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It is often possible to speed up the mixing of a Markov chain $\{ X_{t} \}_{t\in \mathbb{N}}$ on a state space $\Omega$ by \textit{lifting}, that is,running a more efficient Markov chain $\{ \hat{X}_{t} \}_{t \in \mathbb{N}}$ ona larger state space $\hat{\Omega} \supset \Omega$ that projects to $\{ X_{t}\}_{t \in \mathbb{N}}$ in a certain sense. In [CLP99], Chen, Lov{\'a}sz and Pakprove that for Markov chains on finite state spaces, the mixing time of anylift of a Markov chain is at least the square root of the mixing time of theoriginal chain, up to a factor that depends on the stationary measure.Unfortunately, this extra factor makes the bound in [CLP99] very loose forMarkov chains on large state spaces and useless for Markov chains on continuousstate spaces. In this paper, we develop an extension of the evolving set methodthat allows us to refine this extra factor and find bounds for Markov chains oncontinuous state spaces that are analogous to the bounds in [CLP99]. Thesebounds also allow us to improve on the bounds in [CLP99] for some chains onfinite state spaces.

Bounds on Lifting Continuous Markov Chains to Speed Up Mixing