Why the distribution of the first exit time (for an SDE) is the solution of the PDE $\mathcal Au=-1$ on $D$ and $u=0$ on $\partial D$?

Question

Let consider the SDE $$dX_t=b(X_t)+\sqrt{\varepsilon }g(X_t)dW_t,\ X_0=x_0\in D$$ where $D$ is a bounded domain. Let $$\mathcal Av(x)=\varepsilon \sum_{i,j=1}^d a_{ij}(x)\frac{\partial ^2}{\partial x_i\partial x_j}v(x)+b(x)\cdot \nabla v(x),$$

where $a=gg^T$. Let $\tau=\inf\{t>0\mid X_t\in \partial D\}$. Why $\mathbb E^{x_0}[\tau]$ is the unique solution of $$\begin{cases}\mathcal A u=-1&in\ D\\ 0&on\ \partial D.\end{cases}$$ I don't understand where it comes from (it looks a bit magic).