Monotonicity for continuous-time random walks

Consider continuous-time random walks on Cayley graphs where the rate assigned to each edge depends only on the corresponding generator. We show that the limiting speed is monotone increasing in the rates for infinite Cayley graphs that arise from Coxeter systems, but not for all Cayley graphs. On finite Cayley graphs, we show that the distance -- in various senses -- to stationarity is monotone decreasing in the rates for Coxeter systems and for abelian groups, but not for all Cayley graphs. We also find several examples of surprising behaviour in the dependence of the distance to stationarity on the rates. This includes a counterexample to a conjecture on entropy of Benjamini, Lyons, and Schramm. We also show that the expected distance at any fixed time for random walks on $\mathbb{Z}^+$ is monotone increasing in the rates for arbitrary rate functions, which is not true on all of $\mathbb{Z}$. Various intermediate results are also of interest.