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Reference for representation of heat equation with Neumann boundary condition on smooth domain using reflected Brownian motion
We know that the solution of the heat equation $\partial_tu=\frac 12\Delta u$ with Dirichlet boundary condition $u\rvert_{\partial\Omega}=g$ is $u(t,x)=\mathbb{E}[g(B_\tau)\mid B_t=x]$, with $\tau$ ...
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