Negative regularity mixing for random volume preserving diffeomorphisms
Abstract
We consider the negative regularity mixing properties of random volume preserving diffeomorphisms on a compact manifold without boundary. We give general criteria so that the associated random transfer operator mixes observables exponentially fast in (with a deterministic rate), a property that is false in the deterministic setting. The criteria apply to a wide variety of random diffeomorphisms, such as discrete-time iid random diffeomorphisms, the solution maps of suitable classes of stochastic differential equations, and to the case of advection-diffusion by solutions of the stochastic incompressible Navier-Stokes equations on . In the latter case, we show that the zero diffusivity passive scalar with a stochastic source possesses a unique stationary measure describing “ideal” scalar turbulence. The proof is based on techniques inspired by the use of pseudodifferential operators and anisotropic Sobolev spaces in the deterministic setting.
1 Introduction
In this paper we study negative regularity mixing by volume-preserving random diffeomorphisms on a smooth compact Riemannian manifold without boundary. Here, the index set is either or . One of our main motivations comes from fluid mechanics, namely the advection equation
| (1.1) |
where is a scalar field and is a Lipschitz regular, divergence-free, (time-dependent) velocity field with . Naturally, the solution is given by where is the flow map of associated to the velocity field . The solution operator
is often called the transfer operator in the dynamical systems literature.
In the context of Anosov maps, the study of mixing properties of the map is closely related to the spectrum of the transfer operator, known as Ruelle Resonances. Introduced by Ruelle within the framework of thermodynamic formalism [43, 41, 42], these resonances offer refined insights into the decay of correlations. Subsequent work has significantly advanced our understanding of these resonances through the use of anisotropic Sobolev spaces [11, 28, 35, 1, 3, 16, 2] and microlocal analysis [20, 24, 21, 25, 26, 22, 23]. See section 1.2 for more context on some of this work in the context of our main result.
In the case when the velocity field is random (see [7, 9, 14, 17]), it is possible to prove a quenched mixing result (or equivalently almost-sure exponential decay of correlations), namely there exists a deterministic and a random constant satisfying , such that for all mean zero there holds
| (1.2) |
In the case of (1.1) or similar examples, the random constant will also depend on the initial velocity field (but not the initial data ).
Due to the time-reversibility of transport equations, it is relatively easy to conclude that one cannot obtain an almost-sure exponential decay result like (1.2) decay without assuming having some small amount of regularity say . In particular one cannot expect decay in if is also taken in .
The goal of this work is to prove that for many random maps, negative regularity mixing does hold in an averaged sense, namely that for all mean-zero , there exists a such that
for some fixed (independent of ), and also that it holds in a quenched sense but with a constant that depends on (Corollary 1.13).
After we state the main results, we include a more extended discussion of both the results and the motivations, however, one can liken the result to something like the non-random multiplicative ergodic theorem [32]. Indeed, for each individual scalar field (and initial velocity field if applicable) there may exist very specific realizations of which obtain arbitrarily bad decay rates in , but these be so rare that if one averages over the ensemble of velocity fields, they are essentially negligible. The motivation and proof draws significant inspiration from the work on anisotropic Sobolev spaces, especially [20].
1.1 Abstract framework and statement of negative regularity mixing
Here we outline a general abstract setting for our theorem that applies broadly to a whole class of random maps, including iid random diffeomorphisms, stochastic transport on compact manifolds (i.e. particle trajectories solve SDEs) [27, 15], advection by velocity fields generated by the stochastic Navier-Stokes equations, alternating random shear flows [12], and other similar settings (see Section 2). Our abstract framework is similar to that of [12], but more general in order to treat the cases studied in [8, 9, 7, 10]. See Section 2 for explanation on how to connect our abstract result to some of the concrete examples.
Let be a -dimensional, smooth, compact Riemannian manifold without boundary and be a probability space, be a preserving transformation and be a Feller Markov chain on a Polish space with a unique stationary (probability) measure and an associated continuous random dynamical system . We denote the corresponding skew product flow on by with invariant ergodic product measure .
We will consider a -regular (with ), random volume preserving diffeomorphism and denote
the -fold composition along the skew product flow . We emphasize that we do not assume that each map in this composition is iid, so the associated process , is not assumed Markovian in general. However, the joint process on is Markovian.
Remark 1.1.
To include the case of iid random diffeomorphisms in our framework, one simply dispenses with and the relevant Markov process becomes where
This includes the time-1 maps of well-posed, volume-preserving SDEs; see [34].
1.1.1 Lyapunov structure
First, we make some assumptions which provide control on how large the derivatives of can be. In most settings, is the velocity field, and the deviations on are stated in terms of the deviations of . Recall that a function is called a Lyapunov function for if has bounded sublevel sets and the associated Markov kernel for , satisfies a Lyapunov-Foster Drift condition: there exists and such that
| (1.3) |
where .
In general, we will require a stronger Lyapunov structure on and the derivatives of . Specifically, we will assume the existence of a two parameter family of Lyapunov functions defined for and , satisfying the following conditions:
Assumption 1 (Lyapunov Structure).
There exists a two parameter family of Lyapunov functions
satisfying for each , , such that the following condition holds: such that , with and there exists a satisfying such that ,
| (1.4) |
where is given in Definition A.5.
In what follows, we will omit the parameters in contexts where they are not important.
Remark 1.2.
Since we are working on a Riemannian manifold, the definition of is somewhat technical and is defined over a suitable atlas. However, intuitively is a measure of the size of the th derivative of and , heuristically
Remark 1.3.
Note, that upon setting in (1.4), an application of Jensen’s inequality implies that satisfies the super-Lyapunov property, namely that for every there is a such that
This is a much stronger condition than a standard Lyapunov-Foster condition (1.3). Nevertheless, it is satisfied for many semilinear, parabolic stochastic PDEs where the nonlinearity is sub-critical with respect to a conserved quantity (e.g. the 2d Stochastic Navier-Stokes equations).
1.1.2 The linearization and projective process
Next, we will need some assumptions that encode dynamical information. The first is the assumption on the spectral properties of the ‘projective’ Markov semigroup, described below. A prominent role here is played by the inverse transpose of the derivative of the flow map, which we denote by , defined by
where the transpose is taken with respect to the Riemannian metric on . At each , naturally acts on covectors with and is a linear cocycle on since
Remark 1.4.
Under Assumption 1, by the multiplicative ergodic theorem [32, 39, 37, 45], the linear cocycle has a Lyapunov spectrum which have the property that since . This can be related to the Lyapunov spectrum of the linear cocycle on via
(see e.g. [12] Lemma B1). Specifically, if , by volume preservation, it must be that .
In what follows, we consider , with , the induced dynamics on the cotangent bundle . Let denote the Markov semigroup associated to the Markov chain on defined for each bounded measurable by
Additionally denote , with
the “projectivized” dynamics on the unit cosphere bundle and denote the Markov semigroup associated to the Markov chain on defined analogously.
Given a Lyapunov function for , we will denote the space to be the space of continuous functions on with finite weighted Lipschitz norm
| (1.5) |
where for and , with being the metric on and the metric on . It follows by the Feller property of and assumption (1.4) that restricts to a bounded linear operator on the space , at least if one replaces the time-step with and considers the process (see [[9] Lemma 5.2] for more information). By relabeling, we can assume without loss of generality that .
Of particular interest are functions which are homogeneous, i.e.
where is a smooth function on which is homogeneous of degree in . Evaluating on such a symbol, we obtain
where is the ‘twisted’ semigroup defined by
Similar to , forms a semigroup of bounded operators on (again, possibly by increasing the time-step and relabeling) and that when one recovers the projective semigroup . We make the following spectral assumption on .
Assumption 2 (Spectral Gap).
-
(i)
There exists a such that for all , admits a simple dominant eigenvalue on with a spectral gap, namely there exists an such that the spectrum of satisfies
where denotes the open ball of radius centered at .
-
(ii)
The eigenvalue satisfies for .
-
(iii)
Let be the rank-one spectral projector associated with . Then is a dominant eigenfunction satisfying
and is bounded below by a positive constant on bounded sets. That is for each bounded set there is a such that .
Remark 1.5.
Remark 1.6.
Irreducibility and geometric ergodicity of the projective process in a suitable Wasserstein metric, usually proved via a weak Harris’ theorem (see [29]), implies Part (i) and (iii) with a straightforward spectral perturbation argument (see e.g. [12, 7] and the references therein).
Note that also, in particular, by setting Assumption 2 also implies geometric ergodicity (again in a suitable metric) of and hence a unique stationary probability measure .
Remark 1.7.
The quantity is called the moment Lyapunov function and, at least for close to , Part (iii) implies that for -almost every ,
The function is also closely related to the probability of finding far from the expected for large . Particularly, it is related to a large deviation principle for the projective chain . See e.g. [6, 5] for more discussions.
Given the connection between and , Assumption 2 implies that the function
is an eigenfunction of with eigenvalue
The function is the starting point of the symbol of a pseudo-differential operator that plays a key role in our proof below.
1.1.3 Two point geometric ergodicity
Closely related to the behavior of the twisted Markov semi-group and the associated eigenfunctions , is the behavior of the two point motion. Namely, let and be two different trajectories on starting from distinct points and , that is and . In order to avoid reducing to the one point motion, we assume that the starting points do not belong to the diagonal set
We denote the associated Markov chain on by . We remark that is an almost-surely invariant set for the two-point semigroup defined over . As the space is not compact, one is likely to use a Lyapunov function on to control the behavior of the two point motion. As in [9, 7, 12] we consider the Lyapunov function with the following properties:
| (1.6) |
for all . Depending on the setting, this kind of Lyapunov function can be constructed using [9, 7, 12] or using large deviation estimates on the exit times [17, 6]. We will need the following assumption on the exponential decay of the two point motion.
Assumption 3 (Exponential Decay of Two Point Motion).
We assume that for all sufficiently small, for all , and all , satisfying (1.6) which is a Lyapunov function for the two point motion and for which the Markov process is -geometrically ergodic, namely there exists a such that for all and all , there holds
Heuristically, we can consider Assumption 3 as a strict improvement over Assumption 2 above, extending the linearized information to the nonlinear two-point dynamics. Indeed, the proofs of [7, 9, 12] used Assumption 2 together with a Harris’ theorem argument using the Lyapunov function to prove Assumption 3. However, it is easier to state these as separate assumptions, rather than list all of the individual assumptions required for the Harris’ theorem argument to apply.
By a now-standard Borel-Cantelli argument together with a little additional Fourier analysis (see [17] and [9, 14, 12]), Assumption 3 implies quenched mixing estimates such as (1.2). We technically do not use such estimates directly, however, the manner in which we employ Assumption 3 is nevertheless quite similar to a quenched mixing estimate.
1.1.4 Main result
In what follows, denote . Our main result is the following theorem.
Theorem 1.8.
Remark 1.9.
We do not currently know if the result holds for , nor do we know how large one can make ; see Section 1.2 for more discussion.
Remark 1.10.
Remark 1.11.
For distributions that cannot be identified with an function, we can define as the distribution acting on the smooth function .
Remark 1.12.
A related result was proven in [15], in the setting of the Kraichnan model, where the authors prove an exact identity for the averaged exponential decay of negative Sobolev norms. However, the proof in [15] relies on the specific structure of the Kraichnan model and does not extend to the general setting considered here. Additionally, as the self similarity required in [15] makes it challenging to define an associated flow map, their results do no immediately follow from Theorem 1.8.
As a corollary of Theorem 1.8, we obtain the following quenched mixing estimate with a random constant depending on the initial data.
Corollary 1.13 (Quenched Result).
Let and and be as in Theorem 1.8. Then, for all with , and with suitably small, there exists a random constant (depending on and the initial ) such that there holds for all and ,
Moreover, has uniform (in and ) moments , .
Proof.
Clearly we can write
where is defined by
To see that has finite moments we estimate
∎
1.2 Comments on the proof of Theorem 1.8
The proof of Theorem 1.8 is inspired primarily by the pioneering work of Faure, Roy and Sjostrand [20] and the subsequent works [24, 21, 25, 26, 22, 23]. In these works the authors use microlocal analysis to construct special anisotropic Sobolev spaces (i.e. spaces with different regularity in different directions in frequency space) on which one can prove spectral a gap, or at least quasi-compactness, for the transfer operator. This microlocal approach as also been used in the context of Anosov flows generated by a vectorfield [19, 18]. See also [11, 28, 35, 1, 3, 16, 2] for other works using anisotropic Banach spaces of functions to study spectra of the transfer operator.
The motivation of these anisotropic spaces is to find a space with
such that for average-zero observables, the scalars decay exponentially in due to a spectral gap of the transfer operator (or to at least prove a quasi-compactness estimate that implies localization of the essential spectrum)
| (1.10) |
In settings where this is possible, it yields a significantly more precise result than quenched mixing estimates. In the microlocal approach of [20] (also in [18, 19]), these spaces are found by building a Lyapunov function for the linearized process on , , and using it to define a pseudo-differential operator with a suitable quantization procedure. The space one obtains the essential spectrum estimates in is then defined via the norm (at least heuristically)
where denotes the pseudo-differential operator associated to the symbol . In the case of Anosov diffeomorphisms, one chooses for on the tangent to the unstable manifold through and for on the tangent to the stable manifold through (the angle between the tangents is uniformly bounded away from zero by assumption of uniform hyperbolicity). After a suitable regularization procedure and a variable-order Egorov’s theorem, one can use this symbol to construct a norm and prove a Lasota-Yorke-type estimate which implies quasi-compactness of the transfer operator (see [20] for more details).
Again considering the case of Anosov diffeomorphisms, one can observe that a negative regularity mixing estimate such as Theorem 1.8 cannot possibly hold for deterministic maps. Indeed, by concentrating the initial distribution in close to the stable manifolds, one can obtain arbitrarily slow mixing rates for certain sequence of pathological initial data. For the random map case, the difference here is that for each fixed initial distribution, the assumptions in Section 1.1.2 basically rule out the possibility that a positive probability set of results in stable manifolds that line up badly to produce poor decay rates. One can wonder exactly how large of a moment one can take in Theorem 1.8, i.e. under what conditions the set of bad remains negligible; our proof currently yields at most second moments. Similarly, one can wonder how large can be taken, i.e. whether one can take all the way down to or even (going further is clearly impossible as point masses cannot be mixed).
In order to prove Theorem 1.8, we use a Lyapunov function (now again in the stochastic process sense) for the linearized process on defined in Section 1.1.2 and use this as a symbol to obtain exponential decay estimates on the quantity
We use a regularization of the symbol
which by Assumption 2, satisfies a spectral-gap type estimate for the linearized process Markov semigroup. Formally, we could then expect by Egorov’s theorem
Given the lower bounds on in Part (iii) of Assumption 2, by Gårding’s inequality, we can heuristically expect that the quantization of this symbol would satisfy something like (ignoring for a moment)
which would suggest Theorem 1.8. The most obvious way this intuition fails is that both Egorov’s theorem and Gårding’s inequality only control high frequencies and both leave an error in lower regularity, for example the most one can hope for from Gårding’s inequality is something like the following for some constants ,
| (1.11) |
This is analogous to why one only directly obtains Lasota-Yorke type estimates in the deterministic setting, rather than direct exponential decay estimates. These errors will need to be dealt with carefully and in particular, require us to first prove a -type mixing estimate (this is done using Assumption 3; see Section 4.2). Note that even this kind of estimate is false for deterministic maps. However, there are two more reasons the above intuition is naïve: (A) the regularity of is limited to and so the symbol must be suitably regularized and, perhaps most importantly, (B) the positivity and regularity of the symbol depend badly on the derivatives of which are unbounded (measured by the surrogate ), which means that the in the Gårding’s inequality (1.11) and the errors in Egorov’s theorem would become time-dependent (or the regularization would need to be time-dependent). These issues present the main difficulties in proving Theorem 1.8. In Section 3, the regularization procedure of the symbol is presented and the basic properties are verified. In Section 4 the main arguments in Theorem 1.8 are given, namely an -type mixing estimate and its use, together with Assumption 1, to absorb the low frequency error terms coming from Egorov’s theorem and Gårding’s inequality to yield Theorem 1.8.
2 Applications
Here we outline several relevant applications of our general framework to examples of interest, most notably stochastic flow for SDEs and flows generated by the stochastic Navier-Stokes equations. Additionally, we state some important applications to the advection diffusion equation with a stochastic source term and the existence of a unique stationary measure in the zero-diffusivity limit.
2.1 Examples
2.1.1 IID diffeomorphisms and stochastic flows
A simple, but wide, class of examples our theorem applies to are iid random diffeomorphisms, such as the case of the stochastic flow of diffeomorphisms associated to an SDE on a compact, Riemannian manifold . Consider smooth divergence-free vector fields on , then the SDE
defines a stochastic flow of diffeomorphisms (see [34] for details). As the vector fields are divergence free, is volume-preserving almost-surely, i.e. the Riemannian volume measure is almost-surely invariant. The projective process solves a similar SDE on the sphere bundle [4] (i.e. the unit tangent bundle), denoted
with and the ‘lifted’ vector fields satisfy
As we explain in more detail below in Section 2.2, [17] provides checkable sufficient conditions for these flows to satisfy Assumption 2–3, at least when combined with the construction of found in [9]; see also the earlier work of [6, 13, 4]. In particular, a quenched mixing estimate (1.2) was proved in [17], specifically for any , (deterministic)
| (2.1) |
with . Our results now additionally prove that for (the lack of implies we can take ),
Another concrete case of iid random diffeomorphisms are the Pierrehumbert flows [12], namely the case of transport by alternating shear flows on . For example, in 2D, the time-one map random map could be given by , where
where are suitable independent random variables, for example each drawn uniformly from suffices. Assumptions 2–3 were proved in [12], wherein a quenched exponential mixing estimate such as (2.1) was proved and a suitable was constructed. Theorem 1.8 now provides also the estimate (1.9).
2.1.2 Stochastic Navier-Stokes equations
Let us briefly explain how to apply Theorem 1.8 to the setting of stochastic Navier-Stokes in , as in works of [9, 7] and [14]. The PDE in question is given by the following in
| (2.2) |
in 3D the viscosity must be replaced by a suitable hyperviscosity but this case otherwise also works. Let us now explain the operator . Following the convention used in [46], we define the following real Fourier basis for functions on by
where and . We set and define a collection of full rank matrices satisfying , , and . Note that in dimension , is just a vector in and is therefore given by . In dimension three, the matrix defines a pair of orthogonal vectors that span the space perpendicular to . Next, we define the natural Hilbert space on velocity fields by
| (2.3) |
with the natural inner product. Let be a cylindrical Wiener process on with respect to an associated canonical stochastic basis and a Hilbert-Schmidt operator on , diagonalizable with respect the Fourier basis on . In the works [7, 9], the operator was assumed to satisfy the following regularity and non-degeneracy assumption
In the work [14] the lower bound assumption was dropped and replaced with the following much weaker assumption.
Assumption 5 ((Essentially) Assumptions in [14]).
Theorem 1.8 can be applied to both of these cases. We define our primary phase space of interest to be velocity fields with sufficient Sobolev regularity (under Assumption 5 we can choose arbitrarily):
Note we have chosen sufficiently large to ensure that so that . Since we will need to take advantage of the “energy estimates” produced by the vorticity structure of the Navier-Stokes equations in , we find it notationally convenient to define the following dimension dependent norm
| (2.4) |
The following well-posedness theorem is classical (see e.g. [33]).
Proposition 2.1.
For all initial data , there exists a -a.s. unique, global-in-time, -adapted mild solution to (2.2) satisfying . Moreover, defines a Feller Markov process on and the corresponding Markov semigroup has a unique stationary probability measure on .
One then defines the Lagrangian flow map as the solution map to the ODE
| (2.5) |
2.2 Checking the conditions for stochastic Navier-Stokes
In this section we sketch some ideas required to verify the conditions of Theorem 1.8 in the setting of stochastic Navier-Stokes. In the setting of stochastic flows, checking the conditions follow along similar lines and typically much easier to verify (see e.g [27]).
Assumption 1 will be show below in Section 2.2.1. Assumption 3 was proved for Assumption 4 in [9] and for Assumption 5 in [14]. Assumption 2 (i) – (iii) regarding the construction of was proved directly in [9] for the case of Assumption 4. In 2.2.2 below, we briefly explain how to obtain Assumption 2 (i) – (iii) for the degenerate noise Assumption 5.
2.2.1 Checking Assumption 1
We must check Assumption 1 for the Lyapunov function given by
where is a This will be proved by making use of the following super Lypaunov bound proved in [9] (the proof works for both Assumption 4 and 5).
Lemma 2.2 ( [9] Lemma 3.7).
Let be a solution to 2.2. There exists a , such that for all , , , , and where and , there exists a constant such that the following estimate holds
| (2.6) |
Assumption 1 reduces to proving the following:
Lemma 2.3.
Let be a solution to (2.2). For all , , and , the following estimate holds,
Proof.
To apply Lemma 2.2 we must first bound derivatives of in terms of something that can be controlled
Claim 2.4.
For each , there exists a satisfying such that the following estimate holds
Proof.
The proof will be by induction on . Clearly it is true for . We assume it is true for all . Using the equation (2.5) we can estimate using the Faà di Bruno formula
where the sum is over the set
Separating out the leading order term in the outer sum , , and applying Grönwall to the remaining terms, we obtain the following estimate
where we used that in if and hence the product is only up to . Using the induction hypothesis we have for and that
where in the second inequality we used that and define . Substituting this back into the expression for we obtain the desired estimate. ∎
Likewise we obtain a similar estimate for the inverse:
Claim 2.5.
For each , there exists a satisfying such that the following estimate holds
Proof.
To prove this, we will make use of the following estimate on derivatives of the inverse of a diffeomorphism which can be deduced from the Faà di Bruno formula,
where the sum is over the set
Substituting in the estimate for we obtain
for some suitable . Using that , we obtain the desired estimate.
∎
Using these estimates we can now prove the desired result by applying the super-Lyapunov bound (2.6) to the above estimates assuming that the regularity of is sufficiently high .
∎
2.2.2 Construction of with degenerate noise
Likely the simplest way to construct is through a spectral perturbation method applied to , which is the method employed in e.g. [7, 9, 12]. The first step is to prove that is the unique, dominant eigenvector for in (the corresponding eigenvalue is of course ), which amounts to verifying a spectral gap for in the dual Lipschitz metric (i.e. the Wasserstein-1 norm). This is typically done with a weak Harris’ theorem; the proof of this geometric ergodicity found in [9], which closely follows [29], can be used for both Assumption 4 and Assumption 5. Then the observation that in the strong operator topology implies the existence of a satisfying Assumption 2 (i) and (iii) through classical spectral perturbation theory. Verifying (ii) is a standard convexity ‘trick’ which proves that as , where is the Lyapunov exponent. See [9, 12] for expositions of this argument.
2.3 Passive scalars
As a final application of Theorem 1.8 we address specifically the advection-diffusion equation on
| (2.7) |
The transfer operator for this equation can be written as follows
| (2.8) |
where solves the SDE
In (2.8), the notation refers to the expectation with respect only to the Brownian motions (which are of course independent from ). For the case of Pierrehumbert [12] and stochastic Navier-Stokes under Assumption 4 [7]. it was proved in those references that Assumptions 1–3 all hold uniformly in for some small (note that this is far from obvious). As the proof of Theorem 1.8 is quantitative as well, in these cases, Theorem 1.8 also holds uniformly in for solutions to (2.7). It seems plausible that one could carry this out also for Assumption 5 however as of writing, this result has not yet appeared in the literature.
There is one last notable consequence of Theorem 1.8 which pertains to the limiting system arising in Batchelor-regime passive scalar turbulence [10], specifically the system
| (2.9a) | |||
| (2.9b) | |||
| (2.9c) | |||
For all , stationary measure for the joint process, which we denote . In [10] it was proved that any sequence such that has a subsequence which converges weak- to a measure supported on for all which is a stationary measure for the system. One can use Theorem 1.8 to prove that in fact, there is a unique stationary measure for the system supported on (and hence a unique limit for all such convergent subsequences as ).
Theorem 2.6.
Proof.
The proof is based on Birkhoff’s ergodic theorem and the contraction in . Suppose that there exist two measure and for the process on which are stationary for the system. Using ergodic decomposition we may assume that they are ergodic. Hence, by Birkhoff’s ergodic theorem, there exist two sets such that for , and for every bounded Lipschitz on , and every initial data we have
Moreover since both project down to a unique stationary measure for the Navier-Stokes system on , it must be that the projections of are both full measure and hence have a non-empty intersection which is also full measure. It follows that we may choose initial data and , such that and we can write
| (2.10) |
Note that since the velocity is the same for , the difference solves the advection-diffusion equation
and hence by Theorem 1.8 the right-hand side of (2.10) goes to zero as , implying that . ∎
3 Constructing the Symbol
In this section, we use the function from Assumption 2 to construct a pseudo-differential operator which exhibits exponential decay under the map , up to a lower order remainder. This involves a certain regularization scheme of the finite regularity symbol , where we mollify at the scale , where as . This scheme is reminiscent of para-differential calculus [36], although it differs in the details.
3.1 Lower bound on
First we show that assumption that is lower bounded on bounded sets actually implies a lowerbound on for all .
Lemma 3.1.
For all Lyapunov functions with and and , satisfies the following lower bound
Proof.
Let and define for with as above. Now let . Our first step is to show that for each , there exists an big enough so that
| (3.1) |
Such a bound follows relatively easily by standard techniques in Markov chains, which we outline now for completeness. Indeed, the discrete Dynkin formula implies that
By the super Lyapunov property mentioned in Remark 1.3 and the definition of this implies the following exponential estimate on
Hence, taking and using Markov’s inequality gives (3.1).
Next, we note that by the spectral gap condition (Assumption 2) on we have that is given by
| (3.2) |
Indeed, has a dominant simple eigenvalue with a spectral gap. Let be the rank one spectral projector associated with and recall that . Then by the spectral gap assumption and Gelfand’s formula in . It follows that
The limit formula (3.2) implies that for each , we have
which is uniform over in . By Assumption 2 there exists a such that
With this in mind, let be large enough so that
for all and , . Now we can write as
Here, denotes the standard filtration and is the corresponding stopped sigma-algebra, consisting of measurable sets for which for all . In the last line of the above inequality, we used the strong Markov property to conclude
We now bound for as follows. Since for and , . Therefore, by (3.1), for any
| (3.3) | ||||
Taking and using that almost-surely, we conclude by monotone convergence that
Next, we note that upon choosing large enough (and consequently large enough), we have that
where we used the Assumption 1 with in the last line to bound using the Markov property
for some . ∎
3.2 Regularization scheme
Now we describe a regularization scheme that gives rise to a symbol using properties of . The main result of this section, Lemma 3.2, shows that this symbol belongs to the symbol class , and is an approximate eigenfunction of (see appendix A for a definition of the symbol class).
For each , we take to be a family of a mollifications of in the variables up to scale . We impose the following requirements on such a mollification. First, we require that for all , we have
| (3.4) | ||||
| (3.5) |
We emphasize that is independent of . Letting be the geodesic distance between any two points on , and let be the corresponding Lipschitz norm:
| (3.6) | ||||
| (3.7) |
Regarding higher order derivatives, given any parametrization where is an open set, we require that
| (3.8) |
The existence of the family is standard. For instance, when , we can define using convolution by a standard mollifier. One can generalize such a mollification scheme to arbitrary by using the exponential map or a partition of unity.
Using the boundedness of w.r.t. the norm in (1.5), the above conditions imply that for each and ,
| (3.9) |
and
| (3.10) |
for any and described above. Moreover,
| (3.11) |
independently of .
Fix a smooth, non-negative dyadic partition of unity of , such that for each , we have , , and
for all . For convenience, given any , we define
For longer expressions, we shall use the notation . Let . We define
| (3.12) |
For each , this defines an approximation of , which improves as . The parameter determines the rate of convergence. Since the role of is that of an eigenfunction to the operator , we think of as an approximate eigenfunction with error
The following Lemma shows that and belong to appropriate symbol classes from which we can construct pseudo-differential operators, with having strictly lower order than (see Appendix A):
Lemma 3.2.
Let and . For each , , and . For each , the seminorms of (defined in Appendix A.1) are bounded as follows: for any ,
| (3.13) |
On the other hand, there exists (depending only on , and ) such that
| (3.14) |
Proof.
We prove the bounds above in multiple steps:
Step 1: We prove (3.13). Fix a coordinate chart from the atlas as defined in Appendix A and set . It suffices to show that for any such , and any two multi-indices , we have
| (3.15) |
In proving this, we will omit the dependence of implicit constants on the parameters listed above. We will also drop the subscript on the coordinate chart and write . To show this, first observe that
is a -homogeneous function in . Moreover, given any cone , we can parameterize in spherical coordinates using
where and maps an open subset of to . Then, with this parametrization, we have
We call and
so that defines a smooth parametrization of in some open subset. Furthermore, the partial derivatives transform as follows under the change of coordinates:
where is a matrix given by the Moore-Penrose pseudo-inverse of , i.e.
and for any , the range of (as a matrix) is .
Thus, for all such that is in the range of , we use the homogeneity in to bound
| (3.16) | ||||
| (3.17) | ||||
| (3.18) |
By covering by finitely many cones for we which can construct such a smooth map , we recover the bound for all possible . In particular, when (as is the case when is in the support of ) the above is bounded by
On the other hand, using the parametrization once more, we have
Then, it is straightforward to show that for all for which the above expression is nonzero that
We then apply the multivariable chain rule to each term in the sum which defines as in (3.12)
to recover (3.13).
Step 2a: We now show (3.14) for . It suffices to show that for all , and , we have
and then take as defined in Step 2b below. Once again, dependence of constants on the parameters enumerated above is implied. Using the identity , we write
| (3.19) |
The difference appearing above can be expressed as follows:
Since the first term is compactly supported, it is trivial to bound it by . As for the second term, we have
| (3.20) | ||||
| (3.21) | ||||
| (3.22) |
Thus,
We can apply this bound to the same difference under , in conjunction with Assumption 1,
| (3.23) | ||||
| (3.24) |
In summary,
| (3.25) |
On the other hand, we have the uniform bound ,
since is zero near the pole of . Taking the minimum of these two bounds, we conclude (3.2).
Step 2b: As in step 1, we fix a coordinate chart , and suppress the dependence of constants on the parameters. For higher order derivatives of , we let , and use the crude bound
| (3.26) | ||||
| (3.27) |
Then, we recycle (3.13) to bound the above by
| (3.28) | ||||
| (3.29) |
Now, in the latter term, we have
Hence, applying Assumption 1,
| (3.30) | |||
| (3.31) | |||
| (3.32) |
Now, observe that since ,
Combining this with Step 2a, we conclude (3.14) for all . ∎
As an application of Gårding’s inequality (Proposition A.7), the lemma above implies that the “norm” defines the same topology as , provided one has control on lower frequencies.
Corollary 3.3 (Quantitative Gårding’s inequality).
Under the hypotheses of Lemma 3.2, we have
| (3.33) |
4 Low regularity mixing
In this section, we use the results of Section 3 to show prove Theorem 1.8, i.e. that decays exponentially fast in . We break this up into two steps: first, we exhibit time 1 exponential decay of high frequencies of through the pseudo-differential operator as in Lemma 4.1. Second, we show that low frequencies decay exponentially for long times, as in Lemma 4.2. The latter of these two results does not rely on . Rather, it is a consequence of the two-point geometric ergodicity in Assumption 3. We combine these two lemmas to prove Theorem 1.8 in Section 4.3.
4.1 Short time, high frequency decay
Lemma 3.2 and Egorov’s Theorem (Theorem A.8) imply that under conjugation by the time-1 flow map , the operator decays by a factor of , plus a lower order remainder:
Lemma 4.1.
Let , , and . Then there exists some (depending only on and ) such that
| (4.1) |
Proof.
By Egorov’s theorem (Theorem A.8), Lemma 3.2 and Assumption 1,
| (4.2) | |||
| (4.3) |
for some . On the other hand, by the boundedness of pseudo-differential operators (Proposition A.6) and Lemma 3.2 above,
for some . Combining these two bounds, and fact that , we finish the proof choosing to be the bigger of and . ∎
4.2 Long time, low frequency decay
Next, we show how the assumption of 2-point mixing in Assumption 3 implies the exponential decay of in a very low-regularity norm.
Lemma 4.2.
There are absolute constants such that the following holds. For any , and any such that , we have
| (4.4) |
Here, we recall that .
Proof.
We break the proof into two steps. For convenience, we write with .
Step 1: First, we show that there exists such that
| (4.5) |
for all . To prove the bound above, let be the kernel of the operator
In particular, for all . We define
so that is mean zero. Then, since is mean zero,
| (4.6) | ||||
| (4.7) |
Then, estimate (4.5) follows from the two-point mixing bound of Assumption 3: for all sufficiently small, and all ,
for all , and denotes the distance between and , Thus, by taking , we have
Step 2: We show that there exists such that
| (4.8) |
To prove this, we use the cocycle property of and Assumption 1 in the case , and take large enough so that
| (4.9) | ||||
| (4.10) | ||||
| (4.11) | ||||
| (4.12) |
Then, by writing , where is the Laplace-Beltrami operator and denotes the divergence on , we can integrate by parts to get
| (4.13) |
Finally, we note
from which (4.8) follows.
Step 3: In the final step, we interpolate the two estimates to get (4.4). For each , let be a mollification of that satisfies
Then, combining (4.5) and (4.8), we have
| (4.14) | ||||
| (4.15) |
We now choose , and
so that
Taking , this gives us (4.4).
∎
4.3 Proof of the main theorem
Now we are ready to prove the main Theorem 1.8
Proof of Theorem 1.8.
We fix the constants as in Lemma 4.2 and set . In the statement of the main theorem, we shall prove the estimate with .
By Lemma 4.1, there exists some such that for each
| (4.16) |
Using interpolation, there exists such that for any , there is some constant for which the following holds:
Applying Gårding’s inequality, Corollary 3.3, to the first term on the right-hand side, we have
| (4.17) |
We now set
so that . Under this re-scaling, is an absolute constant, while . For brevity sake, since and are now fixed, we will drop the subscripts and write .
Combined with (4.16), this yields
| (4.18) |
We now take and set . Then, by the Markov property,
| (4.19) |
By taking the full expectation of both sides, and iterating the above inequality, we have
Then, by applying Lemma 4.2 to the sum (recalling that ) to get
| (4.20) | ||||
| (4.21) |
We apply Corollary 3.3 once more to deduce
for each . Applying Lemma 4.2 once more to the second term on the right-hand side completes the proof. ∎
Appendix A Pseudo-differential calculus
Here, we review the requisite results on pseudo-differential calculus on compact -dimensional Riemannian manifolds .
In this section we fix a finite atlas , where each , we have that is an open set in , and is an open set in . It will also be convenient to define .
A.1 Quantization on manifolds
Definition A.1.
Fix and . We define the Hörmander symbol classes to be the set of satisfying the following. Given any trivialization , we have
for all multi-indices , and all in the range of . In the above context, the bracket is given by for any , where is defined in terms of the metric .
We define a family of seminorms that determine a Fréchet topology on as follows. For each , we define the seminorms
| (A.1) |
Below, we give a generalization of the Kohn-Nirenberg quantization scheme for manifolds, following [38].
Definition A.2.
Fix to be a non-negative cutoff function satisfying the following: first, in an open neighborhood of ; second, the map defines a diffeomorphism from the support of to an open neighborhood of the diagonal .
Then, given any symbol , we set
for any .
Remark A.3.
We remark that while this quantization depends on the choice of , this construction is unique modulo the addition of infinitely smoothing operators. In contrast to [20], we are only concerned with the principal symbol of our operator, although using this quantization scheme is convenient in Lemma 4.2. An alternative, but equally viable quantization scheme can be found in Theorem 14.1 in [47].
A.2 Function spaces on Riemannian manifolds
Throughout this paper, we shall consider a general compact, smooth, -dimensional Riemannian manifold . Here, we fix the function spaces throughout the paper. We define to be the space on with respect to the usual Riemannian volume form, with inner product
for any two real valued . Here, is shorthand for .
Definition A.4.
We say to be the closure of with respect to the norm
We now define a norm-like quantity on the group of diffeomorphisms in , in order to track the regularity of such maps.
Definition A.5.
Let . Given any diffeomorphism , we define the functional
In the above, we set .
A.3 Estimates on pseudo-differential operators
The first result we need is that the operators defined above are bounded between Sobolev spaces. These results are classical when is replaced with Euclidean space .
Proposition A.6.
Let and . Let . Then for each , is a bounded operator from to . More precisely, for each , there exists some such that
In particular, for any ,
We now state the weak Gårding inequality on .
Proposition A.7.
Let and , and . Let , and suppose that it satisfies
for all , with . Then, for all , and , there exists some such that
We now state a version of Egorov’s theorem on manifolds.
Theorem A.8.
Let and . Let , , and let be a smooth diffeomorphism. Define the cotangent lift of to be the map given by
for any and . Moreover, given any map between manifolds, we define its pullback for scalar valued where the composition is well-defined.
The operator can be decomposed
Here, . Moreover, for all , there exists such that
Remark A.9.
While these results do not appear to be available in the literature as stated, their analogues on are well-known, with the caveat that such results usually do not quantify the implicit constant in terms of seminorms on . See [44, 31]. To recover the results on , one can use a partition of unity and an atlas via a standard construction, and Egorov’s theorem for properly supported pseudo-differential operators on (see for instance [31], Theorem 18.1.17).
The quantitative bounds in terms of the seminorms on , as well as in the case of Theorem A.8, are direct consequences of the proofs of these results, as these seminorms appear in bounding error terms in the asymptotic expansions of pseudo-differential operators (see [31], Theorems 18.1.7 and 18.1.8).
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