The L2-cutoff for reversible Markov processes

We consider the problem of proving the existence of an L2-cutoff for families of ergodic Markov processes started from given initial distributions and associated with reversible (more, generally, normal) Markov semigroups. This includes classical examples such as families of finite reversible Markov chains and Brownian motion on compact Riemannian manifolds. We give conditions that are equivalent to the existence of an L2-cutoff and describe the L2-cutoff time in terms of the spectral decomposition. This is illustrated by several examples including the Ehrenfest process and the biased (p,q)-random walk on the non-negative integers, both started from an arbitrary point.