1 Introduction↩︎

1.1 Setting and motivation↩︎

Let \((X_t)_{t\ge 0}\) be a time-homogeneous Markov process with an invariant probability measure \(\mu\) on a measurable space \(S\). The associated semigroup \((P_t)_{t\geq 0}\) on \(L^2(\mu)\) is given by \(P_t g\,(x) = \mathbb{E}_x[g(X_t)]\), where \(\mathbb{E}_x\) denotes the expectation conditioned on \(X_0 = x\). Assume that \((P_t)_{t\geq 0}\) is strongly continuous, with the generator \(L\) defined on \(\operatorname{Dom}(L)\) by \(L g = \lim_{t \downarrow 0} \frac{P_t g - g}{t}\), where \(\operatorname{Dom}(L)\) consists of \(L^2\)-functions \(g\) for which the limit exists. By the Hille-Yosida theorem, \(L\) is closed and \(\operatorname{Dom}(L)\) is dense in \(L^2(\mu)\). We sometimes consider discrete-time Markov chains as well, in which case the setting will be clarified. Let \[\langle f, g\rangle \mathrel{\vcenter{:}}= \int \bar{f} g \, d\mu\] denote the \(L^2(\mu)\) inner product, and \(\| \cdot \|\) the corresponding norm. The empirical average \(\mu_t g\) defined below can be understood as a Bochner integral, since \(s \mapsto g(X_s)\) is continuous as an \(L^2(\mathbb{P}_\mu)\)-valued function by the strong continuity of \((P_t)_{t \geq 0}\). If the process has jointly measurable sample paths, this coincides with the usual pathwise time integral; for simplicity, we assume joint-measurability. We shall also restrict attention to centered observables \(g\) in \(L^2_0(\mu):=\left\{g\in L^2(\mu): \int g\,d\mu=0\right\}\). Note that for \(g\) in \(L^2(\mu)\), we can always take \(g - \mu g \in L^2_0(\mu)\).

A basic problem in the study of a Markov process is to quantify the speed with which the process converges. Two natural but distinct questions are: how quickly the time-\(t\) law approaches \(\mu\), and how quickly the empirical average \[\mu_t g:=\frac{1}{t}\int_0^t g(X_s)\,ds\] approaches \(\mu g \mathrel{\vcenter{:}}= \int g \, d\mu\), the equilibrium average.

If the Markov process is started from stationarity with \(X_0 \sim \mu\), the convergence of empirical averages is naturally measured by the empirical variance \[\operatorname{Var}_\mu(\mu_t g) \mathrel{\vcenter{:}}= \mathbb{E}_\mu |\mu_t g - \mu g|^2.\] In the context of Markov chain Monte Carlo (MCMC) algorithms, the two questions about convergence raised above target, respectively, the burn-in time and the time required for accurate sampling after (approximate) stationarity is reached; see e.g. [2] [3].

In the present paper, we study the same basic convergence questions but depart from the traditional perspectives in some respects. There is already substantial research on nonreversible Markov chains and processes–see the review in section 1.9 of [1] and section 1.3 below. The interest in such processes has been mainly driven by their accelerated convergence. While most previous studies focus on specific classes of examples, we follow the work on nonreversible finite-state Markov chains in [1] and consider a universal spectral theory for nonreversible Markov processes on arbitrary state spaces. Even without reversibility, we find that a generalization of the spectral gap that we call the singular-value gap remains useful and informative. The main finding is that a positive singular-value gap gives finite-time, observable-independent bounds on empirical variance and naturally relates to other quantities for Markov process convergence.

1.1.1 Spectral gap vs. singular-value gap↩︎

In the reversible setting, different notions of convergence are tied to the same spectral quantity. Since \(-L\) is nonnegative and self-adjoint on \(L^2(\mu)\), its spectrum is contained in \(\mathbb{R}_{\geq 0}\). The spectral gap \(\lambda\) of \(L\) is then defined as \[\lambda \mathrel{\vcenter{:}}= \inf \sigma(-L\mid_{L^2_0(\mu)}),\] where \(\sigma\) denotes the spectrum. For a discrete-time finite-state Markov chain with transition matrix \(P\), the generator is given by \(L = P - I\), in which case \(\lambda\) is usually characterized as one minus the second-largest eigenvalue of \(P\). When \(\lambda > 0\), we have exponential \(L^2\)-contraction of the semigroup: \(\|P_t f\| \leq e^{-\lambda t} \|f\|\) for all centered observables \(f\). This leads to the Poincaré inequality \(\|f\|^2 \le \lambda^{-1}\langle f,-Lf\rangle\), where \(\langle-Lf,f\rangle\) is the Dirichlet form evaluated on \(f\) and is often written as \(\mathcal{E}(f,f)\). In fact, the spectral gap can be characterized as (the inverse of) the optimal constant in the Poincaré inequality: \[\begin{align} \label{reversible32spectral32gap32characterization} \lambda = \inf_{f\in \operatorname{Dom}(L)\cap L^2_0(\mu)\backslash \{0\}} \frac{\langle-Lf,f\rangle}{\|f\|^2}. \end{align}\tag{1}\] Although less emphasized in the continuous-time literature, it is a quick consequence of \(L^2\)-contraction and the standard identity 16 that when the process is reversible, \[\begin{align} \label{reversible32empirical32variance32bound} \operatorname{Var}_\mu ( \mu_t g) \le \frac{2}{\lambda t} \|g\|^2 \end{align}\tag{2}\] for all \(g \in L^2_0(\mu)\). For Markov chains on finite state space, the time-\(n\) empirical average is \[\mu_n g \mathrel{\vcenter{:}}= \frac{1}{n} \sum_{i=0}^{n-1} g(X_i),\] and the corresponding empirical variance bound holds exactly as in 2 for a reversible chain, with continuous \(t\) replaced by discrete \(n\); see Lemma 12.22 in [4]. This perspective of the spectral gap as a canonical convergence quantity is central in the classical theory of reversible Markov chains and Markov diffusion processes; see [4] [5] and references therein.

For nonreversible processes, the situation is more delicate, as the generator is no longer self-adjoint. There are multiple ways to retrieve a spectral theory: one may still consider the spectral gap \(\gamma_s\) of the symmetrized process whose generator is given by \[\begin{align} \label{symmetrized32generator} L_s \mathrel{\vcenter{:}}= \frac{L + L^*}{2}; \end{align}\tag{3}\] see the more precise definition of \(\gamma_s\) in 27 below. Alternatively, one may also consider the real-part spectral gap of \(L\) or resolvent and pseudospectral quantities. There is no longer a single canonical spectral object.

For nonreversible finite-state Markov chains, it is demonstrated in [1] that empirical variance of observables can be uniformly bounded by the singular-value gap \(\gamma\), defined as the second-smallest singular value of \(I-P\), where \(P\) is the transition matrix. Note that \(L = P-I\) is the generator in discrete time, and we may equivalently write \[\begin{align} \label{gap32definition32discrete32time} \gamma:=\inf_{f\in L^2_0(\mu)\backslash \{0\}} \frac{\norm{(P-I)f}}{\norm{f}}. \end{align}\tag{4}\] With the formulation in 4 , it is natural to consider the continuous-time analog \[\begin{align} \label{gap32definition32continuous32time} \gamma:=\inf_{f\in \operatorname{Dom}(L)\cap L^2_0(\mu)\backslash \{0\}} \frac{\norm{Lf}}{\norm{f}}. \end{align}\tag{5}\] Indeed, this singular-value gap has also been studied in the context of continuous-time Markov processes, most recently in [[6]][7]. The terminology “singular-value gap” is literal. Let \(L_0 \mathrel{\vcenter{:}}= L \vert_{L^2_0(\mu)}\), with \(\operatorname{Dom}(L_0) = \operatorname{Dom}(L) \cap L^2_0(\mu)\). Consider the nonnegative self-adjoint operator \(|L_0|:=(L_0^*L_0)^{1/2}\). By the polar decomposition for closed densely defined operators, \(\|L_0 f\|=\||L_0|f\|\) for \(f\in\operatorname{Dom}(L_0)\). Therefore, \[\gamma = \inf_{\substack{f\in \operatorname{Dom}(L_0), \;\|f\|=1}} \|L_0f\| = \inf \sigma(|L_0| ).\] Equivalently, \[\gamma^2=\inf\sigma(L_0^*L_0).\] Thus, in a finite state space, \(\gamma\) is exactly the second-smallest singular value of the matrix \(-L\). When \(L\) is normal, by the spectral theorem, we also have \[\gamma = \mathrm{dist}\big(0, \sigma(L_0)\big),\] but this interpretation can fail without normality.

As we shall show, among various inequivalent metrics of convergence for a nonreversible Markov process, the singular-value gap \(\gamma\) defined in 5 specifically characterizes the convergence for empirical averages. This can be intuitively understood through the following simple example, which distinguishes \(\gamma\) from the spectral gap \(\gamma_s\) of the symmetric part of the generator.

The observations above are consistent with the intuition that a drift of strength \(\alpha\) should accelerate the convergence of empirical averages, but it merely translates the law of \(X_t\) by \(\alpha t\) and does not help with its convergence to the uniform distribution.

1.1.2 Asymptotic variance vs. empirical variance↩︎

Since \(\operatorname{Var}_\mu (\mu_t g)\) often decays as \(t^{-1}\), it is of interest to study the asymptotic variance, which is naturally connected to functional CLT results. For \(g\in L_0^2(\mu)\), the asymptotic variance of \(g\), when it exists, is defined by \[\begin{align} \label{asymptotic32variance32definition} \sigma_g^2 := \lim_{t\to\infty} t\,\operatorname{Var}_\mu(\mu_tg). \end{align}\tag{6}\] We will call \(\sigma_g \geq 0\) the asymptotic standard deviation of \(g\) governing long-time fluctuation: \[\sqrt{\operatorname{Var}_\mu (\mu_t g)} \sim \frac{\sigma_g}{\sqrt{t}}.\] A standard way to analyze \(\sigma_g^2\) is through the Poisson equation. If we can find \(f \in \operatorname{Dom}(L)\) that solves \(-Lf = g\), then \[\begin{align} \label{asymptotic32variance32representation32with32L32inverse} \sigma_g^2 = 2\operatorname{Re}\langle g,f\rangle . \end{align}\tag{7}\] See Lemma 5 for a short derivation of \(\eqref{asymptotic32variance32representation32with32L32inverse}\). When we can solve the Poisson equation for \(g\) and the process satisfies a few other technical assumptions, Dynkin’s formula and central limit theorem for martingales imply that \(\sqrt{t} \, \mu_t g\) converges to a Gaussian with mean zero and variance \(\sigma_g^2\). This is the classical CLT result of [8]. A substantial theory of functional CLT, notably initiated by [9], establishes analogous results without requiring a solution to the Poisson equation. See Chapter 2 of [10] and the references therein.

For the purposes of this paper, given a positive \(\gamma\), we are in the simpler regime that the Poisson equation is solvable. In fact, \(L^{-1}\) exists on \(L^2_0(\mu)\) with \(\|L^{-1}\|_{L^2_0 \to L^2_0} \leq \gamma^{-1}\), thanks to Lemma 2. Therefore, the Poisson equation is solvable for all functions in \(L^2_0(\mu)\). This yields a CLT under extra technical assumptions; for instance, when the process is progressively measurable, by Theorem 2.1 in [8]. Even if we only know that \(\gamma > 0\), we have a simple bound for the asymptotic variance: \[\begin{align} \label{upper32bound32on32asymptotic32variance32by32global32gamma} \sigma_g^2 = 2\operatorname{Re}\langle g,-L^{-1}g\rangle \leq \frac{2}{\gamma} \|g\|^2. \end{align}\tag{8}\] In the above bound, the exact value of \(\sigma_g^2\) may be much smaller and dependent on the particular observable \(g\). The singular-value gap studied in this paper provides observable-class-level bounds on finite-time empirical variance and asymptotic variance. The asymptotic variance is usually studied in the literature to obtain sharp long-time behavior for specific observables. Knowledge of asymptotic variance helps refine bounds on finite-time empirical variance: see Theorem 3 below.

1.2 Main results↩︎

We shall state our main results in a subspace-dependent form. Let \(V \subset L^2_0(\mu)\) be a subspace. Consider the singular-value gap of \(L\) on \(V\) given by \[\begin{align} \label{gap32definition32subspace} \gamma_V \mathrel{\vcenter{:}}= \inf_{f \in \operatorname{Dom}(L) \cap V \backslash \{0\}} \frac{\|Lf\|}{\|f\|}. \end{align}\tag{9}\] Let \(U \mathrel{\vcenter{:}}= L(V \cap \operatorname{Dom}(L))\). Note that since \(\mu\) is invariant, \(\int Lf\,d\mu=0\) for \(f \in \operatorname{Dom}(L) \cap L^2_0 (\mu)\); hence \(U \subset L^2_0(\mu)\). We quantify the worst-case convergence of empirical averages of observables in \(U\) by the quantity \[\begin{align} \label{definition32of32empirical32Delta95t94U} \Delta_t^U :=\sup_{g \in U:\, \|g\| = 1} \sqrt{\operatorname{Var}_\mu (\mu_t g)} = \sup_{g \in U+\mathbb{C} \mathbf{1}: \, \|g - \mu g\| = 1} \sqrt{\operatorname{Var}_\mu(\mu_t g)} \end{align}\tag{10}\] We have the following upper and lower bounds on \(\Delta_t^U\) in terms of \(\gamma_V\):

With the above theorem and the key input of Lemma 2, we obtain control of the empirical standard deviation of all observables in \(L^2(\mu)\), quantified by \[\begin{align} \label{definition32of32empirical32Delta95t} \Delta_t :=\sup_{g \in L^2_0(\mu):\, \|g\| = 1} \sqrt{\operatorname{Var}_\mu (\mu_t g)} = \sup_{g \in L^2(\mu): \, \|g - \mu g\| = 1} \sqrt{\operatorname{Var}_\mu(\mu_t g)} \end{align}\tag{11}\]

Note that for a pure rotation on the unit circle, for instance, the empirical average is exactly the equilibrium average at times when an integer number of rotations are performed. Therefore, for a general nonreversible dynamics, it seems infeasible to obtain lower bounds for empirical variance without taking the worst case over a time interval.

One might also wonder whether the bounds above are optimal. We discuss simple examples in section 2.3 that show that the upper and lower bounds in Corollary 1 are generally sharp up to constants. However, Theorem 3 below gives the following refinement in terms of the asymptotic variance: \[\begin{align} \label{refined32empirical32variance32bound32in32main32results} \frac{\sigma_g^2}{t} - \frac{4}{\gamma^2 t^2} \|g\|^2 \leq \operatorname{Var}_\mu(\mu_t g) \leq \frac{\sigma_g^2}{t} + \frac{4}{\gamma^2 t^2} \|g \|^2 \end{align}\tag{12}\] In particular, when \(\sigma_g^2 > 0\) for observables \(g\) under consideration, the standard deviation of empirical averages of \(g\) decays as \(t^{-1/2}\), the expected CLT scale. For certain dynamics and observables, the empirical standard deviation can decay faster on the order of \(t^{-1}\); we discuss a simple instance in Example 4. The bound 12 shows that such a faster convergence happens exactly when the asymptotic variance vanishes, but we do not explore the separation between the \(t^{-1/2}\) and \(t^{-1}\) regimes in more detail.

1.2.1 Outline↩︎

We contextualize our perspective within the existing literature in section 1.3. In section 2, we prove the local Theorem 1 and deduce the global Corollary 1 from Lemma 2, which gives a bounded inverse generator when \(\gamma > 0\). Using this lemma, we also extend Chatterjee’s original theorem for finite-state Markov chains to arbitrary state spaces, confirming that the singular-value gap \(\gamma\) is a universal quantity for convergence. We then discuss the sharpness of our variance bounds and prove the sharpened version 12 in terms of asymptotic variance.

In section 3, we study in detail the example of a two-dimensional OU process with rotational drift. This example is inspired by [11] and sits within a substantial literature on accelerating the convergence of diffusion processes by adding an irreversible drift. The main point of this example is that irreversibility may accelerate the convergence of empirical averages only for certain subspaces of observables.

We explore the connections between \(\gamma\) and some other common quantities related to mixing in section 4. In section 4.1, we bound \(\gamma\) in terms of \(\gamma_s\), the spectral gap of the symmetrized process. Theorem 4 shows that \(\gamma^{-1}\) provides a lower bound for the mixing time in total-variation distance, and Proposition 5 gives an upper bound on the mixing time in the restricted setting of jump processes on finite state space. We then connect the singular-value gap to the Cheeger constant, also known as the bottleneck ratio. Theorem 5 extends the previous results of [[12]][1] to nonreversible jump processes.

Our theory can be applied under minimal assumptions, but we only survey two more examples in section 5. We consider random walks on binary de Bruijn graphs to show that, as \(\gamma^{-1}\) lower bounds the total-variation mixing time by Theorem 4, the singular-value gap could more sharply record mixing properties compared to some other spectral quantities. We then observe that a compact resolvent criterion in Proposition 6 is enough to show that \(\gamma > 0\) for certain Markov diffusion generators, without explicit knowledge of the value of \(\gamma\). We study a specific hypoelliptic generator on the 2-torus and show that \(\gamma > 0\), hence empirical averages converge with fluctuations of order at most \(t^{-1/2}\). This example provides an interesting bridge between abstract PDE techniques and concrete sampling variance estimates. It would be interesting to find more substantial applications of the singular-value gap beyond the examples given in this paper.

1.3 Related literature↩︎

As mentioned above, the closest predecessor to the present work is the theory of nonreversible spectral gaps developed in [1] for nonreversible finite-state Markov chains. Chatterjee shows that for a chain with transition matrix \(P\), the reciprocal of the gap \(\gamma\) defined as the second-smallest singular value of \(I-P\) characterizes the time scale for the convergence of empirical averages. He also relates this quantity to mixing time, Cheeger constants, and path arguments. Our work follows this generator-singular-value viewpoint but extends it to continuous time, general state space, and localized observable classes.

An earlier precursor is [13], which obtains deviation bounds for the empirical averages of nonreversible chains in terms of the singular-value gap of the discrete generator. More recently, [6] discusses the singular-value gap for nonreversible diffusion lifts, although the main relaxation notion there is an \(L^2\)-contraction time of the semigroup rather than empirical-average variance. These generator singular values should be distinguished from other nonreversible spectral quantities. Notably, for discrete-time chains, Fill’s multiplicative reversibilization [14] and Paulin’s pseudo-spectral gap [15] use singular values of transition operators, or of products such as \((P^*)^kP^k\), to control convergence and concentration. [16] emphasize yet another perspective: for general state-space nonreversible chains, spectral theory for geometric ergodicity is often better formulated in weighted \(L^\infty\)-spaces. In the present paper, we keep our focus on \(L^2(\mu)\).

There is a large literature showing that nonreversibility can improve convergence, hence facilitate sampling. In finite state space, lifting is a central mechanism. [17] shows that lifting can speed up mixing and that square-root-type speedups are optimal in broad settings. [18] analyzes a concrete lifted nonreversible sampler. Analogous bounds for lifting constructions on continuous state space are developed in [19].

For diffusions, [[20]][21] show that adding a divergence-free drift preserving a target distribution can accelerate convergence. [22] studies optimal nonreversible linear drifts, emphasizing both exponential rates and prefactors. [23] gives general comparison results showing that antisymmetric perturbations reduce asymptotic variance for diffusions. There are also more detailed investigations of variance reduction: [24] uses large deviations of the empirical measure to quantify variance reduction for irreversible Langevin samplers, while [11] analyzes variance reduction via the Poisson equation and calculations for linear and quadratic observables. Our results are complementary: rather than considering irreversible perturbations of a certain form or reducing observable-specific asymptotic variance, we ask what can be concluded from a positive singular-value gap of the generator, globally or on an observable subspace. The connections of our perspective with asymptotic variance and central-limit theorems are discussed in section 1.1.2 above.

There is also substantial literature on finite-time deviation bounds for empirical averages. In the reversible setting, variance bounds for stationary empirical averages follow from the spectral gap, as recorded in 2 . [25] proves Chernoff and Berry–Esséen inequalities for empirical means of general ergodic Markov processes with a spectral gap. [15] obtains concentration, variance and Bernstein-type bounds for possibly nonreversible Markov chains through the pseudo-spectral gap. [26] derives nonasymptotic empirical-average error estimates under a positive-curvature assumption. More recently, Bernstein-type inequalities are obtained in [7], using the singular-value gap 4 , which the authors call the iterated Poincaré gap. In continuous time, [7] uses a slightly different characterization of the singular-value gap 5 and proves concentration inequalities for empirical averages of bounded observables, allowing nonstationary starts.

In contrast to some aforementioned studies with MCMC sampling in mind, we work primarily under stationarity. In applications where the chain is only approximately stationary, one can combine burn-in, warm-start, or concentration estimates with the stationary bounds studied here; developing this systematically is an interesting direction beyond the scope of the present paper.

2 Singular-value gap controls empirical variance↩︎

We will study the empirical average \(\mu_t g\) through the Cesàro mean \[\begin{align} \label{Cesaro32mean32definition} A_t g:=\frac{1}{t}\int_0^t P_s g\,ds. \end{align}\tag{13}\] Note that \(A_t g\) is a function on the state space \(S\), whereas \(\mu_t g\) is a \(\mathbb{C}\)-valued random variable. They are related by \[\mathbb{E}_x [\mu_t g ]= A_t g \,(x)\] Consider a subspace \(U \subset L^2_0 (\mu)\). Analogous to \(\Delta_t^U, \Delta_t\) in 10 11 , we define \[\begin{align} \delta_t^U &:= \sup_{g \in U: \, \|g \| = 1} \|A_t g \| = \sup_{g \in U + \mathbb{C}\mathbf{1}: \, \|g - \mu g\| = 1} \| A_t g - \mu g\|, \\ \delta_t &:=\sup_{g \in L^2_0(\mu): \, \|g\| = 1} \| A_t g \| = \sup_{g \in L^2 (\mu): \, \|g - \mu g\| = 1} \|A_t g - \mu g\|. \end{align}\] We record a simple lemma for future use:

2.1 Local singular-value gap↩︎

2.1.1 Upper bound in Theorem 1↩︎

The following proposition is inspired by Lemma 3.1 in [1].

With the above proposition and Lemma 1, we are ready to prove the upper bound on the empirical variance of observables in \(U \subset L^2_0 (\mu)\). The identities 15 16 are standard in the literature on the ergodic average of Markov processes.

2.1.2 Lower bound in Theorem 1↩︎

Similarly to the proof of the upper bound, we control \(\Delta_t^{U+V}\) through \(\delta_t^{U + V}\). It is perhaps unnecessary that the lower bound involves observables in \(V\) as well as \(U\); a more refined understanding of the lower bound would be interesting.

As in [1], the dyadic lower bound also implies a uniform lower bound at small times, showing that empirical averages cannot converge in a time much shorter than \(\gamma^{-1}\). We shall state the global version here for simplicity.

2.2 Global singular-value gap↩︎

If we want a universal understanding of all observables and take \(U = L^2_0(\mu)\), it is a priori unclear from the local statement of Theorem 1 that we should be able to find \(f \in L^2_0(\mu)\) that solves the Poisson equation \(-Lf = g\) for every \(g \in L^2_0(\mu)\). The key observation in Lemma 2 below is that if \(L\) possesses a global singular-value gap \(\gamma > 0\), then \(L\) is in fact invertible on \(L^2_0(\mu)\). In particular, we get a bounded Poisson solver that facilitates our proof strategies. Applying this observation to discrete-time Markov chains, we note that Chatterjee’s main theorem in [1] can be extended to general state space.

2.2.1 Positive gap implies bounded Poisson solver↩︎

We state the following lemma in a general form, so that it can be applied to discrete-time Markov chains and invariant subspaces \(K\). For instance, \(K\) can be an eigensubspace when the generator \(L\) has discrete spectrum.

2.2.2 Variance bounds for Markov chains on general state space↩︎

Here we briefly record that Lemma 2 also admits the extension of Theorem 1.2 in [1] to general state space. Let us first clarify the setting. Let \((X_n)_{n=0}^\infty\) be a time-homogeneous discrete time Markov chain on a measurable space \(S\). The Markov transition kernel \(P(x,dy)\) acts on \(g \in L^2(\mu)\) by \(Pg\,(x) = \int g(y) \,P(x,dy)\). Assume that the chain is stationary with \(X_0 \sim \mu\), where \(\mu\) is an invariant measure satisfying \(\int P(x,dy) \, \mu(dx) = \mu(dy)\). Use the same notation as above and as in [1]: \[\mu g:=\int g\,d\mu, \qquad \qquad \mu_n g:=\frac{1}{n} \sum_{i=0}^{n-1} g(X_i)\] As above, use \(\|g\|\) to denote the \(L^2(\mu)\) norm of a function \(g\) on the state space. The worst-case standard deviation of empirical averages at time \(n\) is captured by \[\Delta_n := \sup_{g \in L^2_0(\mu):\, \|g\| = 1} \sqrt{\operatorname{Var}_\mu (\mu_n g)} = \sup_{g \in L^2 (\mu): \, \|g - \mu g\| = 1} \sqrt{\operatorname{Var}_\mu(\mu_n g)}\] In this case, the generator is \(L = P-I\), and we consider the singular-value gap defined in 4 . Note that since \(P\) is a contraction on \(L^2(\mu)\), \(L\) is a bounded operator and we do not need to worry about its domain.

We have the exact analog of Theorem 1.2 in [1] in this setting. A similar local statement holds with the same proof, but we only state the global theorem for simplicity.

2.3 Sharpness of upper and lower bounds↩︎

Although they show that \(t \gtrsim \gamma^{-1}\) is sufficient and necessary for empirical averages to converge, the upper and lower bounds in Corollary 1 should not be interpreted as a two-sided equivalence for \(\Delta_t\). Rather, they capture two different universal obstructions. The upper bound \[\Delta_t \le \frac{2}{\sqrt{\gamma \mkern1mu t}}\] is sharp in general. In particular, it has the correct order \(t^{-1/2}\) when the observable in question has a finite positive asymptotic variance. On the other hand, the lower bound \[\sup_{s\in[t,2t]}\Delta_s \ge \frac{1}{\frac{3}{2} \gamma \mkern1mu t +3}\] is also sharp as a universal lower bound: without additional assumptions, one cannot replace it by a lower bound of order \(t^{-1/2}\). We first illustrate the sharpness of the upper bound with a familiar example.

The following example shows that the lower bound in Corollary 1 is also sharp. Note that in this case, the lower bound in Theorem 3 below does not imply a \(t^{-1/2}\) order for the empirical standard deviation, because the asymptotic variance vanishes.

2.4 Sharpened bounds with access to asymptotic variance↩︎

The preceding examples show that the singular-value gap alone cannot give a two-sided characterization of \(\Delta_t\) at the order \(t^{-1/2}\). A sharper description is possible when we can locate a set of observables for which the asymptotic variance is computable or can be bounded. In particular, when we have a local gap \(\gamma_K > 0\) on a closed invariant subspace \(K\), the asymptotic variance can be uniformly bounded above by \(\gamma_K^{-1}\). As observed in Remark 2, this provides a constant improvement to the upper bound stated in Theorem 1. More importantly, the lower bound in ?? improves the \(t\)-dependence of the lower bound in Theorem 1 when \(\sigma^2_g > 0\) for an observable \(g\) in question, bounding the finite-time standard deviation of empirical averages within the CLT scale of \(t^{-1/2}\). This leads to an improved observable-class lower bound whenever the worst-case asymptotic variance constant \(\sigma_K^2\) is positive.

Let \(K\subset L_0^2(\mu)\) be a closed invariant subspace to which Lemma 2 applies. Assume \(\gamma_K>0\), so that \(L_K^{-1}:K\to K\) exists and \(\|L_K^{-1}\|\le \gamma_K^{-1}\). For \(g\in K\), consider its asymptotic variance \(\sigma_g^2\) defined in 6 . By Lemma 5, we have \[\sigma_g^2 = 2\operatorname{Re}\langle g,-L_K^{-1}g\rangle .\] We also define the worst-case asymptotic variance constant on \(K\) by \[\begin{align} \label{definition32of32worst32case32asymptotic32variance} \sigma_K^2 := \sup_{\substack{g\in K\\ \|g\|=1}} \sigma_g^2 = 2 \sup_{\substack{g\in K\\ \|g\|=1}} \operatorname{Re}\langle g,-L_K^{-1}g\rangle . \end{align}\tag{19}\] Since \(\|L_K^{-1}\|\le \gamma_K^{-1}\), we have \[\begin{align} \label{uniform32upper32bound32of32asymptotic32variance32by32247gamma} 0\le \sigma_K^2\le \frac{2}{\gamma_K}. \end{align}\tag{20}\]

We note that when the process is nonreversible, the variance of finite-time empirical averages might not always be trivially controlled by the asymptotic variance. For one thing, the lower bound in \(\eqref{bound32using32asymptotic32variance32and32gap4432observable32specific}\) is vacuous when the asymptotic variance vanishes. We saw that the empirical standard deviation can indeed decay faster at the order of \(t^{-1}\) in Example 4. Moreover, the upper bound in ?? can fail without a positive singular-value gap: see Example 5 below. This is in contrast to the reversible scenario, where the finite-time variance is bounded by the asymptotic variance whenever the latter is finite. Since this bound has no direct reference to the spectral gap, we can get a uniform upper bound for finite-time variance of a class of observables if we can bound their asymptotic variance uniformly. The aforementioned bound can be quickly deduced from 16 and the observation that \(P_{r/2} = P_{r/2}^*\) for \(r > 0\) when the process is reversible (cf. Proposition 4.29 in [28] and Lemma 2.3 in [29]): \[\begin{align} \operatorname{Var}_\mu (\mu_t g) &= \frac{2}{t^2} \int_0^t (t-r) \Re \langle g, P_r g \rangle \, dr \nonumber \\ &= \frac{4}{t^2} \int_0^{t/2} (t-2s) \| P_s g \|^2 \, ds \tag{21} \\ &\leq \frac{4}{t} \int_0^\infty \|P_s g\|^2 \, ds \nonumber \\ &= \frac{\sigma_g^2}{t}, \tag{22} \end{align}\] where the last equality in 22 follows from the expression of asymptotic variance obtained by multiplying 21 by \(t\) and taking the limit \(t \to \infty\).

3 Ornstein–Uhlenbeck process with rotation↩︎

In the following example, the generator is normal despite not being self-adjoint in \(L^2(\mu)\). Moreover, the spectrum and eigenfunctions are explicit, so we get exact global and local singular-value gaps on eigensubspaces. In this case, the variance of finite-time empirical averages is computable in the eigenbasis, so our theory in section 2 above strictly speaking does not provide more information. Nevertheless, the theory helps us obtain clean bounds for natural classes of observables, and this example serves as a good illustration for various aspects of the bounds we derived. Let \[J= \begin{pmatrix} 0&-1\\ 1&0 \end{pmatrix},\] and consider the diffusion on \(\mathbb{R}^2\) with a rotation of strength \(\alpha\): \[dX_t=(-X_t+\alpha JX_t)\,dt+\sqrt{2}\,dW_t, \qquad \alpha\in\mathbb{R}.\] The invariant measure is the standard Gaussian measure \[\mu(dx)=\frac{1}{2\pi}e^{-|x|^2/2}\,dx = \frac{1}{2\pi} e^{-r^2/2} \,r \, dr d\theta.\] The generator can be decomposed into symmetric and antisymmetric parts: \[L_\alpha = \Delta-x\cdot\nabla+\alpha\partial_\theta = S + \alpha A,\] where the symmetric part is the usual reversible Ornstein–Uhlenbeck generator \(S=\Delta-x\cdot\nabla\), and the antisymmetric part is \(\alpha A = \alpha \partial_\theta\). Since \(S\) is rotationally invariant, \(S\) commutes with \(A\). \(L_\alpha\) is then normal, and in fact its eigenfunctions can be written explicitly in terms of generalized Laguerre polynomials. See Appendix 6.2 for details. We have \((\psi_{k,m})_{k\ge0,m\in\mathbb{Z}}\) an orthonormal basis of \(L^2(\mu)\), such that \[\begin{align} \label{action32of32symmetric32and32antisymmetric32parts32on32psi95k44m} S\psi_{k,m}=-(2k+|m|)\psi_{k,m}, \qquad \partial_\theta\psi_{k,m}=im\psi_{k,m}, \end{align}\tag{23}\] so that \[L_\alpha\psi_{k,m} = \bigl(-(2k+|m|)+i\alpha m\bigr)\psi_{k,m}.\] This decomposition is a polar-coordinate version of the Hermite chaos decomposition for the Gaussian measure. See [30] for a related treatment of complex OU operators.

3.1 Explicit variance computation↩︎

Since the eigenbasis is explicit, one can also compute the empirical variance mode by mode. Let \[n=2k+|m|, \qquad \omega=\alpha m.\] Then \[L_\alpha\psi_{k,m}=(-n+i\omega)\psi_{k,m}.\] For a normalized eigenfunction \(\psi_{k,m}\), the identity 15 gives \[\begin{align} \label{2d32OU32empirical32variance} \operatorname{Var}_\mu(\mu_t\psi_{k,m}) = \frac{2}{t^2} \operatorname{Re} \left[ \frac{e^{(-n+i\omega)t}-1-(-n+i\omega)t}{(-n+i\omega)^2} \right]. \end{align}\tag{24}\] In particular, \[\begin{align} \label{2d32OU32asymptotic32variance} \sigma^2_{\psi_{k,m}} = \lim_{t\to\infty} t\,\operatorname{Var}_\mu(\mu_t\psi_{k,m}) = \frac{2n}{n^2+\omega^2} = \frac{2(2k+|m|)}{(2k+|m|)^2+\alpha^2m^2}. \end{align}\tag{25}\] For any observable \(g \in L^2_0(\mu)\), if we can write it in the eigenbasis as \(g = \sum_{k,m} c_{k,m} \psi_{k,m}\), we get the empirical variance explicitly \[\operatorname{Var}_\mu (\mu_t g) = \sum_{k,m} |c_{k,m}|^2 \operatorname{Var}_\mu (\mu_t \psi_{k,m})\]

3.2 Uniform acceleration for nonradial observables↩︎

Since \(L_\alpha\) is normal, the singular values of \(L_\alpha\) are the moduli of its eigenvalues. On \(L_0^2(\mu)\), the constant mode \((k,m)=(0,0)\) is excluded, so the global singular-value gap is \[\begin{align} \gamma = \inf_{(k,m)\ne(0,0)} \left| -(2k+|m|)+i\alpha m \right| = \min\{\sqrt{1+\alpha^2},2\}, \end{align}\] where the modes \((k,m)=(0,\pm1)\) give \(\sqrt{1+\alpha^2}\), while the radial mode \((k,m)=(1,0)\) gives \(2\). Consequently, the global singular-value gap is capped at 2 when the antisymmetric rotation strength \(\alpha\) grows. The obstruction is the radial quadratic observable \[\psi_{1,0} \propto |x|^2-2,\] whose empirical average does not converge faster when a rotation is present.

This obstruction shows that the subspace formulation of Theorem 1 could be useful when the irreversible part of the dynamics accelerates the convergence of empirical averages in certain directions but leaves some other observables invariant, in which case we may wish to separately consider different observable subspaces. In this case, to avoid the obstruction presented by \(\psi_{1,0}\), consider \[K_{\mathrm{ang}} := \overline{ \operatorname{span}\{\psi_{k,m}:m\ne0\} } = \left\{f \in L^2_0(\mu): \int_0^{2\pi} f(r,\theta) \, d\theta = 0 \;\text{ for a.e. }r \right\},\] where the second characterization above can be obtained by considering the Fourier decomposition \(f(r,\theta) = \sum_{m \in \mathbb{Z}} f_m(r) e^{i m \theta}\) and noting that \(\psi_{k,m}\) for \(k \geq 0\) form a basis for the \(m\)-th mode. Equivalently, \(K_{\mathrm{ang}}\) is the orthogonal complement of radial functions. \[\begin{align} \gamma_{\mathrm{ang}} = \inf_{k\ge0,\;m\ne0} \sqrt{(2k+|m|)^2+\alpha^2m^2} = \sqrt{1+\alpha^2}. \end{align}\]

On this closed invariant subspace, Theorem 1 and Lemma 2 give \[\begin{align} \label{K95ang32empirical32variance32bound} \operatorname{Var}_\mu(\mu_tg) \le \frac{4}{t\sqrt{1+\alpha^2}}\|g\|^2 \end{align}\tag{26}\] for every \(g\in K_{\mathrm{ang}}\). Thus, rotation gives a uniform empirical-average acceleration for all nonradial observables, clarifying, at the level of a whole observable class, the intuition that [11] verifies explicitly for linear and quadratic observables.

3.3 Gaussian chaos and sharpness of variance bounds↩︎

Recall that for the standard Gaussian measure on \(\mathbb{R}^2\), the \(n\)-th Gaussian chaos \(\mathcal{H}_n\) is the eigenspace of the reversible Ornstein–Uhlenbeck generator \(S=\Delta-x\cdot\nabla\) with eigenvalue \(-n\). Equivalently, \(\mathcal{H}_n\) is spanned by products of one-dimensional Hermite polynomials of total degree \(n\). From 23 we see that \[\mathcal{H}_n = \operatorname{span}\{\psi_{k,m}:2k+|m|=n\}.\] In particular, for a given \(n\), the possible angular frequencies \(m=-n,-n+2,\ldots,n-2,n\) give \(n+1\) modes, which equals \(\dim \mathcal{H}_n\). We spell out the first few eigenspaces, which can be obtained from the explicit form of \(\psi_{k,m}\) derived in Appendix 6.2: \[\begin{align} \mathcal{H}_1 &= \operatorname{span}\{\psi_{0,1},\psi_{0,-1}\} = \operatorname{span}\{x_1,x_2\}, \\ \mathcal{H}_2 &= \operatorname{span}\{\psi_{1,0},\psi_{0,2},\psi_{0,-2}\} = \operatorname{span}\{ |x|^2-2,\, x_1^2-x_2^2,\, 2x_1x_2 \}. \end{align}\] In words, \(\mathcal{H}_1\) is the space of centered linear observables and \(\mathcal{H}_2\) is the space of centered quadratic observables. We compute the gap of \(L_\alpha\) on the \(n\)-th Gaussian chaos: \[\gamma_{\mathcal{H}_n} = \min_{m=-n,-n+2,\ldots,n} \sqrt{n^2+\alpha^2m^2} = \begin{cases} n & n \text{ even}\\[4pt] \sqrt{n^2+\alpha^2} & n \text{ odd}. \end{cases}\] For even \(n\), the obstruction is again the radial sector \(m=0\). If we restrict attention to the angular part of \(\mathcal{H}_n\), the local gap increases with \(\alpha\) as \[\gamma_{\mathcal{H}_n\cap K_{\mathrm{ang}}} = \begin{cases} \sqrt{n^2+4\alpha^2} &n \text{ even nonzero}\\[4pt] \sqrt{n^2+\alpha^2} & n \text{ odd}. \end{cases}\] Similarly, we may use 25 to compute the worst-case asymptotic variance defined in 19 : \[\begin{align} \sigma^2_{\mathcal{H}_n} = \max_{m=-n,-n+2, \ldots, n} \, \frac{2n}{n^2 + \alpha^2 m^2} &= \begin{cases} \frac{2}{n} & n \text{ even}\\[4pt] \frac{2n}{n^2 + \alpha^2} & n \text{ odd} \end{cases} \\ \sigma^2_{\mathcal{H}_n \cap K_{\text{ang}}} &= \begin{cases} \frac{2n}{n^2 +4\alpha^2} & n \text{ even}\\[4pt] \frac{2n}{n^2 + \alpha^2} & n \text{ odd} \end{cases} \end{align}\] In particular, for fixed \(n\), the asymptotic variance of observables in \(\mathcal{H}_n \cap K_{\text{ang}}\) is of order \(\alpha^{-2}\). The upper bound in Theorem 1 gives \[\operatorname{Var}_\mu(\mu_tg) \le \frac{4}{t\sqrt{n^2+\alpha^2}}\|g\|^2\] for all \(g \in \mathcal{H}_n \cap K_{\text{ang}}\), which gives a weaker \(|\alpha|^{-1}\) dependence as \(|\alpha| \to \infty\). On the other hand, Theorem 3 gives the upper bound \[(\Delta_t^{\mathcal{H}_n \cap K_{\text{ang}}})^2 \leq \frac{\sigma^2_{\mathcal{H}_n \cap K_{\text{ang}}}}{t} + \frac{4}{\gamma^2_{\mathcal{H}_n \cap K_{\text{ang}}}t^2} \leq \frac{2n}{(n^2 + \alpha^2)t} + \frac{4}{(n^2 + \alpha^2) t^2},\] which matches the \(\alpha^{-2}\) factor of variance reduction as \(|\alpha| \to \infty\).

On the larger nonradial subspace \(K_{\mathrm{ang}}\), the upper bound in Theorem 1 already sharply captures the \(\alpha\)-dependence of variance reduction. Indeed, using 25 as \(|\alpha| \to \infty\), \[\sigma^2 _{K_{\mathrm{ang}}} = \sup_{\substack{k\ge0\\m\ne0}} \frac{2(2k+|m|)}{(2k+|m|)^2+\alpha^2m^2} \asymp \frac{1}{|\alpha|} \asymp \frac{1}{\gamma_{\mathrm{ang}}}.\] The upper bound above follows from \[\frac{2n}{n^2+\alpha^2m^2} \le \frac{2n}{n^2+\alpha^2} \le \frac{1}{|\alpha|} \qquad (m\ne0),\] and the matching lower bound is obtained by taking \(|m|=1\) and choosing \(n=2k+1\) of order \(|\alpha|\). Therefore, for the infinite-dimensional observable class \(K_{\text{ang}}\), the \(\alpha\)-dependence in the finite-time variance bound of Theorem 1 matches the reduction factor for asymptotic variance.

3.4 Practical implication for variance reduction↩︎

In higher dimensions, we have more freedom to choose the perturbation \(J\) while maintaining the invariance of the Gaussian measure. It has been widely observed in the context of Markov chains and diffusion processes that the addition of an irreversible perturbation accelerates convergence of the law of \(X_t\) and, respectively, convergence of empirical averages. It is therefore of interest to find an optimal perturbation to accelerate convergence.

The observable-class perspective we take here suggests an optimization problem different from the one emphasized in [11]. There, the perturbation is chosen to minimize the asymptotic variance of a fixed observable, and their Section 4.2 solves this problem for linear observables under a Frobenius norm constraint on \(J\). In the present framework, one can instead fix a closed invariant class of observables \(K\), such as \(\mathcal{H}_1\), \(\mathcal{H}_n\), or a finite sum of chaoses, and choose \(J\) to maximize the local singular-value gap \(\gamma_K(J)\), or, often equivalently, use Theorem 3 to minimize the worst-case asymptotic variance \(\sigma^2_K(J)\). Solving this optimization problem gives a uniform convergence guarantee for a class of observables and extends naturally beyond linear and quadratic functions.

4 Relation to other mixing quantities↩︎

4.1 Relation to the reversibilized process↩︎

When we have a bounded Markov generator \(L\), its symmetric part can be written as \[\begin{align} L_s \mathrel{\vcenter{:}}= \frac{L + L^*}{2}, \end{align}\] which is self-adjoint on \(L^2(\mu)\). The Markov process with generator \(L_s\) is reversible. We denote by \(\gamma_s\) the spectral gap of \(L_s\), which coincides with its singular-value gap. If \(L\) is in addition normal, \(\gamma\) is the gap of the spectrum of \(L_0 = L \vert_{L^2_0(\mu)}\)and \(\gamma_s\) is the gap of the real part of the spectrum of \(L_0\), so it is natural to expect that \(\gamma_s\) lower bounds \(\gamma\).

For unbounded \(L\), the operator expression \((L+L^*)/2\) may have domain issues. In this case, we define \(\gamma_s\) through the symmetric form associated with \(L\): \[\begin{align} \label{definition32of32gamma95s32through32symmetric32form} \gamma_s := \inf_{\substack{f\in\operatorname{Dom}(L)\cap L^2_0(\mu), \;\|f\|=1}} \operatorname{Re}\langle f,-Lf\rangle . \end{align}\tag{27}\] This agrees with the previous definition when \(L\) is bounded. Moreover, if there is \[\mathcal{C}\subset \operatorname{Dom}(L)\cap\operatorname{Dom}(L^*)\] that is a core for \(L\) in the graph norm with \(\mathbf{1} \in \mathcal{C}\), then \(\gamma_s\) can be computed on \(\mathcal{C}\) as \[\gamma_s = \inf_{\substack{f\in\mathcal{C}\cap L^2_0(\mu),\;\|f\|=1}} \left\langle f,-\frac{L+L^*}{2}f\right\rangle .\] In the smooth diffusion examples considered in the present paper, this is the usual way to understand the symmetrized generator and compute the symmetrized gap.

It is natural to ask whether an upper bound of \(\gamma\) could be also obtained through \(\gamma_s\). Such a bound cannot hold in the most general setting, as demonstrated by Example 1. That is, diffusion on the unit circle with generator \(L = \partial_\theta^2+ \alpha \partial_\theta\) has \(\gamma_s = 1\) but \(\gamma = \sqrt{1+\alpha ^2}\). From this counterexample it is natural to expect that if we demand the strength of the dynamics (in this case the drift) to be bounded, then \(\gamma\) might be bounded above in terms of \(\gamma_s\). We prove such a bound for jump processes with bounded jump rates.

Let \(S\) be a measurable space. Following the convention of [12], we consider unnormalized jump kernel \(q(x,dy)\): for each \(x\), \(q(x,\cdot)\) is a finite measure on \(S\), and for each measurable \(A\), the map \(x\mapsto q(x,A)\) is measurable. The total mass \[q(x):=q(x,S)\] is the jump rate at \(x\). The interpretation is that \(q(x)\) is the exponential rate at which a jump is proposed from state \(x\), and, conditional on such a jump, the next state is sampled from the normalized kernel \[p(x,dy):=\frac{q(x,dy)}{q(x)} \qquad \text{if } q(x)>0.\] The generator for this process is \[\begin{align} \label{jump32process32generator} Lf(x)=\int (f(y)-f(x))\,q(x,dy). \end{align}\tag{28}\] Since the jump rates are bounded and \(\mu\) is invariant, \(L\) is a bounded operator on \(L^p(\mu)\) for every \(1\le p\le\infty\). Indeed, for \(r\ge q_{\max}\), \[\Pi(x,dy)=\frac{q(x,dy)}{r} +\left(1-\frac{q(x)}{r}\right)\delta_x(dy)\] is a Markov kernel preserving \(\mu\), and \(L=r(\Pi-I)\). Therefore, \(\|\Pi\|_{L^p\to L^p}\le1\), hence \[\|Lf\|_p\le 2r\|f\|_p \qquad \text{ for } \;1 \leq p \leq \infty.\] In particular, \(\operatorname{Dom}_{L^2}(L)=L^2(\mu)\).

4.2 Relation to the mixing time↩︎

Next, we relate the singular-value gap \(\gamma\) to the mixing time in total-variation distance. For a Markov process with an invariant probability measure \(\mu\), write \[d(t) \mathrel{\vcenter{:}}= \sup_{\nu\in\mathcal{P}(S)} \|\nu P_t-\mu\|_{\mathrm{TV}}, \qquad \quad t_{\mathrm{mix}}(\varepsilon) \mathrel{\vcenter{:}}= \inf \{t \geq 0: d(t) \leq \varepsilon\},\] where \(\mathcal{P}(S)\) is the set of probability measures on \(S\). Equivalently, \[d(t)=\sup_{x\in S}\|P_t(x,\cdot)-\mu\|_{\mathrm{TV}}.\] Indeed, the inequality “\(\ge\)” follows by taking \(\nu=\delta_x\). Conversely, for any measurable \(A\), \[(\nu P_t-\mu)(A) = \int (P_t(x,A)-\mu(A))\,\nu(dx).\] Therefore \[|(\nu P_t-\mu)(A)| \le \sup_{x\in S}|P_t(x,A)-\mu(A)| \le \sup_{x\in S}\|P_t(x,\cdot)-\mu\|_{\mathrm{TV}}.\] Taking the supremum over \(A\) proves the reverse inequality.

The above definitions are canonical for finite state space discrete-time chains, where \(t\) takes values in nonnegative integers and \(P_t = P^t\). On a general state space, total variation is one of many metrics for mixing, and it is usually more meaningful when the process is uniformly ergodic in the sense that \(d(t) \to 0\) as \(t \to \infty\).

4.2.1 Lower bound on \(t_{\mathrm{mix}}(\varepsilon)\)↩︎

For a Markov kernel \(\pi(x,dy)\) on a standard Borel space \(S\), write \[\begin{align} \delta(\pi) \mathrel{\vcenter{:}}= \sup_{x,x' \in S} \|\pi(x,\cdot) - \pi(x', \cdot)\|_{\mathrm{TV}}. \end{align}\] Note that \(\delta(P_t) \leq 2d(t)\) by triangular inequality. We need the following lemma from [31]. The original paper considers real-valued functions, but the proof still goes through for complex-valued functions.

4.2.2 Upper bound on \(t_{\mathrm{mix}}(\varepsilon)\)↩︎

We can obtain an upper bound on the mixing time by \(\gamma\) only in the restrictive setting of jump processes on finite state space. Note from the proof below that the upper bound can in fact be stated in terms of \(\gamma_s\).

The finite state space assumption in Proposition 5 above is essential. On a countably infinite state space, a positive spectral gap does not imply finite worst-case total-variation mixing time, even for a reversible jump process with bounded rates.

4.3 Relation to the Cheeger constant↩︎

Consider a stationary Markov jump process on a standard Borel space \(S\) with jump kernel \(q(x,dy)\) and invariant probability measure \(\mu\). Let the jump rate be bounded by \(q_{\mathrm{max}} \mathrel{\vcenter{:}}= \mathrm{ess \;sup}_x q(x,S) < \infty\). Recall that the generator is \[\begin{align} \label{generator32of32jump32process322} Lf(x) = \int \big(f(y)-f(x) \big) \,q(x,dy). \end{align}\tag{36}\] Let \[Q(dx,dy) \mathrel{\vcenter{:}}= \mu(dx) q(x,dy)\] be the stationary flux of the jump process. The Cheeger constant is defined by \[\begin{align} \label{Cheeger32constant32definition} \xi \mathrel{\vcenter{:}}= \inf_{A \subset S:\, 0 < \mu(A) \leq \frac{1}{2}} \frac{Q(A, A^c)}{\mu(A)} \end{align}\tag{37}\]

The constant \(\xi\) is also known as the bottleneck ratio, with several slightly different definitions in the literature. When the jump process is reversible i.e. \(L\) is self-adjoint in \(L^2(\mu)\), we recall the classical result in [12], Theorems 2.1 and 3.5: \[\begin{align} \label{lawler32sokal32cheeger32inequality} \frac{\xi^2}{2q_{\max}} \leq \gamma \leq 2 \xi, \end{align}\tag{38}\] where \(\gamma\) is the usual spectral gap. In the general nonreversible setting, we obtain

Before proving Theorem 5, we first record some basic characterizations of the adjoint jump process. Define the reversed flux measure by \[Q^\dagger(dx,dy):=Q(dy,dx).\] Since \(\mu\) is invariant, for every measurable set \(A\), \[\begin{align} \label{equality32Q40A44S41326132Q40S44A41} 0=\int L\mathbf{1}_A\,d\mu = Q(S,A)-Q(A,S). \end{align}\tag{39}\] Thus, we have the equality of marginals \[q(x) \, \mu(dx) = Q(dx,S) = Q(S,dx) = Q^\dagger(dx,S).\] In particular, since the first marginal of \(Q^\dagger\) is absolutely continuous with respect to \(\mu\) and \(S\) is standard Borel, there exists a finite kernel \(q^\dagger(x,dy)\) such that \[Q^\dagger(dx, dy) = \mu(dx) \, q^\dagger (x,dy).\] Unsurprisingly, time reversal corresponds to \(L^2(\mu)\)-adjoint:

We are now ready to prove Theorem 5. Note that the content of the lower bound is essentially in the proof of [12]. The proof of the upper bound refines the proof idea in Theorem 1.6 of [1] and crucially depends on the connection between the singular-value gap and the empirical variance of observables.

5 Applications↩︎

5.1 Shift-register chain on binary de Bruijn graph↩︎

Let \[B_n=\{-1,1\}^n\] Define a discrete-time Markov chain on \(B_n\) such that from \[x=(x_1,\ldots,x_n)\] the chain moves to \[(x_2,\ldots,x_n,+1) \quad\text{or}\quad (x_2,\ldots,x_n,-1)\] with probability \(1/2\) each. This is the simple random walk on the directed binary de Bruijn graph. The chain is nonreversible and has uniform stationary measure \(\mu\). Indeed, after \(n\) steps, all original coordinates have been forgotten, so \[P^n(x,\cdot)=\mu \qquad\text{for every }x\in B_n.\] On the other hand, for \(k<n\), the law \(P^k(x,\cdot)\) is uniform on only \(2^k\) states, while \(\mu\) is uniform on \(2^n\) states. Hence \[\|P^k(x,\cdot)-\mu\|_{\mathrm{TV}} = 1-2^{k-n}, \qquad 0\le k<n.\] In particular, for every \(\varepsilon<1/2\), \[t_{\mathrm{mix}}(\varepsilon)=n.\]

We now compute the singular-value gap of \(L=P-I\). Let \[\chi_A(x):=\prod_{i\in A}x_i, \qquad A\subseteq\{1,\ldots,n\},\] be the Walsh characters. These form an orthonormal basis of \(L^2(\mu)\), where the constant subspace is spanned by \(\chi_\emptyset=1\); see section 1.3 of [33]. For \(A\neq\emptyset\), \[P\chi_A = \begin{cases} \chi_{A+1}, & n\notin A,\\ 0, & n\in A, \end{cases}\] where \[A+1:=\{i+1:i\in A, \, i < n\}.\] Thus \(P\) decomposes \(L^2_0(\mu)\) into orthogonal nilpotent shift chains. The longest such chain has length \(n\), for example: \[\chi_{\{1\}}\mapsto\chi_{\{2\}}\mapsto\cdots\mapsto\chi_{\{n\}}\mapsto0.\] On a shift chain of length \(\ell\), \(P\) is the nilpotent shift \[\begin{align} \label{shift32chain} S_\ell: \;e_1\mapsto e_2\mapsto\cdots\mapsto e_\ell\mapsto0. \end{align}\tag{40}\] The smallest singular value of \(I-P\) on this \(\ell\)-dimensional subspace is the square root of the smallest eigenvalue of the tridiagonal matrix \((I-S_\ell)^*(I-S_\ell)\): \[2\sin\left(\frac{\pi}{4\ell+2}\right) \asymp \frac{1}{\ell}.\] Therefore, the global singular-value gap is \[\gamma_n = 2\sin\left(\frac{\pi}{4n+2}\right) \asymp \frac{1}{n}.\] Using Theorem 4, we get \[t_{\mathrm{mix}}(\varepsilon) \ge \frac{1-\sqrt{2\varepsilon}}{\gamma_n} \asymp n,\] which has the correct order.

This example is prototypically nonnormal and illustrates a slow-mixing mechanism caused by deterministic or nearly deterministic memory that is not captured by traditional spectral quantities. On \(L^2_0(\mu)\), the transition operator is nilpotent, so all eigenvalues of \(P\) are \(0\), hence all eigenvalues of \(L=P-I\) are \(-1\). Thus, the distance between zero and the spectrum of \(L\) is \(1\). One can also check that the symmetrized gap \(\gamma_s\) has order \(n^{-2}\). Indeed, \(-L_s\) acts on a shift block of length \(\ell\) as \[I - \frac{S_\ell +S^*_\ell}{2},\] with smallest eigenvalue \[1- \cos \frac{\pi}{\ell + 1} \asymp \frac{1}{\ell^2}.\]

By contrast, the singular-value gap of \(L\) is of order \(n^{-1}\), exactly reflecting the length of the longest nilpotent shift chain and giving the correct order for the mixing-time lower bound. In this case, the reason that the singular-value gap “sees” the length of the shift chains can be understood by the expansion \[(I - S_\ell)^{-1} = I + S_\ell + \cdots + S_\ell^{\ell-1},\] from which we see that \(\|(I-S_\ell)^{-1}\|\), which is the inverse of the local singular-value gap by Lemma 2, has order \(\ell\).

5.2 A hypoelliptic shear on the torus↩︎

We first observe a mechanism for a Markov generator to possess a positive singular-value gap.

Hypoelliptic diffusion generators often satisfy the assumptions of the previous proposition. We illustrate this point with the following example. Let \(M=\mathbb{T}^2\) and consider \[L=\partial_y^2+\sin(y)\partial_x .\] An invariant measure \(\mu\) is the normalized Lebesgue measure. The operator is nonreversible: \(\partial_y^2\) is symmetric and \(\sin(y)\partial_x\) is skew-adjoint in \(L^2(\mu)\). Unlike the OU process in section 3, the spectrum of \(L\) is not explicit.

Nevertheless, we can check that \(L\) has a positive singular-value gap on \(L^2_0(\mu)\) by checking the conditions of Proposition 6. We first note that \(L\) satisfies Hörmander’s bracket condition [34]. The relevant vector fields are \[X_1=\partial_y,\qquad X_0=\sin(y)\partial_x.\] Their commutator is \[[X_1,X_0]=\cos(y)\partial_x.\] Since \(\sin(y)\) and \(\cos(y)\) do not vanish simultaneously, the vector fields \(X_1,X_0,[X_1,X_0]\) span the tangent space at every point of \(\mathbb{T}^2\). Therefore, Hörmander’s bracket condition holds, and \(L\) is hypoelliptic.

We now identify the kernel of \(L\). If \(Lf=0\), then hypoellipticity implies that \(f\) is smooth. Taking real parts in the identity \(\langle f,Lf\rangle=0\), we obtain \[0=\operatorname{Re}\langle f,Lf\rangle = -\|\partial_y f\|^2.\] Thus, \(f\) is independent of \(y\), say \(f(x,y)=F(x)\). The equation \(Lf=0\) then gives \[\sin(y)F'(x)=0,\] hence \(F\) is constant. Thus, the kernel of \(L\) consists only of constants.

One may also check that \(L\) has compact resolvent by showing that the embedding \((\operatorname{Dom}(L), \|\cdot\| + \|L \cdot \|) \hookrightarrow L^2(\mathbb{T}^2)\) is compact. See Appendix 6.4 for details. As a result, conditions for Proposition 6 are satisfied and \[\gamma = \inf_{\substack{f\in\operatorname{Dom}(L)\cap L^2_0(\mu) \backslash \{0\}}} \frac{\|Lf\|}{\|f\|} >0.\] Consequently, Corollary 1 yields \[\operatorname{Var}_\mu (\mu_t g) \le \frac{4}{\gamma t}\|g\|^2, \qquad g\in L^2_0(\mu).\]

Although the spectrum of \(L\) is not explicit in this example, we take advantage of the standard tools for hypoelliptic differential operators and the compact-resolvent criterion in Proposition 6 to show that \(\gamma > 0\). Thus, even though the symmetric part \(\partial_y^2\) has no spectral gap on \(L_0^2(\mu)\), the coupling created by the shear drift \(\, \sin(y) \partial_x\) is enough to give a uniform stationary empirical-variance bound.

More generally, the compact-resolvent assumption of Proposition 6 is natural for hypoelliptic operators on compact manifolds. Hörmander’s bracket condition can be checked to prove hypoellipticity for operators of the form \(L = X_0+\sum_i X_i^2\), and subelliptic estimates often imply compactness of the embedding of \(\operatorname{Dom}(L)\) into \(L^2\).

6 Omitted details↩︎

6.1 The Poisson-equation formula for asymptotic variance↩︎

We record a derivation of the asymptotic variance formula 7 . This formula is known in the literature, but often with extra hypotheses, such as when a central limit theorem is sought in addition to a limiting second-moment. Thus, we provide a clear statement and proof in accordance with our operator-theoretic perspective in this paper.

Lemma 5 (Poisson equation formula for asymptotic variance). Consider a strongly continuous Markov semigroup \((P_t)_{t \geq 0}\) on a measurable space with an invariant probability measure \(\mu\). Let \(K\subset L_0^2(\mu)\) be a closed \(P_t\)-invariant subspace, and let \(L_K\) denote the generator of the restricted semigroup \((P_t \vert_K)_{t\ge0}\). Fix \(g\in K\). Suppose there exists \(f\in \operatorname{Dom}(L_K)\) such that \[-L_K f=g.\] Then the asymptotic variance of \(g\) exists and is given by \[\sigma_g^2 := \lim_{t\to\infty} t\,\operatorname{Var}_\mu(\mu_tg) = 2\operatorname{Re}\langle g,f\rangle .\] In particular, if the local gap defined in ?? satisfies \(\gamma_K > 0\), then \[\sigma_g^2 = 2\operatorname{Re}\langle g,-L_K^{-1}g\rangle .\]

Proof. To simplify the notation, denote by \((P_t)_{t \geq 0}\) the semigroup restricted to \(K\), and let \(A_t\) be the corresponding restricted Cesàro-mean operator, defined in 13 . By 16 , \[\operatorname{Var}_\mu(\mu_tg) = \frac{2}{t^2}\operatorname{Re} \int_0^t (t-s)\langle g,P_sg\rangle\,ds .\] Since \(f\in\operatorname{Dom}(L_K)\) and \(-L_Kf=g\), the map \(s\mapsto P_s f\) is differentiable in \(L^2(\mu)\), and \[\frac{d}{ds}P_s f = P_sL_Kf = -P_sg .\] Therefore \[\frac{d}{ds}\langle g,P_s f\rangle = \langle g,-P_sg\rangle = -\langle g,P_sg\rangle .\] Thus \[\begin{align} \int_0^t (t-s)\langle g,P_sg\rangle\,ds &= -\int_0^t (t-s) \frac{d}{ds}\langle g,P_s f\rangle\,ds = t\langle g,f\rangle - \int_0^t \langle g,P_s f\rangle\,ds . \end{align}\] Multiplying by \(2/t\) and using the definition of \(A_t\), we obtain \[\begin{align} \label{intermediate32identity32in32proof32of32lemma325} t\,\operatorname{Var}_\mu(\mu_tg) = 2\operatorname{Re}\langle g,f\rangle - 2\operatorname{Re}\langle g,A_t f\rangle . \end{align}\tag{41}\]

It remains to show that as \(t \to \infty\), \[\langle g,A_t f\rangle\to0 .\] By Theorem V.4.5 and Example V.4.7 in [27], \((P_t)_{t \geq 0}\) is mean ergodic, so the strong limit \[\Pi h :=\lim_{t\to\infty}A_t h\] exists for all \(h \in K\). By Lemma V.4.4 in [27], \(\Pi\) is the mean-ergodic orthogonal projection onto \(\mathrm{Ran}\, \Pi = \ker L_K\). Since \(g = -L_K f\), by 14 we have \[\|A_t g\| = \frac{\|f - P_t f\|}{t} \leq \frac{2}{t}\|f\| \to 0 \qquad \text{as} \quad t \to \infty.\] Therefore, \(\Pi g=0\), hence \(g\perp \operatorname{Ran}\Pi\). In particular, \[\langle g,A_t f\rangle \to \langle g,\Pi f\rangle=0.\] Returning to 41 , we conclude that \[\sigma_g^2 = \lim_{t\to\infty} t\,\operatorname{Var}_\mu(\mu_tg) = 2\operatorname{Re}\langle g,f\rangle .\]

Finally, when \(\gamma_K > 0\), \(L_K\) is invertible by Lemma 2, so we may take \(f = -L_K^{-1} g\). ◻

6.2 Eigenfunctions of the rotating Ornstein–Uhlenbeck generator↩︎

In polar coordinates, the generator can be decomposed as \[L = S + \alpha A, \quad \text{where} \quad S=\Delta-x\cdot\nabla = \partial_{rr} + \left(\frac{1}{r}-r\right)\partial_r + \frac{1}{r^2}\partial_{\theta\theta}, \quad A = \partial_\theta\] We look for eigenfunctions of the form \[\psi(r,\theta)=R(r)e^{im\theta}, \qquad m\in\mathbb{Z}.\] Then \(\partial_\theta\psi=im\psi\), and \(S\psi=-\lambda\psi\) becomes \[R''(r) + \left(\frac{1}{r}-r\right)R'(r) - \frac{m^2}{r^2}R(r) = -\lambda R(r).\] Let \(a=|m|\), \(\rho=r^2/2\), and set \[R(r)=r^a Q(\rho).\] A direct calculation gives \[\begin{align} \label{Laguerre32equation} \rho Q''(\rho)+(a+1-\rho)Q'(\rho) + \frac{\lambda-a}{2}Q(\rho)=0. \end{align}\tag{42}\] This is the generalized Laguerre equation. Polynomial solutions occur when \[\frac{\lambda-a}{2}=k, \qquad k=0,1,2,\ldots,\] so by, say, p.781 of [35], the equation 42 is solved by \[\lambda=2k+a, \qquad Q(\rho)=L_k^{(a)}(\rho).\] Thus \[\psi_{k,m}(r,\theta) = c_{k,m}r^{|m|}L_k^{(|m|)}(r^2/2)e^{im\theta}\] satisfies \[S\psi_{k,m}=-(2k+|m|)\psi_{k,m}, \qquad \partial_\theta\psi_{k,m}=im\psi_{k,m}.\] The normalization constant \(c_{k,m}\) can be computed from the orthogonality relation \[\int_0^\infty \rho^a e^{-\rho} L_k^{(a)}(\rho)L_\ell^{(a)}(\rho)\,d\rho = \frac{\Gamma(k+a+1)}{k!}\mathbf{1}_{k=\ell}.\] Since \(d\mu=(2\pi)^{-1}e^{-r^2/2}r\,dr\,d\theta\), this gives \[c_{k,m} = \left( \frac{k!}{2^{|m|}\Gamma(k+|m|+1)} \right)^{1/2}.\]

6.3 Spectrum of the Jacobi operator for the reflected birth-death chain↩︎

We consider the jump process in Example 7 and compute its spectral gap. Recall that the generator of this process is given by \[\begin{align} (Lf)(0)&=\lambda(f(1)-f(0)), \\ (Lf)(n) &= \lambda(f(n+1)-f(n)) + \mu(f(n-1)-f(n)), \quad n \geq 1. \end{align}\] The invariant measure is given by \(\pi(n)=(1-\rho)\rho^n, \; \rho=\frac{\lambda}{\mu}<1.\) Conjugating \(-L\) from \(L^2(\pi)\) to \(\ell^2(\mathbb{N})\) by the spectrum-preserving unitary isometry \[(Uf)(n)=\sqrt{\pi(n)}f(n)\] gives the self-adjoint operator \(H=U(-L)U^{-1}\) on \(\ell^2(\mathbb{N})\) that satisfies \[(Hg)(0)=\lambda g(0)-\sqrt{\lambda\mu}\,g(1),\] and, for \(n\ge1\), \[(Hg)(n) = (\lambda+\mu)g(n) - \sqrt{\lambda\mu}\,g(n+1) - \sqrt{\lambda\mu}\,g(n-1).\] The vector \((\sqrt{\pi(n)})_{n\ge0}\) is an eigenvector with eigenvalue \(0\). Set \[c:=\sqrt{\lambda\mu}.\] Away from the boundary, \(H\) has constant Jacobi coefficients: diagonal coefficient \(\lambda+\mu\) and off-diagonal coefficient \(-c\). By Theorem 3.19 in [36] \[\begin{align} \label{essential32spectrum32of32Jacobi32operator} \sigma_{\mathrm{ess}}(H) = [\lambda+\mu-2c,\lambda+\mu+2c]. \end{align}\tag{43}\] Therefore, to prove that the spectral gap is \(\gamma = (\sqrt{\mu} - \sqrt{\lambda})^2 = \lambda + \mu - 2c\), it remains to check that there are no eigenvalues in the interval \(\bigl(0,\lambda+\mu-2c \bigr).\) Suppose that \(\theta<\lambda+\mu-2c\) is an eigenvalue of \(H\), with eigenvector \(g\in\ell^2(\mathbb{N})\). For \(n\ge1\), the eigenvalue equation is \[(\lambda+\mu-\theta)g(n) - c\,g(n+1) - c\,g(n-1) = 0.\] This is a system of linear recurrence equations, and the solution is determined by \(g(0)\) and \(g(1)\). With the ansatz \(g(n) = r^n\), we get \[\begin{align} \label{characteristic32recurrence32equation} c\left(r+\frac{1}{r}\right)=\lambda+\mu-\theta. \end{align}\tag{44}\] Since \(\theta<\lambda+\mu-2c\), this equation has one root \(r\in(0,1)\) and one root \(r^{-1}>1\), and we deduce that all solutions must be of the form \(g(n) = Ar^n + Br^{-n}\). The condition \(g\in\ell^2(\mathbb{N})\) forces \(B=0\), so for \(n\ge1\), \[g(n)=A r^n.\] We now use the bulk equation at \(n=1\): \[(\lambda+\mu-\theta)g(1)-c g(2)-c g(0)=0.\] Substituting \(g(1)=Ar\), \(g(2)=Ar^2\), and 44 , we obtain \[c(r+r^{-1})Ar-cAr^2-cg(0)=0,\] hence \(g(0)=A\). Now use the boundary equation at \(n=0\): \[\lambda g(0)-c\,g(1)=\theta g(0).\] Since \(g(1)=rg(0)\), this gives \[\theta=\lambda-cr.\] On the other hand, 44 gives \[\theta=\lambda+\mu-c\left(r+\frac{1}{r}\right).\] Comparing the two expressions for \(\theta\), we obtain \[r=\frac{c}{\mu} = \sqrt{\frac{\lambda}{\mu}}.\] Substituting this into \(\theta=\lambda-cr\) yields \[\theta = \lambda-\sqrt{\lambda\mu}\sqrt{\frac{\lambda}{\mu}} = 0.\] Therefore, the only eigenvalue of \(H\) below \(\lambda+ \mu - 2\sqrt{\lambda \mu}\) is \(0\). Together with 43 and the fact that \(\sigma(H) = \sigma(-L)\), this shows that the spectral gap of \(L\) is \[\gamma = (\sqrt{\mu}-\sqrt{\lambda})^2.\]

6.4 Compactness for the hypoelliptic shear on the torus↩︎

We prove here that for the Markov diffusion generator \[L=\partial_y^2+\sin(y)\partial_x\] on \(L^2(\mathbb{T}^2)\), the embedding \[(\operatorname{Dom}(L), \|\cdot\| + \|L \cdot \|) \hookrightarrow L^2(\mathbb{T}^2)\] is compact. For \(f \in \operatorname{Dom}(L)\), write \[f(x,y)=\sum_{m\in\mathbb{Z}}e^{imx}\phi_m(y).\] Since the coefficients of \(L\) are independent of \(x\), the \(x\)-Fourier modes are invariant: \[L(e^{imx}\phi(y)) = e^{imx}L_m\phi(y), \qquad L_m:=\partial_y^2+im\sin(y).\] By Parseval, \[\|f\|^2+\|Lf\|^2 = \sum_{m\in\mathbb{Z}} \left( \|\phi_m\|_{L^2_y}^2 + \|L_m\phi_m\|_{L^2_y}^2 \right).\] For brevity, set \[E_m := \|\phi_m\|_{L^2_y}^2+\|L_m\phi_m\|_{L^2_y}^2.\] We have the following estimates, whose proof we defer for now.

Lemma 6. With the above notations, we have for some constant \(C\) that \[\begin{align} \|\phi_m\|^2_{H^1_y} & \leq C E_m \\ \|\phi_m\|_{L^2_y}^2 &\le C|m|^{-1/3}E_m , \quad m \neq 0. \end{align}\]

Let \((f_n)\) be a sequence in \(\operatorname{Dom}(L)\) satisfying \[\|f_n\|+\|Lf_n\|\le 1.\] Write \[f_n(x,y)=\sum_{m\in\mathbb{Z}}e^{imx}\phi_{n,m}(y),\] and define \(E_{n,m}\) as above. Then \[\sum_m E_{n,m}\le C\] uniformly in \(n\). The high-\(x\)-frequency estimate in Lemma 6 gives, for \(M\ge1\), \[\begin{align} \label{tail32estimate322} \sum_{|m|>M}\|\phi_{n,m}\|_{L^2_y}^2 \le C M^{-1/3}\sum_{|m|>M}E_{n,m} \le C M^{-1/3}, \end{align}\tag{45}\] uniformly in \(n\). For each fixed \(m\), the estimate \[\|\phi_{n,m}\|_{H^1_y}^2\le C E_{n,m}\le C\] shows that \((\phi_{n,m})_n\) is bounded in \(H^1(\mathbb{T}_y)\). By the Rellich compactness theorem, \(H^1(\mathbb{T}_y)\hookrightarrow L^2(\mathbb{T}_y)\) compactly. Therefore, by a diagonal argument, we can extract a subsequence, reindexed by \(n\), such that \((\phi_{n,m})_n\) converges in \(L^2_y\) for every \(m\). Using the tail estimate 45 , we see that \((f_n)_n\) is Cauchy, hence convergent in \(L^2(\mathbb{T}^2)\). Thus, the graph-norm unit ball \[\{f:\|f\|+\|Lf\| \le1\}\] is relatively compact in \(L^2(\mathbb{T}^2)\). Equivalently, we have a compact embedding \[(\operatorname{Dom}(L),\|\cdot \|+\|L \cdot\|)\hookrightarrow L^2(\mathbb{T}^2).\]

Proof of Lemma 6. Since smooth functions are dense in \(\operatorname{Dom}(L)\), we can assume \(\phi_m\) to be smooth. Then \[\operatorname{Re}\langle \phi_m,L_m\phi_m\rangle = \operatorname{Re}\langle \phi_m,\phi_m''\rangle = -\|\phi_m'\|_{L^2_y}^2.\] Therefore, \[\|\phi_m'\|_{L^2_y}^2 \le |\langle \phi_m,L_m\phi_m\rangle| \le \|\phi_m\|_{L^2_y}\|L_m\phi_m\|_{L^2_y} \le \frac{1}{2}E_m.\] In particular, \[\|\phi_m\|_{H^1_y}^2\le C E_m.\]

Next, for \(|m|\ge1\), we estimate \(\sin(y)\phi_m\). Since \[L_m\phi_m=\phi_m''+im\sin(y)\phi_m,\] we have \[im\|\sin(y)\phi_m\|_{L^2_y}^2 = \langle \sin(y)\phi_m,L_m\phi_m\rangle - \langle \sin(y)\phi_m,\phi_m''\rangle .\] Taking imaginary parts gives \[\begin{align} |m|\|\sin(y)\phi_m\|_{L^2_y}^2 &\le |\langle \sin(y)\phi_m,L_m\phi_m\rangle| + |\operatorname{Im}\langle \sin(y)\phi_m,\phi_m''\rangle| . \end{align}\] The first term is bounded by \[\|\phi_m\|_{L^2_y}\|L_m\phi_m\|_{L^2_y} \le \frac{1}{2} E_m.\] For the second term, integrate by parts: \[\begin{align} \langle \sin(y)\phi_m,\phi_m''\rangle &= -\int_{\mathbb{T}}\cos(y)\overline{\phi_m(y)}\phi_m'(y)\,dy - \int_{\mathbb{T}}\sin(y)|\phi_m'(y)|^2\,dy . \end{align}\] The second integral is real, and hence \[|\operatorname{Im}\langle \sin(y)\phi_m,\phi_m''\rangle| \le \|\phi_m\|_{L^2_y}\|\phi_m'\|_{L^2_y} \le C E_m.\] Thus, \[|m|\|\sin(y)\phi_m\|_{L^2_y}^2 \le C E_m.\]

We now prove that high \(x\)-frequencies have small \(L^2\)-mass. Let \[Z_\delta:=\{y\in\mathbb{T}:|\sin y|\le \delta\}.\] For \(0<\delta\le1/2\), \(|Z_\delta|\le C\delta\). Since \(H^1(\mathbb{T})\hookrightarrow L^\infty(\mathbb{T})\) continuously, \[\int_{Z_\delta}|\phi_m(y)|^2\,dy \le |Z_\delta|\|\phi_m\|_{L^\infty_y}^2 \le C\delta\|\phi_m\|_{H^1_y}^2 \le C\delta E_m.\] On the complement \(Z_\delta^c\), we have \(|\sin y|>\delta\), and so \[\int_{Z_\delta^c}|\phi_m(y)|^2\,dy \le \delta^{-2}\|\sin(y)\phi_m\|_{L^2_y}^2 \le C\delta^{-2}|m|^{-1}E_m.\] Choosing \(\delta=|m|^{-1/3}\) gives, for \(|m|\ge1\), \[\|\phi_m\|_{L^2_y}^2 \le C|m|^{-1/3}E_m.\] ◻

Acknowledgements↩︎

I am grateful to Sourav Chatterjee for the suggestion to look for extensions of his results, and to Amir Dembo for encouraging me in the research process. I thank both of them and Persi Diaconis for helpful feedback.

References↩︎