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 $ \beta\in [0,2]$ and $ T_{\mathbb{H}}$ the exit time from the upper half-plane.

By $ \mathbb{H}$ -conformally invariant we mean that if for $ D\subset \mathbb{H}$ the $ f:D\to \mathbb{H}$ is a conformal map and $ X\in D$ then $ f(X)$ has the law of X but with possibly different variance/time change.

To get a flavor for it set $ \beta=0$ , then we simply set X=2d Brownian motion to obtain $ P_{z}[X_{T_{\mathbb{H}}}\in \mathbb{R}^{-}]=\frac{1}{\pi}arg(z)$ .

Attempts

1)Finding a generator:

The $ f(s):=a\int^{s}_{0}sin(\theta)^{\beta}d\theta$ satisfies the ode

$ $ -\beta f’cot(s)+f”=0,$ $

with boundary $ f(0)=0,f(\pi)=1$ , where $ a:=1/(\int^{\pi}_{0}sin(\theta)^{\beta}d\theta)$ . So one idea is to turn the ode into a pde: $ \Delta f(x,y)=\frac{\beta}{y}f_{y}(x,y)$ with boundary $ 1_{\mathbb{R}^{-}}$ .

Q2: Next we want to apply Feynman Kac to this pde but the boundary data is not continuous, so the rest is just speculation.

The diffusion we get is $ $ dX_{1}=dB_{1}, dX_{2}=\frac{\beta}{X_{2}}dt+dB_{2},$ $

which interestingly has shown up in the literature under the keyword Bessel-Brownian diffusions. As we can see from the pde it is not conformally invariant. However, it would still be interesting if:

Q3: $ $ 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}^{-}],$ $

2)Finding other conformally invariant processes

Q4: Is conformal invariance unique to Brownian motion?

In “Conformal mapping of some non-harmonic functions in transport theory” Bazant identified other pdes that are also conformally invariant:

$ $ a(u) \nabla^{2}u+b(u) |\nabla u|^{2} u=0,$ $ for possibly nonlinear functions a,b.