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:
$$
\pi^G(B_{r+1}\setminus B_r)\geq Q(B_r,B_r^c).
$$
While $\pi^G(B_r)\leq1/2$, the definition of $\Phi_*^G$ therefore gives
$$
\pi^G(B_{r+1})
\geq(1+\Phi_*^G)\pi^G(B_r).
$$
Iteration for $r=0,\ldots,t-1$ proves the formula.


Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-215/4/b/solution