples given are the expected exit times from given points of a cardioid and regular m-gon, as well as bounds on the expected exit time of an infinite wedge. We also calculate the expected exit time of an infinite strip, and in the process obtain a probabilistic derivation of Euler's result that ζ(2) = ∞ n=1 1 n 2 = π 2 6 . We conclude by showing how the formula can be applied to some domains which are not simply connected.

In this paper a version of the Phragmén-Lindelöf principle is proved using probabilistic techniques. In particular, we will show that if the p th moment of the exit time of Brownian motion from a planar domain is finite, then an analytic function on that domain is either bounded by its supremum on the boundary or else goes to ∞ along some sequence more rapidly than e |z| 2p . We also provide a method of constructing domains whose exit time has finite p th moment. This allows us to give a general Phragmén-Lindelöf principle for spiral-like and star-like domains, as well as a new proof of a theorem of Hansen. A number of auxiliary results are presented as well.