Lower bound on contractivity rate via square root of underlying spectral gap (or equivalent upper bound on lift spectral gap)

Determine whether, for every second-order lift (\hat P_t) of a reversible diffusion with generator L and spectral gap gap(L), the quantity 1/ν in the T-delayed exponential contractivity (or exponential contractivity in 2T-average) of the lifted semigroup can be bounded below by a constant multiple of 1/√gap(L). Equivalently, establish an upper bound of the form gap(\hat L) ≤ C·√gap(L) for the spectral gap of the lift’s generator \hat L, with a universal constant C independent of T.

Background

The paper introduces second-order lifts of reversible diffusions and defines non-asymptotic relaxation times via L2 contractivity. A central result (Theorem 4.1) shows that for any lift, (1/ν)+T is bounded below by a constant multiple of 1/√gap(L), implying at most a square-root improvement of relaxation times. However, this lower bound involves T, the delay parameter in the contractivity definition, and does not isolate 1/ν alone.

The authors note that obtaining a lower bound directly on 1/ν in terms of 1/√gap(L) would follow from an upper bound on the spectral gap gap(\hat L) of the lifted generator by a multiple of √gap(L) of the underlying reversible generator. They emphasize that they were unable to prove such an upper bound and relate this difficulty to a known gap in the discrete-time literature, where upper bounds on the spectral gap of lifts are also unknown.