Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 215 / 4 / b / Solution

Put $B_r=B(x,s+r)$. Every transition leaving $B_r$ enters $B_{r+1}$, and stationarity bounds its flow by the stationary mass newly reached:While ${\pi }^G(B_r)\le 1/2$, the definition of ${\Phi }_{\ast }^G$ therefore givesIteration for $r=0,\dots ,t-1$ proves the formula.