Let $D_1$ and $D_2$ be two bounded simply connected Jordan domains in $\mathbb{R}^2$. By Carathéodory's Theorem there exists a homeomorphism $f:\bar{D}_1 \to \bar{D}_2$ such that the restriction $f:D_1 \to D_2$ is conformal. If $E\subset \partial D_1$ is Borel measurable then the harmonic extension of the indicator function $\mathbf{1}_E$ from $\partial D_1$ to $\bar{D}_1$ is called the harmonic measure of $E$ and is denoted by $\omega(x,E;D_1)$ for $x\in \bar{D}_1$. It is well known that $f$ preserves harmonic measure in the sense that $\omega(x,E;D_1)=\omega(f(x),f(E);f(D_1))=\omega(f(x),f(E);D_2)$. Since $\omega(\cdot,E;D_1)$ is really just the solution to the BVP

$\Delta u=0$ in $D_1$

$u=\mathbf{1}_E$ on $\partial D_1$
we can generalize harmonic measure to other elliptic operators $\mathcal{L}$ besides the Laplacian. This generalization is called $\textit{elliptic measure}$ or $\mathcal{L}\textit{harmonic measure}$, see Diffusions and Elliptic Operators by Bass. My question is whether given $D_1,D_2,$ and $\mathcal{L}$ as above does there exist an injective "generalized conformal map" from $D_1$ onto $D_2$ that preserves elliptic measure and obeys a first order CauchyRiemann type system? I'm interested in finding out whether such maps exist and what their CauchyRiemann systems are for constant coefficient operators of the form $\mathcal{L}u=\Delta u+\alpha\cdot \nabla u$. I believe that quasiconformal maps give an affirmative answer for a certain class of elliptic operators but I'm not sure these include what I'm looking for.