For a path $\Gamma$,
$$
\{f(x)-f(y)\}^2
\leq|\Gamma|\sum_{e\in\Gamma}\{\nabla_ef\}^2
$$
by the <Cauchy-Schwarz inequality>. Average over $\nu_{xy}$, multiply by $\widetilde Q(x,y)$, and sum. Reversing the order of summation in the definition of the two <Dirichlet form of a Markov chain>[Dirichlet forms] gives
$$
\mathcal E_{\widetilde P}(f,f)\leq B\mathcal E_P(f,f).
$$
Let $M=\max_x\pi(x)/\widetilde\pi(x)$. The variational formula for variance gives
$$
\operatorname{Var}_\pi f
=\min_c\sum_x\pi(x)(f(x)-c)^2
\leq M\operatorname{Var}_{\widetilde\pi}f.
$$
Take a nonconstant eigenfunction attaining the Rayleigh quotient $\gamma$ for $P$. Then
$$
\widetilde\gamma
\leq\frac{\mathcal E_{\widetilde P}(f,f)}
{\operatorname{Var}_{\widetilde\pi}f}
\leq MB\frac{\mathcal E_P(f,f)}{\operatorname{Var}_\pi f}
=\boxed{MB\gamma}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 215 / 3 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-215/3/a/solution