Title: Simple Conditions for Metastability of Continuous Markov Chains,
Abstract: A family $\{Q_{\beta}\}_{\beta \geq 0}$ of Markov chains is said to exhibit
$\textit{metastable mixing}$ with $\textit{modes}$
$S_{\beta}^{(1)},\ldots,S_{\beta}^{(k)}$ if its spectral gap (or some other
mixing property) is very close to the worst conductance
$\min(\Phi_{\beta}(S_{\beta}^{(1)}), \ldots, \Phi_{\beta}(S_{\beta}^{(k)}))$ of
its modes. We give simple sufficient conditions for a family of Markov chains
to exhibit metastability in this sense, and verify that these conditions hold
for a prototypical Metropolis-Hastings chain targeting a mixture distribution.
Our work differs from existing work on metastability in that, for the class of
examples we are interested in, it gives an asymptotically exact formula for the
spectral gap (rather than a bound that can be very far from sharp) while at the
same time giving technical conditions that are easier to verify for many
statistical examples. Our bounds from this paper are used in a companion paper
to compare the mixing times of the Hamiltonian Monte Carlo algorithm and a
random walk algorithm for multimodal target distributions.