Functional inequalities such as the Poincaré and log-Sobolev inequalities quantify convergence to equilibrium in continuous-time Markov chains by linking generator properties to variance and entropy decay. However, many applications, including multiscale and non-reversible dynamics, require analysing probability measures that are not at equilibrium, where the classical theory tied to steady states no longer applies. We introduce generalised versions of these inequalities for arbitrary positive measures on a finite state space, retaining key structural properties of their classical counterparts. In particular, we prove continuity of the associated constants with respect to the reference measure and establish explicit positive lower bounds. As an application, we derive quantitative coarse-graining error estimates for non-reversible Markov chains, both with and without explicit scale separation, and propose a quantitative criterion for assessing the quality of coarse-graining maps.
Functional inequalities such as the Poincaré inequality (PI) and the log-Sobolev inequality (LSI) are central tools in the quantitative analysis of Markov semigroups and diffusion processes. In the setting of a continuous-time Markov chain with an irreducible generator matrix, these inequalities quantify how fast observables and distributions relax toward the steady state (or towards equilibrium). The PI controls the exponential decay of the variance of observables, thereby determining the rate of convergence to steady state in the L 2 -sense. The LSI provides a stronger, entropy-based control, implying exponential convergence of the law of the process to equilibrium in relative entropy. In both cases, the inequality constants (respectively called the PI and LSI constants) encapsulate the strength of the mixing mechanism in that small(/large) constants correspond to slow(/fast) equilibration. These inequalities have thus become fundamental in the analysis of convergence to equilibrium, concentration of measure, and the stability of stochastic systems, see [DSC96, SC97, AF99, BT06, MT06, EF18, SS19] for a non-exhaustive list. Most of these references focus on the special setting of reversible Markov chains which allow for considerably deeper analysis.
However, the assumptions of reversibility and equilibrium (characterised by the steady state) is restrictive in certain applications. For instance, complex chemical and biomolecular systems are often modelled via high-dimensional, possibly non-reversible, Markov chains constructed from molecular dynamics [PWS + 11, SS13], which are routinely coarse-grained into clusters [DW05, KW07,FSW18]. Inspired by coarse-graining approaches developed for diffusion processes [LL10,Cho03], the recent work [HS24] by two of the authors provides a reduced dynamics on the clusters by defining an effective generator through conditional expectations. The accuracy of this (and related) reduced models depends on the mixing properties within each cluster, governed by measures that are not the steady states. In these contexts, the classical PI and LSI inequalities, defined with the steady state as the reference measure, are no longer applicable and therefore lead to the following natural question.
for reference measures other than the steady state?
To the best of our knowledge, a comprehensive framework for non-equilibrium functional inequalities is not available in the literature. Therefore, in this work, we introduce generalised versions of the Poincaré and log-Sobolev inequalities (called gPI and gLSI) defined for arbitrary reference measures in the class of positive probability measures on the state space of the underlying continuous-time Markov chain, see (21) and ( 22). These inequalities are built upon natural generalisations of the Dirichlet form and the Fisher information. We show that these latter generalisations retain key analytical properties of their classical counterparts: non-negativity, convexity, and continuity in their argument, see Proposition 3.2. Using these generalised functionals, we establish a family of functional inequalities parametrised by the reference probability measure ζ, whose optimal constants are denoted by αgPI(ζ) and αgLSI(ζ), see (23). Beyond the use of non-equilibrium reference measures, a crucial point of this study is that we do not require the Markov chain to be reversible (i.e. to be in detailed balance), which is often assumed when dealing with such functional inequalities.
We demonstrate that these generalised constants possess several desirable properties. As in the equilibrium case, there is a hierarchy for the functional inequalities, cf. [BT06]. Specifically, the gLSI implies the gPI with αgLSI(ζ) ≤ αgPI(ζ), see Proposition 4.1. Furthermore, both the gPI and gLSI inequalities satisfy the tensorisation property, see Proposition 4.2. Beyond these classical properties, we prove that the constants vary continuously with ζ, see Proposition 3.5 and Theorem 4.4. In addition, we provide strictly positive lower bounds for the constants, see Theorem 4.7. Specifically, these lower bounds also hold for the classical PI and LSI constants, by which we establish their positivity for irreducible generators that need not be reversible. These properties require ζ to be strictly positive. However, this is typically guaranteed in applications where ζ is linked to solutions of forward Kolmogorov equations with irreducible generators, see for instance [HPST20,Lemma C.1].
We demonstrate the utility of these generalised inequalities by applying them to two problems where non-equilibrium analysis is essential. First, we derive quantitative stability estimates for the timedependent probability distribution of a Markov chain relative to any other such measure ζ, see Section 5.1. Here ζ is time-dependent, and consequently the gPI and gLSI constants are time-dependent. We apply our continuity result and lower bounds to establish an exponential convergence resu
This content is AI-processed based on open access ArXiv data.