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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 $X=(X_{1,\beta},X_{2,\beta})$ in the upper half-plane such that $$c\int^{arg(z)}_{0}\sin(\theta)^{\beta}d\theta= P_{z}[(X_{1,\beta}(T_{\mathbb{H}}),X_{2,\beta}(T_{\mathbb{H}}) )\in \mathbb{R}^{-}],$$ 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 $P_{z}[X_{T_{\mathbb{H}}}\in \mathbb{R}^{-}]=\frac{1}{\pi}arg(z)$. Then this probability uniquely characterizes SLE(4).

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.

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