THE RELAXATION SPEED IN THE CASE THE FLOW SATISFIES EXPONENTIAL DECAY OF CORRELATIONS

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1 HE RELAXAIO SPEED I HE CASE HE FLOW SAISFIES EXPOEIAL DECAY OF CORRELAIOS Brice Franke, hi-hien guyen o cite this version: Brice Franke, hi-hien guyen HE RELAXAIO SPEED I HE CASE HE FLOW SAISFIES EXPOEIAL DECAY OF CORRELAIOS 6 <hal-7798> HAL I: hal Submitte on 9 Feb 6 HAL is a multi-isciplinary open access archive for the eposit an issemination of scientific research ocuments, whether they are publishe or not he ocuments may come from teaching an research institutions in France or abroa, or from public or private research centers L archive ouverte pluriisciplinaire HAL, est estinée au épôt et à la iffusion e ocuments scientifiques e niveau recherche, publiés ou non, émanant es établissements enseignement et e recherche français ou étrangers, es laboratoires publics ou privés

2 HE RELAXAIO SPEED I HE CASE HE FLOW SAISFIES EXPOEIAL DECAY OF CORRELAIOS BRICE FRAKE AD HI-HIE GUYE Abstract We stuy the convergence spee in L -norm of the iffusion semigroup towar its equilibrium when the unerlying flow satisfies ecay of correlation Our result is some extension of the main theorem given by Constantin, Kiselev, Ryzhik an Zlatoš in [3] Our proof is base on Weyl asymptotic law for the eigenvalues of the Laplace operator, Sobolev imbeing an some assumption on ecay of correlation for the unerlying flow Introuction Let M, g) be a -imensional compact Riemannian manifol without bounary he Riemannian metric g yiels a volume measure on M enote by vol M, a Laplace operator enote by an a covariant erivative enote by Moreover it also yiels a notion of ivergence for C -vector fiels see [] for an introuction to those notions) For a ivergence free vector fiel u an some A R we consier the solution φ A t) of the parabolic partial ifferential equation { t φa t) = Au φ A t) + φ A t), ) φ A ) = φ We are intereste in the asymptotic behavior of the solutions φ A t) when φ satisfies M φ vol M = It is well known that φ A t) KA e ρat φ, where ρ A is the spectral gap of the operator L A = + Au an K A is some positive constant Here an in the following we note the usual L -norm on M with respect to vol M herefore, if A is fixe then φ A t) as t A natural question is to stuy what happens if the times t is fixe an A tens to infinity Franke, Hwang, Pai an Sheu prove in [7] that lim ρ A = inf A { φ vol M, φ =, φ is eigenfunction of u in H } It follows that ρ A iverges to infinity as A, if an only if, the anti-symmetric operator u has no eigenfunctions satisfying H -regularity However, we o not have any control on K A as A Constantin, Kiselev, Ryzhik an Zlatoš prove in [3], that when t is fixe φ A t) as A, if an only if, u has no eigenfunction in H hey call vectorfiels u having this property relaxation enhancing In particular, this property is satisfie when the volume preserving Date: January 6, 6 Key wors an phrases Decay of correlation, relaxation spee, enhancement of iffusivity, non self-ajoint generator, incompressible rift

3 RELAXAIO SPEED UDER DECAY OF CORRELAIOS flow Φ t ) t R which is generate by the evolution equation t Φ tx) = uφ t x)), Φ x) = x is weakly mixing see [] for a efinition) In this article we will make the following ecay of correlation assumption on the flow Φ t ) t R : Assumption Decay of correlation) We suppose that for some κ >, there exist two positive constants C, C such that for all f, f C κ M) an all t >, we have f, f Φ t f,, f C e Ct f C κ f C κ Results on ecay of correlation for Anosov flows on compact manifols where prove for κ = 5 by Dolgopyat in [5] Our main result in this paper is as follows: heorem Let φ A t)) t be the solution of ) with φ = If Φ t ) t R satisfies Assumption, then for any t > there exist three constants A t, Θ t, Ξ > such that φ A t) ] < exp [ Θ t lnξa)) 3+κ+ for all A > A t heorem provies an answer to the question how close the iffusion is to its equilibrium as A grows It thus etermines the spee of the relaxation phenomenon he essential ingreients for the proof are Assumption an Weyl asymptotic law on the eigenvalues of the Laplace operator he constant Θ t an A t epen on the constants in those statements an will be mae explicit in the the proof of the main result In particular those constants become more explicit if we consier the problem on the torus = [, ] see in Section ) For some fixe real value function U efine on R n, Hwang, Hwang-Ma an Sheu prove in [9] that among the vectorfiels satisfying ivue U ) =, the zero vectorfiel yiels the smallest spectral gap for the family of iffusion operators L u = U + u his means that the convergence towar the equilibrium is slowest for the reversible iffusion generate by the self-ajoint operator L = U his has some consequence in Markov Monte Carlo Methos, where usually reversible iffusions are use to approximate a given probability istribution see Geman, Hwang [8]) It was then suggeste in [9] to perturb the self-ajoint generator by aing some antisymmetric operator However, it is then important to measure the improvement mae through this evice For this it might be important to unerstan the relaxation spee in the result of [3] Our result generalizes to iffusions generate by L u as long as the unperturbe self-ajoint operator L has iscrete spectrum an as information on the asymptotics of its eigenvalues is available he paper is organize as follows In Section, we present some known results on eigenvalue istributions, that will be neee in the proof of our main theorem We also prove some result connecte to RAGE theorem, which stans for Ruelle, Amrein, Georgescu an Enss see []) he Proposition 5 will play a central role in the proof of our main result, since it relates the convergence spee in RAGE theorem with the eigenvalues of the Laplacian an the ecay of correlation assumption Our main result, heorem 3, is restate in an equivalent form an prove in Section 3 In the last section, we consier the relaxation spee on torus Preliminaries On the compact manifol M the operator is a self-ajoint positive efinite operator with iscrete spectrum, which is compose of non-negative eigenvalues

4 RELAXAIO SPEED UDER DECAY OF CORRELAIOS 3 < λ λ Let us enote by x) = λ j x the number of eigenvalues, counte with multiplicity, smaller or equal to x We nee the following classical results he etaile proofs can be foun in the references Proposition Corollary 5 [6]) As x +, we have ) x) = π) ω vol M M)x + Ox ), where ω is the volume of the unit isk in R For simplicity of notation, we will enote Ω := π) ω vol M M) For more information on the Ox )/ )-function, one can also consult [] he following corollary is an immeiat consequence Corollary here exists a constant C 3 > such that for all x C 3 ) /Ω, we have Ω ) ) ) x Ω C 3 x x x) Ω + C 3 x x 3 Ω x } Corollary 3 For any x > max {, C3+), we have 9x) x) Proof By Corollary, we have for all x > with x > C 3 + ) /Ω, ) ) 9x) x) Ω C 3 9x) 9x) Ω + C 3 x x Ω = Ω x 3 ) C 3 x 3 + ) x 3 + )Ω x C3 ) We enote the eigenfunctions of the operator associate to the eigenvalues λ, λ, by ϕ, ϕ, hey form some orthogonal base for the Hilbert space { } H := f L M, vol M ) : fvol M = Let us also enote by P the orthogonal projection on the subspace spanne by the first eigenvectors ϕ, ϕ,, ϕ he Sobolev space H m associate with is forme by all vectors ψ = j= c jϕ j H satisfying M ψ H = λ m m j c j < j= he relation between the norms C κ an H m is given through the following result Proposition Sobolev imbeing []) here exists a constant C > such that for all n ϕ n C κ C ϕ n H +κ+ = C λ +κ+ n We now present the following proposition which is central for the proof of our main result

5 RELAXAIO SPEED UDER DECAY OF CORRELAIOS Proposition 5 Uner Assumption one has for any, > an for any function f H with f = that P f Φ t ) C C t λ +κ+ C Remark 6 he Proposition 5 gives an explicit expression for some constant in Lemma 3 from [3] his lemma states that for any, ξ > an any compact set K {f H, f = }, there exists, ξ, K) such that P f Φ t ) t ξ for all, ξ, K) an all f K Accoring to Proposition 5 the explicit choice, ξ) = C C λ +κ+ ξ C implies P f Φ t ) t ξ for all, ξ) It therefore turns out that the constant in Lemma 3 from [3] can be chosen not to epen on K Proof of Proposition 5 he proof follows the proof of RAGE theorem from the book of Cycon, Froese, Kirsch an Simon see []) We use our assumption on ecay of correlation an the explicit expression for the projection operator P to obtain an inequality from the proof presente there For all f H we have the ecomposition f = k= ϕ k, f ϕ k By the above notation, we have P f = ϕ k, f ϕ k Let us efine Q )f = an therefore Q )Q )f) = It follows that 3) = Q ) Q )f) = k= k= j= = k= P f Φ t )) Φ t t hus we have k= ϕ k Φ t, f ϕ k Φ t t, ϕ k Φ t, Q )f) ϕ k Φ t t k= j= k= j= ϕ k Φ t, ϕ j Φ s ϕ j Φ s, f ϕ k Φ t st ϕ k Φ t, ϕ j Φ s st ϕ k ϕ j ϕ k, ϕ j Φ s t st By Assumption, there exist positive constants C, C such that ) ϕ k, ϕ j Φ s t C e C s t ϕ k C κ ϕ j C κ By Proposition, for all n, we have 5) ϕ n C κ C ϕ n H +κ+ = C λ +κ+ n

6 RELAXAIO SPEED UDER DECAY OF CORRELAIOS 5 From 3), ) an 5) we obtain Q ) 6) Moreover, one has 7) k= j= = C C It is obvious that obtain k= λ +κ+ k C e C s t Cλ +κ+ k ) e C s t st e C s t st = C + e C C k= λ +κ+ k 8) Q ) C C C One has for all f with f =, 9) P f Φ t ) t = Combination of 8) an 9) gives λ +κ+ j st < C λ +κ+ Combining with 6) an 7) we = f, P f Φ t ) t λ +κ+ f, P f Φ t )) Φ t t P f Φ t )) Φ t t Q )f f Q ) C C C λ +κ+ We also nee the following classical statement on the Lipshitz norm of the flow: Proposition 7 For all t R one has that Φ t Lip e u Lip t Proof See [3] p 66 3 Main result an proofs Following the approach from [3], we prefer to work with the rescale solution φ A t) = φ ɛ t/ɛ), which satifies the following equation { s φɛ s) = u + ɛ )φ ɛ s), 3) φ ɛ ) = φ he following theorem is then equivalent to our main result, heorem heorem 3 Let φ ɛ s)) s be the solution of 3) with φ = For any > there exit constants A, Θ an Ξ such that φ ɛ ) < exp [ Θ ln Ξ )) ] 3+κ+ for all ɛ < ɛ ɛ A

7 RELAXAIO SPEED UDER DECAY OF CORRELAIOS 6 Remark 3 Some explicit expression for the constants Θ an A is given later in Remark 33 Proof Since our proof relies strongly on the proof of heorem from the paper of Constantin, Kiselev, Ryzhik an Zlatoš see [3]) we have to introuce some of the concepts an notations, which are use there hey prove that, for any given an δ, there exists an ɛ δ) such that for all ɛ < ɛ δ), one has φ ɛ /ɛ) < δ Our purpose here is to explicit the constants involve in this statement; that means to better unerstan the relation between ɛ an δ when is fixe We will prouce some explicit function ɛ expl δ) with ɛ expl δ) < ɛ δ) It then follows that φ ɛexpl δ) /ɛ expl δ)) δ, for all δ he function ɛ expl δ) has some explicit inverse function δ expl ɛ) an it will then follow that φ ɛ /ɛ) < δ expl ɛ), for all ɛ sufficiently small We will first briefly explain how the constant ɛδ) is constructe in [3] ote that some of the constructions presente there are simplifie by the fact that Assumption rules out point spectrum for the operator u hey construct ɛ δ) as follows ɛ δ) = min { δ), λ δ) δ) B t)t where λ δ) is a suitable eigenvalue λ δ) satifying e λδ)/8 < δ an with Bt) = Φ t Lip, see proofs of heorem in [3] an heorem in []) Without loss of generality, we assume that λ δ)+ > λ δ) Moreover, δ) = δ), /, K) where, ξ, K) is a constant satisfying P f Φ t ) t < ξ, for all >, ξ, K) an all f K, where K is a suitably chosen compact subset of the set S := {f H : f = } However we saw in Remark 6 that the constant, ξ, K) can be chosen inepenent from K We therefore rop the K in the notation If we can fin explicit functions expl δ) an λ expl expl δ) satifying δ) > δ) an λ expl δ) > λ δ) then the following function will be an explicit lower boun for ɛ δ) ɛ expl δ) := min expl δ), λ expl δ) expl δ) B t)t We choose the function λ expl 3) λ expl 8 lnδ) δ)) for all δ satisfying 33) } δ) := 7 lnδ)/ From Corollary 3, we have ), 8 lnδ) > max {, C 3 + ) Ω },

8 RELAXAIO SPEED UDER DECAY OF CORRELAIOS 7 his is equivalent to the existence of an eigenvalue λ δ) satisfying 3) 8 lnδ) < λ δ) < 7 lnδ) = λ expl δ) Since we assume λ δ)+ > λ δ), we have by Corollary that for all δ satifying 35) δ) = λ δ) ) 3 Ω λ δ) 8 lnδ) C 3) Ω From Remark 6, we obtain δ) = δ), ) = 8C Cδ) λ /+κ+ δ) C where We efine 8C CΩ λ3/+κ+ δ) C C 5 8 lnδ) C 5 := 9 3/+κ+ Ω 8C C C ) 3/+κ+ =: expl δ) 36) ɛ expl δ) := for all δ satifying the relations 33), 35) an u Lip λ expl δ) exp[ u Lip expl δ)] 37) 8 lnδ) 9 We then have expl δ) = 7 lnδ)) u Lip 7 lnδ) u Lip expl δ) > u Lip λ expl δ) exp[ u Lip expl δ)] Moreover, we obtain with Proposition 7 that expl δ) B t)t an it then follows that λ expl expl δ) δ) expl δ) B t)t herefore, ɛ expl δ) < min expl > e u Lipt t < exp[ u Lip expl δ)] u Lip u Lip λ expl δ) exp[ u Lip expl δ)] δ), λ expl δ) expl δ) B t)t = ɛexpl δ)

9 RELAXAIO SPEED UDER DECAY OF CORRELAIOS 8 From 33) we have λ expl δ) 9 C 5 expl δ) an it follows that 38) ɛ expl δ) > C 5 u Lip 9 expl δ) exp[ u Lip expl δ)] C 5 u Lip 9 exp[ + u Lip ) expl δ)] =: ɛexpl δ) where we use e x > x for x = expl δ) > It follows from 38) that the inverse function is given by 39) δ expl ɛ) = exp [ 8 C 5 + u Lip ) ln C5 u Lip 9 ɛ ) ) ] 3+κ+ he proof is complete with Ξ := C 5 u Lip 9, Θ := 8 an A in the following remark C 5 + u Lip ) ) 3+κ+ Remark 33 he relation between an ɛ expl is euce from 38) an 36) Aing all of conitions 33), 35) an 37) we have 8 lnδ) 3) max {, C 3 + ) Ω, C 3) } Ω, 9 It then follows that with A := ɛ expl δ) = 9 exp[ + u Lip ) expl δ)] C 5 u Lip [ ) ] 3/+κ+ 9 exp + u Lip )C 8 lnδ) 5 = C 5 u Lip A [ 9 exp + u Lip )C 5 max {, C 3 + ) C 5 u Lip Ω, C 3) Ω, } 3/+κ+ ] 9 his conition ensures that C 5 u Lip /9 ɛ) > therefore the formula 39) is well efine the particular case of the torus We consier the problem on the torus = [, ] In this case, we know exactly the eigenvalues of the Laplace operator herefore, Corollary an Corollary 3 will be simplifie by Corollary Moreover, we can give the exact value for the constant C in Proposition, this is provie by Proposition he following are the etails

10 RELAXAIO SPEED UDER DECAY OF CORRELAIOS 9 Corollary In the case of torus = [, ], the number of eigenvalues of the Laplace operator smaller or equal to x is x) = { = # m, n) Z : m + n x } π λ x It is easy to see that with all x > we have x) x/π + ), furthermore x + π) ) x) Proposition For any eigenfunction ϕ n associate to the eigenvalue λ n an for any κ >, we get ϕ n C κ κ/ κλ κ/ n Proposition 5 is base on Propositon which is improve for the case of the torus in Remark 6 If we use this in our proof then we get the following improve Proposition Proposition 3 For any, > an for any function f with f =, we have P f Φ t ) C t κ κ/ λ κ/ C hen we have the concrete case of heorem heorem If Φ t ) t R satisfies ecay of correlation in Assumption, then for any >, we have φ A ) exp C π )) ) 8 C π + 6C u Lip κ κ ln u Lip κ+ A 3π, where A satisfies [ ) ) κ+ ] exp + 6C u Lipκ κ C π 9 + 3π A u Lip References [] RA Aams : Sobolev Spaces, Acaemic Press, ew-york, 978) [] I Chavel : Eigen-Values in Riemannian Geometry, Acaemic Press, ew-york, 98) [3] P Constantin, A Kiselev, L Ryzhik, A Zlatoš : Diffusion an mixing in flui flow, Annals of Mathematics, 68, 63-67, 8) [] H Cycon, R Froese, W Kirsch, B Simon : Schröinger Operators, Springer-Verlag, ew-york, 987) [5] D Dolgopyat : On ecay of correlations in Anosov flows, Annals of Mathematics, 7, , 998) [6] JJ Duistermaat, V Guillemin : he spectrum of positive elliptic operators an perioic bicharacteristics, Inventiones Mathematicae, 9, 39-79, 975) [7] B Franke, C-R Hwang, H-M Pai, S-J Sheu : he behavior of the spectral gap uner growing rift, ransactions of the American Mathematical Society, 36, 35-35, ) [8] S Geman, C-R Hwang : Diffusion for global optimization, SIAM Journal of Control an Optimization,, 3-3, 986) [9] C-R Hwang, S-Y Hwang-Ma, S-J Sheu : Accelerating iffusions, Annals of Applie Probability, 5,33-, 5) [] P Walters : An Introuction to Ergoic heory, Grauate ext in Mathematics, Springer- Verlag, ew-york, Berlin, Heielberg, 98) [] -L Pham : Meilleures estimations asymptotiques es restes e la fonction spectrale et es valeurs propres rélatifs au Laplacien, C R Aca Sci Paris, Sér A, 89, 63-66, 979)

11 RELAXAIO SPEED UDER DECAY OF CORRELAIOS [] W P Ziemer : Weakly Differentiable Functions, Springer-Verlag, ew-york, 989) Laboratiore e Mathématiques e Bretagne Atlantique UMR 65, UFR Sciences et echniques, Université e Bretagne Occientale, 6 Avenue Le Gorgeu, CS 93837, 938 Brest, ceex 3, France aress: bricefranke@univ-brestfr aress: thi-hiennguyen@univ-brestfr