The intersection probability of Brownian motion and SLE (Q277923)

Summary: By using the excursion measure Poisson kernel method, we obtain a second-order differential equation of the intersection probability of a Brownian motion and \(\text{S} \text{L} \text{E}_\kappa\). 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 \(\text{S} \text{L} \text{E}_\kappa\) and the planar Brownian motion started from distinct points in an upper half-plane \(\overline{H}\).