Consider divergence form elliptic pde in smooth boundary domain D

$$ Au:=\sum_{i,j}\partial_{i}(a_{ij}(x)\partial_{i}u(x)), $$

with boundary data $u|_{\partial D}:=1_{A}$ for $A\subset \partial D$.

By Riesz representation there is a measure called the L-measure s.t.

$$u(x)=\int_{\partial D} \phi(y)d\omega_{x}^{L}(y)$$

if $u|_{\partial D}=\phi\in C(\partial D)$. For the harmonic case (Dirichlet problem) we have that the harmonic measure is harmonic and satisfies

$$\omega_{x}(A,\partial D)=P_{x}[B_{T_{\partial D}}\in A],$$

where B is Brownian motion.

Q1: Do we have any such formula for nice enough sets A and pde Au that give a Feynman-Kac formula for some Ito process X: $$u(x)=\omega^{L}(x,A)=P_{x}[X_{T_{\partial D}}\in A].$$

In modern literature they mainly deal with continuous boundary data. For a related question in parabolic pdes see this question.

The particular equation I have is in the upper half plane $\{(x,y):y>0\}$

$$\frac{1}{\beta}y \Delta u+\partial_{y}u=0$$

with $\beta>0$ and boundary data $u|_{R}=1_{R^{-}}$ the indicator on the half line. Indeed as mentioned below as well the divergence form I had in mind was $\nabla(y^{\beta}\nabla u)=0$ (which is the same in our case because $y>0$).

Q2: If we can prove that we can apply Feynman-Kac then we obtain: $$u(x)=P[(X_{1,T_{\mathbb{H}}},X_{2,T_{\mathbb{H}} })\in \mathbb{R}^{-}],$$ where $dX_{1}=dB_{1}, dX_{2}=\frac{\beta}{X_{2}}dt+dB_{2}.$

**Attempts**

1)mainly going through the corresponding proofs for the harmonic measure. In Doob's potential theory book 2.IX.7/13 he proves the harmonic measure statement above.

2)As with dirichlet we have the same representation for continuous boundary data:

$$E_{x}[\phi(X_{T_{\partial D}})]=u(x)=\int_{\partial D} \phi(y)d\omega_{x}^{L}(y).$$

So at least heuristically by approximating the indicator function $1_{R^{-}}$ by continuous functions we obtain something close to the desired formula above.