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Diffusive and Rough Homogenisation in Fractional Noise Field

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Diffusive and Rough Homogenisation in Fractional Noise Field Diffusive and rough homogenisation in fractional noise field Johann Gehringer and Xue-Mei Li Imperial College London * June 23, 2020 Abstract With recently developed tools, we prove a homogenisation theorem for a random ODE with short and long-range dependent fractional noise. The effective dynamics are not necessarily diffusions, they are given by stochastic differential equations driven simultaneously by stochastic processes from both the Gaussian and the non-Gaussian self-similarity universality classes. A key lemma for this is the ‘lifted’ joint functional central and non-central limit theorem in the rough path topology. keywords: passive tracer, fractional noise, multi-scale, functional limit theorems, rough differential equations MSC Subject classification: 34F05, 60F05, 60F17, 60G18, 60G22, 60H05, 60H07, 60H10 Contents 1 Introduction 2 2 Preliminaries 5 2.1 Hermiteprocesses................................ ........ 5 2.2 FractionalOrnstein-Uhlenbeckprocesses . ................ 6 2.3 HermiteRank..................................... ..... 7 2.4 JointfunctionalCLT/non-CLT . .......... 7 2.5 AssumptionsandConventions . .......... 8 3 Lifted joint functional limit theorem 9 arXiv:2006.11544v1 [math.PR] 20 Jun 2020 3.1 Relativecompactnessofiteratedintegrals . ................. 10 3.2 Young integral case (functional non-CLT in rough topology) ................. 14 3.3 Itô integral case (functionalCLT in roughtopology) . ................... 15 3.4 ProofofTheoremB................................. ...... 22 3.5 ProofoftheconditionalintegrabilityoffOU . ................. 22 4 Multi-scale homogenisation theorem 24 5 Appendix 26 5.1 Someroughpaththeory.... .... .... ... .... .... .... .. ........ 26 *[email protected], [email protected] 1 1 Introduction Fractional noise is the ‘derivative’of a fractional Brownian motion. Its covarianceat times separated by a span 2H 2 2H 1 1 s is ̺˜(s) 2H(2H 1) s − +2H s − δs where H is the Hurst parameter taking values in (0, 1) 2 ∼ − | | | |1 1 \{ } and δs is the Dirac measure. The ‘H = ’ case is white noise. If H > , ̺ds˜ = which means that the 2 2 R ∞ noise has non-integrable long range dependence (LRD). If H < 1 , the process is negatively correlated. Just 2 R as white noise is used for modelling noise coming from a large number of independent random components, fractional noise is used for modelling Long range dependence (LRD). LRDs are observed in nature and in time series data. We study the two scale passive tracer problem, this is also called the tagged particle problem, with fractional noise. ε We consider a slow/fast system in which the slow variables are given by a random ODE x˙ t = G(xt,yt ). This touches on two problems. The first is the passive tracer problem modelling the motion of a tagged particle in a disturbed flow, not necessarily incompressible, which allows simulation of the turbulent from the Lagrangian description. The other is the dynamical description for Brownian particles in a liquid at rest. The slow variables evolve in their natural time scale, while the fast random environmentevolves in the microscopic scale ε. The aim is to extract a closed effective dynamics which approximates the slow variables when ε is sufficiently small. This effective dynamics will be obtained from the persistent effects coming from the fast- moving variables through adiabatic transmission. If the environment is stationary strong mixing noise with sufficiently fast rate of convergence, the homogenisation problem is synonymous with ‘diffusion creation’, and is therefore also known as diffusive homogenisation. There have been continuous explorations of the diffusive homogenisation problem, see [Gre51, Has66, Kub57, KV86, LOV00, PK74, Tay21, KLO12] and the references therein. Recently long range dependent noises are also studied in several papers in the context of homogeneous incompressible fluids, however, they inevitably fall within the central limit theorem regimes [FK00, KNR12] and the effective dynamics are either Brownian motions or fractional Brownian motions. We will study a family of vector fields without spatial homogeneity, the resulting dynamics can take the form of a process resembles locally a fractional Brownian motion and more generally they compromise of a larger class of stochastic dynamical systems of the form n N k k dxt = fk(xt) dX + fk(xt)dX , x = x , (1.1) ◦ t t 0 0 k k n X=1 =X+1 k where Xt is a Wiener process for k n and otherwise a Gaussian or a non-Gaussian Hermite process. To our best knowledge, this presents a new≤ effective limit class. In these equations, the symbol denotes the Stratonovich integral and the other integrals are in the sense of Young integrals. ◦ The homogenisation problem we consider is: N ε ε ε x˙ = αk(ε) fk(x ) Gk(y ), t t t (1.2) k X=1 ε x0 = x0, ε where y = y t and yt are the short and long range dependent stationary fractional Ornstein-Uhlenbeck ε 1 processes (fOU) with Hurst parameter H (0, 1) 2 and one time probability distribution µ, the centred p ∈ \{ } 1 real valued functions Gk L (µ) transforms the noise. If fk are in b and Gk are bounded measurable, the ∈ ε N ε ε C solutions to the equations x˙ t = k=1 fk(xt )Gk(yt ) will be approximated by the averaged dynamics which, in this case, is the trivial ODE x˙ t = 0, c.f. [LH19] and [LS]. A homogenisation theorem will then describe the fluctuation around this average,P for this we must rescale the vector fields to arrive at a non-trivial limit. The different scales αk(ε) are reflections of the non-strong mixing property of the noise, they tend to as ∞ 2 ε 0 at a speed tailored to the transformations Gk. These scales determine the local self-similar property of → 2 1 the limit. If G is an L function with Hermite rank m, to be defined below, then m = 2(1 H) is the critical value for the limit to be locally a Brownian motion. If m is smaller, the effective limit is− locally a Hermite process of rank m, otherwise a Wiener process. Our main theorem is the following. We take αk(ε) to be α(ε,H∗(mk)), the latter is defined by (1.3). 1 3 d d pk Theorem A Let H (0, 1) 2 , fk b (R ; R ) and Gk L (R; R,µ) be real valued functions satisfying Assumption∈ 2.10. Then\ { the} solutions∈ C of (1.2) converge weakly∈ in γ, on any finite time interval and for any γ ( 1 , 1 1 ), to the solution of (1.1). C ∈ 3 2 − mink≤n pk The linear contraction in the Langevin equation and the exponential convergence of the solutions would lead to the belief that it mixes as fast as the Ornstein-Uhlenbeckprocess. But, the auto-correlation functions of the increment process, which measures how much the shifted process remembers, exhibits power law decay. 1 For H > 2 , the auto correlation function is not integrable. Conventional tools are not applicable here, we turn to the theory of rough path differential equations and view (1.2) as rough differential equations driven by stochastic processes with a parameter ε. By the continuity theorem for solutions of rough differential equations, it is then sufficient to prove the convergence of these drivers in the rough path topology. For t ε continuous processes this concerns the scaling limits of the path integrals of the form 0 Gk(ys )ds together with their canonical lifts. Using rough path theory for stochastic homongenisation is a recent development, in [KM17, BC17], this was used for diffusive homogenisation. Proving and formulatingR an appropriate functional limit theorem, however, turned out to be one of our main endeavours. For independent identically distributed random variables, the central limit theorems (CLTs) states that 1 n √n k=1 Xk converges to a Gaussian distribution. For correlated random variables, non-Gaussian distribu- tions may appear. One of these was proved by Rosenblatt: Let Zn be a stationary Gaussian sequence with P d 1 2 d 1 correlation ̺(n) n− where d (0, 2 ) and let Yn = (Zn) 1 then n − Yn converges to a non-Gaussian distribution. To emphasise∼ the non-Gaussian∈ nature, those limit− theorems with non-Gaussian limits are re- ferred to ‘non-Central Limit Theorems’ (non-CLTs). A functional limit theorem concerns path integrals of t ε functionals of a stochastic process yt. For a centred function G, it states that limε 0 √ε 0 G(ys)ds con- verges to a Brownian motion. Non-CLTs and functional non-CLTs were extensively→ studied [MT07, BH02, BM83, Taq75], these were then shown to hold for a larger class of functions [CNN20, NP05]R with Malli- avin calculus. In a nutshell, for a class of Gaussian processes and for a centred L2 function G with the t ε scaling constant depending on its Hermite rank m, the limit of α(ε) 0 G(ys )ds will be a BM if the scale is 1 or 1 ; otherwise it is a self-similar Hermite process of degree m with self-similar exponent √ε √ε ln(ε) R | | H∗(m)= m(H 1)+1. We will use functional limit theorems for both cases. − Let α(ε,H∗(m)) be positive constants as follows, they depend on m,H and ε and tend to as ε 0, ∞ → 1 1 √ε , if H∗(m) < 2 , 1 1 α(ε,H (m)) = , if H∗(m)= , (1.3) ∗ √ε ln(ε) 2 H∗|(m) 1| 1 ε − , if H∗(m) > 2 . 1 Observe that H∗ decreases with m andH∗(1) = H. If H 2 we only see the diffusion scale. We state below our key limit theorem, the lifted joint functional limit theorem≤ in the rough path topology, c.f. (5.2), see §3.4. The proof for the main theorem is finalised in §4. 1 Theorem B (Lifted joint functional CLTs/ Non-CLTs) Let H (0, 1) 2 and fix a finite time horizon 2 ∈ \{ } T .
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