# Finding process $(X_{1},X_{2})$ s.t. $\int^{arg(z)}_{0}\sin(\theta)^{\beta}d\theta=P_{arg(z)}[(X_{1},X_{2})\in \mathbb{R}^{-}]$

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.