Additive functionals as rough paths
Abstract
We consider additive functionals of stationary Markov processes and show that under KipnisVaradhan type conditions they converge in rough path topology to a Stratonovich Brownian motion, with a correction to the Lévy area that can be described in terms of the asymmetry (nonreversibility) of the underlying Markov process. We apply this abstract result to three model problems: First we study random walks with random conductances under the annealed law. If we consider the Itô rough path, then we see a correction to the iterated integrals even though the underlying Markov process is reversible. If we consider the Stratonovich rough path, then there is no correction. The second example is a nonreversible OrnsteinUhlenbeck process, while the last example is a diffusion in a periodic environment.
As a technical step we prove an estimate for the variation of stochastic integrals with respect to martingales that can be viewed as an extension of the rough path BurkholderDavisGundy inequality for local martingale rough paths of [FV08, CF19, FZ18] to the case where only the integrator is a local martingale.
1 Introduction
In recent years there has been an increased interest in the link between homogenization and rough paths. It had been observed previously that homogenization often gives rise to nonstandard rough path limits [LL03, FGL15]. The more recent investigations were initiated with the work of Kelly and Melbourne [KM17, KM16, Kel16] who study rough path limits of additive functionals of the form , where is a deterministic dynamical system with suitable mixing conditions. In that way they are able to prove homogenization results for the convergence of deterministic multiscale systems of the type
for which under suitable conditions converges to an autonomous stochastic differential equation. This line of research was picked up and extended for example by [BC17, CFK19b, CFK19a, LS18, LO18]. More recent results also cover discontinuous limits [CFKM19].
Motivated by this problem, as well as by the aim of understanding the invariance principle for random walks in random environment in rough path topology, we want to study rough path invariance principles for additive functionals of Markov processes in generic situations. If we are only interested in a central limit theorem at a fixed time, then there are of course many results of this type and many ways of showing them. See for example [Pel19] for a recent and fairly general result. A particularly successful approach for proving such a central limit theorem and even the functional central limit theorem (invariance principle) is based on Dynkin’s formula and martingale arguments, and it was developed by Kipnis and Varadhan [KV86] for reversible Markov processes and later extended to many other situations; see the nice monograph [KLO12]. Here we extend this approach to the rough path topology and we study some applications to model problems like random walks among random conductances, additive functionals of OrnsteinUhlenbeck processes, and periodic diffusions.
This can also be seen as a complementary direction of research with respect to the recent advances in regularity structures [BHZ19, CH16, BCCH17], where the aim is to find generic convergence results for models associated to singular stochastic PDEs. In those works the equations tend to be extremely complicated, but the approximation of the noise is typically quite simple (the prototypical example is just a mollification of the driving noise, but [CH16] also allow some stationary mixing random fields that converge to the spacetime white noise by the central limit theorem). In our setting the equation that we study is very simple (just a stochastic ODE), but the approximation of the noise is very complicated and (at least for us) it seems difficult to check whether the conditions of [CH16, Theorem 2.34] are satisfied for the kind of examples that we are interested in.
The most interesting model that we study here is probably the random walk among random conductances. Here we distribute i.i.d. conductances on the bonds of (where means and are neighbors). Then we let a continuous time random walk move along , with jump rate from to (resp. from to ). We are interested in the large scale behavior, i.e. we study for . It is well known that the path itself converges in distribution under the annealed law to a Brownian motion with an effective diffusion coefficient. Our contribution is to extend this convergence to the rough path topology, which allows us for example to understand the limit of discrete stochastic differential equations
(1) 
but also of SPDEs driven by . And here we encounter a surprise: Even though is in a certain sense reversible (more precisely the underlying Markov process of the environment as seen from the walker is reversible), the iterated integrals do not converge to , but instead we see a correction: We have
where is a correction given by
for the law of the random conductances. Of course, vanishes if the conductances are deterministic (i.e. if is a Dirac measure). But if the conductances are truly random, then typically the effective diffusion is not just given by the expected conductance, and in one can even show that this is never the case (see the discussion at the top of p.89 of [KLO12]). Therefore, is typically nonzero, and the solution of (1) converges to the solution of
If on the other hand we denote by the linear interpolation of the pure jump path , then converges to the limit that we would naively expect, namely to the Stratonovich rough path above . From the point of view of stochastic calculus this is a bit surprising: After all, there are stability results for Itô integrals [KP91], while the quadratic variation (i.e. the difference between Itô and Stratonovich integrals) is very unstable. In fact, we are not aware of any previous results of this type (naive limit for the Stratonovich rough path, correction for the Itô rough path), but it seems to be a generic phenomenon. The same effect appears for periodic diffusions, and we expect to see it for nearly all models treated in the monograph [KLO12]. On the other hand, for ballistic random walks in random environment, after centering, a correction to the Stratonovich rough path is identified in terms of the expected stochastic area on a regeneration interval [LO18, Theorem 3.3]. Moreover, for random walks in deterministic periodic environments simple examples for nonvanishing corrections are available [LS18, Section 1.2] (or [LO18, Section 4.2]). For processes that can be handled with the KipnisVaradhan approach we generically expect to see a correction to the Stratonovich rough path if and only if the underlying Markov process is nonreversible.
Structure of the paper
In the next section we introduce some basic notions from rough path theory. Section 3 presents our main result Theorem 3.3, the rough path invariance principle for additive functionals of stationary Markov processes, which holds under the same conditions as the abstract result in [KLO12]. The proof is based on recent advances on Itô rough paths with jumps due to Friz and Zhang [FZ18], on stability results for Itô integrals under the so called UCV condition by Kurtz and Protter [KP91], on Lépingle’s BurkholderDavisGundy inequality in variation [Lép75], and on repeated integrations by parts together with a new estimate on the variation of stochastic integrals (Proposition 3.13). In Section 4 we apply our abstract result to three model problems: random walks among random conductances, additive functionals of OrnsteinUhlenbeck processes, and periodic diffusions. Finally, Section 5 contains the proof of Proposition 3.13 which might be of independent interest.
Notation
For two families of real numbers indexed by the notation means that for every where is a constant. Let for . We interpret any function also as a function on via , . For a metric space we write resp. for the continuous resp. càdlàg functions from to . A function is called continuous resp. càdlàg if for all the map on is continuous resp. càdlàg, and we write resp. for the corresponding function spaces.
2 Elements of rough path theory
Here we recall some basic elements of rough path theory for Itô rough paths with jumps. See [FZ18] for much more detail.
Let us write (resp. ) to denote the uniform norm of (resp. ). For and a normed space , we define the variation of (and so in particular of ) by
(2) 
where the supremum is taken over all finite partitions of and the summation is over all intervals . Note that for any , we have that .
Definition 2.1 (variation rough path space).
For , the space (resp. ) of càdlàg (resp. continuous) variation rough paths is defined by the subspace of all functions satisfying Chen’s relation, that is,
(3) 
for , and
(4) 
The variation Skorohod distance on is
where are the strictly increasing bijective functions from onto itself, and . The uniform Skorohod distance is defined similarly, except with the variation respectively variation distance replaced by the uniform distance; see [FZ18, Section 5] for details.
For we use the notation for the leftpoint Riemann integral, that is
whenever this limit is well defined along an implicitly fixed sequence of partitions of with mesh size going to zero. Note that if is a semimartingale and is adapted to the same filtration, then this definition coincides with the Ito integral. We remark also that the iterated integrals
satisfy Chen’s relation (3). Moreover, so do , for any fixed matrix .
Remark 2.2.
Note that by Chen’s relation whenever , and therefore
Consequently, the uniform resp. Skorohod distance of the (oneparameter) paths and controls the uniform resp. Skorohod distance of and .
The following lemma by [FZ18] will be useful in the sequel.
Lemma 2.3.
Let be a sequence of càdlàg rough paths and let . Assume that there exists a càdlàg rough path such that in distribution in the Skorohod (resp. uniform) topology and that the family of real valued random variables is tight. Then in distribution in the variation Skorohod (resp. uniform) topology for all .
Proof.
This follows from a simple interpolation argument, see the proof of Theorem 6.1 in [FZ18]. ∎
Invariance principles for rough path sequences guarantee the convergence of the solutions to rough differential equations where the noise is approximated by the path sequence. Moreover, whenever the second level (the first order ‘iterated integrals’) of the rough path has a correction, the limiting path solves a driftmodified rough equation defined explicitly in terms of the correction. More precisely, [FZ18, Theorem 6.1 and Proposition 6.9] proved the following. Let be a sequence of semimartingales and assume that converges in distribution in variation Skorohod (resp. uniform) distance to a rough path , where is a semimartingale and for . Then the solutions of
converge in distribution in the Skorohod (resp. uniform) topology to the solution of
where denotes rough path integration and is just the Itô integral.
3 Additive functionals as rough paths
Here we present our abstract convergence result for additive functionals of stationary Markov processes. We place ourselves in the context of Chapter 2 in [KLO12]: Let be a càdlàg Markov process in a filtration satisfying the usual conditions, with values in a Polish space , and let be a stationary probability measure for and . We assume that the transition semigroup of can be extended to a strongly continuous contraction semigroup on . We write for the infinitesimal generator of and we assume that is ergodic for , i.e. that is almost surely constant whenever . We also assume that there exists a common core for and , where is the adjoint of , and that contains the constant functions. We write
Notation.
We write or (and or ) for the distribution of the stationary process on the Skorohod space . The notation is reserved for the integration with respect to on the space .
Definition 3.1.
The space is defined as the completion of with respect to the norm
or more precisely we identify if , and is the completion of the equivalence classes. The space is the dual of : We define for
and then is the completion of with respect to .
If takes values in we also write , or , etc.
Note that if then we can take for so that and by sending we see that . Therefore, we get that for all .
Let now . Our aim is to derive a scaling limit for the (absolutely continuous) rough path , where
and the integration with respect to is in the RiemannStieltjes sense. Let us first recall the following result:
Lemma 3.2 ([Klo12], Theorem 2.33).
Assume that is ergodic for . Let and assume that the solution to the resolvent equation with satisfies
(5) 
for some . Then converges in distribution in to a Brownian motion with covariance matrix
Our aim is to extend Lemma 3.2 to the rough path topology. Our main result is:
Theorem 3.3.
For the rest of the section we shall assume without further mention that the conditions of Theorem 3.3 are satisfied.
Remark 3.4.
As is of finite variation the iterated integral “wants” to converge to the Stratonovich integral, and describes the area correction. Note that if is symmetric, i.e. if is reversible, so in that case we indeed obtain the Stratonovich rough path over .
Remark 3.5.
In Lemma 3.2 and Theorem 3.3 the ergodicity of with respect to is only needed for proving the tightness of in the uniform topology. This is relatively subtle because we need tightness of certain martingales for which we only know that , which is insufficient to apply Kolmogorov’s continuity criterion. If we can show for some and for the martingales of Lemma 3.9 below, then we do not need the ergodicity of with respect to (although we do need ergodicity with respect to ).
The strategy for proving Theorem 3.3 is to apply Lemma 2.3, which separates the convergence proof into two problems: Showing tightness of in the variation topology, and identifying the limit in the Skorohod topology. To identify the limit we follow a similar strategy as in [KLO12] and combine it with tools from rough paths together with a simple integration by parts formula.
Let us formally sketch how the correction arises, under the assumption that we can solve the Poisson equation (which is for example the case if has a spectral gap and ). In that case we have
for a sequence of martingales . Therefore,
By the ergodic theorem, the first term on the right hand side converges to
To understand the contribution of the remaining contribution we use integration by parts: Since is of finite variation, we have
Since is stationary, we have , and converges jointly to . And the martingale sequence satisfies the “UCV condition” (see the next section for details), and therefore . After passing to the limit we apply integration by parts once more and deduce that
So overall
3.1 Tightness in variation
The following notion was introduced by KurtzProtter [KP91].
Definition 3.6 (UCV condition).
Let be a sequence of càdlàg local martingales. We say that satisfies the Uniformly Controlled Variation (UCV) condition if
Strictly speaking this is a very particular special case of the definition by Kurtz and Protter, who are much more permissive and consider general semimartingales rather than local martingales, and they allow for localization with stopping times as well as truncation of large jumps. But here we only need the special case above.
The celebrated result of KurtzProtter [KP91] guarantees the convergence in the Skorokhod topology of the stochastic integrals of a sequence of càdlàg local martingales satisfying the UCV condition. Before we state it, recall that a sequence of processes in is called Ctight if it is tight in the Skorohod topology and all limit points are continuous processes.
Theorem 3.7 ([Kp91], Theorem 2.2).
Let be converging in probability in the Skorokhod topology (or jointly in distribution) to a pair . Suppose that is a sequence of local martingales which satisfies the UCV condition. Then, converges to as in probability (or weakly) in . In particular, if in addition , then is tight.
Corollary 3.8.
Let satisfy the same assumptions as in Theorem 3.7. If in addition is a sequence of semimartingales and converges to in probability (or jointly in distribution), where is a semimartingale and is an adapted càdlàg process of finite variation, then
in probability (or weakly) in . In particular, if in addition , then is tight.
Proof.
Using integration by parts, we have
so that the claim follows from the KurzProtter result together with another integration by parts:
∎
Throughout this section we will often use the following representation of additive functionals:
Lemma 3.9.
Let . Then we have for
(7) 
where is a martingale and is a martingale with respect to the backward filtration , such that
(8) 
Assume that is ergodic for . Then under the rescaling and and similarly for both processes converge in distribution in to a Wiener process, and by (8) they satisfy the UCV condition. If and for , then
(9)  
Proof.
The representation (7) is obtained e.g. by applying Dynkin’s formula to and on , and then computing . If is in the domain of , then also (8) follows from Dynkin’s formula; otherwise we use an approximation argument, see p.35 of [KLO12]. For the convergence of and see the proof of Theorem 2.32/2.33 in [KLO12]. The representation for follows by writing the integral against as a limit of Riemann sums – note that is continuous and of finite variation, so the integral is defined pathwise and we do not need to worry about quadratic covariations or the difference between forward and backward integral. ∎
For and we define the modulus of continuity
We will need the following lemma:
Lemma 3.10 ([Js03], Proposition VI.3.26).
The sequence is Ctight if and only if the following two conditions hold:

For all we have

for all we have
If is a sequence of processes in , then these two conditions are equivalent to tightness in the uniform topology.
Since the uniform modulus of continuity is subadditive, i.e. , it follows from this Lemma that the sum of two Ctight sequences is again Ctight. Note that the same is not necessarily true for sequences that are tight in the Skorohod topology on .
Lemma 3.11.
Under the assumptions of Lemma 3.2 the sequence is tight in .
Proof.
By Lemma 3.2 and Remark 2.2 it suffices to show that is tight in . We shall use Lemma 3.10. Since the set is dense in , see Claim B on p.42 of [KLO12], we can find so that . Then eq. (9) from Lemma 3.9 (or rather a slight modification with the inner integral in the second term on the right hand side running from to instead of from to ) gives
By Theorem 3.7 together with Lemma 3.2 the two stochastic integrals are Ctight in (note that is adapted to ). The third term on the right hand side is Ctight by the characterization of Lemma 3.10. It remains to treat the term
By Corollary 3.8 the integral of the two martingales can be handled as before. The remaining term satisfies
by Lemma A.1 in the appendix. Combining all this with the necessity of the conditions in Lemma 3.10, we get
by Chebyshev’s inequality, and similarly we get by bounding :
Hence satisfies the assumptions of Lemma 3.10 and therefore it is tight in