Gregory Tycho Markowsky

On electric resistances for distance-regular graphs

We investigate the behavior of electric potentials on distance-regular graphs, and extend some results of a prior paper, Koolen and Markowsky (2010) [15]. Our main result shows that if the distance between points is measured by electric resistance then all points are close to being equidistant on a distance-regular graph with large valency. In particular, we show that the ratio between resistances between pairs of vertices in a distance-regular graph of diameter 3 or more is bounded by 1+6k, where k is the degree of the graph. We indicate further how this bound can be improved to 1+4k in most cases. A number of auxiliary results are also presented, including a discussion of the diameter 2 case as well as applications to random walks.

The Derivative of the Intersection Local Time of Brownian Motion Through Wiener Chaos

Rosen (Seminaire de Probabilites XXXVIII, 2005) proved the existence of a process known as the derivative of the intersection local time of Brownian motion in one dimension. The purpose of this paper is to use the methods developed in Nualart and Vives (Publicacions Matematiques 36(2):827–836, 1992) in order to give a simple new proof of the existence of this process. Some related theorems and conjectures are discussed.

Proof of a Tanaka-like formula stated by J. Rosen in Séminaire XXXVIII

Let B t be a one dimensional Brownian motion, and let α′ denote the derivative of the intersection local time of B t as defined by J. Rosen in [2]. The object of this paper is to prove the following formula

Birth-death chains and the local time of Brownian motion

\frac{1}{2}\alpha _t^\prime (x) + \frac{1}{2}\operatorname{sgn} (x)t = \int_0^t {L_s^{B_s - x} dB_s - \frac{1}{2}\int_0^t {\operatorname{sgn} (B_t - B_u - x)du} }

A probabilistic proof of the open mapping theorem for analytic functions

(1) which was given as a formal identity in [2] without proof.

Random walks at random times: Convergence to iterated Lévy motion, fractional stable motions, and other self-similar processes

In this paper, we prove a number of related results on distance-regular graphs concerning electric resistance and simple random walk. We begin by proving several results on electric resistance; in particular we prove a sharp constant bounding the ratio of electrical resistances between any two pairs of points and give a counterexample to a conjecture made in a previous paper regarding the growth of resistances with respect to distance. We then show how a number of strong bounds on moments of hitting times, cover times, and related quantities for simple random walk may be deduced from the bound on resistance.

Hölder Continuity and Occupation-Time Formulas for fBm Self-Intersection Local Time and Its Derivative

A connection between Brownian motion and birth-death chains is explored. Several results concerning birth-death chains are shown to be consequences of well known results on Brownian motion. DOI: 10.1017/S000497271100284X

A Conjecture of Biggs Concerning the Resistance of a Distance-Regular Graph

We explore a method which is implicit in a paper of Burkholder of identifying the H 2 Hardy norm of a conformal map with the explicit solution of Dirichlets problem in the complex plane. Using the series form of the Hardy norm, we obtain an identity for the sum of a series obtained from the conformal map. We use this technique to evaluate several hypergeometric sums, as well as several sums that can be expressed as convolutions of the terms in a hypergeometric series. The most easily stated of the identities we obtain are Eulers famous Basel sum, as well as the sum We will be able to obtain the following hypergeometric reduction: A related identity is We will obtain two families of identities depending on a parameter, representative examples of which are and where C(k) is the kth Catalan number. We will also sum two series whose terms are defined by certain recurrence relations, and discuss an extension of the method to maps which are not conformal.