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Consider nice domain $D\subset \mathbb R^d$ and $\Delta u =0$ with $u\big|_{\partial D}=g$. It is well known that $u(x)=E^x[g(B(\tau))]$ where $\tau$ is exit time of $B$ from the domain $D$.

What if we consider Neumann boundary conditions? Is there a stochastic representation?

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Yes, see Section 4.4.2 in "Stochastic Differential Equations, Backward SDEs, Partial Differential Equations" by Pardoux and Rascanu.

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