$B$ is identically zero.

We can easily impose these boundary conditions in practice by modifying our sampling procedure as follows. Suppose that we have performed a sampling as before and computed the steady state (unbiased) $\pi_\alpha$ and $\nu_{\alpha,\beta}$. We can then run another independent sampling where we only consider the cells outside of $A$ and $B$. In these cells, we run trajectories as before (though, as we will see, using re-entry points that are different from the ones calculated before), store their exit points from the cells to build databanks of re-entry points in other cells, and compute $$\label{eq:fluxdefA}
  \nu^A_{\alpha,\beta} = \frac{N^A_{\alpha,\beta}}{T^A_\alpha}$$ where $N^A_{\alpha,\beta}$ is the total number of times the trajectories hit the boundary between $B_\alpha$ and $B_\beta$ and $T^A_\alpha$ is the total simulation time in cell $B_\alpha$. Note that $\nu^A_{\alpha,\beta} $ is only defined by  if $\alpha$ is the index of a cell $B_\alpha$ not inside $A$ and $B$ (the index $\beta$, on the other hand, runs over all the cells, including those forming $A$ and $B$). Consistent with the bias we need to impose to focus on trajectories assigned to $A$, we supplement this by $\nu_{\alpha,\beta}^A=\nu_{\alpha,\beta}$ if $\alpha$ is the index of a cell $B_\alpha$ used to 