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We are trying to build a discrete model for each SLE (Schramm-Loewner evolution) and one key step is solving the following question: Q: Finding a two-dimensional $ \mathbb{H}$ -conformally invariant (details below) process $ X=(X_{1},X_{2})$ in the upper half-plane such that $ $ c\int^{arg(z)}_{0}sin(\theta)^{\beta}d\theta= P_{z}[(X_{1,T_{\mathbb{H}}},X_{2,T_{\mathbb{H}} })\in \mathbb{R}^{-}],$ $ for constant $ c=1/(\int^{\pi}_{0}sin(\theta)^{\beta}d\theta)$ and arbitrary $Read more