Stochastic calculation: conformal mappings and diffusion processes with contour condition

I have a question about a relationship between conforming assignments and diffusion processes with contour condition.

Leave $ D_1 $ be a simply connected domain without problems of $ mathbb {R} ^ 2 cong mathbb {C} $. This can be unlimited.

We can define the usually reflecting the Brownian movement $ X $ in $ bar {D_1} $. We can also describe the Skorohod equation. The generator is the Laplacian. $ Delta $ in $ D_1 $ with Neumann contour condition.

Leave $ D_2 subset mathbb {R} ^ 2 $ be another domain simply connected without problems.
Leave $ Psi: D_1 to D_2 $ Be a conformal mapping
We also assume that $ Psi $ It extends to a homeomorphism from $ bar {D_1} to bar {D_2} $ Y $ Psi ( partial D_1) = partial D_2 $ (I do not know if this assumption is necessary, please tell me if it is unnecessary).

Using this map, we can make the change of variable: $ D_1 ni ( rho, z) mapsto (r, w) in D_2 $, where $ r = text {Re} Psi ( rho, z) $ Y $ z = text {Im} Psi ( rho, z) $.

In $ (r, w) $-to coordinate, $ Delta $ can not take the form $ frac { partial ^ 2} { partial r ^ 2} + frac { partial ^ 2} { partial w ^ 2} $. In $ (r, w) $-to coordinate, $ Delta $ You can become a more general broadcast operator. We denote by $ mathcal {L} $ the operator.

This are my questions:

  • The diffusion process? $ Psi (X) $ corresponds to the operator $ mathcal {L} $?
  • Does the operator $ mathcal {L} $ satisfy the Neumann limit condition in $ partial D_2 $?

You will think that these are trivial. But I do not know how to justify these results.

Should I prove that the diffusion process determined by $ mathcal {L} $ (with Neumann contour condition) and the diffusion process. $ Psi (X) $ coincide?