Ruelle–Pollicott Resonances of Stochastic Systems in Reduced State Space. Part I: Theory

Abstract

A theory of Ruelle–Pollicott (RP) resonances for stochastic differential systems is presented. These resonances are defined as the eigenvalues of the generator (Kolmogorov operator) of a given stochastic system. By relying on the theory of Markov semigroups, decomposition formulas of correlation functions and power spectral densities (PSDs) in terms of RP resonances are then derived. These formulas describe, for a broad class of stochastic differential equations (SDEs), how the RP resonances characterize the decay of correlations as well as the signal’s oscillatory components manifested by peaks in the PSD. It is then shown that a notion reduced RP resonances can be rigorously defined, as soon as the dynamics is partially observed within a reduced state space V. These reduced resonances are obtained from the spectral elements of reduced Markov operators acting on functions of the state space V, and can be estimated from series. They inform us about the spectral elements of some coarse-grained version of the SDE generator. When the time-lag at which the transitions are collected from partial observations in V, is either sufficiently small or large, it is shown that the reduced RP resonances approximate the (weak) RP resonances of the generator of the conditional expectation in V, i.e. the optimal reduced system in V obtained by averaging out the contribution of the unobserved variables. The approach is illustrated on a stochastic slow-fast system for which it is shown that the reduced RP resonances allow for a good reconstruction of the correlation functions and PSDs, even when the time-scale separation is weak. The companions articles, Part II [114] and Part III [113], deal with further practical aspects of the theory presented in this contribution. One important byproduct consists of the diagnosis usefulness of stochastic dynamics that RP resonances provide. This is illustrated in the case of a stochastic Hopf bifurcation in Part II. There, it is shown that such a bifurcation has a clear manifestation in terms of a geometric organization of the RP resonances along discrete parabolas in the left half plane. Such geometric features formed by (reduced) RP resonances are extractable from time series and allow thus for providing an unambiguous “signature” of nonlinear oscillations embedded within a stochastic background. By relying then on the theory of reduced RP resonances presented in this contribution, Part III addresses the question of detection and characterization of such oscillations in a high-dimensional stochastic system, namely the Cane–Zebiak model of El Niño-Southern Oscillation subject to noise modeling fast atmospheric fluctuations.

Elements of Stochastic Analysis

In this appendix we present a short survey of elements of stochastic analysis used in the Main Text. The main objective is to introduce the key concepts and tools of stochastic analysis for stochastic differential equations (SDEs), to a wider audience in the geosciences and macroscopic physics.

1.1 Markov Semigroups

Two approaches dominate the analysis of stochastic dynamics. We are here concerned with the approach rooted in Stochastic Analysis which, contrary to the random dynamical system (RDS) approach [3, 20, 24], does not substitute a deterministic (nonlinear) flow S(t) by a stochastic flow $S(t,\omega )$ acting¹⁰ on the state space ${\mathcal {X}}$ but rather by a family of linear operators $P_t$, acting on a space of observables of the state space, i.e. on functions of ${\mathcal {X}}$. A typical choice of observables is given by ${\mathcal {C}}_b({\mathcal {X}})$, the space of bounded and continuous functions on ${\mathcal {X}}$. In what follows ${\mathcal {X}}$ is a finite-dimensional Polish space.

More precisely, this family $P_t$ reflects the (averaged) action of the stochastic flow at the level of functions and is given as the mapping which to each function $\phi$ in ${\mathcal {C}}_b({\mathcal {X}})$ associates the function:

$$\begin{aligned} P_t \phi (x) ={\mathbb {E}}( \phi (S(t,\cdot )x))=\int _{\varOmega } \phi (S(t,\omega ) x) \, \text {d}{\mathbb {P}} (\omega ), \quad t\ge 0,\; x\in {\mathcal {X}}. \end{aligned}$$

(A.1)

In (A.1), the function $\phi$ is the aforementioned observable. Its physical meaning could be, for instance, the potential vorticity or the temperature of a fluid at a given location or averaged over a volume. The RHS of (A.1) involves averaging over the realizations $\omega$, i.e. expectation. For deterministic flow it reduces to $P_t\phi (x)=\phi (S(t)x)$ and is known as the Koopman operator. Note that $P_t$ such as defined in (A.1) is not limited to stochastic flow, more generally $P_t \phi (x) ={\mathbb {E}}( \phi (X_t^x))$ where $X_t^x$ denotes a stochastic process that solves Eq. (2.1) (as associated with $P_t$) and emanates from x in ${\mathcal {X}}$.

Under general assumptions on F and D, the stochastic process $X_t$ solving Eq. (2.1) is Markovian (i.e. the future is determined only by the present value of the process) which translates at the level of $P_t$ into the following semigroup property

$$\begin{aligned} P_0=\text {Id}, \quad P_t P_s=P_{t+s}, \; \; t,s\ge 0. \end{aligned}$$

(A.2)

A breakdown of (A.2) indicates thus that the underlying stochastic process is non-Markovian.

It is noteworthy to mention that even when $P_t$ satisfies (A.2), it does not ensure that $P_t$ is a strongly continuous semigroup [94] on ${\mathcal {C}}_b({\mathcal {X}})$. Nevertheless, $(P_t)_{t\ge 0}$ is extendable to a strongly continuous semigroup in $L^2_{\mu }$ as soon as $\mu$ is an invariant measure of the Markov semigroup; see Theorem 4 below. The spectral theory of such semigroups [40] is at the core of the description of mixing properties in $L^2_{\mu }$, such as presented in Sect. 2.2 in the Main Text.

1.2 Ergodic Invariant Measures and the Strong Feller–Irreducibility Approach

The Fokker–Planck equation (2.3) may support several weak stationary solutions. An important question, is thus the identification of stationary measures that describe the asymptotic statistical behavior of the solutions of Eq. (2.1), in a typical fashion. The notion of ergodic invariant measures plays a central role in that respect, and relies on the following important characterization of ergodic measures for (stochastically continuous) Markov semigroups [37, Theorem 3.2.4].

Definition 1

An invariant measure is ergodic if one of the following three equivalent statements holds:

• (i) For any $f \in L^2_\mu ({\mathcal {X}})$, if $P_t f =f$, almost surely w.r.t $\mu$ ($\mu$-a.s.) for all $t\ge 0$, then f is constant $\mu$-a.s.

• (ii) For any Borel set $\varGamma$ of ${\mathcal {X}}$, if $P_t \mathbb {1}_{\varGamma } =\mathbb {1}_{\varGamma }$$\mu$-a.s. for all $t\ge 0$, then $\mu (\varGamma )=0$ or 1.

• (iii) For any $f \in L^2_\mu ({\mathcal {X}})$, $\frac{1}{T} \int _0^T P_s f \, \text {d}s \underset{T\rightarrow \infty }{\longrightarrow }\int f \, \text {d}\mu$ in $L^2_\mu ({\mathcal {X}}).$

In practice, an efficient approach to show the existence of an ergodic measure consists of showing the existence of a unique invariant measure, since in this case such an invariant measure is necessarily ergodic [37, Theorem 3.2.6]. Various powerful approaches exist to deal with the existence of a unique invariant measure. The next section discusses the classical approach based on the theory of strong Feller Markov semigroups and irreducibility.

The main interest of the strong Feller–Irreducibility approach lies in its usefulness for checking the conditions of the Doob–Khasminskii Theorem [36, 37, 72], the latter ensuring the existence of at most one ergodic invariant measure. This strategy requires the proof of certain smoothing properties of the associated Markov semigroup, and to show that any point can be (in probability) reached at any time instant by the process regardless of initial data. This property is known as irreducibility. It means that $P_t \mathbb {1}_{U} (x)> 0$ for all x in ${\mathcal {X}}$, every $t>0$, and all non-empty open sets U of ${\mathcal {X}}$, which is equivalent to say that

$$\begin{aligned} {\mathbb {P}}(\Vert S(t,\cdot )x-z\Vert < \epsilon ) >0, \end{aligned}$$

(A.3)

for any z in ${\mathcal {X}}$, $\epsilon >0$ and $t>0$; see [13, p. 67]. In other words the irreducibility condition expresses the idea that any neighborhood of any point z in ${\mathcal {X}}$, is reachable at each time, with a positive probability.

Remarkably, the irreducibility is usually inferred from the controllability of the associated control system ${\dot{x}}=F(x) + D(X) u(t)$; see [19] for a simple illustration. This approach is well-known and based on the support theorem of Stroock and Varadhan [?] (see also [65, Theorem 8.1]) that shows that several properties of the SDEs can be studied and expressed in terms of the control theory of ordinary differential equations (ODEs); see [37, Secns. 7.3 and 7.4] for the case of additive (non-degenerate) noise and [2, 73] for the more general case of nonlinear degenerate noise, i.e. in the case where the noise acts only on part of the system’s equations, corresponding to ker$(Q)\ne \{0\}$.

The strong Feller property means that the Markov semigroup maps bounded measurable functions into bounded continuous functions. This property, related to a regularizing effect of the Markov semigroup $(P_t)_{t\ge 0}$, is a consequence of the hypoellipticity of the Kolmogrorov operator${\mathcal {K}}$ defined on smooth functions $\psi$ (of class $C^2$) as follows when ${\mathcal {X}}={\mathbb {R}}^d$:

$$\begin{aligned} {\mathcal {K}}\psi (x)= \frac{1}{2}\text {Tr}({{\varvec{\varSigma }}} \nabla ^2 \psi (x)) +\langle F(x), \nabla \psi \rangle , \end{aligned}$$

(A.4)

where

$$\begin{aligned} \text {Tr}({{\varvec{\varSigma }}} \nabla ^2 \psi (\cdot ))=\sum _{i,j=1}^d \big [D(x)D(x)^T\big ]_{ij}\partial _{ij}^2 \psi . \end{aligned}$$

(A.5)

Here $\text {Tr}$ denotes the trace of a matrix. Note that hypoelliptic operators include those that are uniformly elliptic for which the Weyl’s smoothing lemma applies; e.g. [28, Theorem 4.7]. Hypoellipticity allows nevertheless for dealing with the case of degenerate noise, which is important in applications.

A very efficient criteria for hypoellipticity is given by Hörmander’s theorem [63, 90]; see also [24, Appendix C1] for a discussion on the related Hörmander’s bracket condition and its implications to the existence of other types of meaningful measures for SDEs, namely the Sinaï–Ruelle–Bowen (SRB) random measures. We refer also to Part II [114], for an instructive verification of the Hörmander’s condition in the case of the Hopf normal form subject to additive noise.

From a geophysical perspective, it is noteworthy to mention that the strong Feller–Irreducibility approach allows for dealing with a broad class of truncations of fluid dynamics models that would be perturbed by noise, possibly degenerate. For instance, in the case of truncations of 2D or 3D Navier–Stokes equations, the strong Feller–Irreducibility approach has been shown to be applicable even for an additive noise that forces only very few modes [4, 99]. The delicate point of the analysis is the verification of the controllability (and thus irreducibility) of the associated control system, by techniques typically adapted from [66] or rooted in chronological calculus as in [4]. Whatever the approach, the analysis requires the appropriate translation into geometrical terms of the cascade of energy in which the nonlinear terms transmit the forcing from the few modes to all the others [100]. We mentioned however [87] for an example of a stochastic dynamical system which has the square of the Euclidean norm as the Lyapunov function, is hypoelliptic with nonzero noise forcing, and that yet fails to be reachable or ergodic.

1.3 Markov Semigroups and Mixing

We recall here standard results about Markov semigroups. It states that any Markov semigroup that is strong Feller and irreducible and for which an invariant measure exists (which is thus unique) is not only ergodic but also strongly mixing for the total variation norm of measures. Given two probability measures $\mu _1$ and $\mu _2$ on ${\mathcal {X}}$, we recall that the latter is defined as [60, Eq. (3.1)]

$$\begin{aligned} \text {TV}(\mu _1,\mu _2)=\sup _{\begin{array}{c} g \in {\mathcal {B}}_b({\mathcal {X}})\\ \Vert g\Vert _{\infty }\le 1 \end{array}} \bigg | \int g \, \, \text {d}\mu _1-\int _{{\mathcal {X}}} g \, \, \text {d}\mu _2\bigg |, \end{aligned}$$

(A.6)

where ${\mathcal {B}}_b({\mathcal {X}})$ denotes the set of Borel measurable and bounded functions on ${\mathcal {X}}.$

Theorem 4

Let $\mu$ be an invariant measure of a Markov semigroup $(P_t)_{t\ge 0}$. For any $p\ge 1$ and $t\ge 0$, $P_t$ is extendable to a linear bounded operator on $L^p_\mu ({\mathcal {X}})$ still denoted by $P_t$. Moreover

• (i) $\Vert P_t\Vert _{{\mathcal {L}}(L^p_\mu ({\mathcal {X}}))} \le 1$

• (ii) $P_t$ is strongly continuous semigroup in $L^p_\mu ({\mathcal {X}})$.

If furthermore $(P_t)_{t\ge 0}$ is strong Feller and irreducible, then $\mu$ is ergodic (and unique) and for any x in ${\mathcal {X}}$ and g in $L^1_{\mu }$

$$\begin{aligned} \underset{T\rightarrow \infty }{\lim }\frac{1}{T} \int _0^T g(X_\tau ^x) \, \text {d}\tau = \int _{{\mathcal {X}}} g(x)\, \text {d}\mu , \; \; {\mathbb {P}}\text {-a.s.}, \end{aligned}$$

(A.7)

where $X_t^x$ denotes the stochastic process solving the SDE associated with $P_t$.

In this case, the invariant measure $\mu$ is also strongly mixing in the sense that for any measure $\nu$ on ${\mathcal {X}}$, we have:

$$\begin{aligned} \text {TV}({\mathcal {L}}_t \nu ,\mu )\underset{t\rightarrow \infty }{\longrightarrow }0. \end{aligned}$$

(A.8)

For the definition of a strongly continuous semigroup also known as $C_0$-semigroup we refer to [40, p. 36]. For an introduction to semigroup theory we refer to [41, 118].

Proof

We prove first (i). The proof is standard and can be found e.g. in [59, Prop. 1.14] but is reproduced here for the reader’s convenience. Let g be in ${\mathcal {C}}_b({\mathcal {X}})$. By the Hölder inequality, we have

$$\begin{aligned} |P_t g (x) |^p \le P_t (|g|^p) (x). \end{aligned}$$

(A.9)

If we now integrate both sides of this inequality with respect to $\mu$, we obtain

$$\begin{aligned} \int _{{\mathcal {X}}} |P_t g (x) |^p \mu (\, \text {d}x) \le \int _{{\mathcal {X}}} P_t (|g |^p) (x) \mu (\, \text {d}x)= \int _{{\mathcal {X}}} | g|^p (x) \mu (\, \text {d}x), \end{aligned}$$

(A.10)

the latter equality resulting from the invariance of $\mu$. Since ${\mathcal {C}}_b({\mathcal {X}})$ is dense in $L^p_\mu ({\mathcal {X}})$, the inequality (A.10) can be extended to any function in $L^p_\mu ({\mathcal {X}})$, and thus $(P_t)_{t\ge 0}$ can be uniquely extended to a contraction semigroup in $L^p_\mu ({\mathcal {X}})$, and property (i) is proved.

Let us show now that $(P_t)_{t\ge 0}$ is strongly continuous in $L^p_\mu ({\mathcal {X}})$. Since $(P_t)_{t\ge 0}$ is a Markov semigroup, for any g in ${\mathcal {C}}_b({\mathcal {X}})$ and x in ${\mathcal {X}}$, we have that the mapping $t \mapsto P_t g (x)$ is continuous. Therefore by the dominated convergence theorem

$$\begin{aligned} \lim _{t\rightarrow 0} \; P_t g =g \text { in } L^p_\mu ({\mathcal {X}}). \end{aligned}$$

(A.11)

The density of ${\mathcal {C}}_b({\mathcal {X}})$ in $L^p_\mu ({\mathcal {X}})$ allows us to conclude that this convergence holds when g is in $L^p_\mu ({\mathcal {X}})$.

The ergodicity of $\mu$ results from the aforementioned Doob’s theorem. The time-average property (A.7) and the mixing property (A.8) can be obtained as a consequence of e.g. [105, Cor. 2.3]; see also [110, Cor. 1]. $\square$

1.4 Generator of a Markov Semigroup

Recall that the generator A of any strongly continuous semigroup $(T(t))_{t\ge 0}$ on a Hilbert space ${\mathcal {H}}$ is defined as the operator $A:D(A)\subset {\mathcal {H}} \rightarrow {\mathcal {H}}$, such that

$$\begin{aligned} A \varphi = \lim _{t\rightarrow 0^+} \frac{1}{t}\big (T(t) \varphi -\varphi \big ), \end{aligned}$$

(A.12)

defined for every $\varphi$ in the domain

$$\begin{aligned} D(A)=\{\varphi \in {\mathcal {H}} \;|\; \lim _{t\rightarrow 0^+} \frac{1}{t}\big (T(t) \varphi -\varphi \big ) \text { exists}\}. \end{aligned}$$

(A.13)

As any generator of a contraction semigroup, given an invariant measure $\mu$, the generator K of the contraction semigroup $(P_t)_{t\ge 0}$ in $L^2_{\mu }$ (Theorem 4(i)) is dissipative, which is equivalent to say, since $L^2_{\mu }$ is a Hilbert space, that

$$\begin{aligned} \mathrm{Re \,}\langle K f,f \rangle _{L^2_{\mu }} \le 0, \; \forall \, f \in D(K), \end{aligned}$$

(A.14)

where D(K) denotes the domain of K; see e.g. [40, Prop. II.3.23]. The domain D(K) is furthermore dense in $L^2_{\mu }$ and K is a closed operator; see [94, Cor. 2.5 p. 5]. The isolated part of the spectrum of K provides the Ruelle–Pollicott resonances; see Sect. 2.2.

1.5 Return to Equilibrium and Spectral Gap

We present here some useful results concerning (i) the exponential return to equilibrium for strong Feller and irreducible Markov semigroups, and (ii) spectral gap in the spectrum of the Markov semigroup generator K; see Theorems 5 and 6 below. Theorem 5 deals with semigroups that become quasi-compact after a finite time, and Theorem 6 addresses the exponential $L^2$-convergence and lower bound of the spectral gap. For Theorem 5, the approach is based on Lyapunov functions such as formulated in [98]. We propose a slightly different presentation for which we provide the main elements of the proof. We refer to [31] for an efficient (and beautiful) generalization of such Lyapunov-type criteria allowing for sub-exponential convergence towards the equilibrium.

Recall that the essential spectral radius$\mathbf{r }_{ess}(T)$ of a linear bounded operator T on a Banach space ${\mathcal {E}}$ satisfies [40, p. 249] the Hadamard formula

$$\begin{aligned} \mathbf{r }_{ess}(T)=\underset{n\rightarrow \infty }{\lim }\Vert T^n\Vert _{ess}^{1/n}, \end{aligned}$$

(A.15)

where

$$\begin{aligned} \Vert T\Vert _{ess}=\inf \Big \{ \Vert T- {\mathcal {C}}\Vert _{{\mathcal {L}}({\mathcal {E}})} \, : \, {\mathcal {C}} \text { is a linear and compact operator of } {\mathcal {E}}\Big \}. \end{aligned}$$

(A.16)

We have then the following convergence result.

Theorem 5

Let ${\mathcal {P}}=(P_t)_{t\ge 0}$ be a strong Feller and irreducible Markov semigroup in $L^2_\mu ({\mathbb {R}}^d)$ (${\mathcal {X}}={\mathbb {R}}^d$) generated by an SDE given by Eq. (2.1) for which F and G are locally Lipschitz. Assume that there exists a Lyapunov function¹¹U and a compact set ${\mathfrak {A}}$ for which there exist $a >0$, $0<\kappa <1$ and $b<\infty$, such that

$$\begin{aligned} {\mathcal {K}} U&\le a U, \end{aligned}$$

(A.17a)

$$\begin{aligned} P_{t_0} U&\le \kappa U + b \mathbb {1}_{{\mathfrak {A}}}, \; \text { for some } t_0>0, \end{aligned}$$

(A.17b)

where ${\mathcal {K}}$ is the Kolmogorov differential operator generating the Markov process associated with ${\mathcal {P}}$. Then for all $t>t_0$, $P_t$ becomes quasi-compact, i.e.

$$\begin{aligned} \mathbf{r }_{ess}(P_t)\le \kappa , \end{aligned}$$

(A.18)

where the essential spectral radius is taken for $P_t$ as acting on ${\mathcal {E}}={\mathcal {F}}_{U}$ given by

$$\begin{aligned} {\mathcal {F}}_{U}=\{f:{\mathbb {R}}^d \rightarrow {\mathbb {R}} \;|\; \text { f Borel measurable and } \Vert f\Vert _{U} <\infty \}, \end{aligned}$$

(A.19)

and endowed with the norm

$$\begin{aligned} \Vert f\Vert _U=\underset{x\in {\mathbb {R}}^d}{\sup }\frac{|f(x)|}{U(x)}. \end{aligned}$$

(A.20)

Furthermore $(P_t)_{t\ge 0}$ has a unique invariant measure $\mu$, and the inequality (A.18) ensures that there exist $C>0$ and $\lambda >0$ such that for all f in ${\mathcal {F}}_{U}$,

$$\begin{aligned} \left| P_t f(x) -\int f \, \text {d}\mu \right| \le C e^{-\lambda t} U(x), \;\; t>t_0, \; \forall \, x\in {\mathbb {R}}^d. \end{aligned}$$

(A.21)

The proof of this result is found in Appendix A.6.

Remark 5

The assumption (A.17b) is sometimes verified from moment estimates in practice. For instance if there exist $k_0>0$ and $k_1>0$ such that

$$\begin{aligned} {\mathbb {E}} |X_t^x| \le k_0 e^{-k_1 t}|x| +c, \; t\ge 0, \end{aligned}$$

(A.22)

then for any $t\ge -\frac{1}{k_1} \log (\frac{1}{4 k_0})$, we have ${\mathbb {E}} (|X_t^x| +1)\le \frac{1}{2} (|x|+1)-\frac{1}{4} |x|+c+\frac{1}{2}$, which leads to

$$\begin{aligned} {\mathbb {E}} (|X_t^x| +1) \le \frac{1}{2} \big (|x|+1\big ) +\big (c+\frac{1}{2} \big )\mathbb {1}_{B_r}, \end{aligned}$$

(A.23)

for all $r>4(c+\frac{1}{2}),$ and thus (A.17b) holds with $U(x)=|x|+1.$

More generally, if

$$\begin{aligned} {\mathcal {K}} U \le -\alpha U +\beta , \; \text { with }\alpha >0, \text { and } \; 0\le \beta <\infty , \end{aligned}$$

(A.24)

then $\frac{\, \text {d}}{\, \text {d}t} P_t U (x)={ P_t {\mathcal {K}} U} (x)\le -\alpha P_t U(x) +\beta$, leading to

$$\begin{aligned} {\mathbb {E}} \big [U(X_t^x)\big ]\le U(x) e^{-\beta t} +\frac{\beta }{\alpha } \big ( 1-e^{-\alpha t}\big ), \;\; t>0, \end{aligned}$$

(A.25)

and similarly (A.17b) holds. In addition, (A.24) implies (A.17a). Note that (A.24) and (A.25) are quite standard; see e.g. [33, Lemma 2.11].

Finally, note also that finding a Lyapunov function may be easier than proving inequalities of the form (A.22). For instance, if there is a Lyapunov function which grows polynomially like $\Vert p\Vert ^q$, then one knows that the process has moments of order q; see [85, 86].

Finally, lower bounds of the spectral gap in $L^2_{\mu }$ may be derived for a broad class of SDEs. Recall that the generator K has a spectral gap in $L^2_{\mu }$ if there exists $\delta >0$ such that

$$\begin{aligned} \sigma (K)\cap \{\lambda \,:\, \text {Re}(\lambda )>-\delta \}=\{0\}. \end{aligned}$$

(A.26)

The largest $\delta >0$ with this property is denoted by $\text {gap}(K)$, namely

$$\begin{aligned} \text {gap}(K)=\sup \{ \delta >0 \text { s.t. }(A.26) \text { holds} \}. \end{aligned}$$

(A.27)

The following result is a consequence in finite dimension of more general convergence results [57, Theorems 2.5 and 2.6]. Since $(P_t)_{t\ge 0}$ is a $\hbox {C}_0$-semigroup in $L^2_{\mu }$, the theory of asymptotic behavior of a semigroup with a strictly dominant, algebraically simple eigenvalue (e.g. [118, Theorem. 3.6.2]) implies the spectral gap property stated in the following.

Theorem 6

Assume that $(P_t)_{t\ge 0}$ is strong Feller and irreducible. Assume furthermore that the following ultimate bound holds for the associated stochastic process $X_t^x$, i.e. there exist $c,k,\alpha >0$ such that

$$\begin{aligned} {\mathbb {E}} \, |X_t^x|^2< k |x|^2e^{- \alpha t} +c, \; \;t\ge 0, \; \; x\in {\mathbb {R}}^d. \end{aligned}$$

(A.28)

Then there exists a unique invariant measure $\mu$ for which the U-uniform ergodicity (A.21) holds with $U(x)=1+|x|^2$, as well as the following exponential $L^2$-convergence

$$\begin{aligned} \Vert P_t \varphi - \int \varphi \, \text {d}\mu \Vert _{L^2_{\mu }} \le C e^{-\lambda t} \Vert \varphi \Vert _{L^2_{\mu }}, \; t\ge 0, \; \varphi \in L^2_{\mu }, \end{aligned}$$

(A.29)

with C and $\lambda$ positive constants independent of $\varphi$; the latter rate of convergence being the same as that of (A.21). Furthermore, one has the following lower bound for the $L^2_{\mu }$-spectrum of the generator K:

$$\begin{aligned} 0<\lambda \le \text {gap}(K). \end{aligned}$$

(A.30)

We will see in Part II [114] of this three-part article that Theorem 6 has important practical consequences. In particular it shows for a broad class of controllable ODEs, perturbed by a white noise process for which the Kolmogorov operator is hypoelliptic, that an $L_{\mu }^2$-spectral gap is naturally induced by the noise whereas in absence of the latter the gap may be zero, leading thus to a form of mixing enhancement by the noise. We finally mention [64] for other conditions, ensuring an $L^2_\mu$-gap based on spectral gaps in Wasserstein distances, verifiable in practice by following the approach of [61].

1.6 Proof of Theorem 5

Proof It is standard from the theory of Lyapunov functions that the existence of a unique invariant measure $\mu$ is ensured by the condition (A.17a) together with the irreducibility and strong Feller properties. The rest of the proof is thus concerned with (A.18) and the exponential convergence (A.21).

Step 1 First, note that the Itô formula gives

$$\begin{aligned} \, \text {d}U = {\mathcal {K}} U \, \text {d}t + \text { ``Martingale''}, \end{aligned}$$

(A.31)

which leads (since ${\mathcal {K}} U \le a U$) to

$$\begin{aligned} {\mathbb {E}}(U(x(t;x))) =P_t U(x)\le e^{a t}U(x), \end{aligned}$$

(A.32)

and therefore $P_t$ is extendable to a linear operator on ${\mathcal {F}}_U$ (defined in (A.19)) with norm $\Vert P_t\Vert \le e^{a t}$.

The second inequality in (A.17) ensures that for any $t>t_0$,

$$\begin{aligned} P_t U (x) \le \kappa U(x) +b, \; \forall \; x\in {\mathbb {R}}^d. \end{aligned}$$

(A.33)

By definition, a Markov semigroup is monotone, thus one may iterate (A.33) to obtain (by using $P_t \mathbb {1}_{{\mathbb {R}}^d}=\mathbb {1}_{{\mathbb {R}}^d}$),

$$\begin{aligned} P_{nt} U(x) \le \kappa ^n U(x) +\frac{b}{1-\kappa }, \; \; n\ge 1. \end{aligned}$$

(A.34)

Consider now an arbitrary compact set ${\mathfrak {B}}$ in ${\mathbb {R}}^d$ and f in ${\mathcal {F}}_{U}$, we have the bound

$$\begin{aligned} | P_{nt}f(x) -\mathbb {1}_{{\mathfrak {B}}}(x) P_{nt} \mathbb {1}_{{\mathfrak {B}}} f(x) |&\le U(x) \underset{y\in {\mathbb {R}}^d\backslash {\mathfrak {B}}}{\sup }\frac{|P_{nt} f(y)|}{U(y)},\nonumber \\&\le U(x) \Vert f\Vert _{U} \sup \frac{|P_{nt} U(y)|}{U(y)},\nonumber \\ \end{aligned}$$

(A.35)

where we have used the basic inequality (A.9) (with $p=1$). This last inequality with (A.34) leads to

$$\begin{aligned} | P_{nt}f(x) -\mathbb {1}_{{\mathfrak {B}}}(x) P_{nt} \mathbb {1}_{{\mathfrak {B}}} f(x) | \le U(x) \Vert f\Vert _{U} \left( \kappa ^n + \frac{b}{1-\kappa } \; \underset{y\in {\mathcal {H}}\backslash {\mathfrak {B}}}{\sup }\; \frac{1}{U(y)}\right) . \end{aligned}$$

(A.36)

Since $\lim _{|x|\rightarrow \infty } U(x)=\infty$, given $\epsilon >0$ and $n>1$ one may thus choose a compact set ${\mathfrak {B}}_n$ such that

$$\begin{aligned} \Vert P_{nt}f -\mathbb {1}_{{\mathfrak {B}}_n}\, P_{nt} \mathbb {1}_{{\mathfrak {B}}_n}f \Vert _U \le \Vert f\Vert _{U} (\kappa +\epsilon )^n, \end{aligned}$$

(A.37)

which leads to

$$\begin{aligned} \Vert P_{nt}-\mathbb {1}_{{\mathfrak {B}}_n} \,P_{nt}\mathbb {1}_{{\mathfrak {B}}_n} \Vert _{{\mathcal {L}}({\mathcal {F}}_U)} \le (\kappa +\epsilon )^n. \end{aligned}$$

(A.38)

Step 2 We show now that the linear operator

$$\begin{aligned} \varLambda =\mathbb {1}_{{\mathfrak {B}}} \,P_{t} \mathbb {1}_{{\mathfrak {B}}}: {\mathcal {F}}_U \longrightarrow {\mathcal {F}}_U, \end{aligned}$$

(A.39)

is compact for any compact set ${\mathfrak {B}}$ of ${\mathbb {R}}^d$. This is equivalent to showing that for any sequence $g_k$ in ${\mathcal {F}}_U$ such that $\Vert g_k\Vert _U\le 1$, one can extract a subsequence such that $\varLambda g_k$ is convergent in ${\mathcal {F}}_U$. Since $P_t$ is strongly Feller and $\mathbb {1}_{{\mathfrak {B}}} g_k$ is bounded for each k, then $P_{t} \mathbb {1}_{{\mathfrak {B}}} g_k$ belongs to ${\mathcal {C}}_b({\mathfrak {B}})$, by definition. Thus the sequence $(\varLambda g_k)$ lies in ${\mathcal {C}}({\mathfrak {B}})$.

We have

$$\begin{aligned} |\varLambda g_k (x)|\le \Vert g_k\Vert _U P_t U(x) \le \kappa U(x) + b \le \kappa \; \underset{y\in {\mathfrak {B}}}{\sup }\; U(y) +b,\;\; x\in {\mathfrak {B}}, \end{aligned}$$

(A.40)

which shows that $\{\varLambda g_k\}$ is equibounded.

Furthermore, since $P_t$ is strong Feller, it has a smooth kernel¹² and we have for all x and $x'$ in ${\mathfrak {B}}$

$$\begin{aligned} \begin{aligned} |\varLambda g_k (x)-\varLambda g_k (x')|&\le \int _{y\in {\mathfrak {B}}}|{\mathfrak {p}}_t(x,y)- {\mathfrak {p}}_t(x',y)| |f(y)|\, \text {d}y,\\&\le |x-x'| \underset{u,v\in {\mathfrak {B}}}{\sup }|\partial _u {\mathfrak {p}}_t(u,v)| \Vert g_k\Vert _U \int _{{\mathfrak {B}}} U(y) \, \text {d}y, \end{aligned} \end{aligned}$$

(A.41)

which shows that $\{\varLambda g_k\}$ is equicontinuous.

Thus, the Ascoli–Arzelà theorem [122, p. 85] applies and guarantees that a subsequence from $\varLambda g_k$ converges in ${\mathcal {C}}({\mathfrak {B}})$ to g. Now since $U\ge 1$, the same extraction from $\varLambda g_k$ converges to $g\mathbb {1}_{{\mathfrak {B}}}$ in ${\mathcal {F}}_U$. We conclude that $\mathbb {1}_{{\mathfrak {B}}} \,P_{t} \mathbb {1}_{{\mathfrak {B}}}$ is a compact mapping for any compact set ${\mathfrak {B}}$ of ${\mathbb {R}}^d$.

Step 3 Let ${\mathfrak {B}}_n$ be a sequence of compact sets satisfying (A.37), and let us consider the compact operators (from Step 2) ${\mathfrak {C}}_n$ defined by $\mathbb {1}_{{\mathfrak {B}}_n} \,P_{n t} \mathbb {1}_{{\mathfrak {B}}_n}$. We have then

$$\begin{aligned} \Vert P_{nt}\Vert _{ess}&=\inf \Big \{ \Vert P_{nt}- {\mathfrak {C}}\Vert _{{\mathcal {L}}({\mathcal {F}}_U)} \, : \, {\mathfrak {C}} \text { is a linear and compact operator of } {\mathcal {F}}_{U}\Big \} \nonumber \\&\le \Vert P_{n t}- {\mathfrak {C}}_n\Vert \le (\kappa +\epsilon )^n. \end{aligned}$$

(A.42)

By applying to $P_t$ the Hadamard formula recalled in (A.15), we have thus for $t>t_0$

$$\begin{aligned} \mathbf{r }_{ess}(P_t)=\lim _{n\rightarrow \infty } \Vert P_{nt}\Vert _{ess}^{1/n}\le \kappa +\epsilon , \end{aligned}$$

(A.43)

for all $\epsilon >0,$ and we deduce (A.18).

The exponential convergence is then ensured by showing that there is no other eigenvalue than 1 on the unit disk (or outside the unit disk) and that 1 is a simple eigenvalue; see [98].

$\square$