Long time dynamics and disorder-induced traveling waves in the stochastic Kuramoto model
aa r X i v : . [ m a t h . P R ] J un LONG TIME DYNAMICS AND DISORDER-INDUCED TRAVELINGWAVES IN THE STOCHASTIC KURAMOTO MODEL
ERIC LUC¸ ON AND CHRISTOPHE POQUET
Abstract.
The aim of the paper is to address the long time behavior of the Kuramotomodel of mean-field coupled phase rotators, subject to white noise and quenched frequen-cies. We analyse the influence of the fluctuations of both thermal noise and frequencies(seen as a disorder) on a large but finite population of N rotators, in the case where thelaw of the disorder is symmetric. On a finite time scale [0 , T ], the system is known to beself-averaging: the empirical measure of the system converges as N → ∞ to the deter-ministic solution of a nonlinear Fokker-Planck equation which exhibits a stable manifoldof synchronized stationary profiles for large interaction. On longer time scales, compe-tition between the finite-size effects of the noise and disorder makes the system deviatefrom this mean-field behavior. In the main result of the paper we show that on a timescale of order √ N the fluctuations of the disorder prevail over the fluctuations of thenoise: we establish the existence of disorder- induced traveling waves for the empiricalmeasure along the stationary manifold. This result is proved for fixed realizations ofthe disorder and emphasis is put on the influence of the asymmetry of these quenchedfrequencies on the direction and speed of rotation of the system. Asymptotics on thedrift are provided in the limit of small disorder. Introduction
Long time dynamics of mean-field interacting particle systems.
The macro-scopic behavior of numerous stochastic interacting particle systems appearing in physicsor biology is usually described by nonlinear partial differential equations. In this context,systems of diffusions in all-to-all interactions, that is mean-field particle systems [32, 33],have attracted much attention in the past years, since they are relevant in many situationsfrom statistical physics (synchronization of oscillators [1, 27, 41]) to biology (emergenceof synchrony in neural networks [3, 9]) and have provided particle approximations for var-ious PDEs (see [31, 10] and references therein). From a statistical physics point of view,a natural extension of these models concerns similar particle systems in a random envi-ronment, that is when the particles obey to the influence of an additional randomness,or disorder , representing inhomogeneous behaviors between particles. Such a modelingis particularly relevant in a biological context, where each particle/diffusion captures thestate of one single individual (activity of a neuron, phase in a circadian rythm) and the dis-order models intrinsic dynamical behavior for each individual (e.g. inhibition or excitationin populations of heterogeneous neurons [3, 9]).The aim of the paper is to address the influence of the disorder on the long time dynamicsof a large but finite population of mean-field interacting diffusions with noise. A crucial
Date : October 4, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Kuramoto synchronization model - mean-field particle systems - disorderedmodels - nonlinear Fokker-Planck PDE - long time dynamics - traveling waves - stochastic partial differentialequations. aspect in this perspective is the notion of self-averaging : in the limit of a large number ofindividuals and/or on a long time scale (in a way that needs to be made precise), is themacroscopic behavior of the system the same for every typical realization of the disorder?If not, is it possible to quantify the influence of the fluctuations of the random environmenton the behavior of the system?It appears that the analysis of such mean-field systems differs significantly depending onthe time scale one considers. On a time scale of order 1 (w.r.t. the size of the population),it is now well-known that the macroscopic behavior of mean-field particle systems are welldescribed by nonlinear PDEs of McKean-Vlasov type [20, 33]. A vast literature existson the links between the microscopic system and its mean-field limit (fluctuations, largedeviations and finite time dynamics) mostly in the non-disordered case (see e.g. [18, 34, 43]and references therein) but also for disordered systems [16, 28].When one considers longer time scales (w.r.t. the size of the population) and for a largebut finite number of particles, some randomness remains in the system so that Brownianfluctuations generally induce microscopic dynamics that may differ significantly from thedynamics of the mean-field equation. For mean-field systems without disorder, a vastliterature exists concerning fluctuations induced by thermal noise. In this respect, thenotion of uniform propagation of chaos has been addressed for several mean-field modelsby many authors (see e.g. [31, 8] for the granular media equation or [25, 39] for ranked-based models). In case the mean-field PDE admits an isolated stable fixed point, due tolarge deviation phenomena, the finite-size system exits from any neighborhood of the fixedpoint at exponential times in N ( N being the size of the population) [17, 35], whereasin case of an unstable fixed point, the system escapes at a time scale of order log N [38].Fewer results exist in the case where the mean-field PDE admits a whole stable curve ofstationary solutions. In [7, 15], the effect of thermal noise is considered for the mean-fieldplane rotators model [6] which is known to admit in the limit as N → ∞ a stable circle ofstationary solutions. In this case, the finite size particle system has Brownian fluctuationson time scales of order N .In the case of disordered systems, we are not aware of any similar analysis on longtime dynamics of mean-field interacting particles. The present work could be seen as afirst result in this direction. In particular, we provide in Theorem 2.3 a rigorous andquantitative justification to a phenomenon already observed by Balmforth and Sassi [4]on the basis of numerical simulations.1.2. The stochastic Kuramoto model with disorder.
We address in this paper thelong time behavior of the Kuramoto model with noise and disorder, which describes theevolution of a population of rotators (the j th rotator being defined by its phase ϕ ωj ( t ) ∈ T := R / π Z ), given by the system of N > ϕ ωj ( t ) = δω j d t − KN N X l =1 sin( ϕ ωj ( t ) − ϕ ωl ( t )) d t + σ d B j ( t ) , j = 1 , . . . , N, t > , (1.1)where ( B j ) j =1 ,...,N is a family of standard independent Brownian motions, K , σ and δ arepositive parameters. In particular, δ > δ >
0, as it relies on perturbation results of the case where δ = 0.The Kuramoto model [1, 27, 41] is the main prototype for synchronization phenomenaand, due to its mathematical tractability, has been studied in details in the past years[6, 14, 21, 22]. ISORDER-INDUCED TRAVELING WAVES IN THE STOCHASTIC KURAMOTO MODEL 3
Remark 1.1.
Note that (1.1) is invariant by rotation: if ( ϕ ωj ( t )) j =1 ,...,N solves (1.1) , thenso does ( ϕ ωj ( t ) + α ) j =1 ,...,N for all α ∈ R . Moreover, by the change of variables t → t/σ ,one can get rid of the coefficient σ in front of the Brownian motions (up to the obviousmodifications δ → δ/σ and K → K/σ ). Hence, with no loss of generality, we suppose σ = 1 in the following. Following the point of view adopted at the beginning of this introduction, the system(1.1) presents two types of noise: in addition to the thermal noise ( B j ), the disorder in(1.1) is given by a sequence ( ω j ) j =1 ,...,N of i.i.d random variables with distribution λ ,independent from the Brownian motions. Each ω j represents an intrinsic inhomogeneousfrequency for the rotator ϕ ωj . The index ω in the notation ϕ ωj is used to emphasize thedependency of the system in the disorder.A crucial aspect in the understanding of the dynamics of (1.1) concerns the (possiblelack of) symmetry of the sequence ( ω j ) j > . First note that, by the obvious changeof variables ϕ ωj ( t ) ϕ ωt ( t ) − E ( ω ) t in (1.1), it is always possible to assume that theexpectation of the disorder E ( ω ) = R R ωλ ( d ω ) is zero (otherwise, we observe macroscopictraveling waves with speed E ( ω )). The asymmetry of the disorder can be given at differentscales. The most simple situation corresponds to a macroscopic asymmetry , that is whenthe law λ itself is asymmetric. With no loss of generality, we can for example assume that,on a macroscopic level, a majority of rotators will be associated to a positive frequencywhereas a minority will have negative frequencies. In the limit of an infinite population,this asymmetry makes the whole system rotate at a constant speed that depends onlyon the law λ and this rotation is noticeable at the scale of the nonlinear Fokker-Planckequation (1.3) associated to (1.1). This case has been the object of a previous paper (see[21], Theorem 2.2 and Section 2.2 below).The present paper is concerned with the situation where the law of the disorder issymmetric. Here, the previous argument cannot be applied since in the limit as N →∞ , the population is equally balanced between positive and negative frequencies: themacroscopic speed of rotation found in [21], Theorem 2.2 vanishes. Hence, the analysis oflong time dynamics of (1.1) requires a deeper understanding of the microscopic asymmetry of the disorder, that is the finite-size fluctuations of the disorder w.r.t. the thermal noise.An informal description of the dynamics of (1.1) is the following (see Figure 2 below): if theconstant K is sufficiently large, the mean-field coupling term leads to synchronization ofthe whole system along a nontrivial density. Even if λ is symmetric, finite-size fluctuationsof the sample ( ω j ) j =1 ,...,N make it not symmetric so that the fluctuations of the disordercompete with the fluctuations of the Brownian motions ( B j ) j =1 ,...,N and make the wholesystem rotate with speed and direction depending on the fixed realization of the disorder( ω j ) (and not only on the law λ itself). The main point of the paper is to give a rigorousmeaning to this phenomenon, noticed numerically in [4]: we will show that at times oforder √ N , the dynamics of (1.1) deviates from its mean-field limit, with the apparitionof synchronized traveling waves induced by the finite-size fluctuations of the disorder. Werefer to Paragraph 1.6 below for a precise description of this phenomenon.We present in the following subsections some well-known properties of (1.1) which areneeded to state our result. We describe in particular its infinite population limit onbounded time intervals and the existence of stationary measures for the limit system incase of symmetric disorder.1.3. Mean-field limit on bounded time intervals.
All the statistical information of(1.1) is contained in the empirical measure ( µ ωN,t ) t > ∈ C ([0 , ∞ ) , M ( T × R )) ( M being ERIC LUC¸ ON AND CHRISTOPHE POQUET the set of probability measures endowed with its weak topology) defined as µ ωN,t := 1 N N X j =1 δ ( ϕ ωj ( t ) ,ω j ) , t > . (1.2)When the distribution λ of the disorder satisfies R | ω | λ ( d ω ) < ∞ and the initial con-dition µ ωN, converges weakly to some p when N → ∞ , it is easy to see ([16, 28])that the empirical measure (1.2) converges weakly on bounded time intervals (that isin C ([0 , T ] , M ( T × R )) for all T >
0) to a deterministic limit measure whose density p t with respect ℓ ⊗ λ (where ℓ denotes the Lebesgue measure on T ) satisfies the followingsystem of nonlinear Fokker-Planck PDEs: ∂ t p t ( θ, ω ) = 12 ∂ θ p t ( θ, ω ) − ∂ θ (cid:16) p t ( θ, ω ) (cid:0) h J ∗ p t i λ ( θ ) + δω (cid:1)(cid:17) , ω ∈ Supp( λ ) , θ ∈ T , t > , (1.3)where J ( θ ) := − K sin( θ ) , (1.4)and h·i λ represents the integration with respect to λ : h J ∗ u i λ ( θ ) = R R R T J ( ψ ) u ( θ − ψ, ω ) d ψλ ( d ω ). We insist on the fact that in (1.3), ω is a real number in the support of λ , while in (1.1) and (1.2), it is an index emphasizing the dependency in the disorder ofthe system.Some properties of system (1.3) are detailed in [21]. In particular, if λ -almost surely, p ( · , ω ) is a probability measure then (1.3) admits a unique solution p t for all t > λ -almost surely, p t ( · , ω ) is also a probability measure, with positive density withrespect to the Lebesgue measure and is an element of C ∞ ((0 , ∞ ) × T , R ).1.4. Symmetric disorder.
As already mentioned, we consider the case where the law λ of the disorder is symmetric . We restrict our analysis to finite disorder: fix d > ω j ) j > take their values in { ω − d , ω − ( d − , . . . , ω d − , ω d } ,where ω i = − ω − i for all i = 0 , . . . , d . We denote as ( λ i ∈ [0 , , i = − d, . . . , d ) theprobability of drawing each ω i and assume that λ i = λ − i for all i = 1 , . . . , d . From nowon, the law of the disorder λ is identified with ( λ − d , . . . , λ d ). Note that we may supposein the following that ω = 0 Supp( λ ). The result still holds with obvious changes innotations.Under this hypothesis, almost surely, for sufficiently large N , each possible value ω i ofthe disorder appears at least once and we can rewrite (1.1) by regrouping the rotatorsinto (2 d + 1) sub-populations: for all i = − d, . . . , d , denote as N i the number of rota-tors ( ϕ ij ( t )) j =1 ,...,N i with frequency ω i . Obviously, N = P di = − d N i and the system (1.1)becomesd ϕ ij ( t ) = δω i d t − KN d X k = − d N k X l =1 sin( ϕ ij ( t ) − ϕ kl ( t )) d t + d B ij ( t ) , j = 1 , . . . , N i , i = − d, . . . , d . (1.5)In this framework, the empirical measure µ ωN,t in (1.2) can be identified with ( µ − dN,t , . . . , µ dN,t ),where µ iN is the empirical measure of the rotators with frequency ω i : µ iN,t = 1 N i N i X j =1 δ ϕ ij ( t ) , t > , i = − d, . . . , d , (1.6) ISORDER-INDUCED TRAVELING WAVES IN THE STOCHASTIC KURAMOTO MODEL 5 and its mean-field limit (1.3) can be identified with p t = ( p − dt , . . . , p dt ), solution to ∂ t p it ( θ ) = 12 ∂ θ p it ( θ ) − ∂ θ p it ( θ ) d X k = − d λ k J ∗ p kt ( θ ) + δω i !! , t > , i = − d, . . . , d . (1.7)1.5. Stationary solutions and phase transition.
A remarkable aspect of the Ku-ramoto model is that one can compute semi-explicitly the stationary solutions of (1.7),when λ is symmetric (see e.g. [40]): each stationary solution to (1.7) is the rotation of aprofile q = ( q − d , . . . , q d ) (i.e. given by q ( · + α ) for some α ∈ T )) of the form q i ( θ ) = S iδ ( θ, Kr ) Z iδ (2 Kr ) , (1.8)where for each i = − d, . . . , d , q i ( · ) is a probability density on T , S iδ ( θ, Kr ) is given by S iδ ( θ, x ) = e x cos θ +2 δω i θ (cid:20) (1 − e πδω i ) Z θ e − x cos u − δω i u d u + e πδω i Z π e − x cos u − δω i u d u (cid:21) , (1.9) Z iδ (2 Kr ) is a normalization constant and r is a solution of the fixed-point problem r = Ψ δ (2 Kr ) , (1.10)with Ψ δ ( x ) = d X k = − d λ k R π cos( θ ) S kδ ( θ, x ) d θZ kδ ( x ) . (1.11)We refer to [40] or [29], p. 75 for more details on this calculation. Computing the so-lution to the fixed-point relation (1.10) enables to exhibit a phase transition for (1.7):the value r = 0 always solves (1.10) and corresponds to the uniform stationary solution q ≡ (1 / π, . . . , / π ). It is the only stationary solution to (1.7) as long as K K c , for acertain critical parameter K c = K c ( δ, ( ω i ) i , ( λ i ) i ) >
1. This characterizes the absence ofsynchrony in case of small interaction. When
K > K c , this flat profile coexists with circlesof synchronized solutions corresponding to positive fixed-points in (1.10): each solution r > q given by (1.8) and to the circleof all its translation q ( · + α ), by invariance by rotation of the system (see Figure 1).However, several circles may coexist when K > K c and these circles may not be locallystable (even the characterization of these circles in full generality is unclear, see e.g. [29], § K > δ : it is indeed proved in [21], Lemma 2.3 that there exists δ = δ ( K ) > δ δ , the fixed-point problem (1.10) admits a uniquepositive solution r δ . We denote by q ,δ the corresponding profile given by (1.8) with r = r δ ,by q ψ,δ its rotation of angle ψ ∈ T (i.e. q ψ,δ ( · ) := q ,δ ( · − ψ )) and by M the correspondingcircle of stationary profiles (see Figure 1): M := { q ψ,δ : ψ ∈ T } . (1.12)It is proved in [21], Theorems 2.2 and 2.5 that the circle M is stable under the evolu-tion (1.7): the solution of (1.7) starting from an initial condition sufficiently close to M converges to a element q ψ,δ of M as t → ∞ . More details about this stability are givenin Section 2.3. Whenever it is clear from the context, we will use the notations q δ or q ψ instead of q ψ,δ , depending on the parameter we want to emphasize. ERIC LUC¸ ON AND CHRISTOPHE POQUET r r > r = 0 π q ψ,δ ( · ) M Ψ δ (2 Kr ) (a) Correspondance betweenfixed-points of Ψ δ ( · ) and sta-tionary solutions to (1.7). θ q ( θ ) q − ( θ ) q ( θ ) q − ( θ ) (b) A synchronized profile with d = 2, q = ( q − , q − , q , q ). Figure 1.
Fixed-point function Ψ δ ( · ) and stationary profiles when K = 5, d = 2, ω = 1, ω = 10 and δ = 0 . , , t = 0 t = 600 , , t = 30 0 2 4 6 t = 74 Figure 2.
Evolution of the marginal of the empirical measure (1.2) on T for a fixedchoice of the disorder ( N = 600, λ = ( δ − + δ ), K = 6). Starting from uniformlydistributed rotators on T ( t = 0), the empirical measure converges to a synchronizedprofile on the manifold M ( t = 6) and then moves (here to the right) at a constantspeed, on a time scale compatible with N / . Long time behavior.
Simulations of (1.5) (Figure 2) suggest an initial transitionof the system from an incoherent state to a synchronized one, during which the empiri-cal measures of the rotators approaches the circle M of synchronized stationary profiles.Secondly, the empirical measure remains close to M and travels at first order at constantspeed (which is random, depending on the realization of the disorder, see Figure 3) along M on the time scale N / t . Let us give some intuition of this phenomenon: to fix ideas,consider the case where d = 1, ω = − ω − = 1 and λ − = λ = . This corresponds to ISORDER-INDUCED TRAVELING WAVES IN THE STOCHASTIC KURAMOTO MODEL 7 − − . .
51 0 5 10 15 20 ce n t e r o f s y n c h r o n i z a t i o n time t Figure 3.
Trajectories of the center of synchronization for different realizations of thedisorder ( λ = ( δ − . + δ . ), K = 4, N = 400). the simplest decomposition in (1.5) between two subpopulations, one naturally rotatingclockwise ( ω i = +1) and the second rotating anti-clockwise ( ω i = − ω , . . . , ω N ) ∈ {± } N may lead, for example, to amajority of +1 with respect to −
1, so that the rotators with positive frequency induce aglobal rotation of the whole system in the direction of the majority. When N is large, thisasymmetry is small, typically of order N − / and is not sufficient to make the empiricalmeasure drift away from the attracting manifold M , but induces a small drift that becomesmacroscopic at times of order N / .The purpose of the paper is precisely to prove the existence of this random travelingwave and show that it is indeed an effect of the fluctuations of the disorder. Our approachconsists in a precise analysis of the dynamics of the empirical measure (1.6), which involvesboth disorder and thermal noise. One of the main difficulties is to control the thermalnoise term and prove that it does not play any role at first order on the N / -time scale.2. Main results and strategy of proof
The result.
Admissible sequence of disorder.
We stress the fact that the random traveling waves de-scribed above is essentially a quenched phenomenon, that is, true for a fixed realization ofthe disorder ( ω i ) i > . In particular, the result does not really depend on the underlyingmechanism that produced the sequence ( ω i ) i > , it only depends on the asymmetry of thissequence. We prove our result for any admissible sequence of disorder ( ω i ) i > , defined asfollows. Definition 2.1.
Fix a sequence ( ω i ) i > taking values in { ω − d , ω − ( d − , . . . , ω d − , ω d } andfor all N > , define the empirical proportions of frequencies in the N -sample ( ω , . . . , ω N ) λ kN := N k N , k = − d, . . . , d , (2.1) ERIC LUC¸ ON AND CHRISTOPHE POQUET where N k is the number of rotators with frequencies equal to ω k (recall Section 1.4). Definealso the fluctuation process associated to ( ω i ) i > by ξ N := ( ξ − dN , . . . , ξ dN ) , where ξ kN := N / ( λ kN − λ k ) , k = − d, . . . , d, N > , (2.2) where ( λ − d , . . . , λ d ) is given in Section 1.4. Note that P dk = − d ξ kN = 0 for all N > . Wesay that the sequence ( ω i ) i > is admissible if the following holds (1) Law of large numbers : for all k = − d, . . . , d , λ kN converges to λ k , as N → ∞ . (2) Central limit behavior : for all ζ > , there exists N (possibly depending on thesequence ( ω i ) i > ) such that for all N > N , max k = − d,...,d (cid:12)(cid:12)(cid:12) ξ kN (cid:12)(cid:12)(cid:12) N ζ . Remark 2.2 (Admissibility for i.i.d. variables) . An easy application of the Borel-CantelliLemma shows that any independent and identically distributed sequence of disorder ( ω i ) i > with law λ is almost surely admissible, in the sense of Definition 2.1.Main result. From now on, we fix once and for all an admissible sequence ( ω i ) i > in thesense of Definition 2.1. A convenient framework for the analysis of the dynamics of (1.6)and (1.7) corresponds to the space H − d , dual of the space H d , which is the closure of theset of vectors ( u − d , . . . , u d ) of regular functions u k with zero mean value on T under thenorm k u k ,d := d X k = − d λ k Z T ( ∂ θ u k ( θ )) d θ ! / . (2.3)Remark that if u is a vector of probability measures on T , then u naturally belongsto H − d , since the family of vectors given by a n,k ( θ ) = (0 , . . . , , √ cos ( nθ ) n √ λ k , , . . . ,
0) and b n,k ( θ ) = (0 , . . . , , √ sin ( nθ ) n √ λ k , , . . . ,
0) form an orthonormal basis of H d and for each suchvector u k u k − ,d = vuut d X k = − d ∞ X n =1 (cid:0) h u, a n,k i + h u, b n,k i (cid:1) π vuut d X k = − d ( λ k ) − . (2.4)More details on the construction of H − d are given in Appendix A. The main result of thepaper is the following. Theorem 2.3.
For all
K > , there exists δ ( K ) such that, for all δ δ ( K ) , there existsa linear form b : R d +1 → R (depending in K , δ , the probability distribution λ and thepossible values of the disorder ω i ) and a real number ε > such that, for any admissiblesequence ( ω i ) i > , any vector of probability measures p satisfying dist H − d ( p , M ) ε such that for all ε > , P (cid:16) k µ N, − p k − ,d > ε (cid:17) → , as N → ∞ , (2.5) then, there exists θ ∈ T (depending on p ) and a constant c such that for each finite time t f > and all ε > , denoting t N = cN − / log N , we have P sup t ∈ [ t N ,t f ] (cid:13)(cid:13)(cid:13) µ N,N / t − q θ + b ( ξ N ) t (cid:13)(cid:13)(cid:13) − ,d > ε ! → , as N → ∞ . (2.6) ISORDER-INDUCED TRAVELING WAVES IN THE STOCHASTIC KURAMOTO MODEL 9
Moreover, ξ b ( ξ ) has the following expansion in δ : for all ξ such that P dk = − d ξ k = 0 ,we have b ( ξ ) = δ d X k = − d ξ k ω k + O ( δ ) . (2.7)Theorem 2.3 is simply saying that, on a time scale of order N / , the empirical measure(1.6) is asymptotically close to a synchronized profile q ∈ M , traveling at speed b ( ξ N )along M . This drift depends on the asymmetry ξ N of the quenched disorder ( ω i ) i > .In (2.6), t N represents the time necessary for the system to get sufficiently close to themanifold M . Some particular cases and extensions.
First remark that the situation where the sample ofthe disorder ( ω i ) i =1 ,...,N is perfectly symmetric corresponds to ξ − iN = ξ iN for all i = 1 , . . . , d .In this case, the drift in (2.6) vanishes: Proposition 2.4.
If for all i = 1 , . . . , d , ξ − i = ξ i , then b ( ξ ) = 0 . In particular, if one chooses the disorder in such a way that ( ω i ) i =1 ,...,N is always symmetric(e.g. choose an even number of particles N and define each ω i to be alternatively ± N to see the first order of the expansion of the empirical measure µ N .Proof of Proposition 2.4 is given in Section 7.1.In case the sequence ( ω i ) i > is i.i.d. with law λ , a standard Central Limit Theoremshows that the drift b ( ξ N ) converges in law to a Gaussian distribution N (0 , v ), where v depends on K , δ , the probability distribution λ and the possible values of the disorder ω i . Proposition 2.5.
The following asymptotic of v holds when δ → : v = δ d X k = − d λ k ( ω k ) + O ( δ ) . (2.8)Proof of Proposition 2.5 is given in Section 7.2. Remark 2.6.
Without much modification in the proof, the result can be easily extendedto sequences ( ω i ) i > with fluctuations of order different from √ N , that is when for some a ∈ (0 , , ξ aN → ξ a , as N → ∞ , (2.9) for some vector ξ a where ξ aN := N a ( λ N − λ ) . In this case, the correct time renormalizationis N a and we obtain a result of the type P sup t ∈ [ t N ,t f ] (cid:13)(cid:13) µ N,N a t − q θ + b ( ξ a ) t (cid:13)(cid:13) − ,d > ε ! → , as N → ∞ . (2.10) Here, we only treat the case a = 1 / for simplicity. For smaller fluctuations of size N − a with a > , the time renormalization should be of order N . Since at this scale the effectsof the thermal noise appear, the limit phase dynamics should be of diffusive type and aprecise analysis of the different terms and symmetries that occur would be necessary to getthe proper drift in this case. Links with existing models.
Symmetric versus non-symmetric disorder.
This work is the natural continuation of [21],Theorems 2.2 and 2.5 in the case of a symmetric disorder. The purpose of [21] was toanalyze the dynamics of the nonlinear Fokker-Planck equation (1.3) for both symmetricand asymmetric law of the disorder. The main point is that understanding (1.3) is notsufficient in itself for the analysis of the finite size system (1.1) in the symmetric case,since it does not account for the finite-size effects of the disorder that are crucial here.As already mentioned, in the case where λ is asymmetric, one observes macroscopictravelling waves with deterministic drift at the scale of the nonlinear Fokker-Planck equa-tion (1.3). It is reasonable to think that an analysis similar to what has been done in thispaper would also show the existence of a finite order correction to this deterministic driftfor a large but finite system with quenched disorder.Some previous results already suggested the possibility of these disorder-induced trav-eling waves in the Kuramoto model. Namely, the purpose of previous work [28] was toprove a quenched fluctuation result for the empirical measure (1.2) around its mean-fieldlimit (1.3) on a finite time horizon [0 , T ]. The main conclusion of [28] was that thesefluctuations are disorder dependent and the long time analysis of the limiting fluctuations[30] suggested a non-self-averaging phenomenon for (1.1) similar to the one observed here. The case δ = 0 . This paper uses techniques previously developed in [7] in the context ofthe stochastic Kuramoto model without disorder, that is when one takes δ = 0 in (1.1):d ϕ j ( t ) = − KN N X l =1 sin( ϕ j ( t ) − ϕ l ( t )) d t + d B j ( t ) , j = 1 , . . . , N , (2.11)associated in the limit N → ∞ to the mean-field PDE ∂ t p t ( θ ) = 12 ∂ θ p t ( θ ) − ∂ θ (cid:16) p t ( θ ) J ∗ p t ( θ ) (cid:17) . (2.12)Similarly to (1.7) in Section 1.5, evolution (2.12) generates a stable circle M of stationarysynchronized profiles when K > K c (0) = 1 (see Section B.1 for further details). The model(2.11)-(2.12) has been the subject of a series of recent papers [6, 7, 22, 23], addressing thelinear and nonlinear stability of the circle of synchronized profiles M as well as the longtime dynamics of the microscopic system (2.11). The analysis of (2.12) strongly relies onthe reversibility of (2.11) (with the existence of a proper Lyapunov functional, see [6] formore details), whereas reversibility is lost when δ > M exponentially fast (that corresponds to the synchronization of the system (2.11)along a stationary profile solving (2.12)) and then stays close to M for a long time withhigh probability, while the phase of its projection on M performs a Brownian motion as N → ∞ which corresponds to a macroscopic effect of the thermal noise. The persistence ofproximity of the empirical measure to M for long times and the convergence of this phaseto a Brownian motion were in fact already established in the unpublished PhD Thesis [15]the authors of [7] were not aware of, using in particular moderate deviations estimates ofthe mean field process. Note that the techniques of [15] do not apply here, since a similaranalysis would involve moderate (or large) deviations in a quenched set-up, result that, tothe best of our knowledge, has not been proven so far (for averaged large deviations, see[16]).A significant difference between [7, 15] and the present analysis is that the Brownianexcursions in [7, 15] occur on a time scale of order N whereas it is sufficient to look at times ISORDER-INDUCED TRAVELING WAVES IN THE STOCHASTIC KURAMOTO MODEL 11 of order N / to see the traveling waves in the disordered case. This will entail significantsimplifications in the analysis of (1.5), since the detailed analysis on the thermal noiseperformed in [7, 15] will not be required here.Note also that, contrary to [7], we do not prove the first step of the phenomenondescribed in Figure 2, that is the initial approach of the system to a neighborhood ofthe manifold M in an exponentially short time, regardless of the initial condition. Thisresult would require a global stability result for the system of PDEs (1.7) which has notbeen proved for the moment, due to the absence of any Lyapunov functional for (1.7)when δ >
0. We prove our result for initial conditions belonging to some macroscopicneighborhood of M (see Section 6 for more details). SPDE models with vanishing noise.
This paper is related to previous works in the contextof SPDE models for phase separation. In [12, 19], the authors studied the Allen-Cahnmodel with symmetric bistable potential and vanishing noise. They showed that for aninitial data close a profile connecting the two phase, the interface performs a Brownianmotion. Some techniques initially introduced in these works, as the discretization of thedynamics in an iterative scheme, were developed in [7] in the context of the Kuramotomodel without disorder (making use of Sobolev spaces with negative exponents to dealwith empirical measures) and will have a central role in our analysis (see Section 2.6).The results of [12, 19] have been extended in [11] by considering small asymmetries in thepotential which induce a drift in the interface dynamics and by considering macroscopi-cally finite volumes [5], with effect a repulsion at the boundary for the phase. Stochasticinterface motions have also been recently studied in the context of the Cahn-Hilliard modelwith vanishing colored noise [2]. In this model, the limit behavior of the interface is givenby a SDE (or system of SDE’s in the case of several interfaces) with drift and diffusioncoefficients depending on coloration of the noise and on the length of the interface.2.3.
Linear stability of stationary solutions.
In the whole paper, we suppose that
K > δ > δ ( K ) >
0. This critical value δ ( K ) isdetermined by δ ( K ) = min( δ ( K ) , δ ( K )), where δ ( K ) ensures the existence of a uniquecircle M of stationary solutions (recall Section 1.5) and where δ ( K ) comes from thestability analysis of this circle (see Appendix B for more details).More precisely, our result relies deeply on the linear stability of the dynamical systeminduced by the limit system of PDEs (1.7) in the neighborhood of the circle of stationaryprofiles M . For ψ ∈ T , δ >
0, consider the operator L ψ,δ of the linearized evolution around q ψ,δ ∈ M given by( L ψ,δ u ) i = 12 ∂ θ u i − δω i ∂ θ u i − ∂ θ u i d X k = − d λ k ( J ∗ q kψ,δ ) + q iψ,δ d X k = − d λ k ( J ∗ u k ) ! , (2.13)for all i = − d, . . . , d with domain (cid:26) u = ( u − d , . . . , u d ) : u i ∈ C ( T ) and Z T u i ( θ ) d θ = 0 , ∀ i = − d, . . . , d (cid:27) . (2.14)Due to the invariance by rotation of the model (1.7), L ψ,δ is linked to L ,δ in an obviousway: L ψ,δ u ψ ( · ) = L ,δ u ( · ), where u ψ ( · ) = u ( · − ψ ), so that the operators ( L ψ,δ ) ψ ∈ T obviously share the same spectral properties. For any operator L , the usual notations σ ( L ) (resp. ρ ( L ) and R ( λ, L )) will be used for the spectrum of L (resp. its resolvent setand its resolvent operator for λ ∈ ρ ( L )). One can prove (see [21], Theorem 2.5 and Appendix B below) that for all 0 δ δ ( K ), L ψ,δ is closable in H − d , sectorial, has 0 for eigenvalue, associated to the eigenvector ∂ θ q ψ,δ ,which belongs to the tangent space of M in q ψ,δ (this reflects the fact that the dynamicsinduced by (1.7) on M is trivial) and that the rest of the spectrum is negative, separatedfrom the eigenvalue 0 by a spectral gap γ L >
0. More details about these questions aregiven in Appendix B.The fact that the eigenvalue 0 is isolated from the rest of the spectrum σ ( L ψ,δ ) r { } implies that H − d can be decomposed into a direct sum T ψ,δ ⊕ N ψ,δ , where T ψ,δ =Span( ∂ θ q ψ,δ ) such that the spectrum of the restriction of L ψ,δ to N ψ,δ (resp. T ψ,δ ) is σ ( L ψ,δ ) r { } (resp. { } ). We denote by P ψ,δ the projection on T ψ,δ along N ψ,δ and P sψ,δ = 1 − P ψ,δ . Both P ψ,δ and P sψ,δ commute with L ψ,δ . In particular, for all ψ ∈ T , δ >
0, there exists a linear form p ψ,δ satisfying, for all u ∈ H − d P ψ,δ u = p ψ,δ ( u ) ∂ θ q ψ,δ . (2.15)We also denote by C P and C L positive constants such that for all u ∈ H − d , t > k P ψ,δ u k − ,d C P k u k − ,d , (2.16) k P sψ,δ u k − ,d C P k u k − ,d , (2.17) (cid:13)(cid:13) e tL ψ,δ P sψ,δ u (cid:13)(cid:13) − ,d C L e − γ L t (cid:13)(cid:13) P sψ,δ u (cid:13)(cid:13) − ,d , (2.18) (cid:13)(cid:13) e tL ψ,δ u (cid:13)(cid:13) − ,d C L (cid:18) √ t (cid:19) k u k − ,d . (2.19)Inequality (2.18) is a consequence of [24], Theorem 1.5.3, p. 30 and (2.19) is proved inProposition B.7 in Appendix B. Once again, we will often drop the dependency in theparameters ψ or δ in P ψ,δ and P sψ,δ for simplicity of notations.A consequence of the contraction (2.18) along the space N ψ,δ is that M is locally stablewith respect to the evolution given by (1.7) (see for example exercise 6 ∗ of the Chapter 6of [24], or Theorem 2.2 of [21] for our particular model): for any p in a neighborhood of M , there exists ψ ∈ T such that the solution of (1.7) converges to q ψ,δ exponentially fast(with rate given by γ L ).2.4. Dynamics of the empirical measure.
The starting point of the proof of Theo-rem 2.3 is to write the semi-martingale decomposition (see Proposition 3.1) of the differencebetween the empirical measure µ N,t defined in (1.6) and any element of q ψ,δ ∈ M . Namely,define the process t ν N,t , t > ν iN,t := µ iN,t − q iψ,δ , i = − d, . . . , d. (2.20)The point is to write a mild formulation of this semi-martingale decomposition that makessense in the space H − d (recall that µ N,t and ν N,t belong to H − d due to (2.4)). This mildformulation involves in particular the semi-group e tL ψ,δ of the operator L ψ,δ (2.13) so thatone can take advantage of the contraction properties of this semi-group in the neighborhoodof the manifold M . Proposition 2.7.
For all
K > , for all δ δ ( K ) , the process ( ν N,t ) t > definedby (2.20) satisfies the following stochastic partial differential equation in C ([0 , + ∞ ) , H − d ) ,written in a mild form: ν N,t = e tL ψ,δ ν N, + Z t e ( t − s ) L ψ,δ ( D N − ∂ θ R N ( ν N,s )) d s + Z N,t , N > , t > , (2.21) ISORDER-INDUCED TRAVELING WAVES IN THE STOCHASTIC KURAMOTO MODEL 13 where D N = D N,ψ,δ := − ∂ θ q ψ,δ d X k = − d ( λ kN − λ k )( J ∗ q kψ,δ ) ! , (2.22) R N ( ν N,s ) = R N,ψ,δ ( ν N,s ) := d X k = − d ( λ kN − λ k ) J ∗ q kψ,δ ! ν N,s + q ψ,δ d X k = − d ( λ kN − λ k )( J ∗ ν kN,s )+ d X k = − d λ kN J ∗ ν kN,s ! ν N,s , (2.23) and Z N,t is the limit in H − d as t ′ ր t of Z N,t,t ′ defined by Z N,t,t ′ ( h ) = d X i = − d λ i N i N i X j =1 Z t ′ ∂ θ (cid:20)(cid:16) e ( t − s ) L ∗ ψ h (cid:17) i (cid:21) ( ϕ ij ( s )) d B ij ( s ) , (2.24) that we denote Z N,t ( h ) = d X i = − d λ i N i N i X j =1 Z t ∂ θ (cid:20)(cid:16) e ( t − s ) L ∗ ψ h (cid:17) i (cid:21) ( ϕ ij ( s )) d B ij ( s ) , (2.25) and where all the terms in (2.21) make sense as elements of C ([0 , ∞ ) , H − d ) . The proof of Proposition 2.7 may be found in Section 3. The term Z N,t in (2.21)represents the effect of the thermal noise on the system. The term involving D N is theone that produces the drift we are after on the time scale N / t , when the empiricalmeasure µ N,t is close to the manifold M . To make this drift appear, we rely on aniterative procedure, as explained in Section 2.6.2.5. Moving closer to the manifold M . We place ourselves in the framework of The-orem 2.3: we fix ε > p ∈ H − d suchthat dist H − d ( p , M ) ε with P (cid:16) k µ N, − p k − ,d > ε (cid:17) → N → ∞ , for all ε > ε will be chosen small enough in Section 6.The first step in proving our result is to show that the empirical measure µ N,t reaches aneighborhood of size N − / in a time of order log N . We use the projection defined in thefollowing lemma, whose proof can be found in Appendix C, along with several regularityresults. Lemma 2.8.
There exists σ > such that for all h such that dist H − d ( h, M ) σ , thereexists a unique phase ψ =: proj M ( h ) ∈ T such that P ψ ( h − q ψ ) = 0 and the mapping h proj M ( h ) is C ∞ . From now on, we fix a sufficiently small constant ζ , more precisely satisfying ζ < . (2.26)We prove the following result: Proposition 2.9.
Under the above hypotheses, there exists a phase θ ∈ T , an event B N such that P ( B N ) → and a constant c > such that for all ε > , for N sufficient large,on the event B N , the projection ψ = ψ N = proj M (cid:0) µ N,N / t N (cid:1) is well-defined and (cid:13)(cid:13)(cid:13) µ N,N / t N − q ψ (cid:13)(cid:13)(cid:13) − ,d N − / ζ , (2.27) and | ψ − θ | ε , (2.28) where t N = cN − / log N . We refer to Section 6 for a proof of this result. Since it relies on a discretization schemesimilar to the one we introduce in the next paragraph, we leave the details to Section 6.2.6.
Dynamics on the manifold M . We now place ourselves on the event B N (seeProposition 2.9), so that on the time N / t N we have k µ N,N / t N − q ψ k − ,d N − / ζ where ψ = proj M (cid:0) µ N,N / t N (cid:1) . The point is to analyse the dynamics of (2.21) on a timescale of order N / , using the knowledge we have on stability of the manifold M (recall(1.12)). The following iterative scheme we introduce is similar to ones used in [7, 12]. The iterative scheme.
We divide the evolution of the dynamics (2.21) in time intervals[ T n , T n +1 ] with T n = N / t N + nT where T is a constant independent of N , satisfying T > e − γ L T C L C P , (2.29)where the constants C L and C P where introduced in Section 2.3. The number of steps n f is chosen as n f := $ N / T ( t f − t N ) % . (2.30)The intuition of this discretization is the following: if for a certain n = 0 , , . . . , n f − µ T n = µ N,T n is close enough to the manifold M , we can define the phase α n of its projection on M by: α n := proj M ( µ N,T n ) . (2.31)This projection is in particular well defined when k µ T n − q α n − k − ,d σ , where the constant σ > M , we introduce the followingstopping couple (where the infimum corresponds to the lexicographic order):( n τ , τ ) = inf { ( n, t ) ∈ { , . . . , n f } × [0 , T ] : k µ T n − + t − q α n − k − ,d > σ } . (2.32)Using (2.32), we can define the following sequence of stopping times ( τ n , n = 1 . . . n f ): τ n := (cid:26) T if n < n τ ,τ if n > n τ , (2.33)and consider the stopped process µ ( n ∧ n τ − T + t ∧ τ n . The projection of this stopped processis well defined on the whole interval [ T , T n f ], so that we can now define rigorously therandom phases ψ n − defined as ψ n − := proj M ( µ ( n ∧ n τ − T + t ∧ τ n ) , n = 1 , . . . , n f . (2.34) ISORDER-INDUCED TRAVELING WAVES IN THE STOCHASTIC KURAMOTO MODEL 15 ψ n − corresponds to the phase of the projection of the process µ unless it has been stoppedand in that case, it is the phase at the stopping time. The object of interest here is theprocess ν n,t defined for n = 1 , . . . , n f as ν n,t := µ ( n ∧ n τ − T + t ∧ τ n − q ψ n − . (2.35)Using (2.21), we see that this process satisfies the mild equation ν n,t = e ( t ∧ τ n ) L ψn − ν n, − Z t ∧ τ n e ( t ∧ τ n − s ) L ψn − ( D ψ n − + R ψ n − ( ν n,s )) d s + Z n,t ∧ τ n , (2.36)where D ψ n − := D N,ψ n − ,δ (recall (2.22)), R ψ n − ( ν n ) := R N,ψ n − ,δ ( ν n ) (recall (2.23)) and Z n,t is defined as Z n,t ( h ) = d X i = − d λ i N i N i X j =1 Z t ∂ θ (cid:20)(cid:16) e ( t − s ) L ∗ ψn − h (cid:17) i (cid:21) ( ϕ ij ( T n − + s )) d B ij ( T n − + s ) . (2.37)Note that we drop here the dependence in N and δ for simplicity. Controlling the noise and a priori bound on the fluctuation process.
A key point in theanalysis of (2.36) is to show that one can control the behavior of the noise part Z n,t in(2.36) along the discretization introduced in the last paragraph. More precisely, for ζ chosen according to (2.26) and some positive constant C Z and defining the event A N = A N ( C Z ) := ( sup n n f sup t T k Z n,t k − ,d C Z r TN N ζ ) , (2.38)the purpose of Section 4 is precisely to prove that P ( A N ) tends to 1 as N → ∞ . With theknowledge of (2.38), one can prove that the process ν n remains a priori bounded: usingthat the sequence of the disorder ( ω i ) i > is admissible (recall Definition 2.1), we prove inProposition 5.1, Section 5, that on the event A N ∩ B N ,sup n n f sup t ∈ [0 ,T ] k ν n,t k − ,d = O ( N − / ζ ) , (2.39)as N → ∞ . Expansion of the dynamics on the manifold M . The last step of the proof consists inlooking at the rescaled dynamics of the phase of the projection of the empirical measureon M , that is the process Ψ Nt := ψ n t , (2.40)where ( ψ n ) n n f is given by (2.34) and n t := $ N / T ( t − t N ) % . (2.41)Namely, we prove in Propositions 5.2 and 5.3 that, with high probability as N → ∞ , thefollowing expansion holds: Ψ Nt = ψ + b ( ξ N ) t + O ( N − / ζ ) , (2.42)where b is the linear form of Theorem 2.3 and that µ N,N / t is close to q Ψ Nt with highprobability. Organization of the rest of the paper.
Section 3 is devoted to prove the mildformulation described in Paragraph 2.4. The control of the noise term in (2.37) is addressedin Section 4. The dynamics on the manifold M and the approach to the manifold arestudied in Sections 5 and 6 respectively. The asymptotics of the drift as δ → δ used throughout the paper.3. Proof of the mild formulation
Define L ,d as the closure of the space of regular test functions ( u − d , . . . , u d ) such that R T u k = 0 for all k = − d, . . . , d under the norm k u k ,d := d X k = − d λ k Z T u k ( θ ) d θ ! / , (3.1)and the space H αd ( α >
0) closure of the same set of test functions under the norm(denoting k · k the L -norm on T ) k u k α,d := (cid:13)(cid:13)(cid:13) ( − ∆ d ) α/ u (cid:13)(cid:13)(cid:13) ,d = d X k = − d λ k (cid:13)(cid:13)(cid:13) ( − ∆) α/ u k (cid:13)(cid:13)(cid:13) ! / , (3.2)where ∆ d denotes the Laplacian on T d +1 . We denote by H − αd the dual space of H αd . Wealso write, for any bounded signed measure m on T , the usual distribution bracket as h m , f i := Z T f ( θ ) m ( d θ ) , and for any vector ( m , . . . , m d ) of such measures h m , F i d := d X i = − d λ i (cid:10) m i , F i (cid:11) = d X i = − d λ i Z T F i ( θ ) m i ( d θ ) , the corresponding bracket weighted w.r.t. the disorder. Obviously, when the above mea-sure coincide with an L function, this expression coincides with the L scalar product h· , ·i ,d associated to (3.1).This section is devoted to prove Proposition 2.7. We begin first with a weak formulationof the SPDE (2.21). Proposition 3.1.
For all
K > , for all δ δ ( K ) , for any ( t, θ ) F t ( θ ) =( F − dt ( θ ) , . . . , F dt ( θ )) ∈ C , ([0 , + ∞ ) × T , R ) such that R T F t ( θ ) d θ = 0 , h ν N,t , F t i d = h ν N, , F i d + Z t (cid:10) ν N,s , ∂ s F s + L ∗ ψ,δ F s (cid:11) d d s + Z t h D N , F s i d d s + Z t h R N ( ν N,s ) , ∂ θ F s i d d s + M FN,t , N > , t > , (3.3) where D N , R N ( ν N ) are respectively defined in (2.22) and (2.23) and M FN,t := d X i = − d λ i N i N i X j =1 Z t ∂ θ F is ( ϕ ij ( s )) d B ij ( s ) . (3.4) ISORDER-INDUCED TRAVELING WAVES IN THE STOCHASTIC KURAMOTO MODEL 17 In (3.3) , the operator L ∗ ψ,δ is the dual in L ,d of the operator L ψ,δ in (2.13) : ( L ∗ ψ,δ v ) i = 12 ∂ θ v i + δω i ∂ θ v i + ( ∂ θ v i ) d X k = − d λ k J ∗ q kψ,δ − Z T ( ∂ θ v i ) d X k = − d λ k J ∗ q kψ,δ ! d θ − d X k = − d λ k J ∗ ( q kψ,δ ∂ θ v k ) . (3.5)We refer to Appendix B (see in particular Propositions B.3 and B.4) for a detailedanalysis of the spectral properties of the operator L ψ,δ and its dual L ∗ ψ,δ . All we need toretain here is that when δ is small, the operator L ψ,δ is sectorial in H − d and generatesa C -semi-group t e tL ψ,δ in this space. Moreover, on the space L ,d , one has that( e tL ψ,δ ) ∗ = e tL ∗ ψ,δ . Since the phase ψ is not relevant in this paragraph, we write forsimplicity q δ , L δ instead of q ψ,δ and L ψ,δ . Proof of Proposition 3.1.
Note that, using the definition of J ( · ) in (1.4) and of the empir-ical measure µ N,t in (1.6), the system (1.5) may be rewritten asd ϕ ij ( t ) = δω i d t + d X k = − d λ kN J ∗ µ kt (cid:0) ϕ ij ( t ) (cid:1) d t + d B ij ( t ) , i = − d, . . . , d . (3.6)Consider ( t, θ ) F t ( θ ) = ( F it ( θ )) i =1 ,...,d ∈ C , ([0 , + ∞ ) × T , R ) d such that for all t > R T F t ( θ ) d θ = 0. An application of Itˆo Formula to (1.5) gives, for i = 1 , . . . , d , j = 1 , . . . , N i , t > F it ( ϕ ij ( t )) = F i ( ϕ ij (0)) + Z t ∂ s F is ( ϕ ij ( s )) d s + 12 Z t ∂ θ F is ( ϕ ij ( s )) d s + Z t ∂ θ F is ( ϕ ij ( s )) δω i + d X k = − d λ kN J ∗ µ kN,s ( ϕ ij ( s )) ! d s + Z t ∂ θ F is ( ϕ ij ( s )) d B ij ( s ) . After summation over j = 1 , . . . , N i , we obtain, for i = 1 , . . . , d , h µ iN,t , F it i = h µ iN, , F i i + Z t * µ iN,s , ∂ s F is + 12 ∂ θ F is + ∂ θ F is δω i + d X k = − d λ kN ( J ∗ µ kN,s ) !+ d s + 1 N i N i X j =1 Z t ∂ θ F is ( ϕ ij ( s )) d B ij ( s ) . (3.7) Replacing µ iN,t by ν iN,t + q iδ in (3.7) (recall (2.20)), we obtain h ν iN,t , F it i + h q iδ , F it i = h ν iN, , F i i + h q iδ , F i i + Z t * ν iN,s + q iδ , ∂ s F is + 12 ∂ θ F is + ∂ θ F is δω i + d X k = − d λ kN J ∗ ( ν kN,s + q kψ ) !+ d s + 1 N i N i X j =1 Z t ∂ θ F is ( ϕ ij ( s )) d B ij ( s )= h ν iN, , F i i + Z t * ν iN,s , ∂ s F is + 12 ∂ θ F is + ∂ θ F is δω i + d X k = − d λ kN J ∗ q kδ !+ d s + Z t * ν iN,s , ∂ θ F is d X k = − d λ kN ( J ∗ ν kN,s ) + d s + Z t * q iδ , ∂ θ F is d X k = − d λ kN ( J ∗ ν kN,s ) + d s + h q iδ , F i i + Z t * q iδ , ∂ s F is + 12 ∂ θ F is + ∂ θ F is δω i + d X k = − d λ kN J ∗ q kδ !+ d s + 1 N i N i X j =1 Z t ∂ θ F is ( ϕ ij ( s )) d B ij ( s ) . (3.8)Since by definition q δ is a stationary solution to (1.7), one easily sees that h q iδ , F it i = h q iδ , F i i + Z t h q iδ , ∂ s F is i d s, i = 1 , . . . , d, t > , (3.9)and 0 = * ∂ θ q iδ − δω i ∂ θ q iδ − ∂ θ q iδ d X k = − d λ k J ∗ q kδ ! , F is + = * q iδ , ∂ θ F is + δω i ∂ θ F is + ∂ θ F is d X k = − d λ k J ∗ q kδ + . (3.10)Summing (3.8) over i = − d, . . . , d and using (3.9) and (3.10), we obtain h ν N,t , F t i d = h ν N, , F i d + Z t * ν N,s , ∂ s F s + 12 ∂ θ F s + δ∂ θ F s ⊗ w + ∂ θ F s d X k = − d λ kN J ∗ q kδ + d d s + Z t * ν N,s , ∂ θ F s d X k = − d λ kN J ∗ ν kN,s + d d s + Z t * q δ , ∂ θ F s d X k = − d λ kN J ∗ ν kN,s + d d s + Z t * q δ d X k = − d ( λ kN − λ k ) J ∗ q kδ , ∂ θ F s + d d s + M FN,t , (3.11) ISORDER-INDUCED TRAVELING WAVES IN THE STOCHASTIC KURAMOTO MODEL 19 where M FN,t is defined in (3.4) and where we have used the notation F ⊗ ω = ( F i ω i ) i ,...,d .Note that * q δ , ∂ θ F s d X k =1 λ k J ∗ ν kN,s + d = d X i =1 d X k = − d λ i λ k h q iδ , ∂ θ F is J ∗ ν kN,s i = − d X i = − d d X k = − d λ i λ k h ν iN,s , J ∗ ( q iδ ∂ θ F is ) i = − * ν s , d X k = − d λ k J ∗ ( q kδ ∂ θ F ks ) + d . (3.12)The result of Proposition 3.1 is a simple reformulation of (3.11) using (3.12) and thedefinition of L ∗ δ in (3.5). (cid:3) We are now in position to prove Proposition 2.7:
Proof of Proposition 2.7.
Let us apply the identity (3.3) of Proposition 3.1 in the case oftest functions F s of the form F s = e ( t − s ) L ∗ δ h, for any test functions h of class C on T . Then ∂ s F s = − L ∗ δ F s and one obtains h ν N,t , h i d = h ν N, , e tL ∗ δ h i d + Z t D D N , e ( t − s ) L ∗ δ h E d d s + Z t D R N ( ν N,s ) , ∂ θ e ( t − s ) L ∗ δ h E d d s + M FN,t . (3.13)We aim at proving that one can write a mild version of this weak equation and that thismild formulation makes sense in H − d . Consider a sequence ( v l ) l > of elements of L ,d converging as l → ∞ in H − d to ν N, ∈ H − d . Then, for h of class C , D v l , e tL ∗ δ h E d = D v l , e tL ∗ δ h E ,d = (cid:10) e tL δ v l , h (cid:11) ,d = (cid:10) e tL δ v l , h (cid:11) d . (3.14)By continuity of e tL δ on H − d , e tL δ v l converges in H − d to e tL δ ν N, , as l → ∞ . In particular,for all h ∈ H d , (cid:12)(cid:12)(cid:10) e tL δ ν N, , h (cid:11) d − (cid:10) e tL δ v l , h (cid:11) d (cid:12)(cid:12) k h k ,d (cid:13)(cid:13) e tL δ ν N, − e tL δ v l (cid:13)(cid:13) − ,d → l →∞ , (3.15)so that, at the limit for l → ∞ , for all t > D ν N, , e tL ∗ δ h E d = (cid:10) e tL δ ν N, , h (cid:11) d . (3.16)Since the function D N defined in (2.22) is regular, it is straightforward to prove in thesame way that D D N , e ( t − s ) L ∗ δ h E d = D e ( t − s ) L δ D N , h E d . (3.17)The continuity of the mapping t e tL δ ν N, and t R t e ( t − s ) L δ D N d s in H − d is immediatefrom the continuity of the semigroup in H − d .We now focus on the term R N ( ν N,s ) defined in (2.23): consider ( w s,l ) l > a sequenceof elements of L ,d converging in H − d to ν N,s (consider for example w s,l = φ l ∗ ν N,s for a regular approximation of identity ( φ l ) l > ) and define R s,l := d X k = − d ( λ kN − λ k ) J ∗ q kδ ! w s,l + q δ d X k = − d ( λ kN − λ k )( J ∗ ν kN,s )+ d X k = − d λ kN J ∗ ν kN,s ! w s,l . (3.18)For any l >
1, the following identity holds: D R s,l , ∂ θ e ( t − s ) L ∗ δ h E d = D R s,l , ∂ θ e ( t − s ) L ∗ δ h E ,d = − D e ( t − s ) L δ ∂ θ R s,l , h E ,d . (3.19)Since h is regular and R s,l converges in H − d to R N ( ν N,s ), the lefthand part of the previousidentity converges as l → ∞ to (cid:10) R N ( ν N,s ) , ∂ θ e ( t − s ) L ∗ δ h (cid:11) d . Moreover, for all h regular, usingthe estimate (B.24) on the regularity of the semigroup e tL δ , (note in particular that e tL δ can be extended to a continuous operator to H − d to H − d , see Proposition B.7 below), (cid:12)(cid:12)(cid:12)D e ( t − s ) L δ ∂ θ ( R s,l − R N ( ν N,s )) , h E d (cid:12)(cid:12)(cid:12) k h k ,d (cid:13)(cid:13)(cid:13) e ( t − s ) L δ ∂ θ ( R s,l − R N ( ν N,s )) (cid:13)(cid:13)(cid:13) − ,d , C k h k ,d (cid:18) √ t − s (cid:19) k ∂ θ ( R s,l − R N ( ν N,s )) k − ,d , C k h k ,d (cid:18) √ t − s (cid:19) k R s,l − R N ( ν N,s ) k − ,d . Since the last estimate is true for all h regular, one obtains that (cid:13)(cid:13)(cid:13) e ( t − s ) L δ ∂ θ ( R s,l − R N ( ν N,s )) (cid:13)(cid:13)(cid:13) − ,d C (cid:18) √ t − s (cid:19) k R s,l − R N ( ν N,s ) k − ,d . (3.20)Since R s,l converges to R N ( ν N,s ) in H − d , one obtains that one can make l → ∞ in (3.19): D R N ( ν N,s ) , ∂ θ e ( t − s ) L ∗ δ h E d = − D e ( t − s ) L δ ∂ θ R N ( ν N,s ) , h E d . The same argument as before shows also that (cid:13)(cid:13)(cid:13) e ( t − s ) L δ ∂ θ R N ( ν N,s ) (cid:13)(cid:13)(cid:13) − ,d C (cid:18) √ t − s (cid:19) k R N ( ν N,s ) k − ,d C (cid:18) √ t − s (cid:19) k ν N,s k − ,d Cπ vuut d X k = − d ( λ k ) − (cid:18) √ t − s (cid:19) , (3.21)where we used (2.4). The inequality (3.21) implies that the integral R t (cid:13)(cid:13) e ( t − s ) L δ ∂ θ R N ( ν N,s ) (cid:13)(cid:13) − ,d d s is almost surely finite. Using [44], Theorem 1, p. 133, we deduce that R t e ( t − s ) L δ ∂ θ R N ( ν N,s ) d s makes sense as a Bochner integral in H − d . The continuity of t R t e ( t − s ) L δ ∂ θ R N ( ν N,s ) d s in H − d is a direct consequence of the bounds found in Proposition B.7.It remains to treat the noise term in (3.13). The precise control of this term is made inSection 4 below (see in particular Proposition 4.1). We prove actually more in Section 4since we have to take into account the dependence in N , which is not important forthis proof. Let us admit for the moment that the proof of Proposition 4.1 is valid. Inparticular, one deduces from (4.28) and an application of the Kolmogorov Lemma thatthe almost-sure limit when t ′ ր t of Z N,t,t ′ defined in (2.24) exists in H − d . The continuity ISORDER-INDUCED TRAVELING WAVES IN THE STOCHASTIC KURAMOTO MODEL 21 of the limiting process t Z N,t in H − d comes from (4.3) and the rest of the proof ofProposition 4.1. It is immediate to see from (3.4) that for all F regular, h Z N,t , F i d = M FN,t ,where we see the term Z n,t as a vector ( Z − dn,t , . . . , Z dn,t ), with Z kn,t ( h ) = λ k Z n,t (ˆ h k ) andˆ h k = (0 , . . . , , h k , , . . . , h regular that h ν N,t , h i d = (cid:10) e tL δ ν N, , h (cid:11) d + (cid:28)Z t (cid:16) e ( t − s ) L δ D N − e ( t − s ) L δ ∂ θ R N ( ν N,s ) (cid:17) d s , h (cid:29) d + h Z N,t , h i d , (3.22)where everything above makes sense as element of H − d . Since this is true for all h regular,the identity (2.21) follows. Proposition 2.7 is proved. (cid:3) Controlling the noise
This section is devoted to control the noise term Z n,t defined in (2.37). More precisely,we prove the following proposition (recall the definition of A N = A N ( C Z ) given in (2.38)). Proposition 4.1.
For all ζ > , there exists a constant C Z such that P ( A N ) → , as N → ∞ . To prove Proposition 4.1, we rely on the two following lemmas:
Lemma 4.2 (Garsia-Rademich-Rumsey) . Let χ and Ψ be continuous, strictly increasingfunctions on (0 , ∞ ) such that χ (0) = Ψ(0) = 0 and lim t ր∞ Ψ( t ) = ∞ . Given T > and φ continuous on (0 , T ) and taking its values in a Banach space ( E, k . k ) , if Z T Z T Ψ (cid:18) k φ ( t ) − φ ( s ) k χ ( | t − s | ) (cid:19) d s d t B < ∞ , (4.1) then for s t T , k φ ( t ) − φ ( s ) k Z t − s Ψ − (cid:18) Bu (cid:19) χ ( d u ) . (4.2)Proof of Lemma 4.2 may be found in [42], Theorem 2.1.3. The second result estimatesthe moments of the process Z n,t : Lemma 4.3.
For all ε > and all integer m > , there exists a positive constant C m,ε such that for all s < t T , E k Z n,t − Z n,s k m − ,d C m,ε N m (cid:16) ( t − s ) m (1 / − ε ) + ( t − s ) m (cid:17) . (4.3)Let us first prove Proposition 4.1, relying on these two lemmas. Proof of Proposition 4.1.
Using Lemma 4.3, we can apply Lemma 4.2 with the choices φ ( t ) = Z n,t , χ ( u ) = u ζ m and Ψ( u ) = u m , (4.4)which implies that there exist a constant C (depending in m , ε and ζ ) and a positiverandom variable B such that for every 0 s < t T : k Z n,t − Z n,s k m − ,d ≤ C ( t − s ) ζ B , (4.5)where B satisfies E ( B ) ≤ CN m Z T Z T (cid:16) | t − s | m (1 / − ε ) − − ζ + | t − s | m − − ζ (cid:17) d s d t . (4.6) A simple integration shows that E ( B ) CN m (cid:0) T m (1 / − ε ) − ζ + T m − ζ (cid:1) , whenever m (1 / − ε ) − ζ > m − ζ >
1, that is when m > ζ )1 − ε . We can fix for example ε = 1 / m such that m > ζ ). Since T >
1, we have E ( B ) C T m − ζ N m andwe obtain: E sup s
Proof of Lemma 4.3.
Our aim here is to get the appropriate bounds for the process Z .We follow mostly the ideas of [21]. Recall that Z n,t ( h ) = d X i = − d λ i N i N i X j =1 Z T n − + tT n − ∂ θ (cid:20)(cid:16) e ( t − u ) L ∗ ψn − h (cid:17) i (cid:21) (cid:0) ϕ ij ( u ) (cid:1) d B ij ( u ) . (4.10)Let us define the process Z n,t,t ′ for 0 < t ′ < t as Z in,t,t ′ ( h ) = d X i = − d λ i N i N i X j =1 Z T n − + t ′ T n − ∂ θ (cid:20)(cid:16) e ( t − u ) L ∗ ψn − h (cid:17) i (cid:21) (cid:0) ϕ ij ( u ) (cid:1) d B ij ( u ) . (4.11)Our aim is to estimate for 0 < s ′ < s < t , s ′ < t ′ < t and for all integers m > E ( k Z n,t,t ′ − Z n,s,s ′ k m − ,d ). We can decompose Z n,t,t ′ − Z n,s,s ′ as follows: Z n,t,t ′ − Z n,s,s ′ = M n,s ′ ,s,t + M n,s ′ ,t ′ ,t , (4.12)where M n,s ′ ,s,t ( h ) = d X i = − d λ i N i N i X j =1 Z T n − + s ′ T n − ∂ θ (cid:20)(cid:16)(cid:16) e ( t − u ) L ∗ ψn − − e ( s − u ) L ∗ ψn − (cid:17) h (cid:17) i (cid:21) (cid:0) ϕ ij ( u ) (cid:1) d B ij ( u ) , (4.13)and M n,s ′ ,t ′ ,t ( h ) = d X i = − d λ i N i N i X j =1 Z T n − + t ′ T n − + s ′ ∂ θ (cid:20)(cid:16) e ( t − u ) L ∗ ψn − h (cid:17) i (cid:21) (cid:0) ϕ ij ( u ) (cid:1) d B ij ( u ) . (4.14) ISORDER-INDUCED TRAVELING WAVES IN THE STOCHASTIC KURAMOTO MODEL 23
The processes ( M n,s ′ ,s,t ( h )) s ′ ∈ [0 ,s ) and ( M n,s ′ ,t ′ ,t ( h )) t ′ ∈ ( s ′ ,t ) are martingales, with Itˆo brack-ets (cid:2) M n, · ,s,t ( h ) (cid:3) s ′ = d X i = − d N i X j =1 Z T n − + s ′ T n − (cid:16) U ,i,jn,u,s,t ( h ) (cid:17) d u , (4.15)and (cid:2) M n,s ′ , · ,t ( h ) (cid:3) t ′ = d X i = − d N i X j =1 Z T n − + t ′ T n − + s ′ (cid:16) U ,i,jn,u,t ( h ) (cid:17) d u , (4.16)where we have used the notations U ,i,jn,u,s,t ( h ) = λ i N i ∂ θ (cid:20)(cid:16)(cid:16) e ( t − u ) L ∗ ψn − − e ( s − u ) L ∗ ψn − (cid:17) h (cid:17) i (cid:21) (cid:0) ϕ ij ( u ) (cid:1) , (4.17)and U ,i,jn,u,t ( h ) = λ i N i ∂ θ (cid:20)(cid:16) e ( t − u ) L ∗ ψn − h (cid:17) i (cid:21) (cid:0) ϕ ij ( u ) (cid:1) . (4.18)Let ( h l ) l > be a complete orthonormal basis in H d . Using Parseval’s identity, we obtain E k Z n,t,t ′ − Z n,s,s ′ k − ,d = ∞ X l =1 E | ( Z n,t,t ′ − Z n,s,s ′ )( h l ) | ∞ X l =1 E | M n,s ′ ,s,t ( h l ) | + 2 ∞ X l =1 E | M n,s ′ ,t ′ ,t ( h l ) | ∞ X l =1 d X i = − d N i X j =1 Z T n − + s ′ T n − E (cid:16) U ,i,jn,u,s,t ( h l ) (cid:17) d u +2 ∞ X l =1 d X i = − d N i X j =1 Z T n − + t ′ T n − + s ′ E (cid:16) U ,i,jn,u,t ( h l ) (cid:17) d u d X i = − d N i X j =1 Z T n − + s ′ T n − E k U ,i,jn,u,s,t k − ,d d u + 2 d X i = − d N i X j =1 Z T n − + t ′ T n − + s ′ E k U ,i,jn,u,t k − ,d d u . (4.19)For m >
1, we have E k Z n,t,t ′ − Z n,s,s ′ k m − ,d = E ∞ X l =1 | ( Z n,t,t ′ − Z n,s,s ′ )( h l ) | ! m m E ∞ X l =1 | M n,s ′ ,s,t ( h l ) | ! m + m E ∞ X l =1 | M n,s ′ ,t ′ ,t ( h l ) | ! m , (4.20) and using H¨older and Burkholder-Davis-Gundy inequalities, we obtain for the terms in-volving M E ∞ X l =1 | M n,s ′ ,s,t ( h l ) | ! m = ∞ X l ,l ,...,l m =1 E | M n,s ′ ,s,t ( h l ) | . . . | M n,s ′ ,s,t ( h l m ) | ∞ X l ,l ,...,l m =1 ( E | M n,s ′ ,s,t ( h l ) | m ) /m . . . ( E | M n,s ′ ,s,t ( h l m ) | m ) /m C m ∞ X l ,l ,...,l m =1 E (cid:2) M n, · ,s,t ( h l ) (cid:3) s ′ . . . E (cid:2) M n, · ,s,t ( h l m ) (cid:3) s ′ C m ∞ X l ,l ,...,l m =1 d X i = − d N i X j =1 Z T n − + s ′ T n − E (cid:16) U ,i,jn,u,s,t ( h l ) (cid:17) d u . . . d X i = − d N i X j =1 Z T n − + s ′ T n − E (cid:16) U ,i,jn,u,s,t ( h l m ) (cid:17) d u = C m ∞ X l =1 d X i = − d N i X j =1 Z T n − + s ′ T n − E (cid:16) U ,i,jn,u,s,t ( h l ) (cid:17) d u m = C m d X i = − d N i X j =1 Z T n − + s ′ T n − E k U ,i,jn,u,s,t k − ,d d u m . (4.21)The same work can be done for the terms involving M , which leads to E k Z n,t,t ′ − Z n,s,s ′ k m − ,d C ′ m d X i = − d N i X j =1 Z T n − + s ′ T n − E k U ,i,jn,u,s,t k − ,d d u m + C ′ m d X i = − d N i X j =1 Z T n − + t ′ T n − + s ′ E k U ,i,jn,u,t k − ,d d u m . (4.22)It remains now to find appropriate bounds for E k U ,i,jn,u,s,t k − ,d and E k U ,i,jn,u,t k − ,d . On onehand, for h ∈ H d , since δ θ ∈ H − / − ε for all ε >
0, we have | U ,i,jn,u,t ( h ) | CN i (cid:13)(cid:13)(cid:13)(cid:13) ∂ θ (cid:20)(cid:16) e ( t − u ) L ∗ δ h (cid:17) i (cid:21)(cid:13)(cid:13)(cid:13)(cid:13) / ε,d CN i (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) e ( t − u ) L ∗ δ h (cid:17) i (cid:13)(cid:13)(cid:13)(cid:13) / ε,d . (4.23)For the rest of the proof, we set ε = (any ε ∈ (0 , /
4) would be sufficient). ApplyingProposition B.6 with β = 1 / ε/
2, we obtain, for any 0 < γ < γ L ∗ δ , | U ,i,jn,u,t ( h ) | CN i (cid:16) e − γ ( t − u ) ( t − u ) − / − ε/ (cid:17) k h k ,d , (4.24)which means that k U ,i,jn,u,t k − ,d CN i (cid:0) e − γ ( t − u ) ( t − u ) − / − ε/ (cid:1) . On the other hand,proceeding as before, we get the bound: | U ,i,jn,u,s,t ( h ) | CN i (cid:13)(cid:13)(cid:13)(cid:13)(cid:16)h e ( t − s ) L ∗ δ − i e ( s − u ) L ∗ δ h (cid:17) i (cid:13)(cid:13)(cid:13)(cid:13) / ε,d . (4.25) ISORDER-INDUCED TRAVELING WAVES IN THE STOCHASTIC KURAMOTO MODEL 25
Applying Proposition B.6 with β ′ = 1 / ε/ β = 1 / − ε , we get for all e h ∈ H − εd , k [ e ( t − s ) L ∗ δ − e h k / ε,d C ε ( t − s ) / − ε k (1 − P , ∗ ) e h k − ε,d . (4.26)For e h = e ( s − u ) L ∗ δ h and using again Proposition B.6 with this time β = 1 / − ε/
2, thisleads to | U ,i,jn,u,s,t ( h ) | CN i ( t − s ) / − ε ( s − u ) − / ε/ e − γ ( s − u ) k h k ,d , (4.27)which means that k U ,i,jn,u,s,t k − ,d CN i ( t − s ) / − ε ( s − u ) − / ε/ e − γ ( s − u ) . We can nowestimate (4.22): using that N cN i CN , we obtain E k Z n,t,t ′ − Z n,s,s ′ k m − ,d C ′′ m N m ( t − s ) / − ε Z s ′ ( s − u ) − ε e − γ ( s − u ) d u ! m + C ′′ m N m Z t ′ s ′ (cid:16) e − γ ( t − u ) ( t − u ) − / − ε (cid:17) d u ! m C ′′′ m N m (cid:0) ( t − s ) m (1 / − ε ) + ( t ′ − s ′ ) m (1 / − ε ) + ( t ′ − s ′ ) m (cid:1) . (4.28)Taking t ′ ր t and s ′ ր s and using Fatou Lemma, we deduce the result. (cid:3) Dynamics on the manifold M The purpose of this section is to prove the results described in Section 2.6 concerningthe process ν n defined in (2.35).Recall that the scheme defined in Section 2.6 starts at a time t N = O ( N − / log N ),such that there exists an event B N with P ( B N ) → B N if we denote ψ = proj M ( µ N,N / t N ) then k µ N,N / t N − q ψ k − ,d N − / ζ . In other words, theinitial condition of the scheme satisfies k ν , k − ,d N − / ζ on B N .The existence ofthese times t N and event B N will be proved in the Section 6. The first result provesestimate (2.39): Proposition 5.1.
There exists an event Ω N with P (Ω N ) → as N → ∞ such that,almost surely on Ω N , sup n n f sup t ∈ [0 ,T ] k ν n,t k − ,d = O ( N − / ζ ) , (5.1) where the error O ( N − / ζ ) is uniform on Ω N .Proof of Proposition 5.1. Recall the definition of the event A N in (2.38) and define Ω N := A N ∩ B N . Since the purpose of Section 4 was precisely to prove that P ( A N ) →
1, weobviously have that P (Ω N ) →
1, as N → ∞ .Throughout this proof we work on the event Ω N and proceed by induction. We alreadyknow that k ν , k − ,d N − / ζ . If we suppose that k ν n, k − ,d N − / ζ , then fromthe mild formulation (2.36), from (2.18) and (2.19) and from the estimates on the noiseterm Z n,t on Ω N ⊂ A N , we obtain k ν n,t k − ,d C L e − γ L t ∧ τ n N − / ζ + 2 T C L k D ψ n − k − ,d + C L ( T + 2 T / ) sup s t k R ψ n − ( ν n,s ) k − ,d + T / N − / ζ . (5.2) Since the sequence ( ω i ) i > is admissible (recall Definition 2.1), we have k D ψ n − k − ,d CN − / max k = − d,...,d | ξ kN | CN − / ζ . (5.3)Define the time t ∗ as t ∗ := inf n t ∈ [0 , T ] : k ν n,t k − ,d > C L N − / ζ o . (5.4)Obviously t ∗ > t t ∗ , one readily sees from (2.23) thatsup s t k R ψ n − ( ν n,s ) k − ,d C (cid:18) sup s t k ν n,s k − ,d + N − / max k = − d,...,d | ξ kN | sup s t k ν n,s k − ,d (cid:19) CN − ζ . (5.5)Putting together (5.2), (5.3) and (5.5) gives that t ∗ = T if N is large enough. Consequently,by construction of the stopping time τ n in (2.33), one has that τ n = T and the choice of T (recall (2.29)) implies that k ν n,T k − ,d C P N − / ζ . (5.6)To conclude the recursion it remains to show that k ν n +1 , k − ,d N − / ζ . To do this,let us write ν n +1 , in terms of ν n,T : ν n +1 , = q ψ n − + ν n,T − q ψ n . (5.7)Since P sψ n ν n +1 , = ν n +1 , , where we recall that P sψ n is the projection on the space N ψ n , wecan rewrite it as ν n +1 , = P sψ n ( q ψ n − + ν n,T − q ψ n )= P sψ n ( q ψ n − − q ψ n ) + ( P sψ n − P sψ n − ) ν n,T + P sψ n − ν n,T . (5.8)Since q ψ n − − q ψ n = ( ψ n − − ψ n ) q ′ ψ n + O (( ψ n − ψ n − ) ) (and this estimate makes sensein H − d ) and P sψ n ∂ θ q ψ n = 0, the first term of the second line of (5.8) is of order O (( ψ n − ψ n − ) ). Using the smoothness of the projection proj M (Lemma 2.8), | ψ n − ψ n − | = | proj M ( µ ( n ∧ n τ − T + t ∧ τ n ) − proj M ( µ (( n − ∧ n τ − T + t ∧ τ n − ) | C k µ ( n ∧ n τ − T + t ∧ τ n − µ (( n − ∧ n τ − T + t ∧ τ n − k − ,d C k ν n − ,T k − ,d + C k ν n − , k − ,d CN − / ζ . (5.9)Combining the last two arguments, we obtain that the first term of the second line of (5.8)is of order O ( N − ζ ). For the second term, the smoothness of the mapping ψ P sψ gives k ( P sψ n − P sψ n − ) ν n,T k − ,d C | ψ n − ψ n − |k ν n,T k − ,d CN − ζ . (5.10)Taking the H − d norm on the two sides in (5.8), we obtain k ν n +1 , k − ,d k P sψ n − ν n,T k − ,d + O ( N − ζ ) N − / ζ + O ( N − ζ ) , (5.11)which implies the result for N large enough. (cid:3) We are interested in the rescaled dynamics of the phase of the projection of the empiricalmeasure on M and in particular use the rescaled discretization of this phase dynamics givenby the process Ψ Nt (recall (2.42)). ISORDER-INDUCED TRAVELING WAVES IN THE STOCHASTIC KURAMOTO MODEL 27
Proposition 5.2.
There exist a linear form b : R d +1 → R and an event Ω N satisfying P (Ω N ) → as N → ∞ such that on the event Ω N we have for t ∈ [ t N , t f ] : Ψ Nt = ψ + b ( ξ N ) t + O ( N − / ζ ) , (5.12) where the O ( N − / ζ ) is uniform on Ω N .Proof of Proposition 5.2. We work for the moment on the event Ω N defined in the proofof Proposition 5.1. Using Proposition 5.1, Lemma C.1 below and the fact that ψ n =proj M ( q ψ n − + ν n,T ), we have the following first order expansion of Ψ Nt in (5.12) (recallthe definition of p in (2.15) and note that there are O ( N / ) terms in the sum): Ψ Nt := ψ + n t X n =1 p ψ n − ( ν n,T ) + O ( N − / ζ ) . (5.13)Let us now decompose the term p ψ n − ( ν n,T ), using the mild formulation (2.36). Re-mark that p ψ n − ( e tL ψn − ν n, ) = p ψ n − ( ν n, ) = 0 and that p ψ n − ( e ( t − s ) L ψn − D ψ n − ) = p ψ n − ( D ψ n − ). Note that Proposition 5.1 shows that τ n f = T on Ω N , so that the timeintegration in the mild formulation (2.36) does not involve any stopping time. Hence itremains, since D ψ n − has no dependency in time, p ψ n − ( ν n,T ) = T p ψ n − ( D ψ n − ) − Z T p ψ n − (cid:16) e ( t − s ) L ψn − ∂ θ R ψ n − ( ν n,s ) (cid:17) d s + p ψ n − ( Z n,T ) . (5.14)Using (2.19) and (5.5) (cid:12)(cid:12)(cid:12)(cid:12)Z T p ψ n − (cid:16) e ( t − s ) L ψn − ∂ θ R ψ n − ( ν n,s ) (cid:17) d s (cid:12)(cid:12)(cid:12)(cid:12) Z T (cid:13)(cid:13)(cid:13) e ( t − s ) L ψn − ∂ θ R ψ n − ( ν n,s ) (cid:13)(cid:13)(cid:13) − ,d d s C Z T (cid:18) √ t − s (cid:19) (cid:13)(cid:13) R ψ n − ( ν n,s ) (cid:13)(cid:13) − ,d d s C ( T + √ T ) N − ζ , (5.15)which leads to p ψ n − ( ν n,T ) = T p ψ n − ( D ψ n − ) + p ψ n − ( Z n,T ) + O ( N − ζ ) . (5.16)We would like to keep only T p ψ n − ( D ψ n − ), since the sum of these terms produce thedrift we are looking for, but unfortunately at each step p ψ n − ( Z n,T ) has the same orderas T p ψ n − ( D ψ n − ). To get rid of this extra term p ψ n − ( Z n,T ), we use the fact that it isan increment of a martingale and thus averages to 0 under summation. More precisely,denoting z n := p ψ n − ( Z n,T ∧ τ n ) and using Doob’s inequality we obtain, P sup m n f (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n m z n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > N − / ζ ! N / − ζ E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n n f z n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (5.17)and we have the following decomposition: E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n n f z n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E X n n f − E (cid:2) | z n +1 | |F T n (cid:3) C X n n f − E (cid:2) k Z n,T ∧ τ n k − ,d (cid:3) Cn f T N − , (5.18) where we have used (4.3). Since n f is of order N / , the probability in (5.17) tends to0 when N → ∞ and recalling (5.16), we deduce that there exists an event Ω N satisfying P (Ω N ) → N →∞ N Ψ Nt = ψ + T n t X n =1 p ψ n − ( D ψ n − ) + O ( N − / ζ ) . (5.19)The quantity p ψ n − ( D ψ n − ) = N − / p ψ n − (cid:0) − ∂ θ (cid:0) ξ N · ( J ∗ q ψ n − ) q ψ n − (cid:1)(cid:1) depends linearlyin ξ N and since the model is invariant by rotation, the projection does not depend on ψ n − . So we can write it as N − / b ( ξ N ), where the linear form b is given by b ( ξ ) := p ( − ∂ θ ( ξ · ( J ∗ q ) q )) = p − ∂ θ q d X k = − d ξ k ( J ∗ q k ) !! . (5.20)We can rewrite (5.19) as Ψ Nt = ψ + TN / $ N / T ( t − t N ) % b ( ξ N ) + O ( N − / ζ ) . (5.21)Since (cid:12)(cid:12)(cid:12) t − t N − TN / j N / T ( t − t N ) k(cid:12)(cid:12)(cid:12) TN / and b ( ξ N ) = O ( N ζ ), we deduce Ψ Nt = ψ + b ( ξ N )( t − t N ) + O ( N − / ζ ) , (5.22)which implies the result, since t N = O ( N − / log N ). Proposition 5.2 is proved. (cid:3) We can now prove the following result, which together with Proposition 2.9 impliesdirectly Theorem 2.3:
Proposition 5.3.
There exists N sufficiently large such that, on the event Ω N , sup t ∈ [ t N ,t f ] (cid:13)(cid:13)(cid:13) µ N,N / t − q ψ + b ( ξ N ) t (cid:13)(cid:13)(cid:13) − ,d = O ( N − / ζ ) , (5.23) where the error O ( N − / ζ ) is uniform on Ω N .Proof of Proposition 5.3. We place ourselves on the event Ω N introduced in the proof ofProposition 5.2. For each t such that N / t ∈ [ T n , T n +1 ] we can decompose µ N,N / t as µ N,N / t = q ψ n + ν n +1 ,N / t − T n . (5.24)But Proposition 5.1 implies that ν n +1 ,N / t − T n = O ( N − / ζ ) and for such time t wehave q ψ n = q Ψ Nt = q ψ + b ( ξ N ) t + O ( N − / ζ ) , (5.25)where we have used Proposition 5.2. (cid:3) Approaching the manifold
The purpose of this section is to prove Proposition 2.9. We follow here the same ideas asin [7], Section 5. From now on, we fix ε > p ∈ H − d such that dist H − d ( p , M ) ε .The parameter ε will be chosen sufficiently small in the following. We proceed in threesteps:(1) We rely on the convergence in finite time of the empirical measure µ N,t to thesolution p t of (1.7) starting from p in order to show that µ N,t approaches M (upto a distance of order ε ). This step requires a time interval of order log ε . ISORDER-INDUCED TRAVELING WAVES IN THE STOCHASTIC KURAMOTO MODEL 29 (2) We use the linear stability of M under (1.7) and control the noise terms of thedynamics to show that the empirical measure approaches M up to a distance oforder N − / ζ . This step requires a time interval of order log N .(3) We show that the empirical measure stays at distance N − / ζ from M up to thetime t N . First step.
As explained in Section 2.3, the stability of M implies that if ε is smallenough the deterministic solution p t of the limit PDE (1.7) with initial condition p con-verges to a q θ ∈ M . In particular, after a time s , p t satisfies k p s − q θ k − ,d ε . Dueto the linear stability of M , this time s is of order − γ L log ε .In order to show that the empirical measure is close to the deterministic trajectory p t when N is large, we use a mild formulation similar to the one obtained in Section 3, butthis time relying on the (2 d + 1)-dimensional Laplacian operator ∆ d . More precisely usingsimilar argument as in Section 3, one can obtain the following equality in H − d : µ N,t − p t = e t ∆ d ( µ N, − p ) − Z t e t − s ∆ d " ∂ θ µ N,s ⊗ ω + µ N,t d X k = − d λ kN J ∗ µ kN,s ! − ∂ θ p s ⊗ ω + p s d X k = − d λ k J ∗ p ks ! d s + z t , (6.1)where z t satisfies, for all test function f = ( f − d , . . . , f d ) z t ( f ) = d X i = − d λ i N i N i X j =1 Z t ∂ θ (cid:20)(cid:16) e t − s ∆ d f (cid:17) i (cid:21) ( ϕ ij ( s )) d B ij ( s ) . (6.2)Since ∆ d is simply the classical one-dimensional Laplacian operator ∆ on each coordinate,it is sectorial (in fact self-adjoint) with negative spectrum. Using the classical bound k e t ∆ f k − C √ t k f k − for the one-dimensional Laplacian operator, we directly obtain k e t ∆ d f k − ,d C √ t k f k − ,d , (6.3)and with similar estimates as the one used in Section 4, one can show that the event B N defined as B N := ( sup t s k z t k − ,d r t N N ζ ) (6.4)satisfies P ( B N ) → N → ∞ . Let us write the shortcut U N,s,t := e t − s ∆ d " ∂ θ µ N,s ⊗ ω + µ N,t d X k = − d λ kN J ∗ µ kN,s ! − ∂ θ p s ⊗ ω + p s d X k = − d λ k J ∗ p ks ! , for the term within the integral in (6.1). Note that the mapping ( µ, ν ) ∂ θ ( µJ ∗ ν )satisfies (see [7], Lemma A.3 for a proof) k ∂ θ ( µJ ∗ ν ) k − C k µ k − k ν k − . (6.5) Using (6.3) and (6.5), we obtain k U N,s,t k − ,d = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) e t − s ∆ d ∂ θ µ N,s d X k = − d λ kN J ∗ µ kN,s ! − ∂ θ p s d X k = − d λ k J ∗ p ks !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − ,d C √ t − s (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∂ θ µ N,s d X k = − d λ kN J ∗ µ kN,s ! − ∂ θ p s d X k = − d λ k J ∗ p ks !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − ,d C √ t − s d X i = − d d X k = − d λ k (cid:13)(cid:13)(cid:13) ∂ θ (cid:16) µ iN,s J ∗ µ kN,s (cid:17) − ∂ θ (cid:16) p is J ∗ p ks (cid:17)(cid:13)(cid:13)(cid:13) − (6.6)+ C √ t − s d X i = − d d X k = − d (cid:12)(cid:12)(cid:12) λ kN − λ k (cid:12)(cid:12)(cid:12) (cid:13)(cid:13)(cid:13) ∂ θ (cid:16) µ iN,s J ∗ µ kN,s (cid:17)(cid:13)(cid:13)(cid:13) − (6.7) C √ t − s ( k p s k − ,d + k µ N,s k − ,d ) k µ N,s − p s k − ,d + C √ t − s N − / ζ k µ N,s k − ,d (6.8) C ′ √ t − s (cid:16) k µ N,s − p s k − ,d + N − / ζ (cid:17) , (6.9)where we have used in particular (2.4), since both p s and µ N,s are probabilities. Let usplace ourselves on the event B N := n k µ N, − p k − ,d ε o ∩ B N , (6.10)which satisfies obviously P ( B N ) → N → ∞ . Then, for all t s , (6.3) and (6.6)imply that (6.1) can be rewritten on the event B N as k µ N,t − p t k − ,d ε C r s N N ζ + C Z t √ t − s k µ N,s − p s k − ,d d s , (6.11)so applying the Gronwall-Henry inequality ([24], Lemma 7.1.1 and Exercise 1), one obtainsthat for some a > N and ε ), on the event B N and for all t s k µ N,t − p t k − ,d (cid:18) ε C r s N N ζ (cid:19) e as . (6.12)We deduce that for N large enough, the projection ψ := proj M ( µ N,s )is well defined and k µ N,s − p s k − ,d ε on B N , which means that | ψ − θ | Cε and k µ N,s − q θ k ε . Second step.
Now that we know that dist( µ N,s , M ) ε with increasing probabilityas N → ∞ , we can use a similar scheme as the one defined in Section 2.6 to show that theempirical measure approaches M up to a distance N − / ζ with high probability. Sincethis part is very similar to the work done in Section 5, we do not specify all the details.We consider the evolution of the dynamics on time intervals [ e T n , e T n +1 ] with e T n = s + n e T where e T is such that e − γ L e T C L C P . We consider also a sequence of real numbers h n satisfying h = 2 ε and h n +1 = h n and take this time the number of step e n f of our schemeas e n f := inf n n : h n N − / ζ o . (6.13) ISORDER-INDUCED TRAVELING WAVES IN THE STOCHASTIC KURAMOTO MODEL 31
It is clear that e n f is of order O (log N ). To ensure the existence of the projections of theprocess on M at each step, we introduce, as in Section 2.6, the stopping time( e n τ , e τ ) = inf { ( n, t ) ∈ { , . . . , e n f } × [0 , e T ] : k µ e T n − + t − q α n − k − ,d > σ } , (6.14)where α n = proj M ( µ e T n ) when it exists. This allows us to define the random phases e ψ n − defined as e ψ n − := proj M ( µ ( n ∧ e n τ − e T + t ∧ e τ n ) , (6.15)and the processes e ν n,t defined for n = 1 , . . . , e n f as e ν n,t := µ ( n ∧ e n τ − e T + t ∧ e τ n − q e ψ n − . (6.16)This last process satisfies the mild equation e ν n,t = e ( t ∧ e τ n ) L e ψn − e ν n, − Z t ∧ e τ n e ( t ∧ e τ n − s ) L e ψn − ( D e ψ n − + R e ψ n − ( e ν n,s )) d s + e Z n,t ∧ e τ n , (6.17)where e Z n,t is defined as e Z n,t ( f ) = d X i = − d N i N i X j =1 Z t ∂ θ "(cid:18) e ( t − s ) L ∗ e ψn − f (cid:19) i ( ϕ ij ( e T n − + s )) d B ij ( e T n − + s ) . (6.18)Section 4 shows that the event e A N = ( sup n e n f sup t ∈ [0 , e T ] k e Z n,t k − ,d e T / N − / ζ ) , (6.19)satisfies P ( e A N ) → N → ∞ .In the first step of this proof we have shown, since e ψ = ψ , that, on the event B N (recall (6.10)), we have k e ν , k − ,d = k µ N,s − q ψ k − ,d h . Our aim is to prove that onthe event B N defined as B N := e A N ∩ B N , (6.20)we have k e ν n, k − ,d h n for all n = 1 , . . . , e n f . This would imply, using the notations s = e T n f and ψ = proj M ( µ N,s ), that k e µ N,s − q ψ k − ,d N − / ζ . We place ourselveson the event B N . From the mild formulation (6.17), if n < e n f and k e ν n, k − ,d h n we get k e ν n,t k − ,d C L e − γ L t ∧ e τ n h n + 2 C L e T k D e ψ n − k − ,d + C L (cid:16) e T + 2 e T / (cid:17) sup s t k R e ψ n − ( e ν n,s ) k − ,d + e T / N − / ζ . (6.21)Consider the time e t ∗ defined as e t ∗ := inf n t ∈ [0 , e T ] : k e ν n,t k − ,d > C L h n o . (6.22) For all t e t ∗ we havesup s t k R e ψ n − ( e ν n,s ) k − ,d C (cid:18) sup s t k e ν n,s k − ,d + N − / max k = − d,...,d | ξ kN | sup s t k e ν n,s k − ,d (cid:19) C ( C L h n + C L N − / ζ h n ) . (6.23)The last quantity is smaller than C ( N, ε ) h n , where C ( N, ε ) → N → ∞ and ε → k D e ψ n − k − ,d CN − / ζ . Since n < e n f we have h N > C L N − / ζ , which means that N − / ζ is negligible with respect to h n for N large enough. So for N large enough, e t ∗ > e T and we have (recall that e − λ e T C L C P ) k e ν n, e T k − ,d C P h n + o ( h n ) C P h n , (6.24)when ε is small enough. It remains to show that k e ν n +1 , k − ,d h n to conclude therecursion. We do not prove it in details, since it can be done by proceeding exactly as inthe proof of Proposition 5.1, decomposing k e ν n, e T k − ,d and showing that it can be writtenas k e ν n +1 , k − ,d k P s e ψ n − e ν n, e T k − ,d + O ( h n ) , (6.25)which implies that k e ν n +1 , k − ,d h n + O ( h n ) h n on the event B N when ε is smallenough and concludes the recursion. Note that the estimate for e ψ n − e ψ n − obtained in(5.9) leads to | ψ − ψ | n f X n =1 | e ψ n − e ψ n − | C n f X n =1 h n Ch Cε , (6.26)on the event B N , which gives (cid:12)(cid:12) ψ − θ (cid:12)(cid:12) C ′ ε for some C ′ . Third step.
In the previous step, we have constructed a time s such that s − λ log ε + C log N for some constant C and such that k e µ N,s − q ψ k − ,d N − / ζ withhigh probability. We can now consider a time s = c log N for c = C + 1, which doesnot depend in ε . For N large enough, we obviously have s > s . In order to prove that k e µ N,s − q ψ k − ,d N − / ζ with high probability, where ψ = proj M ( µ N,s ), it sufficesto decompose the dynamics on the interval [ s , s ] according to an iterative scheme withtime step ˆ T satisfying e − γ L ˆ T C L C P as does T and apply exactly the same procedure asin Proposition 5.1.This last step induces a phase shift | ψ − ψ | CN − / ζ log N Cε , for N largeenough. This concludes the proof, with t N = N − / s .7. Estimates on the drift b The case of a symmetric disorder.
We prove here Proposition 2.4 and dropfor simplicity the dependency in ψ and δ . We consider ξ = ( ξ − d , . . . , ξ d ) such that ξ − i = ξ i for all i = 1 , . . . , d and aim at proving that b ( ξ ) = 0, where the drift b ( ξ ) = p (cid:16) − ∂ θ (cid:16)nP dk = − d ξ k ( J ∗ q k ) o q (cid:17)(cid:17) is given by (5.20).The space of regular ( C , say) test functions f = ( f − d , f − ( d − , . . . , f d − , f d ) can benaturally decomposed into the direct sum of the space O (resp. E ) of odd (resp. even) test ISORDER-INDUCED TRAVELING WAVES IN THE STOCHASTIC KURAMOTO MODEL 33 function in both variables ( θ, i ), that is f ∈ O (resp. f ∈ E ) if and only if f − i ( − θ ) = − f i ( θ )(resp. f − i ( − θ ) = f i ( θ )) for all θ ∈ T and i = 0 , . . . , d . One easily sees from the definition of J ( · ) in (1.4) and the definition of q in (1.8) that q ∈ E and (cid:0) ( J ∗ q − d ) , . . . , ( J ∗ q d ) (cid:1) ∈ O .Let us denote Q ( θ ) := P dk = − d ξ k ( J ∗ q k )( θ ). Using that ξ − i = ξ i , one obtains that Q ( θ ) = ξ ( J ∗ q )( θ ) + P dk =1 ξ k (cid:0) ( J ∗ q k )( θ ) + ( J ∗ q − k )( θ ) (cid:1) , so that we deduce that Q isan odd function of θ and that θ Q ( θ ) q ( θ ) ∈ cO . Consequently ∂ θ ( Q ( θ ) q ( θ )) ∈ E .Hence, in order to prove Proposition 2.4, it suffices to prove that ∀ h ∈ E , p ( h ) = 0 . (7.1)This is indeed the case since one easily sees from the definition (2.13) of the operator L = L ψ,δ that L ( E ) ⊂ E and L ( O ) ⊂ O and since p is the projection on the eigenfunction ∂ θ q ∈ O . Proposition 2.4 is proved.7.2. Small δ asymptotics of the drift. Our aim here is to prove Proposition 2.5 thatgives the first order expansion of the drift b ( ξ ) defined in (5.20) as δ →
0. Due to therotational invariance of the system, we can work with the stationary solution q ,δ that wedenote q δ throughout this section. We denote p δ as p ψ =0 ,δ (recall (2.15)) and D δ ( ξ ) as D N, ,δ , (recall (2.22)). With these notations the drift b is given by b ( ξ ) = p δ ( D δ ( ξ )) . (7.2)When δ = 0, it is straightforward to see that q δ = ( q − dδ , . . . , q dδ ) is equal to ( q , . . . , q ),where q is the stationary solution of the nonlinear Fokker-Planck equation without dis-order (2.12). We refer to Section B.1 below for precise definitions (see in particular (B.2)and (B.3) where the normalisation factor Z and the fixed-point parameter r are defined).The following result (proved in Appendix D) provides the next order of the approximationof q δ as δ → Lemma 7.1.
For i = − d, . . . , d we have q iδ ( θ ) = q ( θ ) + δω i κ ( θ ) q ( θ ) + O ( δ ) , (7.3) where κ ( θ ) = 2 θ + 4 π R πθ e − Kr cos u d u Z − R π e Kr cos u u d u Z − π R π e Kr cos u R πu e − Kr cos v d v d u Z , (7.4) and where the error O ( δ ) is uniform in θ ∈ T . The projection p δ also converges in some sense to the projection p on the tangent spaceof the stable circle of stationary profiles of (2.12) at q . Moreover, the system given by(2.12) admits a nice Hilbertian structure, which allows to know p explicitly. This allowsus to obtain the following first order expansion of p δ , whose proof is given in Appendix D. Lemma 7.2.
For all coordinate by coordinate primitive ( U − d , . . . , U d ) of u smooth, wehave p δ ( u ) = Z Z − π d X k = − d λ k Z T (cid:18) − π Z q (cid:19) U k + O ( δ k u k − ,d ) . (7.5)We have now the tools required to obtain the first order expansion of the drift b ( ξ ).The result we want to prove is Proposition 7.3.
For all ξ such that P dk = − d ξ k = 0 we have b ( ξ ) = δ d X k = − d ξ k ω k + O ( δ ) . (7.6) Proof of Proposition 7.3.
First remark that when δ = 0, we obtain, using Lemma 7.1, thatfor all i = − d, . . . , d : D i ( ξ ) = ∂ θ " q d X k = − d ξ k J ∗ q = 0 , (7.7)since P dk = − d ξ k = 0. We deduce, using again Lemma 7.1, the following expansion for D iδ ( ξ ): D iδ ( ξ ) = δω i ∂ θ " κq d X k = − d ξ k J ∗ q + δ∂ θ " q d X k = − d ξ k ω k J ∗ ( κq ) + O ( δ )= δ∂ θ " q d X k = − d ξ k ω k J ∗ ( κq ) + O ( δ ) , (7.8)where we have used again the fact that P dk = − d ξ k = 0. Applying Lemma 7.2, we deduce b ( ξ ) = δ (cid:20) Z Z − π Z T (cid:18) − π Z q (cid:19) q J ∗ ( κq ) (cid:21) d X i = − d d X k = − d λ i ξ k ω k + O ( δ ) , (7.9)and recalling that P di = − d λ i = 1 and denoting c b := Z Z − π Z T (cid:18) − π Z q (cid:19) q J ∗ ( κq ) , (7.10)we simply obtain b ( ξ ) = δc b d X k = − d ξ k ω k + O ( δ ) . (7.11)It remains to show that c b = 1. Now using the fact that J ( θ − θ ′ ) = − K sin θ cos θ ′ + K cos θ sin θ ′ , R π sin( θ ) q ( θ ) = 0 and R π cos( θ ) q ( θ ) = r , we obtain Z π q ( θ ) J ∗ ( κq )( θ ) d θ = Kr Z π sin( θ ′ ) κ ( θ ′ ) q ( θ ′ ) d θ ′ , (7.12)and Z π J ∗ ( κq )( θ ) d θ = 0 . (7.13)So the constant c b can be simplified as follows c b = Kr Z Z − π Z π sin( θ ) κ ( θ ) q ( θ ) d θ , (7.14)which leads to c b = 2 Kr Z Z − π " Z π sin θ e Kr cos θ Z (cid:18) θ + 2 π R πθ e − Kr cos u d u Z (cid:19) d θ . (7.15) ISORDER-INDUCED TRAVELING WAVES IN THE STOCHASTIC KURAMOTO MODEL 35
Integrating by parts and using the fact that ∂ θ [ e Kr cos( θ ) ] = − Kr sin θe Kr cos θ , weobtain Z π θ sin θe Kr cos θ d θ = − πe Kr Kr + Z Kr , (7.16)and Z π sin θe Kr cos θ Z πθ e − Kr cos u d u = e Kr Z Kr − πKr , (7.17)which implies that c b = Kr Z Z − π (cid:16) Kr − π Kr Z (cid:17) = 1. Proposition 7.3 is proved. (cid:3) Using Proposition 7.3, we can now compute the first order of the variance v of thelimiting normal distribution of b ( ξ N ) when the disorder is i.i.d: Proof of Proposition 2.5.
From the Central Limit Theorem, we know that ξ N convergesas N → ∞ to a Gaussian distribution with mean 0 and covariance matrix Σ satisfying (cid:26) Σ k,k = λ k (1 − λ k ) k ∈ {− d, . . . , d } , Σ k,l = − λ k λ l k, l ∈ {− d, . . . , d } , k = l . (7.18)Applying Proposition 7.3 we obtain v = δ X k ∈{− d,...,d } λ k (1 − λ k )( ω k ) − X k,l ∈{− d,...,d } , k = l λ k λ l ω k ω l + O ( δ ) , (7.19)and since λ − k = λ k and ω − k = − ω k the terms with l = − k cancel in the second sum,which gives the result. (cid:3) Appendix A. Construction of rigged-spaces
We specify here the construction of the Hilbert distributions spaces we work with inthis paper. It is based on the notion of rigged Hilbert spaces (see [13], p. 81).A.1.
Functional spaces on T . Consider L := (cid:8) u ∈ L , R T u ( θ ) d θ = 0 (cid:9) , the spaceof square integrable functions with zero mean value, endowed with the norm k u k := (cid:0)R T u ( θ ) d θ (cid:1) . We call a weight any strictly positive function θ w ( θ ) on T . For anyweight w on T , define H w as the closure of (cid:8) u ∈ C ( T ) , R T u ( θ ) d θ = 0 (cid:9) w.r.t. the norm k u k ,w := (cid:18)Z T ( ∂ θ u ( θ )) w ( θ ) d θ (cid:19) . There is a continuous and dense injection of H w into L and the corresponding dual spacecan be identified as H − /w , that is the closure of (cid:8) u ∈ C ( T ) , R T u ( θ ) d θ = 0 (cid:9) under thenorm k u k − , /w := (cid:18)Z T U ( θ ) w ( θ ) d θ (cid:19) , where U is the primitive of u such that R T U w = 0. A.2.
Functional spaces on T × R . The correct set-up of the paper is to consider testfunctions of both oscillators and frequencies, that is ( θ, ω ) u ( θ, ω ), where θ ∈ T and ω ∈ R . Since the disorder is assumed to take a finite number of values (cid:8) ω − d , . . . , ω d (cid:9) ,it is equivalent to consider vector-valued test functions θ ( u − d ( θ ) , . . . , u d ( θ )) and it isstraightforward to define the counterparts of the norms defined in the last paragraph forthese vector-valued functions: Consider L ,d := (cid:0) L (cid:1) d endowed with the product norm k u k ,d := d X k = − d λ k (cid:13)(cid:13)(cid:13) u k (cid:13)(cid:13)(cid:13) ! . In the same way, consider the space H w,d , closure of (cid:8) ( u − d , . . . , u d ) ∈ C ( T ) , R T u k ( θ ) d θ = 0 (cid:9) under the norm k u k ,w,d := d X k = − d λ k (cid:13)(cid:13)(cid:13) u k (cid:13)(cid:13)(cid:13) ,w ! , (A.1)as well as the space H − /w,d endowed with the norm k u k − , /w,d := d X k = − d λ k (cid:13)(cid:13)(cid:13) u k (cid:13)(cid:13)(cid:13) − , /w ! . (A.2)Note that if w and w are bounded weights, the norms k·k ,w and k·k ,w (resp. k·k − , /w and k·k − , /w ) are equivalent. The same holds for the (2 d + 1)-dimensional norms.A.3. Fractional spaces.
Define also the fractional norm k · k α,d (where α > d the Laplacian operator on each coordinate, k · k the L -norm on T and k u k ,d = P k λ k k u k k and define k u k α,d = k (1 − ∆ d ) α/ u k ,d = d X k = − d λ k k (1 − ∆) α/ u k k . (A.3)We denote as H αd the closure of regular functions with zero mean-value on T under theprevious norm and H − αd the corresponding dual space. Appendix B. Spectral estimates and regularity results on semigroups
The purpose of this paragraph is to establish spectral estimates on L ψ,δ and its adjointas well as regularity estimates on their semigroups e tL ψ,δ and e tL ∗ ψ,δ .B.1. The case δ = 0 . The analysis of the dynamics of (1.5) and (1.7) is based on pertur-bations argument on the mean-field plane rotators system (2.11) and (2.12). The proofrelies in particular strongly on the fact that (2.11) is reversible, with an explicit free en-ergy [6, 15]. However, one should note that the limit as δ → δ → ∂ t q it ( θ ) = 12 ∂ θ q it ( θ ) − ∂ θ q it ( θ ) d X k = − d p k J ∗ q kt ( θ ) !! , i = − d, . . . , d , (B.1)which corresponds to the situation where the disorder is no longer present but wherethe rotators have been (artificially) separated in different subpopulations. Following the ISORDER-INDUCED TRAVELING WAVES IN THE STOCHASTIC KURAMOTO MODEL 37 terminology of [21] where (B.1) has been already encountered, we call this system the non-disordered system . It is shown in [21], Section 2.1, that the non-disordered system (B.1)presents most of the properties of the mean field plane rotators model (2.12). In particular,for all
K >
1, one can show that (B.1) admits a unique circle M ,nd of synchronized profiles,that is stable as t → ∞ . M ,nd is given by the translations of the profile q ,nd = ( q , . . . , q ),where q is the profile generating the stable circle M of non trivial solutions of (2.12),namely q ( θ ) := e Kr cos θ Z (2 Kr ) , (B.2)where Z ( x ) = 2 πI ( x ), I ( x ) = π R π e x cos( θ ) d θ is the standard modified Bessel functionof order 0 and r is the unique positive solution of the fixed-point problem r = Ψ (2 Kr ) , with Ψ ( x ) := R π cos( θ ) e x cos θ d θ Z ( x ) . (B.3)The derivation of these stationary solutions is highly similar to the procedure describedin Section 1.5 and we refer to the aforementioned references for more details. Note thatone can draw a simple correspondance between the present definitions and the definitionsof Section 1.5 in the case of δ = 0: namely, one readily sees that, for any i = − d, . . . , d , S i ( θ, x ) = e x cos( θ ) Z ( x ) (recall (1.9)) and Z i ( x ) = Z ( x ) , so that the definition of Ψ δ when δ = 0 (recall (1.11)) coincides with Ψ given in (B.3).B.2. Spectral estimates when δ = 0 . Define the linearized operator around any sta-tionary solution q ,nd ∈ M ,nd :( Au ) i = 12 ∂ θ u i − ∂ θ ( J ∗ q ) u i + q d X k =1 J ∗ u k ! , i = − d, . . . , d , (B.4)with domain D ( A ) = (cid:8) ( u − d , . . . , u d ) ∈ C ( T ) d +1 , R T u k ( θ ) d θ = 0 , k = − d, . . . , d (cid:9) . Werecall the following result (see [21], Proposition 2.1): Proposition B.1.
A is essentially self-adjoint with compact resolvent in H − /q ,d . Itsspectrum lies in ( −∞ , , is a simple eigenvalue, with eigenspace spanned by ∂ θ q ,nd .The spectral gap between and the rest of the spectrum is denoted as γ A . One can deduce from Proposition B.1 similar spectral properties of its dual A ∗ in L ,d :( A ∗ v ) i := 12 ∂ θ v i + ( J ∗ q ) ∂ θ v i − Z T (cid:0) ( J ∗ q ) ∂ θ v i (cid:1) d θ − d X k =1 λ k J ∗ ( q ∂ θ v k ) , i = − d, . . . , d , (B.5)with domain D ( A ∗ ) = D ( A ). Proposition B.2. A ∗ is essentially self-adjoint with compact resolvent in H /q ,d . Itsspectrum lies in ( −∞ , , and is a simple eigenvalue and its spectral gap γ A ∗ is equal to γ A .Proof of Proposition B.2. Let us introduce the operator U defined from H q ,d to H − /q ,d as U f ( θ ) := − ∂ θ ( q ( θ ) ∂ θ f ( θ )) . U is an isometry between H q ,d and H − /q ,d : U realizes a bijection from { u ∈ C ∞ ( T ) d , R T u k ( θ ) d θ = 0 , k = 1 , . . . , d } into itself and for every f, g ∈ H q ,d , h U f, U g i − , /q ,d = X k Z T (cid:0) q ( θ ) ∂ θ f k ( θ ) (cid:1) (cid:0) q ( θ ) ∂ θ g k ( θ ) (cid:1) q ( θ ) d θ = X k Z T q ( θ ) ∂ θ f k ( θ ) ∂ θ g k ( θ ) d θ = h f, g i ,q ,d . (B.6)Moreover, the following identity holds: A ∗ = U − AU , (B.7)so the operators A on H − /q ,d and A ∗ on H /q ,d have the same structural and spectralproperties. (cid:3) B.3.
Spectral estimates of L ψ,δ and its adjoint. We are in position to deduce spectralestimates on the disordered operators L δ and its adjoint L ∗ δ in L ,d (we drop the index ψ in this section for simplicity). Proposition B.3.
The adjoint L ∗ δ of L δ in L ,d is given by for all i = 1 , . . . , d ( L ∗ δ v ) i = 12 ∂ θ v i + δω i ∂ θ v i + ( ∂ θ v i ) d X k = − d λ k J ∗ q kδ − Z T ( ∂ θ v i ) d X k = − d λ k J ∗ q kδ ! d θ − d X k = − d λ k J ∗ ( q kδ ∂ θ v k ) , (B.8) with domain D ( L ∗ δ ) = D ( A ) .Proof of Proposition B.3. For all regular u and v , h L ∗ δ v, u i ,d = h v, L δ u i ,d = d X i = − d λ i * v i , ∂ θ u i − δω i ∂ θ u i − ∂ θ u i d X k = − d λ k J ∗ q kδ + q iδ d X k = − d λ k J ∗ u k !+ ,d = d X i = − d λ i * ∂ θ v i + δω i ∂ θ v i + ∂ θ v i d X k = − d λ k J ∗ q kδ − Z T ∂ θ v i d X k = − d J ∗ q kδ ! d θ, u i + ,d + d X i = − d d X k = − d λ i λ k h q iδ ∂ θ v i , J ∗ u k i ,d = * ∂ θ v i + δω i ∂ θ v i + ( ∂ θ v i ) d X k = − d λ k J ∗ q kδ − Z T ( ∂ θ v i ) d X k = − d λ k J ∗ q kδ ! d θ, u i + ,d − * d X k = − d λ k J ∗ ( q kδ ∂ θ v k ) , u i + ,d , which precisely gives (3.5). (cid:3) The main result of this section is the following
ISORDER-INDUCED TRAVELING WAVES IN THE STOCHASTIC KURAMOTO MODEL 39
Proposition B.4.
There exists δ = δ ( K ) > such that for all δ δ , everything thatfollows is true: the operator L ∗ δ (resp. L δ ) is sectorial in H q ,d (resp. H − /q ,d ), its spectrumlies in a sector of the type { λ ∈ C : | arg( λ ) | > π/ α } for some α > and is anisolated eigenvalue for L ∗ δ (resp. L δ ), at a distance from the rest of the spectrum denotedby γ L ∗ δ (resp. γ L δ ). Moreover, both L δ and L ∗ δ generate a C -semigroup t e tL δ (resp. t e tL ∗ δ ) in L ,d and e tL ∗ δ = (cid:0) e tL δ (cid:1) ∗ .Proof of Proposition B.4. The result concerning the operator L δ has been proved in [21],Th. 2.5. For the sake of completeness, we recall here the main arguments concerning L ∗ δ in H q ,d but we refer to [21], Section 6.2 for precise details. Note that we need a precisecontrol of the spectrum of L ∗ δ around the origin. In particular, one has to ensure thatthe spectrum of L ∗ δ remains in the negative part of the complex plane. We write L ∗ δ as aperturbation for small disorder of the non-disordered case: L ∗ δ = A ∗ + B δ , (B.9)where A ∗ is given in (B.5) and B δ is a small perturbation as δ →
0. More precisely,following the exact same strategy as in [21], Proposition 6.5, p. 356, one obtains that theoperator B δ is A ∗ -bounded: there exist constants a δ and b δ (only depending on δ and K )such that for all u in the domain of (the closure of) A ∗ k B δ u k ,q ,d a δ k u k ,q ,d + b δ k A ∗ u k ,q ,d , (B.10)with a δ = O ( δ ) and b δ = O ( δ ), as δ →
0. Note that the only things that differs betweenthis result and [21], Proposition 6.5 is that we work here with an H -norm whereas theresult in [21] concerns an H − -norm.Fix some ε > L ∗ δ,ε := L ∗ δ − ε and A ε := A − ε ,so that L ∗ δ,ε = A ∗ ε + B δ . Fix α ∈ (0 , π ) and introduce the following subset of the complexplane Σ α := n λ ∈ C , | arg( λ ) | < π α o ∪ { } . The operator A ε (as A itself) is self-adjoint in H − , /q and hence, sectorial. In particular,there exists M > k R ( λ, A ε ) k H − /q ,d M | λ | , for all λ ∈ Σ α . Note that theconstant M is indeed independent of ε > A in place of A ε (see [21], (6.12)). Using (B.7), one obtains that k R ( λ, A ∗ ε ) k H q ,d M | λ | .For λ ∈ Σ α , u ∈ H q ,d , k B δ R ( λ, A ∗ ) u k ,q ,d a δ k R ( λ, A ∗ ) u k ,q ,d + b δ k A ∗ R ( λ, A ∗ ) u k ,q ,d , M a δ | λ | k u k ,q ,d + ( M + 1) b δ k u k ,q ,d , Choose δ sufficiently small so that b δ (1 + M ) and a δ Mε . Then for | λ | > ε > M a δ ,we have k B δ R ( λ, A ∗ ) u k k u k so that the operator 1 − B δ R ( λ, A ∗ ) is invertible from H q ,d into itself, with norm smaller than 2. A simple computation shows that in this case( λ − ( A ∗ + B δ )) − = R ( λ, A ∗ )(1 − B δ R ( λ, A ∗ )) − , which gives that, for λ ∈ Σ α , | λ | > ε , k R ( λ, L ∗ δ ) k H q ,d M | λ | . Consequently, the spectrumof L ∗ δ is contained inΘ α,ε := (cid:26) λ ∈ C , π α arg( λ ) π − α (cid:27) ∪ { λ ∈ C , | λ | ε } . In particular, 0 ∈ ρ ( L ∗ δ, ε ) and for all λ ∈ C with ℜ ( λ ) > | λ | < | λ + 2 ε | ), (cid:13)(cid:13)(cid:13) R ( λ, L ∗ δ, ε ) (cid:13)(cid:13)(cid:13) H q ,d M | λ +2 ε | M | λ | . The fact that this estimate can be extended to some Σ α ′ for some α ′ is a consequence of a Taylor’s expansion argument (see [21], Proposition 6.2),so that L ∗ δ, ε (and L ∗ δ ) is indeed sectorial.At this point, we cannot rule out the possibility that some elements of the spectrum of L ∗ δ may lie in Θ ε,α ∩ { λ ∈ C , ℜ ( λ ) > } . The last point of the proof is to show that one canchoose ε and a smaller δ such that this situation does not hold: choose ε = γ A >
0, where γ A is the spectral gap of A . In particular, the circle centered in 0 with radius ε separatesthe eigenvalue 0 (of multiplicity 1) from the rest of the spectrum of A ∗ . An applicationof [26], Theorem IV-3.18, p. 214, shows that one can choose δ sufficiently small so thatthe spectrum of the perturbed operator L ∗ δ is likewise separated by this circle: for such δ ,there is a unique eigenvalue (with multiplicity 1) within the boundary of this circle). Butwe know already that 0 is an eigenvalue for the perturbed operator L ∗ δ . By uniqueness, weconclude that there is no eigenvalue in the positive part of the complex plane. We leavethe details of this argument to [21], Section 6.2.5.Using [36], Corollary 10.6, p. 41, L ∗ δ is the generator of the adjoint of t e tL δ in L ,d ,which is a C -semigroup. This concludes the proof of Proposition B.4. (cid:3) B.4.
Equivalence of norms.
For any 0 β
1, consider the interpolation norm k · k V β associated to the sectorial operator 1 − L ∗ δ defined as k u k V β = k (1 − L ∗ δ ) β u k ,q ,d . (B.11)Recall also the definition of the fractional norm in (A.3). Lemma B.5.
Under the assumptions of Proposition B.4, for any β , there exists c , C > such that for all u , c k u k β,d k u k V β C k u k β,d . (B.12) Proof of Lemma B.5.
We can decompose L ∗ δ as follows: L ∗ δ = 12 ∆ d + R , (B.13)where, for all i = 1 , . . . , d ( Rv ) i = δω i ∂ θ v i + ∂ θ v i d X k = − d λ k J ∗ q k − d X k = − d λ k J ∗ ( q k ( ∂ θ v k )) − Z T ∂ θ v i ( θ ) d X k = − d λ k J ∗ q k ( θ ) ! d θ . (B.14)Since R only contains first order derivatives and J and q k are smooth, it is easy to seethat for all u ∈ H d , we have k Ru k ,d C k u k ,d . (B.15)One deduces immediately from this estimate that there exists a constant C > u ∈ H d , k [2(1 − L ∗ δ ) − (1 − ∆ d )] u k ,d C k u k ,d . (B.16)Consequently, the operator [2(1 − L ∗ δ ) − (1 − ∆ d )](1 − ∆ d ) − / is bounded in H d . Since1 − L ∗ δ is sectorial in H d with the same domain as ∆ d , an application of [24], Theorem ISORDER-INDUCED TRAVELING WAVES IN THE STOCHASTIC KURAMOTO MODEL 41 (cid:13)(cid:13) (1 − L ∗ δ ) β · (cid:13)(cid:13) ,d and (cid:13)(cid:13) (1 − ∆ d ) β · (cid:13)(cid:13) ,d are equivalent. The normequivalence (B.12) follows directly from the definitions (B.11) and (A.3). (cid:3) B.5.
Regularity of semigroups.
Recall here the definition of the projection P ψ,δ on thekernel Span( ∂ θ q ψ,δ ) of L ψ,δ defined in Section 2.3. We drop here the dependance on ψ forsimplicity. The corresponding projection on the kernel of L ∗ δ is given by P , ∗ δ . This kernelis one-dimensional, spanned by some θ v ( θ ) and there exists a linear form e p , boundedon H d such that, for all u ∈ H d , P , ∗ δ u = e p ( u ) v . Note that it is easy to see that v is aregular ( C ∞ ) function on T . Proposition B.6.
Suppose the assumptions of Proposition B.4 are true. For any γ ∈ [0 , γ L ∗ δ ) , any β ∈ [0 , and all t > , u ∈ H d , (cid:13)(cid:13)(cid:13) e tL ∗ δ (1 − P , ∗ δ ) u (cid:13)(cid:13)(cid:13) β,d C e − γt t β k (1 − P , ∗ δ ) u k ,d , (B.17) and (cid:13)(cid:13)(cid:13) e tL ∗ δ u (cid:13)(cid:13)(cid:13) β,d C (cid:18) e − γt t β (cid:19) k u k ,d , (B.18) and for all β > , β ′ > such that β + β ′ and all h ∈ H β +2 β ′ d , (cid:13)(cid:13)(cid:13)(cid:16) e tL ∗ δ − (cid:17) h (cid:13)(cid:13)(cid:13) β ′ ,d t β k (1 − P , ∗ δ ) h k β ′ +2 β,d . (B.19) Proof of Proposition B.6.
Following Proposition B.4, L ∗ δ P , ∗ δ = 0 and L ∗ δ (1 − P , ∗ δ ) is sec-torial in H d , with spectrum lying in { λ ∈ C : | arg( λ ) | > π/ ε ′ } − γ L ∗ δ for some ε ′ > γ < γ L ∗ δ ),we obtain that for all t > k ( − L ∗ δ ) β e tL ∗ δ (1 − P , ∗ δ ) u k ,d C β t − β e − γt k u k ,d . (B.20)Now as in the proof of Lemma B.5, we can apply [24], Theorem 1.4.8 to show that thenorms induced by ( − L ∗ δ ) β and (1 − L ∗ δ ) β are equivalent on the range of (1 − P , ∗ δ ) and weobtain for all u ∈ H d k e tL ∗ δ u k β,d C k (1 − L ∗ δ ) β e tL ∗ δ ( P , ∗ δ u + (1 − P , ∗ δ ) u ) k ,d C ′ β (1 + e − γt t − β ) k u k ,d . (B.21)We have used here in particular the fact that for all u ∈ H d , (cid:13)(cid:13)(cid:13) e tL ∗ δ P , ∗ δ u (cid:13)(cid:13)(cid:13) β,d | e p ( u ) | (cid:13)(cid:13)(cid:13) e tL ∗ δ v (cid:13)(cid:13)(cid:13) β,d = | e p ( u ) | k v k β,d C k u k ,d , since k v k β < + ∞ . Concerning (B.19), remark that e tL ∗ δ − (cid:16) e tL ∗ δ (1 − P , ∗ δ ) − (cid:17) (1 − P , ∗ δ ) , (B.22)so applying Theorem 1.8.4 of [24] we have (cid:13)(cid:13)(cid:13)(cid:16) e tL ∗ δ − (cid:17) h (cid:13)(cid:13)(cid:13) β ′ ,d C (cid:13)(cid:13)(cid:13) (1 − L ∗ δ ) β ′ (cid:16) e tL ∗ δ (1 − P , ∗ δ ) − (cid:17) (1 − P , ∗ δ ) h (cid:13)(cid:13)(cid:13) ,d C ′′ β t β k (1 − L ∗ δ ) β ′ + β (1 − P , ∗ δ ) h k ,d C ′′′ β t β k (1 − P , ∗ δ ) h k β ′ +2 β,d . (B.23)This concludes the proof of Proposition B.6. (cid:3) One can deduce from Proposition B.6 a similar regularity result concerning the semi-group t e tL δ : Proposition B.7.
For all K > , all δ < δ ( K ) , the semigroup t e tL δ is continuousfrom H − d to H − d : for all h ∈ H − d , t > , (cid:13)(cid:13) e tL δ h (cid:13)(cid:13) − ,d C (cid:18) √ t (cid:19) k h k − ,d , (B.24) and for all ε ∈ (0 , / , t > , u > , (cid:13)(cid:13)(cid:13) e ( t + u ) L δ h − e tL δ h (cid:13)(cid:13)(cid:13) − ,d Cu ε (cid:18) t / ε (cid:19) k h k − ,d . (B.25) Proof of Proposition B.7.
Let β ∈ [0 , t > h ∈ H − d and v a regular test function.Consider ( h l ) l > a sequence of elements of L ,d converging to h in H − d . For all l > (cid:12)(cid:12)(cid:10) e tL δ h l , v (cid:11) d (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:10) e tL δ h l , v (cid:11) ,d (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)D h l , e tL ∗ δ v E ,d (cid:12)(cid:12)(cid:12)(cid:12) , k h l k − (1+2 β ) ,d (cid:13)(cid:13)(cid:13) e tL ∗ δ v (cid:13)(cid:13)(cid:13) β,d C k h l k − (1+2 β ) ,d (cid:18) t β (cid:19) k v k ,d , where we used (B.18) in the last inequality. Since h l converges to h in H − , one can make l → ∞ in the previous inequality and obtain (cid:12)(cid:12)(cid:10) e tL δ h , v (cid:11) d (cid:12)(cid:12) C k h k − (1+2 β ) ,d (cid:0) t β (cid:1) k v k ,d and since this is true for all regular v , one deduces that (cid:13)(cid:13) e tL δ h (cid:13)(cid:13) − ,d C (cid:18) t β (cid:19) k h k − (1+2 β ) ,d , (B.26)which gives (B.17) when β = . In the same way, an immediate corollary of (B.19) is thatfor all β > , β ′ > β + β ′
1, for all t > (cid:13)(cid:13)(cid:0) e tL δ − (cid:1) h (cid:13)(cid:13) − (1+2 β +2 β ′ ) ,d t β k h k − (1+2 β ′ ) ,d . (B.27)We now turn to the proof of (B.25). Fix ε ∈ (0 , /
2) and apply (B.26) for β = 1 / ε and (B.27) for β = ε and β ′ = , (cid:13)(cid:13)(cid:13) e ( t + u ) L δ h − e tL δ h (cid:13)(cid:13)(cid:13) − ,d C (cid:18) t / ε (cid:19) (cid:13)(cid:13)(cid:0) e uL δ − (cid:1) h (cid:13)(cid:13) − (2+2 ε ) ,d , Cu ε (cid:18) t / ε (cid:19) k h k − ,d . This concludes the proof of Proposition B.7. (cid:3)
Appendix C. Projections
The purpose of this section is to prove several regularity results concerning the pro-jection P ψ,δ u = p ψ,δ ( u ) ∂ θ q ψ,δ (recall Section 2.3 and (2.15)) and the projection on themanifold M proj M ( · ) defined in Lemma 2.8. Proof of Lemma 2.8.
We first prove that ψ p ψ is smooth. This follows from the factthat the whole operator L ψ is regular in ψ ∈ T : we prove indeed that the mapping ψ L ψ is in fact real holomorphic, in the sense of Kato [26], p.375. Since the problem is invariantby rotation, it suffices to study the regularity of L ψ is a neighborhood of ψ = 0. From ISORDER-INDUCED TRAVELING WAVES IN THE STOCHASTIC KURAMOTO MODEL 43 the definition of the stationary solution q in (1.8), it is straightforward to see that one canexpand q ψ in series of ψ around ψ = 0: q ψ ( θ ) = q ( θ ) + X k > ψ k k ! ∂ kψ q ψ | ψ =0 ( θ ) . From this expansion, one deduces a similar expansion for L ψ around ψ = 0: for all f regular L ψ f = L f + X k > ψ k U k f, where each U k is a differential operator of order 1, so that each U k is relatively-boundedw.r.t L . In particular the hypotheses of [26], Theorem 2.6, p. 377 are satisfied. Inparticular, ( L ψ ) ψ forms a real-holomorphic family. In particular, the mapping ψ P ψ is also regular ([26], Theorem 1.7, p. 368), and so is the mapping ψ p ψ . Then themapping f ( ψ, h ) = p ψ ( h − q ψ ) satisfies for each fixed ψ , f ( ψ , q ψ ) = 0 and ∂ ψ f ( ψ , q ψ ) = − p ψ ∂ ψ q ψ = −
1. So by the implicit function theorem, for all h in a certain neighborhoodof q ψ , there exists a unique ψ =: proj M ( h ) such that f ( ψ, h ) = 0 and h proj M ( h ) issmooth. (cid:3) The next result states that the first order of the projection proj M around q ψ is givenby the linear form p ψ defined in (2.15). Lemma C.1.
For ψ ∈ T , h ∈ H − d such that proj M ( q ψ + h ) is well-defined, we have proj M ( q ψ + h ) = ψ + p ψ ( h ) + O ( k h k − ,d ) . (C.1) Proof of Lemma C.1.
Consider the real u such that proj M ( q ψ + h ) = ψ + u . Due to thesmoothness of proj M , we have u = O ( k h k − ,d ). The real number u satisfies p ψ + u ( q ψ + h − q ψ + u ) = 0 . (C.2)A first order expansion leads to p ψ ( h − u∂ ψ q ψ ) = O ( u ) , (C.3)which gives the result, since p ψ ( ∂ ψ q ψ ) = 1. (cid:3) Appendix D. Expansions in δ The aim of this section is to obtain first order asymptotic of the drift in Theorem 2.3for small δ . We use the notations q δ , p δ as in Section 7.2, putting the emphasis on thedependency of the different terms in δ . We denote also as r δ > r δ = Ψ δ (2 Kr δ ) (recall (1.10)). We begin with a resultconcerning r δ as δ → Lemma D.1.
The mapping δ r δ is C ∞ and its derivative r ′ (0) at δ = 0 is zero, sothat as δ → : r δ = r + O ( δ ) , (D.1) where r is the unique non-trivial solution of the fixed-point problem without disorder (B.3) . Proof of Lemma D.1.
Consider the C ∞ mapping g ( r, δ ) = Ψ δ (2 Kr ) − r . This mappingsatisfies ∂ r g ( r ,
0) = 2 K∂ x Ψ (2 Kr ) −
1. The fixed-point function r Ψ (2 Kr ) is strictlyconvex when K > r > K∂ x Ψ (2 Kr ), this shows that ∂ r g ( r , < δ r δ is C ∞ . Using (1.10), one obtainsthat r ′ (0) = ∂ δ Ψ δ | δ =0 (2 Kr ) + r ′ (0)2 K∂ x Ψ (2 Kr ) . (D.2)Since 2 K∂ x Ψ (2 Kr ) <
1, the proof of Lemma D.1 will be finished once we have provedthat ∂ δ Ψ δ | δ =0 (2 Kr ) = 0. One has (recall the definition of Z in Section B.1) ∂ δ Ψ δ | δ =0 (2 Kr ) = d X k = − d λ k R π cos( θ ) ∂ δ S kδ | δ =0 ( θ, Kr ) d θ Z (2 Kr ) − R π cos( θ ) S ( θ, Kr ) Z (2 Kr ) ∂ δ Z kδ | δ =0 (2 Kr ) ! . (D.3)Some straightforward calculations show that, for all k = − d, . . . , d , θ ∈ T ∂ δ S kδ | δ =0 ( θ, Kr ) = 2 ω k e Kr cos( θ ) θ Z π e Kr cos( u ) d u + 2 π Z πθ e − Kr cos( u ) d u − Z π ue − Kr cos( u ) d u ! (D.4)and ∂ δ Z kδ | δ =0 (2 Kr ) = 2 ω k π Z π e Kr cos( θ ) Z πθ e − Kr cos( u ) d u d θ + Z (2 Kr ) Z π u (cid:16) e Kr cos( u ) − e − Kr cos( u ) (cid:17) d u ! , = 4 πω k Z π e Kr cos( θ ) Z πθ e − Kr cos( u ) d u d θ . (D.5)Since P dk = − d λ k ω k = 0, one obtains from (D.3), (D.4) and (D.5) that ∂ δ Ψ δ | δ =0 (2 Kr ) = 0.This concludes the proof of Lemma D.1. (cid:3) We now turn to the proof of Lemma 7.1:
Proof of Lemma 7.1.
Obviously, for θ ∈ T , q iδ ( θ ) = q ( θ ) + δ∂ δ q δ | δ =0 ( θ ) + O ( δ ) , where the error O ( δ ) does not depend on θ ∈ T . The fact that r ′ (0) = 0 (Lemma D.1)implies that ∂ δ q δ | δ =0 ( θ ) only depends on the derivatives of S δ and Z δ w.r.t. δ , not w.r.t. x . Namely, ∂ δ q iδ | δ =0 ( θ ) = ∂ δ S iδ | δ =0 ( θ, Kr ) Z (2 Kr ) − ∂ δ Z iδ | δ =0 (2 Kr ) S i ( θ, Kr ) Z (2 Kr ) . The expansion found in (7.3) is a simple consequence of (D.4), (D.5) and the expressionof Z in Section B.1. (cid:3) ISORDER-INDUCED TRAVELING WAVES IN THE STOCHASTIC KURAMOTO MODEL 45
Proof of Lemma 7.2.
In the case δ = 0, the projection p defined in (2.15) is given by P ( u ) = p ( u )( ∂ θ q , . . . , ∂ θ q ) = p ( u ) ∂ θ q ,nd . Since in this case, the operator L = A defined in (B.4) is essentially self-adjoint in H − /q ,d (Proposition B.1), the projection p as a natural representation in terms of the scalar product h· , ·i − , /q ,d associated to thenorm defined in (A.2), namely p ( u ) = h ∂ θ q ,nd , u i − , /q ,d k ∂ θ q ,nd k − , /q ,d . (D.6)Using the notations of Section A, we deduce that k ∂ θ q ,nd k − , /q ,d = Z π (cid:16) q ( θ ) − π Z (cid:17) q d θ = 1 − π Z , and h ∂ θ q ,nd , u i − , /q ,d = d X k = − d λ k Z π U k ( θ ) (cid:16) q ( θ ) − π Z (cid:17) q ( θ ) d θ = d X k = − d λ k Z π U k ( θ ) (cid:18) − π Z q ( θ ) (cid:19) d θ , which precisely gives the first order of (7.5). The validity of (7.5) comes from the definitionof the projection P δ in (2.15) and the fact that L δ is a relatively bounded perturbation oforder δ of the operator L = A . (cid:3) Acknowledgments
C.P. acknowledges the support of the ERC Advanced Grant “Malady” (246953). Wethank D. Bl¨omker and G. Giacomin for fruitful discussions and valuable advice.
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Laboratoire MAP5 (UMR CNRS 8145), Universit´e Paris Descartes, Sorbonne Paris Cit´e,75270 Paris, France, [email protected] .Universit`a degli Studi di Roma Tor Vergata, Dipartimento di matematica, I-00133 Roma,Italia, [email protected]@mat.uniroma2.it