If $Q$ is the generator of a well-behaved continuous-time Markov process on a finite state space and $p$ is the invariant distribution, the corresponding Dirichlet form is $\mathcal{D}_Q(f) := \frac{1}{2} \sum_{j,k} p_j Q_{jk} (f_j - f_k)^2$. Write $Var_p(f)$ for the variance of $f$. In studies of convergence it is of great interest to find the infimum of $\mathcal{D}_Q/Var_p$ (where the denominator is restricted to be nonzero). W/l/o/g it suffices to consider $f$ for which $\sum_j p_j f_j = 0$, so that $Var_p(f) = \sum_j p_j f_j^2$.

If $Q^{(m)} = c_m Q$ for $c_m > 0$ and $Q^\otimes = \sum_m I^{\otimes(m-1)} \otimes Q^{(m)} \otimes I^{\otimes(N-m)}$, it can be shown that $\mathcal{D}_{Q^\otimes}(f^{\otimes N})/Var_{p^{\otimes N}}(f^{\otimes N}) = N \langle c \rangle \mathcal{D}_Q(f)/Var_{p}(f)$, where the arithmetic mean is indicated. Which is great and all, but: 

> Is the infimum of
> $\mathcal{D}_{Q^\otimes}/Var_{p^{\otimes
> N}}$ actually attained for an argument of the
> form $f^{\otimes N}$?

I feel like the answer should be trivial but for some reason I'm not seeing it. 

**UPDATE:** Given my answer below, I will award the bounty to the best answer that deals with some combination of finding/addressing any errors in this answer and reconciling the implied result for mixing times of tuples of Markov processes with [this paper][1] ([PDF][2]) of Ycart and coworkers (viz., if the cutoff and mixing times are equivalent as claimed, why do they appear to grow differently with $N$?).


  [1]: http://dx.doi.org/10.1016/j.spa.2006.03.003
  [2]: http://ljk.imag.fr/membres/Bernard.Ycart/publis/cutoffSPA.pdf