g-function and h-function of the complement of a slit

Abstract

In potential theory, h-functions and g functions are key concepts. These two functions are increasing and take values on the interval [0,1]. For a region Ω with a basepoint z0, the h-function h(r) is the probability that a Brownian particle starting at z0 inside Ω first hits the boundary of the region within distance r of z0 before it hits anywhere else on the boundary ∂Ω. The g-function g(r) is the prob ability that a Brownian traveler, starting from z0, will first exit the connected component Ω∗ r through the part of the boundary ∂Ω∗ r that is strictly less than the distance r from z0, where Ω∗ r is the con nected component of Ω ∩B(z0,r) that contains z0. In literature, the functions h and g have been docu mented for the complement of a ray with different locations of basepoint. However, these functions have not been studied for the complement of a sin gle slit. In this study, we compare the h-function and the g-function of the region Ω = C\[1,2] with basepoint z0 = 0. Also, we explain the asymptotic behaviour of these functions. To compute h(r) of the region Ω, we transform the region Ω to a half plane via the M¨obius map followed by the square root transformation. Then we evaluate the angle of sight, which is the angle subtended by the image of ∂Ω∩B(z0,r) at the image of z0 in the halfplane. To evaluate g(r) for r ≤ 2, we map the region Ω∗ r to a halfplane by using the sequence of suitable conformal maps. Then we evaluate the angle of sight, which is the angle subtended by the image of ∂Ω∗ r ∩ B(z0,r) at the image of z0 in the halfplane. For r > 2, the region Ω∗ r becomes a doubly connected region. In this case, we transform Ω∗ r to a concentric annulus, and the harmonic measure at the preimage of the basepoint gives the formula of g(r) for r > 2. In summary, we have explained the computation of h(r) and g(r) of Ω = C\[1,2] with basepoint z0 = 0. In the future, we are interested in investigating g(r) for multiply connected slit regions.