FACTORIZATION OF NON-SYMMETRIC OPERATORS AND EXPONENTIAL H-THEOREM. Preliminary version of January 3, 2016

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1 FACTORIZATION OF NON-SYMMETRIC OPERATORS AND EXPONENTIAL H-THEOREM M.P. GUALDANI, S. MISCHLER, C. MOUHOT Preliminary version of January 3, 216 Abstract. We present an abstract metho for eriving ecay estimates on the resolvents an semigroups of non-symmetric operators in Banach spaces in terms of estimates in another smaller reference Banach space. This applies to a class of operators writing as A + B where A is boune, B is issipative an the two parts satisfy a semigroup commutator conition of regularization. The core of the metho is a high-orer uantitative factorization argument on the resolvents an semigroups. We then apply this approach to the Fokker-Planck euation, to the kinetic Fokker-Planck euation in the torus, an to the linearize Boltzmann euation in the torus. We finally use this information on the linearize Boltzmann semigroup to stuy perturbative solutions for the nonlinear Boltzmann euation. We introuce a non-symmetric energy metho to prove nonlinear stability in this context in L 1 v L x (1+ v k, k > 2, with sharp rate of ecay in time. Our result rastically improves the class of functions consiere in the literature, it also provies optimal rate of convergence an our proof is constructive. As a conseuence of these results, we obtain the first constructive proof of exponential ecay, with sharp rate, towars global euilibrium for the full nonlinear Boltzmann euation for har spheres, conitionally to some smoothness an (polynomial moment estimates. This improves the result in [4] where polynomial rates at any orer were obtaine, an solves the conjecture raise in [16, 37, 98] about the optimal ecay rate of the relative entropy in the H-theorem. Mathematics Subject Classification (2: 47D6 One-parameter semigroups an linear evolution euations [See also 34G1, 34K3], 35P15 Estimation of eigenvalues, upper an lower bouns, 47H2 Semigroups of nonlinear operators [See also 37L5, 47J35, 54H15, 58D7], 35Q84 Fokker-Planck euations, 76P5 Rarefie gas flows, Boltzmann euation [See also 82B4, 82C4, 82D5]. Keywors: spectral gap; semigroup; spectral mapping theorem; uantitative; Plancherel theorem; coercivity; hypocoercivity; issipativity; hypoissipativity; Fokker-Planck euation; Boltzmann euation; H-theorem; exponential rate; stretche exponential weight; Povzner estimate; averaging lemma; thermalization; entropy. Contents 1. Introuction The problem at han Motivation Main results Acknowlegments 4 2. Factorization an uantitative spectral mapping theorems Notation an efinitions Factorization an spectral analysis Hypoissipativity Factorization an uantitative spectral mapping theorems The Fokker-Planck euation 18 1

2 2 M.P. GUALDANI, S. MISCHLER, C. MOUHOT 3.1. The Fokker-Planck euation: moel an results Proof of the main results The kinetic Fokker-Planck euation in a perioic box Summary of the results The linearize Boltzmann euation Review of the ecay results on the semigroup The main hypoissipativity results Strategy of the proof Integral estimates with polynomial weight on the remainer Pointwise estimates on the remainer Dissipativity estimate on the coercive part Regularization estimates in the velocity variable Iterate averaging lemma Proof of the main hypoissipativity result Structure of singularities for the linearize flow The nonlinear Boltzmann euation The main results Strategy of the Proof of Theorem Proof of Theorem 5.3, part (I Proof of Theorem 5.3, parts (II an (III Proof of Theorem Proof of Theorem Structure of singularities for the nonlinear flow Open uestions 78 References Introuction 1.1. The problem at han. This paper eals with (i the stuy of resolvent estimates an ecay properties for a class of generators an associate semigroups in general Banach spaces, an (ii the stuy of relaxation to euilibrium for some kinetic evolution euations, which makes use of the previous abstract tools. Let us give a brief sketch of the first problem. Consier two Banach spaces E E, an two C - semigroup generators L an L respectively on E an E with spectrum Σ(L, Σ(L C. Denote S(t an S(t the two associate semigroups respectively in E an E. Further assume that L E = L, an E is ense in E. The theoretical uestion we aress in this work is the following: Can one erive uantitative informations on Σ(L an S(t in terms of informations on Σ(L an S(t? We provie here an answer for a class of operators L which split as L = A + B, where the spectrum of B is well localize an the iterate convolution (AS B n maps E to E with proper time-ecay control for some n N. We then prove that (i L inherits most of the spectral gap properties of L, (ii explicit estimates on the rate of ecay of the semigroup S(t can be compute from the ones on S(t. The core of the propose metho is a robust factorization argument on the resolvents an semigroups, reminiscent of the Dyson series. In a secon part of this paper, we then show that the kinetic Fokker-Planck operator an the linearize Boltzmann operator for har sphere interactions satisfy the above abstract assumptions, an we thus exten the known spectral-gap properties from the stanar linearization space (an L 2 space with Gaussian weight prescribe by the euilibrium to larger Banach spaces (for example L p with polynomial ecay. It is worth mentioning that the propose metho provies optimal rate of ecay an there is no loss of accuracy in the extension process from E to E (as woul be the case in, say, interpolation approaches.

3 FACTORIZATION OF NON-SYMMETRIC OPERATORS... 3 Proving the abstract assumption reuires significant technical efforts for the Boltzmann euation an leas to the introuction of new tools: some specific estimates on the collision operator, some iterate averaging lemma an a nonlinear non-symmetric energy metho. All together, we are able to prove nonlinear stability of Gaussian euilibrium an of space homogeneous solutions for the Boltzmann euation for har spheres interactions in the torus in a L 1 vl x (1 + v k, k > 2, framework with sharp rate of ecay in time. That result rastically improves the class of functions consiere in the literature since the seminal work by Ukai [17] an provies (for the very first time optimal rate of ecay. The metho of proof is also completely constructive Motivation. The motivation for the abstract part of this paper, i.e. enlarging the functional space where spectral properties are known to hol for a linear operator, comes from nonlinear PDE analysis. The first motivation is when the linearize stability theory of a nonlinear PDE is not compatible with the nonlinear theory. More precisely, the natural function space where the linearize euation is well-pose an stable, with nice symmetric or skew-symmetric properties for instance, is too small for the nonlinear PDE in the sense that no well-poseness theorem is known (an conjecture to be false in such a space. This is the case for the classical Boltzmann euation an therefore it is a key obstacle in obtaining perturbative result in natural physical spaces an connecting the nonlinear results to the perturbative theory. This is relate to the famous H-theorem of Boltzmann. The natural uestion of unerstaning mathematically the H-theorem was emphasize by Truesell an Muncaster [16, pp ] thirty years ago: Much effort has been spent towar proof that place-epenent solutions exist for all time. [... ] The main problem is really to iscover an specify the circumstances that give rise to solutions which persist forever. Only after having one that can we expect to construct proofs that such solutions exist, are uniue, an are regular. The precise issue of the rate of convergence in the H-theorem was then put forwar by Cercignani [37] (see also [38] when he conjecture a linear relationship between the entropy prouction functional an the relative entropy functional, in the spatially homogeneous case. While this conjecture has been shown to be false in general [17], it gave a formiable impulse to the works on the Boltzmann euation in the last two ecaes [32, 31, 11, 17, 11]. It has been shown to be almost true in [11], in the sense that polynomial ineualities relating the relative entropy an the entropy prouction hol for powers close to 1, an it was an important inspiration for the work [4] in the spatially inhomogeneous case. However, ue to the fact that Cercignani s conjecture is false for physical moels [17], these important progresses in the far from euilibrium regime were unable to answer the natural conjecture about the correct timescale in the H-theorem, in orer to prove the exponential ecay in time of the relative entropy. Proving this exponential rate of relaxation was thus pointe out as a key open problem in the lecture notes [98, Subsection 1.8, page 62]. This has motivate the work [87] which answers this uestion, but only in the spatially homogeneous case. In the present paper we answer this uestion for the full Boltzmann euation for har spheres in the torus. We work in the same setting as in [4], that is uner some a priori regularity assumptions (Sobolev norms an polynomial moments bouns. We are able to connect the nonlinear theory in [4] with the perturbative stability theory first iscovere in [17] an then revisite with uantitative energy estimates in several works incluing [59] an [89]. This connexion relies on the evelopment of a perturbative stability theory in natural physical spaces thanks to the abstract extension metho. Let us also mention here the important papers [8, 9, 114, 115] which prove for instance nonlinear stability in (1 + v k with s > 3/p an k > large enough, by non-constructive methos. We emphasize the ramatic gap between the spatially homogeneous situation consiere in [87] an the spatially inhomogeneous one stuie here. In the first case the linearize euation is coercive an the linearize semigroup is self-ajoint or sectorial, whereas in the secon case the euation is hypocoercive an the linearize semigroup is neither sectorial, nor even hypoelliptic. The secon main motivation for the abstract metho evelope here is consiere in other papers [76, 1]. It concerns the existence, uniueness an stability of stationary solutions for egenerate perturbations of a known reference euation, when the perturbation makes the steay solutions leave the natural linearization space of the reference euation. Taking avantage of the theory evelope in the present spaces of the form L 1 vw s,p x

4 4 M.P. GUALDANI, S. MISCHLER, C. MOUHOT work, the first existence result in a collisional regime for spatially inhomogeneous granular gases has been recently obtaine in [13]. More generally, the present work has inspire a large number of papers, among which [29, 14, 34, 33, 15, 8, 45, 35, 25, 24, 28, 36, 79, 81] Main results. We can summarize the main results establishe in this paper as follows: Section 2. We prove an abstract theory for enlarging (Theorem 2.1 the space where the spectral gap an the iscrete part of the spectrum is known for a certain class of unboune close operators. We then prove a corresponing abstract theory for enlarging (Theorem 2.13 the space where explicit ecay semigroup estimates are known, for this class of operators. This can also be seen as a theory for obtaining uantitative spectral mapping theorems in this setting, an it works in general Banach spaces. Section 3. We prove a set of results concerning Fokker-Planck euations. The main outcome is the proof of an explicit spectral gap estimate on the semigroup in L 1 x,v(1 + v k, k > as small as wante, for the kinetic Fokker-Planck euation in the torus with super-harmonic potential (see Theorems 3.1 an Section 4. We prove a set of results concerning the linearize Boltzmann euation. The main outcome is the proof of explicit spectral gap estimates on the linearize semigroup in L 1 an L with polynomial moments (see Theorem 4.2. More generally we prove explicit spectral gap estimates in any space of the form Wv σ, Wx s,p (m, σ s, with polynomial or stretche exponential weight m, incluing the borerline cases L x,v(1 + v 5+ an L 1 vl x (1 + v 2+. We also make use of the factorization metho in orer to stuy the structure of singularities of the linearize flow (see Subsection 4.1. Section 5. We finally prove a set of results concerning the nonlinear Boltzmann euation in perturbative setting. The main outcomes of this section are: (1 The construction of perturbative solutions close to (m, s > 6/p with polynomial or stretche exponential weight m, incluing the borerline cases L x,v(1 + v 5+ an L 1 vl x (1 + v 2+ without assumption on the erivatives: see Theorem 5.3 in a close-to-euilibrium setting, an Theorem 5.5 in a close-to-spatially-homogeneous setting. (2 We give a proof of the exponential H-theorem: we show exponential ecay in time of the relative entropy of solutions to the fully nonlinear Boltzmann euation, conitionnally to some regularity an moment bouns. Such rate is proven to be sharp. This answers the conjecture in [4, 98] (see Theorem 5.7. We also finally apply the factorization metho an the Duhamel principle to stuy the structure of singularities of the nonlinear flow in perturbative regime (see Subsection 5.7. the euilibrium or close to the spatially homogeneous case in W σ, v W s,p x Below we give a precise statement the main result establishe in this paper. Theorem 1.1. The Boltzmann euation t f + v x f = Q(f, f, t, x T 3, v R 3, [ ] Q(f, f := f(x, v f(x, v f(x, v f(x, v v v v σ R 3 S 2 v = v + v + σ v v, v = v + v σ v v with har spheres collision kernel an perioic bounary conitions is globally well-pose for non-negative initial ata close enough to the Maxwellian euilibrium µ or to a spatially homogeneous profile, in the space L 1 vl x (1 + v k, k > 2. The corresponing solutions ecay exponentially fast in time with constructive estimates an with the same rate as the linearize flow in the space L 1 vl x (1 + v k. For k large enough (with explicit threshol this rate is the sharp rate λ > given by the spectral gap of the linearize flow in L 2 (µ 1/2. Moreover any solution that is a priori boune uniformly in time in Hx,v(1 s + v k with some large s, k satisfies the exponential ecay in time with sharp rate O(e λ t in L 1 norm, as well as in relative entropy Acknowlegments. We thank Claue Baros, José Alfréo Cañizo, Miguel Escobeo, Bertran Los, Mustapha Mokhtar-Kharroubi, Robert Strain for fruitful comments an iscussions. The thir author also wishes to thank Thierry Gallay for numerous stimulating iscussions about the spectral theory of non-self-ajoint operators, an also for pointing out the recent preprint [61]. The authors wish

5 FACTORIZATION OF NON-SYMMETRIC OPERATORS... 5 to thank the funing of the ANR project MADCOF for the visit of MPG in Université Paris-Dauphine in spring 29 where this work was starte. The thir author s work is supporte by the ERC starting grant MATKIT. The first author is supporte by NSF-DMS Support from IPAM (University of California Los Angeles an ICES (The University of Texas at Austin is also gratefully acknowlege. 2. Factorization an uantitative spectral mapping theorems 2.1. Notation an efinitions. For a given real number a R, we efine the half complex plane a := {z C, Re z > a}. For some given Banach spaces (E, E an (E, E we enote by B(E, E the space of boune linear operators from E to E an we enote by B(E,E or E E the associate norm operator. We write B(E = B(E, E when E = E. We enote by C (E, E the space of close unboune linear operators from E to E with ense omain, an C (E = C (E, E in the case E = E. For a Banach space X an a generator Λ on X, we enote by S Λ (t, t, its semigroup, by Dom(Λ its omain, by N(Λ its null space an by R(Λ its range. We also enote by Σ(Λ its spectrum, so that for any z belonging to the resolvent set ρ(λ := C\Σ(Λ the operator Λ z is invertible an the resolvent operator R Λ (z := (Λ z 1 is well-efine, belongs to B(X an has range eual to D(Λ. We recall that ξ Σ(Λ is sai to be an eigenvalue if N(Λ ξ {}. Moreover an eigenvalue ξ Σ(Λ is sai to be isolate if Σ(Λ {z C, z ξ r} = {ξ} for some r >. In the case when ξ is an isolate eigenvalue we may efine Π Λ,ξ B(X the associate spectral projector by (2.1 Π Λ,ξ := 1 R Λ (z z 2iπ z ξ =r with < r < r. Note that this efinition is inepenent of the value of r as the application C \ Σ(Λ B(X, z R Λ (z is holomorphic. For any ξ Σ(Λ isolate, it is well-known (see [68, III-(6.19] that Π 2 Λ,ξ = Π Λ,ξ, so that Π Λ,ξ is inee a projector, which commutes with S Λ. When moreover the algebraic eigenspace M(Λ ξ := R(Π Λ,ξ is finite imensional, we say that ξ is a iscrete eigenvalue, written as ξ Σ (Λ. In that case, R Λ is a meromorphic function on a neighborhoo of ξ, with non-removable finite-orer pole ξ, an there exists α N such that Finally for any a R such that M(Λ ξ = N(Λ ξ α = N(Λ ξ α for any α α. Σ(Λ a = {ξ 1,..., ξ k } where ξ 1,..., ξ k are istinct iscrete eigenvalues, we efine without any risk of ambiguity Π Λ,a := Π Λ,ξ1 + + Π Λ,ξk Factorization an spectral analysis. The main abstract factorization an enlargement result is: Theorem 2.1 (Enlargement of the functional space. Consier two Banach spaces E an E such that E E with continuous embeing an E is ense in E. Consier an operator L C (E such that L := (L E C (E. Finally consier a set a as efine above. We assume: (H1 Localization of the spectrum in E. There are some istinct complex numbers ξ 1,..., ξ k a, k N (with the convention {ξ 1,..., ξ k } = if k = such that Σ(L a = {ξ 1,..., ξ k } Σ (L (istinct iscrete eigenvalues. (H2 Decomposition. There exist A, B operators efine on E such that L = A + B, Dom(B = Dom(L an:

6 6 M.P. GUALDANI, S. MISCHLER, C. MOUHOT (i B C (E is such that R B (z is boune in B(E uniformly on z a an R B (z B(E as Re z, in particular Σ(B a = ; (ii A B(E is a boune operator on E; (iii There is n 1 such that the operator (AR B (z n is boune in B(E, E uniformly on z a. Then we have in E: (i The spectrum of L satisfies: Σ(L a = {ξ 1,..., ξ k }. (ii For any z a \ {ξ 1,..., ξ k } the resolvent satisfies: n 1 (2.2 R L (z = ( 1 l R B (z (AR B (z l + ( 1 n R L (z (AR B (z n. l= (iii For any ξ i Σ(L a = Σ(L a, i = 1,..., k, we have (Π L,ξi E = Π L,ξi m 1, N(L ξ i m = N(L ξ i m an M(L ξ i = M(L ξ i. an hence also Remarks 2.2. (1 In wors, assumption (H1 is a weak formulation of a spectral gap in the initial functional space E. The assumption (H2 is better unerstoo in the simplest case n = 1, where it means that one may ecompose L into a regularizing part A (in the generalize sense of the change of space A B(E, E an another part B whose spectrum is well localize in E: for instance when B a is issipative with a < a then the assumption (H2-(i is satisfie. (2 There are many variants of sets of hypothesis for the ecomposition assumption. In particular, assumptions (H2-(i an (H2-(iii coul be weakene. However, (1 these assumptions are always fulfille by the operators we have in min, (2 when we weaken (H2-(i an/or (H2- (iii we have to compensate them by making other structure assumptions. We present later, after the proof, a possible variant of Theorem 2.1. (3 One may relax (H2-(i into Σ(B a {ξ 1,..., ξ k } an the boun in (H2-(iii coul be aske merely locally uniformly in z a \{ξ 1,..., ξ k }. (4 One may replace a \ {ξ 1,..., ξ k } in the statement by any nonempty open connecte set Ω C. (5 This theorem an the next ones in this section can also be extene to the case where E is not necessarily inclue in E. This will be stuie an applie to some PDE problems in future works. Proof of Theorem 2.1. Let us enote Ω := a \ {ξ 1,..., ξ k } an let us efine for z Ω n 1 U(z := ( 1 l R B (z (AR B (z l + ( 1 n R L (z (AR B (z n. l= Observe that thanks to the assumption (H2, the operator U(z is well-efine an boune on E. Step 1. U(z is a right-inverse of (L z on Ω. For any z Ω, we compute (L zu(z = = n 1 ( 1 l (A + (B z R B (z (AR B (z l + ( 1 n (L z R L (z (AR B (z n l= n 1 n 1 ( 1 l (AR B (z l+1 + ( 1 l (AR B (z l + ( 1 n (AR B (z n = I E. l= l= Step 2. (L z is invertible on Ω. First we observe that there exists z Ω such that (L z is invertible in E. Inee, we write L z = (A R B (z + I E (B z with A R B (z < 1 for z Ω, Rez large enough, thanks to assumption (H2-(i. As a conseuence (A R B (z + I E is invertible an so is L z as the prouct of two invertible operators.

7 FACTORIZATION OF NON-SYMMETRIC OPERATORS... 7 Since we assume that (L z is invertible in E for some z Ω, we have R L (z = U(z. An if R L (z B(E = U(z B(E C for some C (,, then (L z is invertible on the isc B(z, 1/C with (2.3 z B(z, 1/C, R L (z = R L (z (z z n R L (z n, an then again, arguing as before, R L (z = U(z on B(z, 1/C since U(z is a left-inverse of (L z for any z Ω. Then in orer to prove that (L z is invertible for any z Ω, we argue as follows. For a given z 1 Ω we consier a continuous path Γ from z to z 1 inclue in Ω, i.e. a continuous function Γ : [, 1] Ω such that Γ( = z, Γ(1 = z 1. Because of assumption (H2 we know that (AR B (z l, 1 l n 1, an R L (z(ar B (z n are locally uniformly boune in B(E on Ω, which implies n= sup U(z B(E := C <. z Γ([,1] Since (L z is invertible we euce that (L z is invertible with R L (z locally boune aroun z with a boun C which is uniform along Γ (an a similar series expansion as in (2.3. By a continuation argument we hence obtain that (L z is invertible in E all along the path Γ with R L (z = U(z an R L (z B(E = U(z B(E C. Hence we conclue that (L z 1 is invertible with R L (z 1 = U(z 1. This completes the proof of this step an proves Σ(L a {ξ 1,..., ξ k } together with the point (ii of the conclusion. Step 3. Spectrum, eigenspaces an spectral projectors. On the one han, we have N(L ξ j α N(L ξ j α, j = 1,..., k, α N, so that {ξ 1,..., ξ k } Σ(L a. The other inclusion was prove in the previous step, so that these two sets are euals. We have prove Σ(L a = Σ(L a. Now, we consier a given eigenvalue ξ j of L in E. We know (see [68, paragraph I.3] that in E the following Laurent series hols R L (z = + l= l (z ξ j l C l, C l = (L ξ j l 1 Π L,ξj, l l 1, for z close to ξ j an for some boune operators C l B(E, l. The operators C 1,..., C l satisfy the range inclusions R(C 2,..., R(C l R(C 1. This Laurent series is convergent on B(ξ j, r\{ξ j } a. The Cauchy formula for meromorphic functions applie to the circle {z, z ξ j = r} with r small enough thus implies that Π L,ξj = C 1 so that C 1 since ξ j is a iscrete eigenvalue. Using the efinition of the spectral projection operator (2.1, the above expansions an the Cauchy theorem we get for any small r > Π L,ξj := ( 1n+1 2iπ = ( 1n+1 2iπ + ( 1n+1 2iπ z ξ j =r z ξ j =r R L (z (A R B (z n z 1 z ξ j =r l= C l (z ξ j l (A R B (z n z l= l C l (z ξ j l (A R B (z n z,

8 8 M.P. GUALDANI, S. MISCHLER, C. MOUHOT where the first integral has range inclue in R(C 1 an the secon integral vanishes in the limit r. We euce that M(L ξ j = R(Π L,ξj R(C 1 = R(Π L,ξj = M(L ξ j. Together with M(L ξ j = N(L ξ j α N(L ξ j α M(L ξ j for some α 1 we conclue that M(L ξ j = M(L ξ j an N((L ξ j α = N((L ξ j α for any j = 1,..., k an α 1. Finally, the proof of Π L,ξj E = Π L,ξj is straightfowar from the euality R L (zf = R L (zf when f E an the integral formula (2.1 efining the projection operator. Let us shortly present a variant of the latter result where the assumption (H2 is replace by a more algebraic one. The proof is then purely base on the factorization metho an somehow simpler. The rawback is that it reuires some aitional assumption on B at the level of the small space (which however is not so restrictive for a PDE s application perspective but can be painful to check. Theorem 2.3 (Enlargement of the functional space, purely algebraic version. Consier the same setting as in Theorem 2.1, assumption (H1, an where assumption (H2 is replace by (H2 Decomposition. There exist operators A, B on E such that L = A + B (with corresponing extensions A, B on E an (i B an B are close unboune operators on E an E, Dom(B = Dom(L, Dom(B = Dom(L, an Σ(B a = Σ(B a =. (ii A B(E is a boune operator on E. (iii There is n 1 such that the operator (AR B (z n is boune from E to E for any z a. Then the same conclusions as in Theorem 2.1 hol. Remark 2.4. Actually there is no nee in the proof that (B z 1 for z a is a boune operator, an therefore assumption (H2 coul be further relaxe to assuming only (B z 1 (E Dom(L E (bijectivity is alreay known in E from the invertibility of (B z. However these subtleties are not use at the level of the applications we have in min. Proof of Theorem 2.3. The Step 1 is unchange, only the proofs of Steps 2 an 3 are moifie: Step 2. (L z is invertible on Ω. Consier z Ω. First observe that if the operator (L z is bijective, then composing to the left the euation (L z U(z = I E by (L z 1 = R L (z yiels R L (z = U(z an we euce that the inverse map is boune (i.e. (L z is an invertible operator in E together with the esire formula for the resolvent. Since (L z has a right-inverse it is surjective. Let us prove that it is injective. Consier f N(L z E: (L z f = an thus (I + G(z (B z f = with G(z := AR B (z. We enote f := (B z f E an obtain f = G(z f f = ( 1 n G(z n f an therefore, from assumption (H2, we euce that f E. Finally f = R B (z f = R B (z f Dom(L E. Since (L z is injective we conclue that f =. This completes the proof of this step an proves Σ(L a {ξ 1,..., ξ k } together with the point (ii of the conclusion.

9 FACTORIZATION OF NON-SYMMETRIC OPERATORS... 9 Step 3. Spectrum, eigenspaces an spectral projectors. On the one han N(L ξ j N(L ξ j, j = 1,..., k, so that Σ(L a {ξ 1,..., ξ k }. Since the other inclusion was prove in the previous step, we conclue that We euce from E E that more generally an we claim that the inverse inclusions Σ(L a = Σ(L a. N(L ξ j α N(L ξ j α, j = 1,..., k, α N, N(L ξ j α N(L ξ j α, j = 1,..., k, α N, hol true. We argue by inuction on α. We first remark that, similarly as in the previous step, for any ξ a, g E an f E such that (L ξf = g, there hols f = G(ξ f + g with f := (B ξf. By iterating this formula we euce n 1 f = ( 1 l G(ξ j l g + ( 1 n G(ξ j n f E. l= That proves the claim for α = 1, because if f N(L ξ j E, we have (L ξ j f = E, an then f E. Next, we assume that the claim is prove at orer α an we consier f N(L ξ j α+1. We may write (L ξ j α [(L ξ j f] = (L ξ j α+1 f =, an by the inuction hypothesis, we get (L ξ j f E, which in turn implies f E. That conclue the proof of the inverse inclusions, an thus N(L ξ j α = N(L ξ j α α 1, j = 1,..., k. Finally, the relation Π L,ξj E = Π L,ξj follows from R L (z(f = R L (z(f when f E an the formula (2.1 for the projector Hypoissipativity. Let us first introuce the notion of hypoissipative operators an iscuss its relation with the classical notions of issipative operators an coercive operators as well as its relation with the recent terminology of hypocoercive operators (see mainly [112] an then [89, 62, 44] for relate references. Definition 2.5 (Hypoissipativity. Consier a Banach space (X, X an some operator Λ C (X. We say that (Λ a is hypoissipative on X if there exists some norm X on X euivalent to the initial norm X such that (2.4 f D(Λ, ϕ F (f s.t. Re ϕ, (Λ a f, where, is the uality bracket for the uality in X an X an F (f X is the ual set of f efine by F (f = F (f := { ϕ X ; ϕ, f = f 2 X = ϕ 2 X }. Remarks 2.6. (1 An hypoissipative operator Λ such that X = X in the above efinition is nothing but a issipative operator, or in other wors, Λ is an accretive operator. (2 When X is an Hilbert norm on X, we have F (f = {f} an (2.4 writes (2.5 f D(Λ, Re ((Λf, f X a f 2 X, where ((, X is the scalar prouct associate to X. In this Hilbert setting such a hypoissipative operator shall be calle euivalently hypocoercive. (3 When X = X is an Hilbert norm on X, the above efinition correspons to the classical efinition of a coercive operator. (4 In other wors, in a Banach space (resp. an Hilbert space X, an operator Λ C (X is hypoissipative (resp. hypocoercive on X if Λ is issipative (resp. coercive on X enowe with a norm (resp. an Hilbert norm euivalent to the initial one. Therefore the notions of hypoissipativity an hypocoercivity are invariant uner change of euivalent norm.

10 1 M.P. GUALDANI, S. MISCHLER, C. MOUHOT The concept of hypoissipativity seems to us interesting since it clarifies the terminology an raws a brige between works in the PDE community, in the semigroup community an in the spectral analysis community. For convenience such links are summarize in the theorem below. This theorem is a non stanar formulation of the classical Hille-Yosia theorem on m-issipative operators an semigroups, an therefore we omit the proof. Theorem 2.7. Consier X a Banach space an Λ the generator of a C -semigroup S Λ. We enote by R Λ its resolvent. For given constants a R, M > the following assertions are euivalent: (i Λ a is hypoissipative; (ii the semigroup satisfies the growth estimate (iii Σ(Λ a = an t, S Λ (t B(X M e a t ; z a, R Λ (z n M (Re z a n ; (iv Σ(Λ (a, = an there exists some norm on X euivalent to the norm : such that Remarks 2.8. f X λ > a, f D(Λ, (1 We recall that Λ a is maximal if f f M f, (Λ λ f (λ a f. R(Λ a = X. This further conition leas to the notion of m-hypoissipative, m-issipative, m-hypocoercive, m-coercive operators. (2 The Hille-Yosia theorem is classically presente as the necessary an sufficient conitions for an operator to be the generator of a semigroup. Then one assumes, aitionally to the above conitions, that Λ b is maximal for some given b R. Here in our statement, the existence of the semigroup being assume, the maximality conition is automatic, an Theorem 2.7 etails how the operator s, resolvent s an the associate semigroup s estimates are linke. (3 In other wors, the notion of hypoissipativity is just another formulation of the minimal assumption for estimating the growth of a semigroup. Its avantage is that it is arguably more natural from a PDE viewpoint. (4 The euivalence (i (iv is for instance a conseuence of [94, Chap 1, Theorem 4.2] an [94, Chap 1, Theorem 5.3]. All the other implications are also prove in [94, Chap 1]. Let us now give a synthetic statement aapte to our purpose. We omit the proof which is a straightforwar conseuence of the Lumer-Philipps or Hille-Yosia theorems together with basic matrix linear algebra on the finite-imensional eigenspaces. The classical reference for this topic is [68]. Theorem 2.9. Consier a Banach space X, a generator Λ C (X of a C -semigroup S Λ, a R an istinct ξ 1,..., ξ k a, k 1. The following assertions are euivalent: (i There exist g 1,..., g m linearly inepenent vectors so that the subspace Span{g 1,..., g m } is invariant uner the action of Λ, an i {1,..., m}, j {1,..., k}, g i M(Λ ξ j. Moreover there exist ϕ 1,..., ϕ m linearly inepenent vectors so that the subspace Span{ϕ 1,..., ϕ m } is invariant uner the action of Λ. These two families satisfy the orthogonality conitions ϕ i, g j = δ ij an the operator Λ a is hypoissipative on Span{ϕ 1,..., ϕ m } : f m Ker(ϕ i D(Λ, f F (f, Re f, (Λ af. n=1

11 FACTORIZATION OF NON-SYMMETRIC OPERATORS (ii There exists a ecomposition X = X X k where (1 X an (X 1 + +X k are invariant by the action of Λ, (2 for any j = 1,..., k X j is a finite-imensional space inclue in M(Λ ξ j, an (3 Λ a is hypoissipative on X : f D(Λ X, f F (f, Re f, (Λ af. (iii There exist some finite-imensional projection operators Π 1,..., Π k which commute with Λ an such that Π i Π j = if i j, an some operators T j = ξ j I Yj +N j with Y j := R(Π j, N j B(Y j nilpotent, so that the following estimate hols k (2.6 t, S Λ (t e t Tj B(X Π j C a e a t, for some constant C a 1. (iv The spectrum of Λ satisfies j=1 Σ(Λ a = {ξ 1,..., ξ k } Σ (Λ an Λ a is hypoissipative on R(I Π Λ,a. Moreover, if one (an then all of these assertions is true, we have X = R(I Π Λ,a, (istinct iscrete eigenvalues X j = Y j = M(Λ ξ j, Π Λ,ξj = Π j, T j = ΛΠ Λ,ξj. As a conseuence, we may write R Λ (z = R (z + R 1 (z, where R is holomorphic an boune on a for any a > a an k R 1 (z = Π β j j Nj n + z ξ j (z ξ j n Π j. j=1 Remark 2.1. When X is a Hilbert space an Λ is a self-ajoint operator, the assumption (i is satisfie with k = 1, ξ 1 =, as soon as there exist g 1,..., g k X normalize such that g i g j if i j, Λg i = for all i = 1,..., k, an n=2 f X := Span{g 1,..., g k }, Λf, f a f, f Factorization an uantitative spectral mapping theorems. The goal of this subsection is to establish uantitative ecay estimates on the semigroup in the larger space E. Let us recall the key notions of spectral boun of an operator L on E: s(l := sup{re ξ : ξ Σ(L} an of growth boun of its associate semigroup 1 w(l := inf t> t S 1 L(t = lim t + t S L(t. It is always true that s(l w(l but we are intereste in proving the euality with uantitative estimates, in the larger space E. Proving such a result is a particular case of a spectral mapping theorem. Let us first observe that in view of our previous factorization result the natural control obtaine straightforwarly on the resolvent in the larger functional space E is a uniform control on vertical lines. It is a classical fact that this kin of control is not sufficient in general for inverting the Laplace transform an recovering spectral gap estimates on a semigroup from it. Inee for semigroups in Banach spaces the euality between the spectral boun an the growth boun is false in general when assuming solely that the resolvent is uniformly boune in any a with a > s(l (with boun epening on a. A classical counterexample [46, Chap. 5, 1.26] is the erivation operator

12 12 M.P. GUALDANI, S. MISCHLER, C. MOUHOT Lf = f on the Banach space C (R + L 1 (R +, e s s of continuous functions that vanish at infinity an are integrable for e s s enowe with the norm f = sup f(s + s + f(s e s s. Another simple counterexample can be foun in [3]: consier 1 p < < an the C -semigroup on L p (1, L (1, efine by (T (tf(s = e t/ f(se t, t >, s > 1. However for semigroups in Hilbert spaces, the Gerhart-Herbst-Prüss-Greiner theorem [51, 64, 96, 4] (see also [46] asserts that the expecte semigroup ecay w(l = s(l is in fact true, uner this sole pointwise control on the resolvent. While the constants seem to be non-constructive in the first versions of this theorem, Engel an Nagel gave a comprehensive an elementary proof with constructive constant in [46, Theorem 1.1; chapter V]. Let us also mention on the same subject subseuent works like Yao [119] an Blake [13], an more recently [61]. The main iea in the proof of [46, Theorem 1.1, chapter V], which is also use in [61], is to use a Plancherel ientity on the resolvent in Hilbert spaces in orer to obtain explicit rates of ecay on the semigroup in terms of bouns on the resolvent. We will present in a remark how this interesting argument can be use in our case, but instea our proof will use a more robust argument vali in Banach spaces, which is mae possible by the aitional factorization structure we have. The key iea is to translate the factorization structure at the level of the semigroups. We shall nee the following efinition on the convolution of semigroup (corresponing to composition at the level of the resolvent operators. Definition 2.11 (Convolution of semigroups. Consier some Banach spaces X 1, X 2, X 3. one-parameter families of operators S 1 L 1 (R + ; B(X 1, X 2 an S 2 L 1 (R + ; B(X 2, X 3, we efine the convolution S 2 S 1 L 1 (R + ; B(X 1, X 3 by t, (S 2 S 1 (t := t S 2 (s S 1 (t s s. For two When S 1 = S 2 an X 1 = X 2 = X 3, we efine recursively S ( = I an S ( l = S S ( (l 1 for any l 1. Remarks (1 Note that this prouct law is in general not commutative. (2 A simple calculation shows that if S i satisfies t, for some a i R, α i N, C i (,, then S i (t B(Xi,X i+1 C i t αi e ai t t, S 1 S 2 (t B(X1,X 2 C 1 C 2 α 1! α 2! (α 1 + α 2! tα1+α2+1 e max(a1,a2 t. Theorem 2.13 (Enlargement of the functional space of the semigroup ecay. Let E, E be two Banach spaces with E E ense with continuous embeing, an consier L C (E, L C (E with L E = L an a R. We assume the following: (A1 L generates a semigroup e tl on E, L a is hypoissipative on R(I Π L,a an Σ(L a := {ξ 1,..., ξ k } Σ (L (istinct iscrete eigenvalues (with {ξ 1,..., ξ k } = if k =. (A2 There exist A, B C (E such that L = A + B (with corresponing restrictions A, B on E, some n 1 an some constant C a > so that (i (B a is hypoissipative on E; (ii A B(E an A B(E;

13 FACTORIZATION OF NON-SYMMETRIC OPERATORS (iii T n := (A S B ( n satisfies T n (t B(E,E C a e a t. Then L is hypoissipative in E with (2.7 t, S L(t k S L (t Π L,ξj C a max{1, t n 1 } e a t, B(E j=1 for some explicit constant C a > epening on the constants in the assumptions. Moreover we have the following factorization formula on the semigroup S L on E: (2.8 S L (t = k n 1 S L (t Π L,ξj + (I Π L,a S B (AS B l (t + [ ] (I Π L,a S L (ASB n (t. j=1 l= Remarks (1 It is part of the result that B generates a semigroup on E so that (A2-(iii makes sense. Except for the assumption that L generates a semigroup, all the other assumptions are pure functional, either on the iscrete eigenvalues of L or on L, B, A, A an T n, an o not reuire maximality conitions. (2 Assumption (A1 coul be alternatively formulate by mean of any of the euivalent assertions liste in Theorem 2.9. Proof of Theorem We split the proof into four steps. Step 1. First remark that since B = L A, A B(E, an L is m-hypoissipative then B is m-hypoissipative an generates a strongly continuous semigroup S B on E. Because of the hypoissipativity of B, we can exten this semigroup from E to E an we obtain that B generates a semigroup S B on E. To see this, we may argue as follows. We enote by E a norm euivalent to E so that B b is issipative in (E, E an E a norm euivalent to E so that B b is issipative in (E, E, for some b R large enough. We introuce the new norm f ɛ := f E + ɛ f E on E so that ɛ is euivalent to E for any ɛ >. Since B b is m-issipative in (E, ɛ, the Lumer- Phillips theorem shows that the operator B b generates a semigroups of contractions on (E, ɛ, an in particular f E, t, S (B b (tf E + ɛ S (B b (tf E f E + ɛ f E. Letting ɛ going to zero, we obtain f E, t, S B (tf E e t b f E. Because of the continuous an ense embeing E E, we euce that we may exten S B (t from E to E as a family of operators S(t which satisfies the same estimate. We easily conclue that S(t is a semigroup with generator B, or in other wors, B generates a semigroup S B = S on E. Finally, since L = A + B an A B(E, we euce that L generates a semigroup. Step 2. We have from (A2-(i that (2.9 t, S B (t E E C e at an we easily euce (by iteration that T l := (A S B ( l, l 1, satisfies (2.1 t, l 1, T l (t B(E C l t l 1 e at for some constants C l >. Let us efine U l := (I E Π L,a S B (A S B ( l, l n 1. From (2.9 an (2.1 an the bouneness of Π L,a, we get (2.11 t, U l (t B(E C l t l e at, l n 1.

14 14 M.P. GUALDANI, S. MISCHLER, C. MOUHOT By applying stanar results on Laplace transform, we have for any f E z a, + e zt U l (tf t = ( 1 l+1 (I E Π L,a R B (z (A R B (z l f. Then the inverse Laplace theorem implies that for l =,..., n 1 an for all a > a, it hols (2.12 U l (tf = ( 1l+1 2iπ ( 1 l+1 := lim M 2iπ (I E Π L,a a +i (I E Π L,a a i a +im e zt R B (z (A R B (z l f z a im e zt R B (z (A R B (z l f z, where the integral along the complex line {a + iy, y R} may not be absolutely convergent, but is efine as the above limit. Let us now consier the case l = n an efine [ U n (t := (I E Π L,a S L (A S B ( n] = [(I E Π L,a S L ] (A S B ( n. Observe that this one-parameter family of operators is well-efine an boune on E since (A S B ( n is boune from E to E by the assumption (A2-(iii. Moreover for f E, the assumption (A3-(iii implies (A SB ( n (tf E C a e at f E an since from (A1 [(I E Π L,a S L ] (tg E C a e at g E for g E, we euce, together with E E, (2.13 U n (tf E C a e at f E (for some constants C a, C a, C a >. Finally observe that z a, + e zt (I E Π L,a S L (t t = (I E Π L,a R L (z by classical results of spectral ecomposition. Therefore the inverse Laplace theorem implies that for any a > a close enough to a (so that a < min{re ξ 1,..., Re ξ k }, it hols U n (tf := lim U n,m (tf, M with (2.14 U n,m (tf := ( 1n+1 2iπ (I E Π L,a a +im a im e zt R L (z (A R B (z n f z. Step 3. Let us prove that the following representation formula hols k n (2.15 f E, t, S L (tf = S L,ξj (t f + U l (t f, where S L,ξj (t = S L (tπ L,ξj an Π L,ξj is the spectral projection as efine in (2.1. Consier f D(L an efine f t = S L (tf. From (A2 there exists b R an C b (, so that (2.16 t f t C 1 (R + ; E an f t E C b e b t f E, an we may assume b > a (otherwise the proof is finishe. Therefore the inverse Laplace theorem implies for b > b (2.17 z b, r(z := is well-efine as an element of E, an (2.18 t, f t = 1 2iπ b +i b i + j=1 e zt r(z z := l= f t e z t t = R L (z f lim M 1 b +im e zt r(z z. 2iπ b im

15 FACTORIZATION OF NON-SYMMETRIC OPERATORS Combining the efinition of f t together with (2.18 an (2.17, we get (2.19 S L (tf = lim M I b,m, where c R \ Re(Σ(L, I c,m := 1 c+im e zt R L (z f z. 2iπ c im Now from (A2-(iii, we have that (A R B (z n efine as (2.2 ( 1 n (A R B (z n = e z t T n (t t is holomorphic on a with values in B(E, E. Hence the assumptions (H1-(H2 of Theorem 2.1 are satisfie. We euce that Σ(L a = Σ(L a, with the same eigenspaces for the iscrete eigenvalues ξ 1,..., ξ k. Moreover, thanks to (A1 an (A2-(i we have sup R L (z B(E C a,ε, z K a,ε with a > a, ε >, sup R B (z B(E C a, z a K a,ε := a \ ( B(ξ 1, ε... B(ξ k, ε. As a conseuence of the factorization formula (2.2, we get Thanks to the ientity a > a, ε >, sup z K a,ε R L (z B(E C a,ε. z / Σ(L, R L (z = z 1 [ I + R L (z L] an the above boun, we have (remember that f D(L (2.21 sup z; Im z M, Re z a R L (z f B(E M. We then choose a (a, b close enough to a an ε > small enough so that B(ξ 1, ε... B(ξ k, ε a. Since R L is a meromorphic function on a with poles ξ 1,..., ξ k, we compute by Cauchy s theorem (2.22 I b,m = I a,m with [ 1 ε 1 (M = 2iπ b a k S L,ξj f + ε 1 (M, j=1 e (x+iy t R L (x + iy f x as M + thanks to (2.21. On the other han, because of Theorem 2.1, we may ecompose (2.23 I a,m = 1 2iπ a +im n 1 e zt a im l= ( 1 l R B (z (A R B (z l f z+ ( 1n 2iπ ] y=m y= M a +im a im e zt R L (z (A R B (z n f z.

16 16 M.P. GUALDANI, S. MISCHLER, C. MOUHOT Note that the limit in (2.23 as M goes to infinity is well efine. Hence (2.19, (2.22 an (2.23 yiel S L (tf = k j=1 S L,ξj (t f + 1 2iπ a +i n 1 e zt a i l= ( 1 l+1 R B (z (A R B (z l f z + ( 1n+1 2iπ a +i a i e zt R L (z (A R B (z n f z. Since k j=1 S L,ξ j (t = Π L,a S L (t we euce that the sum of the last two terms in the euation above belongs to R(I E Π L,a. Hence S L (tf = k j=1 S L,ξj (t f + 1 2iπ a +i n 1 e zt a i l= + ( 1n+1 2iπ ( 1 l+1 (I E Π L,a R B (z (A R B (z l f z a +i As a conseuence of (2.12 an (2.14, we euce that f D(L, t, S L (tf = a i e zt (I E Π L,a R L (z (A R B (z n f z. k S L,ξj (t f + j=1 n U l (t f. Then using the ensity of D(L E, we obtain the representation formula (2.15. We have thus establishe (2.8. Step 4. Conclusion. We finally obtain the time ecay (2.7 by plugging the ecay estimates (2.11 an (2.13 into the representation formula (2.15. Remark There is another way to interpret the factorization formula at the level of semigroups. Consier the evolution euation t f = Lf an introuce the spliting l= with k f = S L,ξi f in + f f n+2, i=1 t f 1 = Bf 1, fin 1 = (I Π L,a f in, t f l = Bf l + Af l 1, fin l =, 2 l n, t f n+1 = Lf n+1 + (I Π L,a Af n, f n+1 in =, t f n+2 = Lf n+2 + Π L,a Af n, f n+2 in =. This system of euations on (f l 1 l n+2 is compatible with the euation satisfie by f, an it is possible to estimate the ecay in time inuctively for f l (for the last euation one uses f n+2 = Π L,a f n+2 = Π L,a (f f n+1 an the ecay of the previous terms. We mae the choice to present the factorization theory from the viewpoint of prouct of resolvents an convolution proucts of semigroups as it reveals the algebraic structure in a much clearer way, an also is more convenient for obtaining properties of the spectrum an precise controls on the resolvent in the large space. Let us finally give a lemma which provies a practical criterion for proving assumptions (A2-(iii in the enlargement theorem 2.13: Lemma Let E, E be two Banach spaces with E E ense with continuous embeing, an consier L C (E, L C (E with L E = L an a R. We assume:

17 FACTORIZATION OF NON-SYMMETRIC OPERATORS (A3 There exist some intermeiate spaces (not necessarily orere E = E J, E J 1,..., E 2, E 1 = E, J 2, euippe with there norm enote by Ej such that (i (B Ej a is hypoissipative an A Ej is boune on E j for each 1 j J. (ii There are some constants l N, C 1, K R, α [, 1 such that t, T l (t B(Ej,E j+1 C ekt t α, for 1 j J 1, with the notation T l := (A S B ( l. Then for any a > a, there exist some explicit constants n N, C a 1 such that t, T n (t B(E,E C a e a t. Proof of Lemma On the one han, hypothesis (A3-(i implies for 1 j J 1 that (2.24 T 1 (t B(Ej C a e at an next (2.25 T l B(Ej C a t l 1 e at l 1. with On the other han, for n = p l, p N, we write T n (t = (T l T l (t }{{} p times = t tp 1 t p 1 t p 2... t2 t 1 T l (δ p... T l (δ 1 δ 1 = t 1, δ 2 = t 2 t 1,..., δ p 1 = t p 1 t p 2 an δ p = t t p 1. We claim that for p > J, there exist at least J 1 increments δ r1,..., δ rj 1 such that δ rj t/(p J for any 1 j J 1; inee, assuming that there exist δ 1,..., δ p J such that δ j > t/(p J, one arrives at the contraiction t t = δ δ p δ δ p J > (p J p J = t. Using (A3(ii to estimate T l (δ rj B(Ej,E j+1 an (2.25 to boun the other terms T l (δ r B(Er in the appropriate space, we have, with Q := {r 1,..., r J 1 }, T n (t B(E,E t tp 1 t p 1 t p 2... C a t l(p J C J e a t e K J t p J t2 t t 1 r / Q C a δr l 1 e a δr tp 1 t p 1 t p 2... Q t2 1 up 1 C e (a+ K J p J t t l(p J+p Jα u p 1 u p 2... C t 1 u2 ek δ δ α J 1 δ α j=1 r j p 1 1 u 1 (u j+1 u j α, with the convention u p = 1. Since the last integral is finite for any p N, we conclue by taking p (an then n large enough so that a + KJ/(p J < a. j=1

18 18 M.P. GUALDANI, S. MISCHLER, C. MOUHOT (3.1 Consier the Fokker-Planck euation 3. The Fokker-Planck euation t f = Lf := v ( v f + F f, f ( = f in (, on the ensity f = f t (v, t, v R an where the (exterior force fiel F = F (v R takes the form (3.2 F = v φ + U, with confinement potential φ : R R of class C 2 an non graient force fiel perturbation U : R R of class C 1 so that (3.3 v R, v (U(v e φ(v =. It is then clear that a stationary solution is µ(v := e φ(v. In orer for µ to be the global euilibrium we make the following aitional classical assumptions on the φ an U: (FP1 The Borel measure associate to the function µ an enote in the same way, µ(v := e φ(v v, is a probability measure an the function φ is C 2 an satisfies one of the two following large velocity asymptotic conitions (3.4 lim inf v or (3.5 ν (, 1 lim inf v ( v v vφ(v > while the force fiel U satisfies the growth conition ( ν v φ 2 v φ > v R, U(v C (1 + v φ(v. It is crucial to observe that (FP1 implies that the measure µ satisfies the Poincaré ineuality ( f 2 (3.6 v µ(v 2 λ P f R µ R 2 µ 1 (v for f v =, R for some constant λ P >. We refer to the recent paper [11] for an introuction to this important subject as well as to the references therein for further evelopments. Actually the above hypothesis (FP1 coul be replace by assuming irectly that (3.6 hols. However, the conitions (3.4 an (3.5 are more concrete an yiel criterion that can be checke for a given potential. The funamental example of a suitable confinement potential φ C 2 (R which satisfies our assumptions is when (3.7 φ(v α v γ an φ(v α γ v v γ 2 as v + for some constants α > an γ 1. For instance, the harmonic potential φ(v = v 2 /2 (/2 ln(2π correspons to the normalise Maxwellian euilibrium µ(v = (2π /2 exp( v 2 / The Fokker-Planck euation: moel an results. For some given Borel weight function m = m(v > on R, let us efine L p (m, 1 p 2, as the Lebesgue space associate to the norm ( 1/p f L p (m := f m L p = f p (v m(v p v. R For any given positive weight, we efine the efect weight function (3.8 ψ m,p := (p 1 m 2 m 2 + m ( m iv F F m p m. Observe that ψ µ 1/2,2 = : ψ m,p uantifies some error to this reference case. Let us enounce two more assumptions:

19 FACTORIZATION OF NON-SYMMETRIC OPERATORS (FP2 The weight m satisfies L 2 (µ 1/2 L p (m (recall p [1, 2] an the conition lim sup v ψ m,p = a m,p <. (FP3 There exists a positive Borel weight m such that L 2 (µ 1/2 L (m for any [1, 2] an there exists b R so that sup ψ m, b, [1,2], v R ( m sup m 2 x R m m 2 b. The typical weights m satisfying these assumptions are m(v e κ φ with κ [, 1/2], m(v = e κ v β β [, 1] an κ > appropriately chosen, or m(v v k, at large velocities. with Here is our main result on the Fokker-Planck euation. Theorem 3.1. Assume that F satisfies (FP1 an consier a C 2 weight function m > an an exponent p [1, 2] so that (FP2 hols if p = 2 an (FP2-(FP3 hols if p [1, 2. Then for any initial atum f in L p (m, the associate solution f t to (3.1 satisfies the following ecay estimate (3.9 t, f t µ f in Lp (m C e λm,p t f in µ f in Lp (m, with λ m,p := λ P if λ P < a m,p, an λ m,p < a m,p as close as wante to a m,p else, an where we use the notation f in := f in v. R Remarks 3.2. (1 Note that this statement implies in particular that the spectrum of L in L p (m satisfies for a as above: Σ(L {z C Re(z a} {}, an that the null space of L is exactly Rµ. (2 When m = m(φ an iv U = U φ =, an alternative choice for the efect weight function associate to the weight m an p [1, 2] coul be ψ m,p =: ψm,p 1 + ψm,p 2 with ψ 1 m,p = ψ 2 m,p = 1 p m 2 µ p v (p 1 p [ µ p m 2p 2 v ( 1 [ m 2p 2 v m 2p 4 µ m 2p 4 v ] ( 1 m 2 µ Notice that again ψ µ 1/2,2 =. The first part ψ 1 m,p is relate to the change in the Lebesgue exponent from 2 to p, an the secon part ψ 2 m,p is relate to the change of weight from µ 1/2 to m. (3 Concerning the weight function m, other technical assumptions coul have been chosen for the function m(v, however the formulation (FP2-(FP3 seems to us the most natural one since it is base on the comparison of the Fokker-Planck operators for two ifferent force fiel. In the case U =, p = 2 an m = e φ/2 the conition (FP2 is nothing but the classical conition (3.5 with ν = 1/2. In any case, the core iea in the ecomposition is that a coercive B in E is obtaine by a negative local perturbation of the whole operator. (4 By mollification the C 2 smoothness assumption of m coul be relaxe: if m(v is not smooth but m(v is smooth, satisfies (FP2-(FP3 an is such that c 1 m(v < m(v c 2 m(v, then it hols ]. f t µ E C f t µ L p ( m C e λ t f in µ Lp ( m C e λ t f in µ E.