We consider billiards with a single cusp where the walls meeting at the vertex of the cusp have zero one-sided curvature, thus forming a flat point at the vertex. For Hölder continuous observables, we show that properly normalized Birkhoff sums, with respect to the billiard map, converge in law to a totally skewed α-stable law. * Received date:

We consider billiards with several possibly non-isometric and asymmetric cusps at flat points; the case of a single symmetric cusp was studied previously in [Zha17] and [JZ18]. In particular, we show that properly normalized Birkhoff sums of Hölder observables, with respect to the billiard map, converge in Skorokhod's M 1 -topology to an α-stable Lévy motion, where α depends on the 'curvature' of the flattest points and the skewness parameter ξ depends on the values of the observable at those same points. Previously, [JZ18] proved convergence of the one-point marginals to totally skewed α-stable distributions for a symmetric cusp. The limits we prove here are stronger, since they are in the functional sense, but also allow for more varied behaviour due to the presence of multiple cusps. In particular, the general limits we obtain allow for any skewness parameter, as opposed to just the totally skewed cases. We also show that convergence in the stronger J 1 -topology is not possible.

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].