Additive Functionals As Rough Paths

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Weierstraß-Institutfür Angewandte Analysis und StochastikLeibniz-Institut im Forschungsverbund Berlin e. V.PreprintISSN 2198-5855Additive functionals as rough pathsJean–Dominique Deuschel1 , Tal Orenshtein1,2 , Nicolas Perkowski3submitted: February 5, 20201TU BerlinStraße des 17. Juni 13610623 BerlinGermanyE-Mail: deuschel@math.tu-berlin.de32Weierstrass InstituteMohrenstr. 3910117 BerlinGermanyE-Mail: tal.orenshtein@wias-berlin.deFU BerlinArnimallee 714195 BerlinGermanyE-Mail: perkowski@math.fu-berlin.deNo. 2685Berlin 20202010 Mathematics Subject Classification. 60F17, 60H10, 60J55, 60K37.Key words and phrases. Rough paths, invariance principles in the rough path topology, additive functionals of Markov processes, Kipnis–Varadhan theory, homogenization, random conductance model, random walks with random conductances.We gratefully acknowledge financial support by the DFG via Research Unit FOR2402 — Rough paths, SPDEs and relatedtopics. The main part of the work of T.O. was carried out while he was employed at HU Berlin and TU Berlin. The mainpart of the work of N.P. was carried out while he was employed at MPI MIS Leipzig. N.P. gratefully acknowledges financialsupport by the DFG via the Heisenberg program.

Edited byWeierstraß-Institut für Angewandte Analysis und Stochastik (WIAS)Leibniz-Institut im Forschungsverbund Berlin e. V.Mohrenstraße 3910117 BerlinGermanyFax:E-Mail:World Wide Web: 49 30 rlin.de/

Additive functionals as rough pathsJean–Dominique Deuschel, Tal Orenshtein, Nicolas PerkowskiAbstractWe 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 weconsider the Itô rough path, then we see a correction to the iterated integrals even though theunderlying Markov process is reversible. If we consider the Stratonovich rough path, then thereis no correction. The second example is a non-reversible Ornstein-Uhlenbeck process, while thelast example is a diffusion in a periodic environment.As a technical step we prove an estimate for the p-variation of stochastic integrals with respectto martingales that can be viewed as an extension of the rough path Burkholder-Davis-Gundyinequality for local martingale rough paths of [FV08, CF19, FZ18] to the case where only theintegrator is a local martingale.1IntroductionIn recent years there has been an increased interest in the link between homogenization and roughpaths. It had been observed previously that homogenization often gives rise to non-standard roughpath limits [LL03, FGL15]. The more recent investigations were initiated with the work of Kelly andMelbourne[KM17, KM16, Kel16] who study rough path limits of additive functionals of the form R n·n 0 f (Ys )ds, where Y is a deterministic dynamical system with suitable mixing conditions. Inthat way they are able to prove homogenization results for the convergence of deterministic multiscalesystems of the type dX n nb(Xtn , Ytn )dt,dY n nf (Ytn )dt,for which under suitable conditions X n converges to an autonomous stochastic differential equation.This line of research was picked up and extended for example by [BC17, CFK 19b, CFK 19a, 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 randomwalks in random environment inRrough path topology, we want to study rough path invariance princi n·ples for additive functionals n 0 f (Xs )ds of Markov processes X in generic situations. If we areonly interested in a central limit theorem at a fixed time, then there are of course many results of thistype and many ways of showing them. See for example [Pel19] for a recent and fairly general result. Aparticularly 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 itwas developed by Kipnis and Varadhan [KV86] for reversible Markov processes and later extended toDOI 10.20347/WIAS.PREPRINT.2685Berlin 2020

J.–D. Deuschel, T. Orenshtein, N. Perkowski2many other situations; see the nice monograph [KLO12]. Here we extend this approach to the roughpath topology and we study some applications to model problems like random walks among randomconductances, additive functionals of Ornstein-Uhlenbeck processes, and periodic diffusions.This can also be seen as a complementary direction of research with respect to the recent advancesin regularity structures [BHZ19, CH16, BCCH17], where the aim is to find generic convergence resultsfor models associated to singular stochastic PDEs. In those works the equations tend to be extremelycomplicated, but the approximation of the noise is typically quite simple (the prototypical example isjust a mollification of the driving noise, but [CH16] also allow some stationary mixing random fields thatconverge to the space-time white noise by the central limit theorem). In our setting the equation thatwe study is very simple (just a stochastic ODE), but the approximation of the noise is very complicatedand (at least for us) it seems difficult to check whether the conditions of [CH16, Theorem 2.34] aresatisfied 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 (η({x, y}))x,y Zd :x y on the bonds of Zd (wherex y means x and y are neighbors). Then we let a continuous time random walk move along Zd ,with jump rate η({x, y}) from x to y (resp. from y to x). We are interested in the large scale behavior,i.e. we study n 1/2 Xnt for n . It is well known that the path itself converges in distribution underthe annealed law to a Brownian motion B with an effective diffusion coefficient. Our contribution is toextend this convergence to the rough path topology, which allows us for example to understand thelimit of discrete stochastic differential equationsndYtn σ(Yt )dXtn ,(1)but also of SPDEs driven by X n . And here we encounter a surprise: Even though X is in a certainsense reversible (more precisely the underlyingMarkov process of the environmentas seen from theR·R·walker is reversible), the iterated integrals 0 Xsn dXsn do not converge to 0 Bs dBs , but insteadwe see a correction: We have Z · Z ·nnnX ,Xs dXs B,Bs dBs Γt ,00where Γ is a correction given by11Γ hB, Bi1 Eπ [η({0, e1 })]Id (hB, Bi1 Eπ [η({0, e1 }) η({0, e1 })]Id )22for the law π of the random conductances. Of course, Γ vanishes if the conductances are deterministic(i.e. if π is a Dirac measure). But if the conductances are truly random, then typically the effectivediffusion is not just given by the expected conductance, and in d 1 one can even show that this isnever the case (see the discussion at the top of p.89 of [KLO12]). Therefore, Γ is typically non-zero,and the solution Y n of (1) converges to the solution Y ofdYt σ(Yt )dBt X k σ·j (Yt )σk (Yt )Γj dt.j,k, If on Rthe other hand we denote by X̃ n the linear interpolation of the pure jump path X n , then(X̃ n , 0 X̃sn dX̃sn ) converges to the limit that we would naively expect, namely to the Stratonovichrough path above B . 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 resultsDOI 10.20347/WIAS.PREPRINT.2685Berlin 2020

Additive functionals as rough paths3of this type (naive limit for the Stratonovich rough path, correction for the Itô rough path), but it seemsto be a generic phenomenon. The same effect appears for periodic diffusions, and we expect to see itfor nearly all models treated in the monograph [KLO12]. On the other hand, for ballistic random walksin random environment, after centering, a correction to the Stratonovich rough path is identified interms of the expected stochastic area on a regeneration interval [LO18, Theorem 3.3]. Moreover, forrandom walks in deterministic periodic environments simple examples for non-vanishing correctionsare available [LS18, Section 1.2] (or [LO18, Section 4.2]). For processes that can be handled with theKipnis-Varadhan approach we generically expect to see a correction to the Stratonovich rough path ifand only if the underlying Markov process is non-reversible.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 resultin [KLO12]. The proof is based on recent advances on Itô rough paths with jumps due to Friz andZhang [FZ18], on stability results for Itô integrals under the so called UCV condition by Kurtz and Protter [KP91], on Lépingle’s Burkholder-Davis-Gundy inequality in p-variation [Lép75], and on repeatedintegrations by parts together with a new estimate on the p-variation of stochastic integrals (Proposition 3.13). In Section 4 we apply our abstract result to three model problems: random walks amongrandom conductances, additive functionals of Ornstein-Uhlenbeck processes, and periodic diffusions.Finally, Section 5 contains the proof of Proposition 3.13 which might be of independent interest.Notation For two families (ai )i I , (bi )i I of real numbers indexed by I the notation ai . bi meansthat ai c bi for every i I where c (0, ) is a constant. Let T : {s, t [0, T ] : s t} forT 0. We interpret any function X : [0, T ] Rd also as a function on T via Xs,t : Xt Xs ,(s, t) T . For a metric space (E, d) we write C([0, T ], E) resp. D([0, T ], E) for the continuousresp. càdlàg functions from [0, T ] to E . A function X : T E is called continuous resp. càdlàgif for all s [0, T ) the map t 7 Xs,t on [s, T ] is continuous resp. càdlàg, and we write C( T , E)resp. D( T , E) for the corresponding function spaces.2Elements of rough path theoryHere we recall some basic elements of rough path theory for Itô rough paths with jumps. See [FZ18]for much more detail.Let us write kXk ,[0,T ] : supt [0,T ] Xt (resp. kXk ,[0,T ] : sup(s,t) T Xs,t ) to denote theuniform norm of X D([0, T ], Rd ) (resp. X D( T , Rd d )). For 0 p and a normedspace (E, · E ), we define the p-variation of Ξ : T E (and so in particular of Ξ : [0, T ] E )by kΞkp,[0,T ] : supPXp Ξs,t 1/p [0, ],(2)[s,t] Pwhere the supremum is taken over all finite partitions P of [0, T ] and the summation is over all intervals[s, t] P . Note that for any 0 p q , we have that kΞkq,[0,T ] kΞkp,[0,T ] .Definition 2.1 (p-variation rough path space). For p [2, 3), the space Dp-var ([0, T ], Rd Rd d )(resp. Cp-var ([0, T ], Rd Rd d )) of càdlàg (resp. continuous) p-variation rough paths is defined byDOI 10.20347/WIAS.PREPRINT.2685Berlin 2020

J.–D. Deuschel, T. Orenshtein, N. Perkowski4the subspace of all functions (X, X) D([0, T ], Rd ) D( T , Rd d ) satisfying Chen’s relation,that is,Xr,t Xr,s Xs,t Xr,s Xr,t(3)for 0 r s t T , and1/2 X0 kXkp,[0,T ] kXkp/2,[0,T ] .(4)The p-variation Skorohod distance on Dp-var ([0, T ], (Rd , Rd d )) is σp,[0,T ] ((X, X), (Y, Y)) : inf { λ kX λ Y λkp,[0,T ] kX (λ, λ) Y (λ, λ)kp/2,[0,T ] },λ ΛTwhere ΛT are the strictly increasing bijective functions from [0, T ] onto itself, and λ supt [0,T ] λ(t) t . The uniform Skorohod distance is defined similarly, except with the p-variation respectively p/2variation distance replaced by the uniform distance; see [FZ18, Section 5] for details.For X, Y D([0, T ], Rd ) we use the notationisZ tZYs dXs : 0Rt0Ys dXs for the left-point Riemann integral, thatYs dXs : lim Xn (0,t]Yu Xv Xu ,[u,v] Pnwhenever this limit is well defined along an implicitly fixed sequence of partitions (Pn ) of [0, t] withmesh size going to zero. Note that if X is a semimartingale and Y is adapted to the same filtration,then this definition coincides with the Ito integral. We remark also that the iterated integralsZtZXs,u dXu Xs,t : sXs,u dXu ,(s,t]satisfy Chen’s relation (3). Moreover, so do X̃s,t : Xs,t (t s)Γ, for any fixed matrix Γ.Remark 2.2. Note that by Chen’s relation Xs,t X0,t X0,s X0,s Xs,t whenever 0 6 s 6 t 6 T ,and thereforekX Yk ,[0,T ] sup Xs,t Ys,t 06s t6T. kX0,· Y0,· k ,[0,T ] (kX0,· k ,[0,T ] kY0,· k ,[0,T ] )kX0,· Y0,· k ,[0,T ] .Consequently, the uniform resp. Skorohod distance of the (one-parameter) paths (X0,· , Xn0,· ) and(Y0,· , Y0,· ) controls the uniform resp. Skorohod distance of (X, X) and (Y, Y).The following lemma by [FZ18] will be useful in the sequel.Lemma 2.3. Let (Z n , Zn ) be a sequence of càdlàg rough paths and let p 3. Assume that thereexists a càdlàg rough path (Z, Z) such that (Z n , Zn ) (Z, Z) in distribution in the Skorohod (resp.uniform) topology and that the family of real valued random variables (k(Z n , Zn )kp,[0,T ] )n is tight.Then (Z n , Zn ) (Z, Z) in distribution in the p0 -variation Skorohod (resp. uniform) topology for allp0 (p, 3).Proof. This follows from a simple interpolation argument, see the proof of Theorem 6.1 in [FZ18].DOI 10.20347/WIAS.PREPRINT.2685Berlin 2020

Additive functionals as rough paths5Invariance principles for rough path sequences guarantee the convergence of the solutions to roughdif

Rough paths, invariance principles in the rough path topology, additive functionals of Markov pro-cesses, Kipnis–Varadhan theory, homogenization, random conductance model, random walks with random conductances. We gratefully acknowledge financial support by the DFG via Research Unit FOR2402 — Rough paths, SPDEs and related topics. The main part of the work of T.O. was carried out while he .

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