Consider $$ Au:=\operatorname{div}\left(y^{\beta}\nabla u\right) \text{ for } (x,y)\in \mathbb{H} $$ and $u|_{\mathbb{R}}(x,0)=\phi(x)$ and some $\beta\in (0,1)$. For $\phi\in L^{2}(\mathbb{R},dx)$ (in fact less is required) we have a unique solution with $$\int_{0}^{\infty}\|\nabla_{(x,y)}u\|^{2}y dy<\infty$$ (see Boundary value problems for degenerate elliptic equations and systems ).

I am interested to know if there is a corresponding diffusion $X_{t}=(X_{1,t},X_{2,t})$ s.t. $$u(z)=E_{z}[\phi(X_{\tau_{\partial \mathbb{H}}})].$$

Q: Is my following approach correct? If not or there is a simpler approach, please only provide hints to help me learn.

- The corresponding process is $(B_{t}, R_{t}(\beta))=(\mbox{Brownian motion}, \mbox{Bessel}(\beta))$ because it has the same generator $$\Delta_{(x,y)}u(x,y)+\frac{\beta}{y}u_{y}=0.$$
- In "On the martingale problem for degenerate", they show that for the degenerate pde $$y\Delta_{(x,y)}u(x,y)+\beta u_{y}=0,$$

there is a corresponding unique-in-law process that solves the martingale problee. Up to a time change, we get the above process. - Since the semi-elliptic $Au=0$ has a unique solution, we get that $u(z)=E_{z}[\phi(X_{\tau_{\partial \mathbb{H}}})]$ solves because it solves the martingale problem and taking expectation of Dynkin's formula (see Oksendal pg. 181 "stochastic Dirichlet problem" ).

Q2: The main issue here is whether the L2-boundary data causes any issues with uniqueness.

The Bessel diffusion is not an Ito diffusion but it is equal in law to an Ito diffusion process (see Oksendal on Bessel processes). So the most uniqueness we can hope for is in law.

isan issue (one needs to impose appropriate bounds on $u$ "at infinity"), and for this reason taking expectation in Dynkin's formula is somewhat problematic (the exit time need not have finite mean, $u$ is not in the Feller generator of the diffusion, etc.). $\endgroup$