stochastic calculation – Conforming mappings and diffusion process with boundary condition

I have a question about a relationship between conformal matches and diffusion processes with boundary conditions.

Let $ D_1 $ to be a simply connected smooth domain of $ mathbb {R} ^ 2 cong mathbb {C} $. This can be unlimited.

We can define the normally reflecting Brownian motion $ X $ sure $ bar {D_1} $. We can also describe the Skorohod equation. The generator is the Laplacian $ Delta $ sure $ D_1 $ with Neumann limit condition.

Let $ D_2 subset mathbb {R} ^ 2 $ to be another smooth domain simply connected.
Let $ Psi: D_1 to D_2 $ be a conforming map.
We also assume that $ Psi $ is extended to a homeomorphism of $ bar {D_1} to bar {D_2} $ and $ Psi ( partial D_1) = partial D_2 $ (I do not know if this assumption is necessary.Please tell me if it is useless.)

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

In $ (r, w) $-to coordinate, $ Delta $ can not take shape $ frac { partial ^ 2} { partial r ^ 2} + frac { partial ^ 2} { partial w ^ 2} $. In $ (r, w) $-to coordinate, $ Delta $ become a more general broadcast operator. We note by $ mathcal {L} $ l & # 39; operator.

My questions are:

  • Is the dissemination process $ Psi (X) $ correspond to the operator $ mathcal {L} $?
  • Does the operator $ mathcal {L} $ meet Neumann's limit condition on $ partial D_2 $?

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

Do I have to prove that the diffusion process determined by $ mathcal {L} $ (with boundary condition of Neumann) and the diffusion process $ Psi (X) $ coincide?