On hyperboundedness and spectrum of Markov operators

Abstract

Consider an ergodic Markov operator $M$ reversible with respect to a probability measure $\mu$ on a general measurable space. It is shown that if $M$ is bounded from $\mathbb {L}^2(\mu )$ to $\mathbb {L}^p(\mu )$, where $p>2$, then it admits a spectral gap. This result answers positively a conjecture raised by Høegh-Krohn and Simon (J. Funct. Anal. 9:121–80, 1972) in the more restricted semi-group context. The proof is based on isoperimetric considerations and especially on Cheeger inequalities of higher order for weighted finite graphs recently obtained by Lee et al. (Proceedings of the 2012 ACM Symposium on Theory of Computing, 1131–1140, ACM, New York, 2012). It provides a quantitative link between hyperboundedness and an eigenvalue different from the spectral gap in general. In addition, the usual Cheeger inequality is extended to the higher eigenvalues in the compact Riemannian setting and the exponential behaviors of the small eigenvalues of Witten Laplacians at small temperature are recovered.

Appendix: On principal Dirichlet eigenvalues at small temperature

The goal of this appendix is to check the assertions (17), (18), (19) and (20) presented in Sect. 4. Concerning the three first relations, we will adapt the proofs of Holley et al. [16] for the corresponding results relative to the spectral gap of a Witten Laplacian on a compact Riemannian manifold without boundary. Their computations are based on the following ideas. The upper bound is obtained by considering a function approximating the indicator function of a well whose height is maximum among those not intersecting a fixed global minimum of the potential. For the Dirichlet eigenvalues, the argument is even simpler, by using a well of maximum height included in $A$ or a small ball if there is no such well. Concerning the lower bound, Holley et al. [16] consider paths with minimal elevation connecting generical points of the manifold with a fixed global minimum of the potential. This approach can in principle be applied to the Dirichlet eigenvalues by using paths with minimal elevation connecting generical points of $A$ to $\partial A$. But technically it requires some curvature bounds on $\partial A$, so we preferred to resort to a modified elevation and to paths linking generical points of $A$ to some nice interior points of $A^{\mathrm {c}}$. The advantage is that the existence of such points is not really restrictive when one is computing Dirichlet connectivity spectra, as shown by (20).

The notations are those introduced in Sect. 4. We begin by considering the simplest bounds:

Proof of (18) and (19)

It is based, on one hand on the observation that for any fixed $\beta \ge 0$, the mapping $\widehat{{\mathcal {D}}}_1\ni A\mapsto \lambda _0(A,L_\beta )$ is non-increasing, when $\widehat{{\mathcal {D}}}_1$ is endowed with the inclusion order. And on the other hand on the fact that for any $\beta \ge 0$ and $B\in \widehat{{\mathcal {D}}}_1$,

$$\begin{aligned} \exp \Big (-\beta \Big (\sup _{B}U-\inf _B U\Big )\Big )\lambda _0(L_0,B)\ \le \ \lambda _0(L_\beta ,B)\\ \le \ \exp \Big (\beta \Big (\sup _{B}U-\inf _B U\Big )\Big )\lambda _0(L_0,B) \end{aligned}$$

If $B$ is a cycle, we have $h(B)=\sup _{B}U-\inf _B U$, so (18) follows at once. If $B$ is a ball of radius $r>0$, $\sup _{B}U-\inf _B U\le 2r\left\| \nabla U\right\| _\infty \le 2r$, so that we get for any $A\in \widehat{{\mathcal {D}}}_1$,

$$\begin{aligned} \lambda _0(L_\beta ,A)&\le e^{2}\max \{\lambda _0(L_0,B)\,:\,B\subset A\hbox { is a ball of radius less than }1/ \beta \} \end{aligned}$$

To conclude to (19), it remains to remark that by compactness of $S$, there exists a constant $k_S^{(3)}\ge 1$, depending only on the Riemannian structure of $S$, such that

$$\begin{aligned} \forall x\in S,\,\forall r\in (0,1],\qquad (k_S^{(3)})^{-1}r^{-2}\ \le \ \lambda _0(L_0,B(x,r))\ \le \ k_S^{(3)}r^{-2} \end{aligned}$$

$\square$

The following arguments mainly follow those of Holley et al. [16]. We will adopt their notations (sometimes with a $A$ in index when the corresponding notions differ), so that we can refer directly to their proof.

Proof of (17)

Let $\beta \ge 1$ be fixed as well as a set $A\in \widehat{{\mathcal {D}}}_1(1/\beta )$. We modify the potential $U$ by defining

$$\begin{aligned} \forall x\in S,\qquad U_A(x)&:= \left\{ \begin{array}{l@{\quad }l} U(x), &{}\hbox {if }x\in A\sqcup \partial A\\ -\infty , &{}\hbox {otherwise} \end{array}\right. \end{aligned}$$

The elevation $E_A(\gamma )$ of a path $\gamma \!\in \! {\mathcal {C}}([0,1], S)$ is $E_A(\gamma )\!=\!\max _{t\in [0,1]}U_A(\gamma (t))$ and for $x,y\in S$ define

$$\begin{aligned} H_A(x,y)&:= \inf \{ E_A(\gamma )\,:\,\gamma \in {\mathcal {C}}([0,1], S),\, \gamma (0)=x,\, \gamma (1)=y\} \end{aligned}$$

There is no difficulty in checking that

$$\begin{aligned} h(A)&= \max \{H_A(x,y)-U(x)\,:\,x\in A, y\not \in A\}\\&= \max _{x\in A}\min _{j\in [\![J]\!]} H_A(x,y_j)-U(x) \end{aligned}$$

where $J$ is the number of connected components of $A^{\mathrm {c}}$ (there are only a finite number of them by the restrictions imposed on $\widehat{{\mathcal {D}}}_1$) and where the $y_j$, $j\in [\![J]\!]$, are any choice of points in each of them, but we take them satisfying

$$\begin{aligned} \mu (B(y_j,1/\beta )\cap A^{\mathrm {c}})&\ge \frac{\mu (B(y_j,1/\beta ))}{2} \end{aligned}$$

For any $\beta \ge 1$, there exists a finite cover of $A$ by balls $\{B(x_k, 1/\beta )\,:\,k\in [\![N_\beta ]\!]\}$, where the $x_k$, $k\in [\![N_\beta ]\!]$, are points of $A$, where $N_\beta \le k_S^{(4)}\beta ^{\dim (S)}$, with $k_S^{(4)}<+\infty$ is a constant depending only on $S$ and not on $A$. Denote $Z_\beta :=\int \exp (-\beta U)\, d\mu$.

Considering any function $\phi \in {\mathcal {C}}^1(S)$ vanishing on $A^{\mathrm {c}}$, we can write

$$\begin{aligned} \mu _\beta [\phi ^2]&\le \frac{1}{Z_\beta }\sum _{k\in [\![N_\beta ]\!]}\int \limits _{B(x_k,1/\beta )} \exp (-\beta U(x))\phi ^2(x)\, \mu (dx)\\&= \frac{1}{Z_\beta \mu (B(y_{j(k)},1/\beta )\cap A^{\mathrm {c}})}\sum _{k\in [\![N_\beta ]\!]}\int \limits _{B(x_k,1/\beta )\times ( B(y_{j(k)},1/\beta )\cap A^{\mathrm {c}})}\\&\quad \exp (-\beta U(x))(\phi (x)-\phi (y))^2\, \mu (dx)\mu (dy)\\&\le \frac{2}{Z_\beta \mu (B(y_{j(k)},1/\beta ))}\sum _{k\in [\![N_\beta ]\!]}\int \limits _{B(x_k,1/\beta )\times B(y_{j(k)},1/\beta )}\\&\quad \exp (-\beta U(x))(\phi (x)-\phi (y))^2\, \mu (dx)\mu (dy)\\&\le \frac{2ek_S^{(5)}\beta ^{\dim (S)}}{Z_\beta }\sum _{k\in [\![N_\beta ]\!]}\exp (-\beta U(x_k))\int \limits _{B(x_k,1/\beta )\times B(y_{j(k)},1/\beta )}\\&\quad (\phi (x)-\phi (y))^2\, \mu (dx)\mu (dy) \end{aligned}$$

where $j(k)\in [\![J]\!]$ is such that

$$\begin{aligned} H_A(x_k,y_{j(k)})&= \min _{j\in [\![J]\!]} H_A(x_k,y_j) \end{aligned}$$

and where $k_S^{(5)}\ge 1$ is a constant only depending on the structure of $S$ such that

$$\begin{aligned} \forall x\in S,\,\forall r\in (0,1],\qquad (k_S^{(5)})^{-1}r^{\dim (S)}\ \le \ \mu (B(x,r))\ \le \ k_S^{(5)}r^{\dim (S)} \end{aligned}$$

With these preliminaries, the proof is now identical to that of Holley et al. developed in pages 338–340 of [16]. They find a constant $k^{(6)}_S>0$, only depending on $S$, such that for any $\beta \ge 1$ and $\phi$ as above,

$$\begin{aligned}&\sup \left\{ \exp (-\beta U(x))\int \limits _{B(x,1/\beta )\times B(y,1/\beta )} (\phi (w)-\phi (v))^2\, \mu (dv)\mu (dw)\right. \\&\qquad \qquad \qquad \qquad \left. \,:\,x\in A, y\not \in A\right\} \\&\quad \qquad \qquad \qquad \le k^{(6)}_S\beta ^{2(\dim (S)-1)}Z_\beta \exp (\beta h(A))\int \left\langle \nabla \phi ,\nabla \phi \right\rangle d\mu _\beta \end{aligned}$$

Putting together the above computations, we end up with the Poincaré inequality

$$\begin{aligned} \mu _\beta [\phi ^2]&\le 2ek^{(4)}_Sk^{(5)}_S k^{(6)}_S\beta ^{4\dim (S)-2} \exp (\beta h(A))\int \left\langle \nabla \phi ,\nabla \phi \right\rangle d\mu _\beta \end{aligned}$$

whose validity for all ${\mathcal {C}}^1$ functions $\phi$ vanishing outside $A$ implies (17).$\square$

The proof of the remaining bound will justify the restrictions imposed on $\widehat{{\mathcal {D}}}_n$, which could have looked strange at first view.

Proof of (20)

So let be given $n\ge 2$, $\beta \ge 1$ and $(A_1, \ldots , A_n)\in \widehat{\mathcal {D}}_n$ such that $A_1$ does not belong to $\widehat{{\mathcal {D}}}_1(1/\beta )$, we are going to prove that there exists $k\in [\![2,n]\!]$ with

$$\begin{aligned} \lambda _0(L_\beta ,A_k)&\ge {k''}_{\!\!\!S}\beta ^2 \end{aligned}$$

(28)

where ${k''}_{\!\!\!S}>0$ is a constant depending only on the Riemannian structure of $S$. Indeed, by definition of $\widehat{{\mathcal {D}}}_1(1/\beta )$, there exists a connected component of $A^{\mathrm {c}}$ which is such that all its points $x$ satisfy

$$\begin{aligned} \mu (B(x,1/\beta )\cap A_1)&> \frac{\mu (B(x,1/\beta ))}{2} \end{aligned}$$

By definition of $\widehat{\mathcal {D}}_n$, this connected component contains a subset $A_k$, with $k\in [\![2,n]\!]$, it is the one that will satisfy (28). Consider $(X^{(\beta )}_t)_{t\ge 0}$ a diffusion process of generator $L_\beta$ and denote $\tau :=\inf \{t\ge 0\,:\,X^{(\beta )}_t\not \in A_k\}$. Since $L_\beta$ is elliptic and $A_k$ is a connected open set, $\lambda _0(L_\beta ,A_k)$ is the asymptotic rate of getting out of $A_k$:

$$\begin{aligned} \forall x\in A_k,\quad \lambda _0(L_\beta ,A_k)&= -\lim _{t\rightarrow +\infty }\frac{1}{t}\ln (\mathbb {P}_{x}[\tau >t]) \end{aligned}$$

where the $x$ in $\mathbb {P}_x$ indicates that the diffusion is starting from $x$: $\mathbb {P}_x$-almost surely, $X^{(\beta )}_0=x$. Thus taking into account the Markov property, to get (28), it is sufficient to find another constant $k_S^{(7)}>0$ depending only on $S$ such that

$$\begin{aligned} \forall x\in A_k,\quad \mathbb {P}_x[\tau \le 1/\beta ^2]&\ge k_S^{(7)} \end{aligned}$$

It is even enough to show that

$$\begin{aligned} \forall x\in A_k,\quad \mathbb {P}_x[X^{(\beta )}_{1/\beta ^2}\in A_1]&\ge k_S^{(7)} \end{aligned}$$

Denote by $(p_t^{(\beta )}(x,y))_{t>0, x,y\in S}$ the kernels corresponding to the semi-group associated to the generator $L_\beta$, so we can write

$$\begin{aligned} \mathbb {P}_x[X^{(\beta )}_{1/\beta ^2}\in A_1]&= \int _{A_1}p^{(\beta )}_{1/\beta ^2}(x,y)\, \mu (dy) \end{aligned}$$

From Theorem 3.1 of Wang [43], we have that for any $t>0$, $\sigma >0$ and $x,y\in S$,

$$\begin{aligned} p_t^{(\beta )}(x,y)\ge k_S^{(8)}(2\pi t)^{-\dim (S)/2}\exp \left[ -\left( \frac{1}{2t}+\frac{\sigma }{3\sqrt{t}}\right) \rho ^2(x,y)-\frac{b^2t}{8}\right. \\ \left. -\left( \frac{b^2}{4\sigma }+\frac{2\dim (S)\sigma }{3}\right) \sqrt{t}\right] \end{aligned}$$

where $k_S^{(8)}>0$ is the volume of $S$ (with respect to the unnormalized Riemannian measure of $S$), where $\rho (x,y)$ is the Riemannian distance between $x$ and $y$ and where $b:=K_S+\beta$, $K_S\ge 0$ being such that the Ricci curvature of $S$ is bounded below by $-K_S$. In particular if we choose $\sigma =\beta$, it appears there exists a constant $k_S^{(9)}>0$, only depending on $S$, such that

$$\begin{aligned} \forall x\in S,\,\forall y\in B(x,1/\beta ),\qquad p_{1/\beta ^2}^{(\beta )}(x,y)&\ge k_S^{(9)}\beta ^{\dim (S)}\end{aligned}$$

It follows that for all $x\in A_k$,

$$\begin{aligned} \int _{A_1}p^{(\beta )}_{1/\beta ^2}(x,y)\, \mu (dy)&\ge \int _{A_1\cap B(x,1\beta )}p^{(\beta )}_{1/\beta ^2}(x,y)\, \mu (dy)\\&\ge k_S^{(9)}\beta ^{\dim (S)}\mu (A_1\cap B(x,1\beta ))\\&\ge \frac{k_S^{(9)}}{2}\beta ^{\dim (S)}\mu ( B(x,1\beta ))\\&\ge \frac{k_S^{(9)}}{2k_S^{(5)}} \end{aligned}$$

as required.$\square$