Geometric ergodicity and the spectral gap of non-reversible Markov chains

We argue that the spectral theory of non-reversible Markov chains may often be more effectively cast within the framework of the naturally associated weighted-L∞ space $${L_\infty^V}$$ , instead of the usual Hilbert space L2 = L2(π), where π is the invariant measure of the chain. This observation is, in part, based on the following results. A discrete-time Markov chain with values in a general state space is geometrically ergodic if and only if its transition kernel admits a spectral gap in $${L_\infty^V}$$ . If the chain is reversible, the same equivalence holds with L2 in place of $${L_\infty^V}$$ . In the absence of reversibility it fails: There are (necessarily non-reversible, geometrically ergodic) chains that admit a spectral gap in $${L_\infty^V}$$ but not in L2. Moreover, if a chain admits a spectral gap in L2, then for any $${h\in L_2}$$ there exists a Lyapunov function $${V_h\in L_1}$$ such that Vh dominates h and the chain admits a spectral gap in $${L_\infty^{V_h}}$$ . The relationship between the size of the spectral gap in $${L_\infty^V}$$ or L2, and the rate at which the chain converges to equilibrium is also briefly discussed.