Central limit theorems for additive functionals of ergodic Markov diffusions processes

Introduction

The law of large numbers (1.1) yields $\lim_{t\to\infty}t^{-1}{{\left<M\right>}}_{t}=\int\!\Gamma(g)\,d\mu$. As a consequence, for a prescribed $f$, if the Poisson equation $Lg=f$ admits a mild enough solution $g$ then

This suggests to deduce (CLT) from a CLT for martingales. We will revisit this strategy. Beyond (CLT), we say that ${(S_{t})}_{t\geq 0}$ satisfies to a Functional Central Limit Theorem (FCLT) or Invariance Principle when for every finite sequence $0<t_{1}\leq\cdots\leq t_{n}<\infty$,

The literature on central limit theorems for discrete or continuous Markov processes is immense and possesses many connected components. Some instructive entry points for ergodic Markov processes are given by [DL01a, DL01b, DL03, CL09, HP04, KM03, Kut04, KM05, GM96, PV01, PV03, PV05, Lan03]. We refer to [KLO] and [HL03] for null recurrent Markov processes. Central limit theorems for additive functionals of Markov chains can be traced back to the works of Kolmogorov and Doeblin [Doe38]. The discrete time allows to decompose the sample paths into excursions. The link with stationary sequences goes back to Gordin [Gor69], see also Ibragimov and Linnik [IL65] and Nagaev [Nag57] (only stable laws can appear at the limit). The link with martingales goes back to Gordin and Lifsic [GL78]. For diffusions, the martingale method was developed by Kipnis and Varadhan [KV86], see also [Hel82] (the Poisson equation is solved via the resolvent).

Outline. Section 2 provides some notations and preliminaries including a discussion on the variance of $S_{t}$. Section 3 is devoted to FCLT at equilibrium and contains a lot of known results. We recall how to use the Poisson equation and compare with the known results on stationary sequences, which seems more powerful. In particular, we give in section 3.1 a direct new proof of the renowned FCLT of Kipnis and Varadhan [KV86, Corollary 1.9] in the reversible case. In section 4.3 we provide a non-reversible version of the Kipnis-Varadhan theorem. Actually some of the results of section 4 are written in the CLT situation, but under mild assumptions, they can be extended to a general FCLT (see Proposition 8.1). All these general results are illustrated by the examples discussed in Section 5. In sections 6 and 7 we exhibit a particularly interesting behavior, i.e. a possible anomalous rate of convergence to a Gaussian limit. This behavior is a consequence of a not too slow decay to equilibrium in the ergodic theorem. Finally we give in the next section some results concerning fluctuations out of equilibrium.

This work benefited from discussions with N. Ben Abdallah, M. Puel and S. Motsch, in the Institut de Mathématiques de Toulouse.

The framework

is a square integrable local martingale for all probability measure on $E$. Its bracket is

and the corresponding semigroup $(P_{t}^{*})_{t\geq 0}$. We shall mainly be interested by diffusion processes with generator of the form

where $(B_{t})_{t\geq 0}$ is a $d$-dimensional standard Brownian Motion, and we have also

Note that since the process admits a unique invariant probability measure $\mu$, the process is positive recurrent. We say that the invariant probability measure $\mu$ is reversible when $L=L^{*}$ (and thus $P_{t}=P_{t}^{*}$ for all $t\geq 0$).

In practice, the initial data consists in the operator $L$. We give below a criterion on $L$ ensuring the existence of a unique probability measure and thus positive recurrence.

Let $\varphi:[1,+\infty[\to\,]0,\infty[$. We say that $V\in D_{e}(L)$ (the extended domain of the generator, see [CL96, DM87]) is a $\varphi$-Lyapunov function if $V\geq 1$ and if there exist a constant $\kappa$ and a closed petite set $C$ such that for all $x$

The quantity $4V$ is the asymptotic variance of the scaled additive functional $\frac{1}{t}S_{t}$.

By using the Markov property, and the invariance of $\mu$, we can write

This implies the first property. The second property follows from the Cesàro rule and

If $\mu$ is not reversible, we do not even know whether $\int\!P^{*}_{s}fP_{s}f\,d\mu$ is non-negative or not. Nevertheless we may define $V_{-}$ and $V_{+}$ by

Poisson equation and martingale approximation

The Poisson equation (3.1) corresponds to a so called coboundary in ergodic theory. If (3.1) admits a regular enough solution $g$, then by Itô’s formula, for every $t\geq 0$ and $\varepsilon>0$,

where ${(M_{t}^{\varepsilon})}_{t\geq 0}$ is a local martingale with brackets

for all $t\geq 0$, where $v$ and $h$ are deterministic functions which may depend on $f$ via $g$, then

solve the Poisson equation $Lg=f$ in the $g$ variable

control the regularity of $g$ in order to use Itô’s formula (3.2)

check the convergence to of $g(X_{\varepsilon^{-1}t})$ as $\varepsilon\to 0$ in an appropriate way.

Let us start with a simple proposition which follows from the discussion above.

so that, if the process is strongly ergodic, $\lim_{t\to\infty}P_{t}g=\int\!g\,d\mu=0$, and thus

If $\mu$ is reversible then $f\in D(L^{-1})$ if and only if

and in this case the Poisson equation (3.1) has a unique solution $g$ given by (3.8).

Moreover, condition (3.11) implies condition (3.12).

so that $(g_{T})_{T}$ is Cauchy, hence convergent, if and only if (3.12) is satisfied. In addition, taking $T=0$ above gives

In this section we assume that $\mu$ is reversible. Corollary 3.10 states that (2.9) (equivalent to the existence of the asymptotic variance) is not sufficient to solve the Poisson equation, even in a weak sense. Nevertheless it is enough to get (FCLT), the result below is Corollary 1.9 of [KV86].

For $T>0$ introduce $g_{T}$ by (3.9), and the corresponding family $((\nabla^{n}g_{T}))_{T>0}$ (recall (2.1)). We thus have $Lg_{T}=f-P_{T}f$ and, for all $S\leq T$,

where ${(M_{t}^{T})}_{t\geq 0}$ is a martingale with brackets ${{\left<M^{T}\right>}}_{t}=\int_{0}^{t/\varepsilon}\!\Gamma(g_{T})(X_{s})\,ds$ (recall (2.2)).

According to what precedes and the framework (recall (2.1)) we may replace ${(M_{t}^{T})}_{t\geq 0}$ by another martingale ${(N_{t}^{h})}_{t\geq 0}$ with brackets ${{\left<N^{h}\right>}}_{t}=\int_{0}^{t/\varepsilon}\!|h|^{2}(X_{s})\,ds$ such that

In addition the ergodic theorem tells us that

Thus we may again apply Rebolledo’s FCLT, taking first the limit in $T$ and then in $\varepsilon$. It remains to control the others terms. But

Since $\lim_{T\to\infty}\int_{T}^{\infty}\!{{\left(\int\!P^{2}_{u}f\,d\mu\right)}}\,du=0$ according to (2.9), we have, uniformly in $\varepsilon$,

Our proof is different from the original one by Kipnis and Varadhan and is perhaps simpler. Indeed we have chosen to use the natural approximation of what should be the solution of the Poisson equation (i.e $g_{t}$), rather than the approximating $R_{\varepsilon}$ resolvent as in [KV86]. Let us mention at this point the work by Holzmann [Hol05] giving a necessary and sufficient condition for the so called “martingale approximation” property (we get some in our proof), thanks to an approximation procedure using the resolvent.

The condition (2.9) is satisfied if Assumption (1.14) in [KV86] is satisfied i.e. there exists a constant $c_{f}$ such that for all $F$ in the domain of $\mathcal{E}$,

Indeed, if we define $\varphi(t):=-\int\!f\,g_{t}\,d\mu$ where as usual $g_{t}=-\int_{0}^{t}P_{s}f\,ds$, and if we take $F=g_{t}$, then $-LF=-Lg_{t}=P_{t}f-f$, and using (3.17) we get $\varphi^{2}(t)\leq c_{f}^{2}(2\varphi(t)-\varphi(2t))$. Using that $\varphi(2t)\geq 0$ we obtain $2c_{f}^{2}\varphi(t)-\varphi^{2}(t)\geq 0$ which implies that $\varphi$ is bounded hence $\varphi(+\infty)<+\infty$. Taking the limit as $t\to\infty$ and using $2V(f)=\varphi(+\infty)$, we obtain

for an ad-hoc constant $c$. Indeed, for some constant $C>0$,

We shall come back later to the method we used in the previous proof, for more general situations including anomalous rate of convergence.

sufficient smoothness of $g$ in order to apply Itô’s formula,

control the brackets i.e. give a sense to the following quantities

For any $r\geq p\geq 1$ and $t\geq 0$ we define

The uniform decay rate is $\alpha:=\alpha_{2,\infty}$. We denote by $\alpha^{*}$ the uniform decay rate of $L^{*}$. We say that the process is uniformly ergodic if $\lim_{t\to\infty}\alpha(t)=0$.

We shall discuss later how to get some estimates on these decay rates.

In the reversible case, we recover a version of the Kipnis-Varadhan statement implying a stronger result (the existence of a solution of the Poisson equation). The results of this section are mainly interesting in the non-reversible situation.

The best choice is $m=p/(q-1)$. We then proceed as in the proof of proposition 3.19. ∎

$L$ is given by (2.4) with smooth coefficients and is hypoelliptic

$\mu$ has positive Lesbegue density $\frac{d\mu}{dx}=e^{-U}$ for some locally bounded $U$

The only thing to do is to show that $g$ (obtained in proposition 3.19) satisfies $Lg=f$ in the Schwartz space of distributions $\mathcal{D}^{\prime}$. To see the latter just write for $h\in\mathcal{D}$,

Since the infinitesimal generator of the process $s\mapsto X_{t-s}$ (for $s\leq t$) is given by $L^{*}$ we can use the previous strategy replacing $L$ by $L^{*}$ and the process $X_{.}$ by its time reversal up to time $t$. It is then known that, similarly to the standard forward decomposition (2.1), one can associate a backward one

where $\left((M^{*})_{t}-(M^{*})_{t-s}\right)_{0\leq s\leq t}$ is a backward martingale with the same brackets as $M$ (in the reversible case this is just the time reversal of $M$). The solution to the dual Poisson equation $L^{*}g=f$ thus furnishes a triangular array of local martingales to which Rebolledo’s FCLT applies. Thus, all the results we have shown with the solution of the Poisson equation are still true with the dual Poisson equation, at least in the uniformly ergodic case. The previous remark yields another possible improvement, which is a standard tool in the reversible case, namely the so called Lyons-Zheng decomposition. If $g$ is smooth enough, summing up the standard forward decomposition (2.1) and the backward decomposition (3.26), we obtain the forward-backward decomposition

so that if one can solve the Poisson equation for the symmetrized operator $L^{S}:=L+L^{*}$ the previous decomposition can be used to study the behavior of our additive functional. This is done in e.g. [Wu99], but of course what can be obtained is only a tightness result since the addition is not compatible with convergence in distribution. However, the forward-backward decomposition will be useful in the sequel.

Comparison with general results on stationary sequences

At the process level we may similarly consider the random variables $S_{[nt]}$ where $[\cdot]$ denotes the integer part again, and for $n\leq(1/\varepsilon)<(n+1)$. The remainder $S_{t/\varepsilon}-S_{[nt]}$ multiplied by a quantity going to 0 will converge to 0 in probability, so that for any $k$-uple of times $t_{1},\ldots,t_{k}$ we will obtain the convergence (in distribution) of the corresponding $k$-uple, provided the usual FCLT holds for $S_{[nt]}$.

This has been improved for chains [CL09]. For (FCLT) we recall [MPU06, Cor. 12]:

Following [CG08] (Section 3, Proposition 3.4), let $\mathcal{F}_{s}$ (resp. $\mathcal{G}_{s}$) be the $\sigma$-field generated by $(X_{u})_{u\leq s}$ (resp. $(X_{u})_{u\geq s}$ ). The strong mixing coefficient $\alpha_{mix}(r)$ is

where the sup runs over $s$ and $F$ (resp. $G$) $\mathcal{F}_{s}$ (resp. $\mathcal{G}_{s+r}$) measurable, non-negative and bounded by 1. If $\lim_{r\to\infty}\alpha_{mix}(r)=0$ then we say that the process is strongly mixing.

Let $\alpha$ be as in definition 3.18. The following correspondence holds :

Hence the process is strongly mixing if and only if it is uniformly ergodic (or equivalently if and only if its dual is uniformly ergodic).

For the first inequality, it suffices to take $F=P_{r}f(X_{0})$ and $G=f(X_{r})$ (respectively $F=f(X_{0})$ and $G=P_{r}^{*}f(X_{r})$) for $f$ $\mu$-centered and bounded by $1$. For the second inequality, let $F$ and $G$ be centered and bounded by $1$, respectively $\mathcal{F}_{s}$ and $\mathcal{G}_{s+r}$ measurable. We may apply the Markov property to get

The preceding proposition implies the following comparison:

In particular if we know that $\alpha^{*}$ is “slowly” decreasing (i.e. there exists $c>0$ such that $\alpha^{*}(t)\leq c\,\alpha^{*}(2t)$), then $\alpha(t)\geq(1/c)\,\alpha^{*}(2t)\geq(1/c^{2})\,\alpha^{*}(t)$. If both $\alpha$ and $\alpha^{*}$ are slowly decreasing, then they are of the same order. More generally, for $t\geq 2$ (for instance)

so that $\alpha(t)\leq c_{1}\,(\alpha^{*}(t))^{1/2}$. Plugging this new bound in the previous inequality we obtain

i.e. $\alpha(t)\leq c_{2}\,(\alpha^{*}(t))^{3/4}$. By induction, for all $\varepsilon>0$ there exists a constant $c_{\varepsilon}$ such that

We shall compare all these results with the one obtained in the previous section later, in particular by giving some explicit comparison results between $\alpha$ and $\alpha_{p,q}$ introduced in definition 3.18. But we shall below give some others nice consequences of mixing.

2. Self normalization with the variance and uniform integrability

The following characterization of the CLT goes back at least to [Den86]. The FCLT seems to be less understood [MPU06, MP06].

If the (possibly infinite) limit exists we denote $\lim_{t\to+\infty}h(t)=2V\leq+\infty$.

We shall say that (Hpos) is satisfied if $\beta(s)\geq 0$ for all $s$ large enough.

Assumption (Hpos) is satisfied is the reversible case, in the non reversible case we only know that $\int_{0}^{t}\,\eta(s)\,ds>0$. Notice that if (Hpos) is satisfied

for $t$ large enough similarly to the reversible case, so that

Denker’s theorem 4.8 allows us to obtain new results, at least CLTs, using the natural symmetrization of the generator and the forward-backward martingale decomposition.

To this end recall first that uniform integrability of a family $F_{t}$ is equivalent (La Vallée-Poussin theorem) to the existence of a non-decreasing convex function $\gamma$ such that $\lim_{u\to+\infty}\,\gamma(u)/u=+\infty$ and

Recall now the following strong version of Burkholder-Davis-Gundy inequalities (see [DM80], chap. VII, Theorem 92 p.304)

In addition Doob’s inequality tells us that the Orlicz norms of $N_{t}^{*}$ and $N_{t}$ are equivalent (with constants independent of $t$).

3. A non-reversible version of Kipnis-Varadhan result

We thus have an approximate forward-backward decomposition

We first look at the corresponding forward martingale $M_{t}^{T}$ whose bracket is given by

We then have for a convex function $\gamma$,

But we know that $P_{T}^{S}f$ goes to 0 in $L^{2}(\mu)$. So there exists some $\gamma$ such that $\gamma((P_{T}^{S}f)^{2})$ is uniformly integrable. Up to a subsequence (we already work with subsequences) we may assume that the convergence holds true $\mu$ almost surely, applying Vitali’s convergence theorem we thus have (we may choose $\gamma(0)=0$) that

We thus may apply Cesàro’s theorem, which furnishes some non-decreasing function $T(t)$ such that $\sup_{t}A(T(t),t)<+\infty$.

We may now conclude as for the proof of Proposition 4.16, obtaining the following reinforcement which is some non-reversible version of Kipnis-Varadhan theorem (at the CLT level), since we already proved that

Complements and examples

In this section we shall first discuss in a quite “general” framework how to compare all the results described in the preceding two sections. This will be done by studying the asymptotic behavior of $P_{t}$. Next we shall describe explicit examples

The uniform decay rate furnishes a first $p,r$-decay rate as follows

The proof is adapted from [CG09]. Pick some $K>1$ and define $g_{K}=g\wedge K\vee-K$. Since $\int gd\mu=0$, defining $m_{K}=\int g_{K}\,d\mu$ it holds

the latter being a consequence of ${{\left\|g\right\|}}_{r}=1$ and $K>1$. It follows

It remains to optimize in $K$. Actually up to a factor 2 we know that the optimum is attained for $\alpha(t)\,K=2\,K^{-(r-p)/p}$ i.e. for $K=(2/\alpha(t))^{p/r}$ (which is larger than one), hence the first result.

The second one is immediate since for $p\geq 2$, $\alpha_{p,\infty}(t)\leq\alpha^{\frac{2}{p}}(t)$, and we may follow the same proof without introducing the variance. ∎

Note that up to a factor 2 due to the proof, the result is coherent for $r=+\infty$.

We can complete the result by the following well known consequence of the semigroup property

For $r=p\geq 1$, either $\alpha_{p,p}(t)=1$ for all $t\geq 0$, or there exist positive constants $c_{p}$ and $C_{p}$ such that $\alpha_{p,p}(t)\leq C(p)\,e^{-c_{p}t}$.

When the second statement is in force we shall (abusively in the non-reversible case) say that $L$ has a spectral gap. We shall discuss in the next section conditions for the existence of a spectral gap or for the obtention of the optimal uniform decay rate.

Specialists in interpolation theory certainly will use Riesz-Thorin theorem in order to evaluate $\alpha_{p,r}$. Let us see what happens.

In section 3.2 we used $\alpha_{2,p}$ for $p>2$. It seems that in full generality the relation $\alpha_{2,p}(t)=c_{p}\,\alpha^{\frac{p-2}{p}}(t)$ is the best possible. However it is interesting to notice the following duality result

For all pair $1\leq p<r\leq+\infty$ there exists $c(p,r)$ such that

For $1<p\leq 2$, $\alpha_{1,p}(t)\leq c(p)\,\left(\alpha^{*}(t)\right)^{\frac{2(p-1)}{p}}$.

This result is of course much better (up to a square) than the one obtained in lemma 5.1 in this situation, since we know that for slowly decreasing $\alpha$ and $\alpha^{*}$ these functions are equivalent (up to some constants). It can also be compared with similar results obtained in [CG09].

These results allow us to compare conditions obtained in Proposition 3.19, Proposition 3.21 on one hand, and Theorem 4.3 or Proposition 4.7 on the other hand.

For example, if we use the bound obtained in lemma 5.1, proposition 3.21 tells that convergence to a brownian motion holds provided

(Remark that it is exactly the condition in [Jon04] Theorem 5). Notice that as soon as $\alpha(t)\,\alpha^{*}(t)\,<\,1/t$ this bound is worse than the one in proposition 4.7, so that the mixing approach seems to be at least as interesting as the usual one.

However, in the diffusion case we shall obtain in proposition 5.10 below a better bound for $\alpha^{*}_{2,p}$. Combined with remark 4.6, it yields (under the appropriate hypotheses) the condition

for some $\varepsilon\geq 0$ ( is allowed in the slowly decreasing case), which is better than the mixing condition in proposition 4.5 as long as $\alpha^{*}(t)>(1/t)^{(\frac{p-1}{p-2})-\eta}$ for some $\eta\geq 0$. $\diamondsuit$

2. Rate of convergence for diffusions

In “non degenerate” situations, $\alpha$ is given by weak Poincaré inequalities:

$\mu$ satisfies a weak Poincaré inequality (WPI) for $\Gamma$ with rate $\beta$ if for all $s>0$ and all $f$ in the domain of $\Gamma$ (or some core) the following holds,

([RW01] Theorem 2.1 and Theorem 2.3) If $\mu$ satisfies (WPI) with rate $\beta$ then both $\alpha(t)$ and $\alpha^{*}(t)$ are less than $2\,\xi^{\frac{1}{2}}(t)$ where $\xi(t)=\inf\{s>0,\beta(s)\,\log(1/s)\leq t\}$.

If $L$ is $\mu$-reversible (or more generally normal) some converse holds, i.e. decay with uniform decay rate $\alpha$ implies some corresponding (WPI).

It is actually quite hard to check, in the reversible case, whether starting with some (WPI) one obtains a $\xi$ which in return furnishes the same (WPI) (see the quite intricate expression of $\beta$ in [RW01] Theorem 2.3). It seems that in general one can loose some slowly varying term (like a $\log$ for instance).

Notice that (WPI) implies the following: $\mathcal{E}(f,f)=0\,\Rightarrow\,f\textrm{ constant}$ i.e. the Dirichlet form is non degenerate. In the degenerate case of course, the uniform decay rate cannot be controlled via a functional inequality. The most studied situation being the diffusion case we now focus on it.

First we recall the following explicit control proved in [BCG08] Theorem 2.1 (using the main result of [DFG09])

Then, if $\lim_{u\to+\infty}\varphi^{\prime}(u)=0$,

If $\varphi$ is linear, $\alpha^{*}(t)$ and $\alpha(t)$ are decaying like $e^{-\lambda t}$ for some $\lambda>0$ (see [DMT95, BCG08, BBCG08]).

Note that the latter bound is better than the general one obtained in lemma 5.1. Of course we may use either remark 3.25 (telling that we may use $\alpha^{*}$ instead of $\alpha$) or Remark 4.6 (comparing both rates) to apply this result.

In the same spirit we shall also recall a beautiful result due to Glynn and Meyn [GM96] or more precisely the version obtained in Gao-Guillin-Wu [GGW10]:

We introduce the Lyapunov control condition, as in [GM96, GGW10]

there exist a positive function $F$, a compact set $C$, a constant $b$ and a (smooth) function $\theta$, going to infinity at infinity such that

Then we have the following (Theorem 3.2 in [GM96] and its refined version Lemma 6.2 in [GGW10])

The authors get the FCLT in Theorem 4.3 of [GM96], but we know how to do in this situation.

We shall continue this section by providing several families of examples, starting with the one-dimensional case. These examples are then extended to $n$-dimensional reversible Langevin stochastic differential equations using Lyapunov conditions and results of [BCG08, BBCG08, CGGR10] to recover Poincaré inequalities or weak Poincaré inequalities through the use of Lyapunov conditions, and so the rate $\alpha^{*}$ or $\alpha$. We will then consider elliptic (non necessarilly reversible) examples for which result of [DFG09], recalled in Proposition 5.10, furnishes the rate $\alpha^{*}$ and then existence of the solution of Poisson equation and CLT where the usual Kipnis-Varadhan condition cannot be used. Comparisons with the recent results of Pardoux-Veretennikov [PV01] will be made. We will end with some hypoelliptic cases such as the kinetic Fokker-Planck equation or oscillator chains for which results of [DFG09, BCG08] still apply, and results of [PV05] are harder to consider. It is of particular interest in PDE theory. One of the main strategy to get explicit convergence controls are Lyapunov conditions as explained before.

3. Reversible case in dimension one

We recall here results of [BCR05] giving necessary and sufficient conditions for a one dimensional measure $d\mu(x)=e^{-V(x)}dx$, associated to the one dimensional diffusion

then $1/4\max(b_{-},b_{+})\leq C\leq 12\max(B_{+},B_{-})$ where, with $m$ a median for $\mu$

and the corresponding ones for $b_{-},B_{-}$ with the left hand side of the median.

3.2. A first particular family : general Cauchy laws

Consider the diffusion process on the line

for some parameters $\alpha>1$ and $\beta\geq 0$. The model is slightly more general than the usual Cauchy laws considering $\beta=0$, but the difference allows interesting behaviors. The corresponding generator is

It is immediate that $V(x)=x^{2}$ satisfies

hence verifies the assumption in proposition 2.7. So the process defined by (5.14) does not explode (is conservative if one prefers), and is ergodic with unique invariant measure $\mu$, which satisfies a local Poincaré inequality on any interval.

The rate $\alpha_{2,\infty}$ is known in this situation. Indeed, according to Proposition 5.13, $\mu$ satisfies a weak Poincaré inequality (recall definition 5.8) with optimal rate

According to Proposition 5.9 (and its converse in the reversible case), for large $t$,

In the sequel we shall only consider bounded functions $f$.

If $\alpha>3$ or $\alpha=3$ and $\beta>2$, $\alpha_{2,\infty}^{2}$ is integrable, and so we may apply Kipnis-Varadhan theorem to all bounded functions $f$.

Interesting cases are $\alpha=3$ and $\beta\leq 2$.

If $\beta>1$, $\theta(x)=|x|$ for large $|x|$’s satisfies the assumptions in Theorem 5.12, and accordingly the usual FCLT holds provided $|f(x)|\leq c/|x|$ at infinity. If $\beta\leq 1$ a similar result holds but this time for $|f(x)|\leq c/|x|^{1+\varepsilon}$ at infinity, for any $\varepsilon>0$.

But it should be interesting to know what happens for bounded $f$’s that do not go to 0 at infinity.

3.3. A second general family: subexponential laws

It is well known the process does not explode and ergodic with unique invariant measure $\mu$. By Proposition 5.13, one easily gets that $\nu_{\alpha}$ satisfies a weak Poincaré inequality with $\beta(s)=k_{\alpha}\log(2/s)^{\frac{2}{\alpha}-2}$. According to Proposition 5.9 (and its converse in the reversible case), for large $t$,

4. Reversible case in general

reversible with respect to $\mu$. In fact one may use as in the one dimensional case Lyapunov functions $W(x)=|x|^{k}$ for large $|x|$ so that for large $|x|$

so that to get a Lyapunov condition we have to impose the compatibility condition $\alpha>n+k-2$. Use now Theorems 2.8 and 5.1 in [CGGR10] to get a weak Poincaré inequality with $\beta(s)=c(n,\alpha)s^{-\frac{2}{\alpha-n}}$ leading to

4.2. Subexponential measures

associated to the $\nu_{\alpha}$-reversible generator

With $W(x)=e^{a|x|^{\alpha}}$ for large $|x|$, one easily gets that for large $|x|$

so that by Theorems 2.8 and 5.1 in [CGGR10], we get that $\nu_{\alpha}$ verifies a weak Poincaré inequality with $\beta(s)=k_{n,\alpha}\log(2/s)^{\frac{2}{\alpha}-2}$. We may then mimic the results given in the one dimensional case.

5. Beyond reversible diffusions

and $a=\sigma\sigma^{*}/2.$ We will suppose that $\sigma$ is bounded and $b,\sigma$ locally (bounded) Lipschitz functions. We assume moreover a condition on the diffusion matrix

Note that Pardoux and Veretennikov also impose an ellipticity condition in [PV01], or a local Doeblin condition in [PV05] preventing however too degenerate models like kinetic Fokker-Planck ones. We also introduce the following family of recurrence conditions

We suppose $M>0$, $\alpha\geq-1$, and when $\alpha=-1$, that the process does not explode (it will be a consequence of the Lyapunov conditions given later). We also define when $\alpha=-1$, $r_{0}=(r-\Lambda n)/2)/\lambda_{+}$. We may then use the results of [DMT95, DFG09] and [PV01] to get that

for some (usually non explicit) constants $C,c>0$. Note that these results are obtained using Lyapunov functions $W_{1}(x)=e^{a|x|}$, $W_{2}(x)=e^{a|x|^{1+\alpha}}$ and $W_{3}(x)=1+|x|^{2k+2}$ respectively, for some $a<\frac{2r}{\lambda_{+}(1+\alpha)}$ whenever $\alpha>-1$). Namely outside a large ball, for some positive $\lambda$

Note however that we have no ellipticity assumption, and we refer to examples in the next paragraph, which cannot be obtained using the results of Pardoux-Veretennikov. Remark finally that our results do not only apply to the existence of the solution of the Poisson equation but also to the FCLT, with a finite variance, which is not at all ensured by Pardoux-Veretennikov’s results. In this perspective, if we want to use Pardoux-Veretennikov result to get a finite variance, we will have to impose that there exists $p\geq 1$ such that $\max(p\beta,\frac{p}{p-1}(\beta+2))<2r_{0}-1$, which will imply that for $p\geq 2$ one has to impose $(r_{0}-1/2)(p-2)>p$ which is slightly stronger than our conditions.

Note also that no additional ellipticity condition is supposed, and even in the subsequent work [PV05], the local Doeblin condition and condition $(A_{T})$ (see [PV05, Page 1113] seems to be verified in only slightly degenerate case. We will then give here particular examples that may be reached through our work.

6. Kinetic models

If the initial law of $(x_{0},v_{0})$ is $\nu$ we denote by $P(t,\nu,dx,dv)$ the law at time $t$ of the process. A standard scaling (see e.g. [DM08]) is to consider

i.e. the law of the scaled process $(\varepsilon\,x_{t/\varepsilon^{2}}\,,\,v_{t/\varepsilon^{2}})$ (also rescale the initial law), solution of

The FCLT with $v(\varepsilon)=\sqrt{\varepsilon}$, if it holds, combined with a standard argument of propagation of chaos (see [CCM10] for more details) implies that as $\varepsilon$ goes to , $P^{\varepsilon}(t,dx,dv)$ converges to the product $N(t,dx)\,M(dv)$ where $M(dv)$ is the projection of the invariant measure of the diffusion on the velocities space and $N(t,dx)$ is the solution of the appropriate (depending on the asymptotic variance) heat equation on the positions space.

Let us present more concrete examples where we can use the results of the paper just using $f(v)=v$ or $f(x,v)=v$, as well as the possible necessity of using another scaling in space (anomalous rate of convergence), via explicit speed of convergence obtained as previously via Lyapunov conditions.

Kinetic Fokker-Planck equation. Let us consider the following stochastic differential system

If $F(x)$ behaves like $|x|^{p}$ for large $|x|$ with $0<p<1$ then one can build a Lyapunov function $W(x,v)$ behaving at infinity as $e^{a(|v|^{2}+|x|^{p})}$ (for $s$ sufficiently small) and such that outside a large ball (see [DFG09, BCG08])

We may thus apply the results explained in the previous case $-1<\alpha<0$.

Oscillator chains. We present here the model studied by Hairer-Mattingly [HM09]: 3-oscillator chains

where $B^{0}$ and $B^{2}$ are two independent brownian motions. Then by Theorem 5.6 in [HM09], if $k>3/2$, one can give a Lyapunov function $W$ for which $LW\leq-\lambda W^{r}+C$ for some $r<1$ so that we may use the results presented before in the polynomial rate case.

An example of anomalous rate of convergence

In all the examples developed before, the asymptotic variance was existing. We shall try now to investigate the possible anomalous rates of convergence, i.e. cases where the variance of $S_{t}$ is super-linear. Instead of studying the full generality, we shall first focus on a simple example, namely the one discussed in section 5.3.2.

We consider the generator $L$ defined in (5.15) in the critical situation $\alpha=3$ and $\beta\leq 2$ or the supercritical one i.e $\alpha<3$ (but $\alpha>1$). For simplicity we shall here directly introduce the function $g$ and choose $g(x)=x^{2}$, so that $f=Lg$ is bounded but does not go to at infinity (hence we cannot use Theorem 5.12).

According to Remark 3.24 we may thus apply Kipnis-Varadhan result, so that from now on these cases are excluded. Remark that for this particular case, Kipnis-Varadhan result applies for $\beta>1$, while for the general bounded case (i.e. $f$ bounded) we have to assume that $\beta>2$. This is presumably due to the non exact correspondence between (WPI) and the decay rate $\xi$ as noticed just after Proposition 5.9.

In the sequel, $c$ will denote a universal constant that may change from place to place.

For $K>0$ we introduce a truncation function $\psi_{K}$ such that, $\mathbf{1}_{[-K,K]}\leq\psi^{\prime}_{K}\leq\mathbf{1}_{[-K-1,K+1]}$ and all $\psi^{\prime\prime}_{K}$ are bounded by $c$ ($\psi_{K}$ is thus an approximation of $x\wedge K\vee-K$).

We then define $g_{K}=\psi_{K}(g)$, $f_{K}=Lg_{K}$ which is still bounded by $c$ and such that

In what follows, we shall use repeatedly the fact that, for large $K$

These estimates follow easily by integrating by parts (integrate $x^{a}$ and differentiate the $\log$).

Now we can write (we are using the notation in section 4.2, in particular (4.10) and (4.9)):

Indeed for $K>K_{0}$ where $K_{0}$ is large enough,

with $\varphi(K)=K^{3-\alpha}\,\log^{-\beta}(K)$ if $\alpha\neq 3$, $\varphi(K)=\log^{1-\beta}(K)$ if $\alpha=3$ and $\beta\neq 1$, and finally $\varphi(K)=\log\log(K)$ if $\alpha=3$ and $\beta=1$ . Note that similarly

Hence, according to (6.5) and (6.6) as well as (6.4) and (6.7) we need for $(\alpha,\beta)\neq(3,1)$

We immediately see that the unique favorable situation is obtained for

So we now consider the cases $\alpha=3$ and $\beta\leq 1$.

Notice that it corresponds to the rate of convergence described in the next section 7.

is uniformly integrable, according to Proposition 4.14. Due to the form of $g_{K}$ it is thus enough to show that

To this end, denote by $u(x,M)=|x|^{2}\,\mathbf{1}_{|x|\leq 1+M}$ for $M\geq 1$, and $\bar{u}(x,M)=u(x,M)-\int u(.,M)\,d\mu$, and $U(t,X,M)=\int_{0}^{t}\,u(X_{s},M)\,ds$.

We know that if $\beta\leq 1$, and $t>1$ for instance,

Recall that $\alpha^{2}(s)=\alpha^{2}_{2,\infty}(s)$ is the mixing coefficient whose expression is recalled in section 5.3.2, i.e. $\alpha^{2}(s)\simeq\log^{1-\beta}(s)\,s^{-1}$.

A direct calculation thus yields (for $t\geq 1$)

Hence if we choose $M(t)=t^{a}$ with $a<1/4$,

i.e. is asymptotically equivalent to the mean of $U(t,X,K(t))$, so that

It follows that $U(t,X,t^{a})/t\,\log^{1-\beta}(t)$ or $U(t,X,t^{a})/t\,\log\log(t)$ when $\beta=1$, is uniformly integrable.

so that it is uniformly integrable. According to what precedes, it immediately follows that $H(t,X,K(t))=U(t,X,K(t))/t\,\log^{1-\beta}(t)$ (with the ad hoc normalization if $\beta=1$) is also uniformly integrable.

It remains to prove our claim. For simplicity we choose $K(t)=t^{1/2}$ (any allowed $K(t)$ furnishes the result but calculations are easier). Since $U(t,X,K(t))-U(t,X,t^{a})\geq 0$ it is enough to calculate for large $t$

If $\beta\neq 1$, the right hand side is equal to

Our claim immediately follows in both cases.

Let us collect the results we have obtained:

be a probability measure on the line and $L_{\beta}={\partial}_{x^{2}}^{2}+\nabla(\log p_{\beta})\,{\partial}_{x}$ the associated diffusion generator for which $\mu_{\beta}$ is reversible and ergodic. $X^{\beta}_{.}$ denotes the associated diffusion process.

For $g(x)=x^{2}$, $f_{\beta}=L_{\beta}g$ is a bounded function with $\mu$-mean equal to 0. We consider the associated additive functional $S_{t}^{f_{\beta}}=\int_{0}^{t}\,f_{\beta}(X_{s}^{\beta})\,ds$.

If $\beta>1$ we may apply Kipnis-Varadhan result (Theorem 3.13).

The previous theorem is really satisfactory and in a sense generic. We shall try in the next sections to exhibit general properties yielding to an anomalous rate of convergence.

Anomalous rate of convergence. Some hints

The standard strategy we used for the CLT is to reduce the problem to the use of the ergodic theorem for the brackets of a well chosen martingale. This requires to approximate the solution of the Poisson equation, i.e. to obtain a decomposition of $S_{t}$ into some martingale terms, whose brackets may be controlled, and remaining but negligible “boundary” terms. In this section we shall address the problem of using this strategy for super-linear variance. Hence we have to choose a correct approximation of the solution of the Poisson equation, and to replace the ergodic theorem for the martingale brackets, by some uniform integrability property. Again we are using the notation (4.9) and (4.10).

As before, for $T>0$ depending on $t$ to be chosen later, introduce again $g_{T}=-\int_{0}^{T}\,P_{s}f\,ds$. We thus have $Lg_{T}=f-P_{T}f$ and using Itô’s formula

We already saw that in the reversible case

A similar estimate holds in the non-reversible case provided (Hpos) holds. This time we see that the good situation is the one where $t\ll T$.

3. The martingale brackets

It remains to calculate the expectation of the martingale brackets $\langle M^{T}\rangle_{t}$.

Hence we certainly need $\left(2\,\eta(T/2)-\eta(T)\right)/\eta(t/4)$ to be bounded. As for the first term this requires at least that $t$ is of the same order as $T$.

4. The good rates

According to what precedes, we have to consider the case when $T$ and $t$ are comparable. For simplicity we shall choose $T=t/2$, so that the final condition in section 7.3 will be automatically satisfied. The final condition in section 7.2 becomes

while the discussion in section 7.1 yields to

It is thus interesting to get a family of $\beta^{\prime}s$ satisfying (7.7) and (7.6). Actually since $\beta$ is non increasing,

Functions satisfying this property are known, according to Karamata’s theory (see [BGT87] chapter 1). Recall the definition

A non-negative function $l$ is slowly varying if for all $u>0$,

Using the direct half of Karamata’s theorem (see [BGT87] Proposition 1.5.8 and equation (1.5.8)) for (7.8) to hold it is enough that

Indeed if (7.10) holds, $\int_{0}^{t}\,s\,\beta(s)\,ds\sim\,t\,l(t)$ so that (7.8) is equivalent to

which is exactly [BGT87] Proposition 1.5.9a.

The converse half of Karamata’s theorem ([BGT87] Theorem 1.6.1) indicates that this condition is not far to be necessary too.

Of course if we replace (7.7) by (7.5) we do not need the full strength of (7.10) since (7.6) is satisfied as soon as $\eta$ is slowly varying.

If the process is reversible and strongly mixing and if $\eta$ given in (4.9) is slowly varying (in particular if (7.10) is satisfied), then there is an equivalence between

$\displaystyle{\frac{S_{t}}{2\sqrt{t\eta(t)}}}$ converges in distribution to a standard Gaussian law as $t\to+\infty$,

$\displaystyle{{{\left(\frac{1}{t\eta(t)}\int_{0}^{t}\!\Gamma(g_{t/2})(X_{s})\,ds\right)}}_{t\geq 1}}$ is uniformly integrable, where $g_{t/2}:=-\int_{0}^{t/2}\!P_{s}f\,ds$.

We shall say (as Denker himself said when writing his theorem) that the previous proposition is not really tractable. Indeed in general we do not know any explicit expression for the semigroup (hence for $g_{t}$). The main interest of the previous discussion is perhaps contained in the feeling that anomalous rate shall only occur when (7.10) is satisfied.

In the next section we shall even go further in explaining:

6. Why is it delicate?

The previous theorem reduces the problem to show that

The first idea is to use the convexity of $\gamma$, yielding

so that our problem reduces to show that $\Gamma(g_{t})/h(2t)$ is $\mu$ uniformly integrable, or, since we assume that $\eta$ is slowly varying, that $\Gamma(g_{t})/\eta(t)$ is $\mu$ uniformly integrable.

It is so delicate that we shall see a natural generic obstruction. In what follows we assume that $\eta(t)\to+\infty$ as $t\to+\infty$.

For simplicity we consider the one dimensional situation with

Of course we may replace $f$ by $P_{\varepsilon}f$ for any $\varepsilon\geq 0$ up to an error term going to 0. Thanks to (hypo-)ellipticity we know that $P_{\varepsilon}f$ is $C^{\infty}$, hence we may and will assume that $f$ is $C^{\infty}$, so that $g_{t}$ is $C^{\infty}$ too.

and we assumed that $\eta$ goes to infinity, while for the second term we know that $p\,|{\partial}_{x}g_{t_{n}}|^{2}/\eta(t_{n})$ converges to $\nu$ and that ${\partial}_{x}p$ is smooth.

Hence, contrary to all the cases we have discussed before, anomalous rate of convergence cannot be uniquely described by the behavior of the semigroup. We need to use pathwise properties of the process. (This sentence may look strange since the semigroup uniquely determines the process, but the important word here is “path”.)

In the situation of lemma 3.23 the good strategy is to use some cut-off of $g$ as we did in the previous section, which in a sense is generic for this situation.

Fluctuations out of equilibrium

In this section we shall mainly discuss the CLT and FCLT out of equilibrium. But before, we shall show that in the strong mixing case (i.e. uniformly ergodic situation), the (CLT) ensures the (FCLT).

As usual we denote by $\bar{H}$ the centered $H-\mu(H)$.

The main difficulty here is that $t_{1}=s_{2}=s$. We introduce an auxiliary time

according to the CLT. For the two remaining terms we have

hence goes to as $\varepsilon\to 0$. Similarly

hence goes to as $\varepsilon\to 0$ exactly as $A_{2,\varepsilon}$. The proof is completed. ∎

[DMT95] Thm 5.2.c, and [DFG09] Thm 3.10 and Thm 3.12.

Under the assumptions of Proposition 5.10, there exists a positive constant $c$ such that for all $x$,

where ${{\left\|\cdot\right\|}}_{TV}$ is the total variation distance and $\psi$ (which goes to at infinity) is defined as follows: $\psi(t)=1/(\varphi\circ H^{-1}_{\varphi})(t)$ for $H_{\varphi}(t)=\int_{1}^{t}\,(1/\varphi(s))ds$, if $\lim_{u\to+\infty}\varphi^{\prime}(u)=0$ and $\psi(t)=e^{-\lambda t}$ for a well chosen $\lambda>0$ if $\varphi$ is linear.

The second result is mentioned (in the case of a brownian motion with a drift) in [CGG07] and proved for a stopped diffusion in dimension one in [CCL+09] Theorem 2.3. The proof given there extends immediately to the uniformly elliptic case below thanks to the standard Gaussian estimates for the density at time $t$ of such a diffusion, details are left to the reader

In the diffusion situation (2.4), assume that the diffusion matrix $a$ is uniformly elliptic and bounded. Assume in addition that the invariant measure $\mu(dx)=e^{-W(x)}\,dx$ is reversible, and that $2\Gamma(W,W)(x)-LW(x)\geq-c>-\infty$.

2. Fluctuations out of equilibrium

Let $\nu$ be a given initial distribution. A direct application of the Markov property shows that

Let $u(\varepsilon)>\varepsilon$ going to as $\varepsilon$ goes to . For any bounded $H_{1},...,H_{k}$, denote $H(Z_{.})=\otimes H_{i}(Z_{t_{i}})$. Then

$\nu$ is absolutely continuous w.r.t. $\mu$

$\nu=\delta_{x}$ for $\mu$ almost all $x$,

Choose $u(\varepsilon)$ such that $u(\varepsilon)\to 0$ as $\varepsilon\to 0$, but with $u(\varepsilon)\gg v(\varepsilon)$. We may apply the previous lemma and to conclude it is enough to show that

If $L$ given by (2.4) is elliptic or more generally hypoelliptic, the previous theorem applies to all initial $\nu$ satisfying the assumptions of Theorem 8.4 or Theorem 8.3. In particular it applies to $\nu=\delta_{x}$ for all $x$.

References