Search

The Loewner differential equation is a partial differential equation that generates a Loewner chain. An estimation of the solution of the dipolar Loewner differential equation for time-direction is first derived by using Bieberbach theorem. Based on the reverse-time Loewner equation the difference of two solutions to the dipolar Loewner equation with two different deriving functions is estimated in terms of the supremum norm of the difference of the two driving functions. This generalizes some related results for the chordal and radial Loewner equations to the dipolar setting. [ABSTRACT FROM AUTHOR]

By using excursion measure Poisson kernel method, we obtain a second-order differential equation of the intersection probability of Brownian motion and SLEκ. Moreover, we find a transformation such that the second-order differential equation transforms into a hypergeometric differential equation. Then, by solving the hypergeometric differential equation, we obtain the explicit formula of the intersection probability for the trace of the chordal SLEκ and planar Brownian motion started from distinct points in an upper half-plane H-. [ABSTRACT FROM AUTHOR]