# 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?