# 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?