Essential m-dissipativity and hypocoercivity of Langevin dynamics with multiplicative noise
aa r X i v : . [ m a t h . F A ] F e b Essential m-dissipativity and hypocoercivityof Langevin dynamics with multiplicativenoise
Alexander Bertram ∗,†
Martin Grothaus ∗,‡
February 16, 2021
Abstract . We provide a complete elaboration of the L -Hilbert space hypoco-ercivity theorem for the degenerate Langevin dynamics with multiplicative noise,studying the longtime behaviour of the strongly continuous contraction semigroupsolving the abstract Cauchy problem for the associated backward Kolmogorov op-erator. Hypocoercivity for the Langevin dynamics with constant diffusion matrixwas proven previously by Dolbeault, Mouhot and Schmeiser in the correspondingFokker-Planck framework, and made rigorous in the Kolmogorov backwards settingby Grothaus and Stilgenbauer. We extend these results to weakly differentiablediffusion coefficient matrices, introducing multiplicative noise for the correspondingstochastic differential equation. The rate of convergence is explicitly computed de-pending on the choice of these coefficients and the potential giving the outer force.In order to obtain a solution to the abstract Cauchy problem, we first prove essen-tial self-adjointness of non-degenerate elliptic Dirichlet operators on Hilbert spaces,using prior elliptic regularity results and techniques from Bogachev, Krylov andRöckner. We apply operator perturbation theory to obtain essential m-dissipativityof the Kolmogorov operator, extending the m-dissipativity results from Conradand Grothaus. We emphasize that the chosen Kolmogorov approach is natural,as the theory of generalized Dirichlet forms implies a stochastic representation ofthe Langevin semigroup as the transition kernel of a diffusion process which providesa martingale solution to the Langevin equation with multiplicative noise. Moreover,we show that even a weak solution is obtained this way. AMS subject classification (2020):
Keywords:
Langevin equation, multiplicative noise, hypocoercivity, essential m-dissipativity,essential self-adjointness, Fokker-Planck equation ∗ Department of Mathematics, TU Kaiserslautern, PO Box 3049, 67653 Kaiserslautern † [email protected] (corresponding author) ‡ [email protected] Introduction
We study the exponential decay to equilibrium of Langevin dynamics with multiplica-tive noise. The corresponding evolution equation is given by the following stochasticdifferential equation on R d , d ∈ N , as dX t = V t d t,dV t = b ( V t )d t − ∇ Φ( X t )d t + √ σ ( V t )d B t , (1.1)where Φ : R d → R is a suitable potential whose properties are specified later, B = ( B t ) t ≥ is a standard d -dimensional Brownian motion, σ : R d → R d × d a variable diffusion matrixwith at least weakly differentiable coefficients, and b : R d → R d given by b i ( v ) = d X j =1 ∂ j a ij ( v ) − a ij ( v ) v j , where a ij = Σ ij with Σ = σσ T .This equation describes the evolution of a particle described via its position ( X t ) t ≥ and velocity ( V t ) t ≥ coordinates, which is subject to friction, stochastic perturbationdepending on its velocity, and some outer force ∇ Φ. To simplify notation, we split R d into the two components x, v ∈ R d corresponding to position and velocity respectively.This extends to differential operators ∇ x , ∇ v , and the Hessian matrix H v .Using Itô’s formula, we obtain the associated Kolmogorov operator L as L = tr (Σ H v ) + b · ∇ v + v · ∇ x − ∇ Φ · ∇ v . Here a · b or alternatively ( a, b ) euc denotes the standard inner product of a, b ∈ R d . Weintroduce the measure µ = µ Σ , Φ on ( R d , B ( R d )) as µ Σ , Φ = (2 π ) − d e − Φ( x ) − v d x ⊗ d v = .. e − Φ( x ) ⊗ ν, i.e. ν is the normalized standard Gaussian measure on R d . We consider the operator L on the Hilbert space H .. = L ( R d , µ ).We note that the results below on exponential convergence to equilibrium can also betranslated to a corresponding Fokker-Planck setting, with the differential operator L FP given as the adjoint, restricted to sufficiently smooth functions, of L in L ( R d , d( x, v )).The considered Hilbert space there is ˜ H .. = L ( E, ˜ µ ), where˜ µ .. = (2 π ) − d e Φ( x )+ v d x ⊗ d v. Indeed, this is the space in which hypocoercivity of the kinetic Fokker-Planck equationassociated with the classical Langevin dynamics was proven in [DMS15]. The rigorousconnection to the Kolmogorov backwards setting considered throughout this paper and2onvergence behaviour of solutions to the abstract Cauchy problem ∂ t f ( t ) = L FP f ( t )are discussed in Section 5.3.The concept of hypocoercivity was first introduced in the memoirs of Cédric Villani([Vil06]), which is recommended as further literature to the interested reader. The ap-proach we use here was introduced algebraically by Dolbeault, Mouhot and Schmeiser(see [DMS09] and [DMS15]), and then made rigorous including domain issues in [GS14]by Grothaus and Stilgenbauer, where it was applied to show exponential convergence toequilibrium of a Fiber laydown process on the unit sphere. This setting was further gen-eralized by Wang and Grothaus in [GW17], where the coercivity assumptions involvingin part the classical Poincaré inequality for Gaussian measures were replaced by weakPoincaré inequalities, allowing for more general measures for both the spatial and thevelocity component. In this case, the authors still obtained explicit, but subexponentialrates of convergence. The specific case of hypocoercivity for Langevin dynamics on theposition space R d has been further explored in [GS16] and serves as the basis for ourhypocoercivity result.In contrast to the mentioned source, we do not know if our operator ( L, C ∞ c ( R d )) isessentially m-dissipative, and are therefore left to prove that first. This property of theLangevin operator has been shown by Helffer and Nier in [HN05] for smooth potentialsand generalized to locally Lipschitz-continuous potentials by Conrad and Grothaus in[CG10, Corollary 2.3]. However, a corresponding result for a non-constant second ordercoefficient matrix Σ is not known to the authors.Moreover, the symmetric part S of our operator L does not commute with the linearoperator B as in [GS16], hence the boundedness of the auxiliary operator BS needs tobe shown in a different way, which we do in Proposition 3.10.In Theorem 3.4, we show under fairly light assumptions on the coefficients and the poten-tial that the operator ( L, C ∞ c ( R d )) is essentially m-dissipative and therefore generatesa strongly continuous contraction semigroup on H . The proof is given in Section 4 andfollows the main ideas as in the proof of [CG10, Theorem 2.1], where a correspondingresult for Σ = I was obtained.For that proof we rely on perturbation theory of m-dissipative operators, starting withessential m-dissipativity of the symmetric part of L . To that end, we state an essen-tial self-adjointness result for a set of non-degenerate elliptic Dirichlet differential op-erators ( S, C ∞ c ( R d )) on L -spaces where the measure is absolutely continuous wrt. theLebesgue measure. This result is stated in Theorem 4.5 and combines regularity resultsfrom [BKR01] and [BGS13] with the approach to show essential self-adjointness from[BKR97].Finally, our main hypocoercivity result reads as follows: Theorem 1.1.
Let d ∈ N . Assume that Σ : R d → R d × d is a symmetric matrix ofcoefficients a ij : R d → R which is uniformly strictly elliptic with ellipticity constant c Σ . Moreover, let each a ij be bounded and locally Lipschitz-continuous, which implies ij ∈ H ,p loc ( R d , ν ) ∩ L ∞ ( R d ) for each p ≥ . Assume the growth behaviour of ∂ j a ij to bebounded either by | ∂ j a ij ( v ) | ≤ M | v | β for ν -almost all v ∈ R d and some M < ∞ , β ∈ ( −∞ , or by | ∂ j a ij ( v ) | ≤ M ( B (0) ( v ) + | v | β ) for ν -almost all v ∈ R d and some M < ∞ , β ∈ (0 , . Define N Σ in the first case as N Σ .. = q M + ( B Σ ∨ M ) and in the second case as N Σ .. = q M + B + dM , where M Σ .. = max {k a ij k ∞ | ≤ i, j ≤ d } and B Σ .. = max n | ∂ j a ij ( v ) | : v ∈ B (0) , ≤ i, j ≤ d o . Let further
Φ : R d → R be bounded from below, satisfy Φ ∈ C ( R d ) and that e − Φ( x ) d x isa probability measure on ( R d , B ( R d )) which satisfies a Poincaré inequality of the form k∇ f k L (e − Φ( x ) d x ) ≥ Λ (cid:13)(cid:13)(cid:13)(cid:13) f − Z R d f e − Φ( x ) d x (cid:13)(cid:13)(cid:13)(cid:13) L (e − Φ( x ) d x ) for some Λ ∈ (0 , ∞ ) and all f ∈ C c ( R d ) . Furthermore assume the existence of a constant c < ∞ such that | H Φ( x ) | ≤ c (1 + |∇ Φ( x ) | ) for all x ∈ R d , where H denotes the Hessian matrix and | H Φ | the Euclidian matrix norm. If β > − ,then also assume that there are constants N < ∞ , γ < β such that |∇ Φ( x ) | ≤ N (1 + | x | γ ) for all x ∈ R d . Then our Langevin operator ( L, C ∞ c ( R d )) is closable on H and its closure ( L, D ( L )) generates a strongly continuous contraction semigroup ( T t ) t ≥ on H . Further, it holdsthat for each θ ∈ (1 , ∞ ) , there is some θ ∈ (0 , ∞ ) such that k T t g − ( g, H k H ≤ θ e − θ t k g − ( g, H k H for all g ∈ H and all t ≥ . In particular, θ can be specified as θ = θ − θ c Σ n + n N Σ + n N , and the coefficients n i ∈ (0 , ∞ ) only depend on the choice of Φ . Finally, our main results may be summarized by the following list:• Essential m-dissipativity (equivalently essential self-adjointness) of non-degenerateelliptic Dirichlet differential operators with domain C ∞ c ( R d ) on Hilbert spaces withmeasure absolutely continuous wrt. the d -dimensional Lebesgue measure is proved,see Theorem 4.5. 4 Essential m-dissipativity of the backwards Kolmogorov operator ( L, C ∞ c ( R d )) as-sociated with the Langevin equation with multiplicative noise (1.1) on the Hilbertspace H under weak assumptions on the coefficient matrix Σ and the potential Φ,in particular not requiring smoothness, is shown, see Theorem 3.4.• Exponential convergence to a stationary state of the corresponding solutions to theabstract Cauchy problem ∂ t u ( t ) = Lu ( t ), see (5.1) on the Hilbert space H withexplicitly computable rate of convergence, as stated in Theorem 1.1, is proved.• Adaptation of this convergence result to the equivalent formulation as a Fokker-Planck PDE on the appropriate Hilbert space ˜ H .. = L ( E, ˜ µ ) is provided. Inparticular, this yields exponential convergence of the solutions to the abstractFokker-Planck Cauchy problem ∂ t u ( t ) = L FP u ( t ), with L FP given by (5.3), to astationary state, see Section 5.3.• A stochastic interpretation of the semigroup as a transition kernel for a diffusionprocess is worked out. Moreover, we prove this diffusion process to be a weaksolution to the Langevin SDE (1.1) and derive for it strong mixing properties withexplicit rates of convergence, see Section 5.2. We start by recalling some basic facts about closed unbounded operators on Hilbertspaces:
Lemma 2.1.
Let ( T, D ( T )) be a densely defined linear operator on H and let L be abounded linear operator with domain H .(i) The adjoint operator ( T ∗ , D ( T ∗ )) exists and is closed. If D ( T ∗ ) is dense in H ,then ( T, D ( T )) is closable and for the closure ( T , D ( T )) it holds T = T ∗∗ .(ii) L ∗ is bounded and k L ∗ k = k L k .(iii) If ( T, D ( T )) is close, then D ( T ∗ ) is automatically dense in H . Consequently by(i), T = T ∗∗ .(iv) Let ( T, D ( T )) be closed. Then the operator T L with domain D ( T L ) = { f ∈ H | Lf ∈ D ( T ) } is also closed.(v) LT with domain D ( T ) need not be closed, however ( LT ) ∗ = T ∗ L ∗ . Let us now briefly state the abstract setting for the hypocoercivity method as in [GS14].5 ata conditions (D).
We require the following conditions which are henceforth as-sumed without further mention.(D1)
The Hilbert space:
Let ( E, F , µ ) be some probability space and define H to be H = L ( E, µ ) equipped with the standard inner product ( · , · ) H .(D2) The C -semigroup and its generator: ( L, D ( L )) is some linear operator on H gen-erating a strongly continuous contraction semigroup ( T t ) t ≥ .(D3) Core property of L : Let D ⊂ D ( L ) be a dense subspace of H which is a core for( L, D ( L )).(D4) Decomposition of L : Let (
S, D ( S ))) be symmetric, ( A, D ( A )) be closed and anti-symmetric on H such that D ⊂ D ( S ) ∩ D ( A ) as well as L | D = S − A .(D5) Orthogonal projections:
Let P : H → H be an orthogonal projection satisfying P ( H ) ⊂ D ( S ) , SP = 0 as well as P ( D ) ⊂ D ( A ) , AP ( D ) ⊂ D ( A ). Moreover, let P S : H → H be defined as P S f .. = P f + ( f, H , f ∈ H. (D6) Invariant measure:
Let µ be invariant for ( L, D ) in the sense that(
Lf, H = Z E Lf d µ = 0 for all f ∈ D. (D7) Conservativity:
It holds that 1 ∈ D ( L ) and L A, D ( A )) is closed, ( AP, D ( AP )) is also closed and densely defined. Hence by vonNeumann’s theorem, the operator I + ( AP ) ∗ ( AP ) : D (( AP ) ∗ AP ) → H, where D (( AP ) ∗ AP ) = { f ∈ D ( AP ) | AP f ∈ D (( AP ) ∗ ) } , is bijective and admits abounded inverse. We therefore define the operator ( B, D (( AP ) ∗ )) via B .. = ( I + ( AP ) ∗ AP ) − ( AP ) ∗ Then B extends to a bounded operator on H .As in the given source, we also require the following assumptions: Assumption (H1).
Algebraic relation:
It holds that
P AP | D = 0. Assumption (H2).
Microscopic coercivity:
There exists some Λ m > − ( Sf, f ) H ≥ Λ m k ( I − P S ) f k for all f ∈ D. ssumption (H3). Macroscopic coercivity:
Define (
G, D ) via G = P A P on D . As-sume that ( G, D ) is essentially self-adjoint on H . Moreover, assume that there is someΛ M > k AP f k ≥ Λ M k P f k for all f ∈ D. Assumption (H4).
Boundedness of auxiliary operators:
The operators (
BS, D ) and( BA ( I − P ) , D ) are bounded and there exist constants c , c < ∞ such that k BSf k ≤ c k ( I − P ) f k and k BA ( I − P ) f k ≤ c k ( I − P ) f k hold for all f ∈ D .We now state the central abstract hypocoercivity theorem as in [GS14]: Theorem 2.2.
Assume that (D) and (H1)-(H4) hold. Then there exist strictly positiveconstants κ , κ < ∞ which are explicitly computable in terms of Λ m , Λ M , c and c suchthat for all g ∈ H we have k T t g − ( g, H k ≤ κ e − κ t k g − ( g, H k for all t ≥ . More specifically, if there exist δ > , ε ∈ (0 , and < κ < ∞ such that for all g ∈ D ( L ) , t ≥ , it holds κ k f t k ≤ (cid:18) Λ m − ε (1 + c + c ) (cid:18) δ (cid:19)(cid:19) k ( I − P ) f t k + ε (cid:18) Λ M M − (1 + c + c ) δ (cid:19) k P f t k , (2.1) where f t .. = T t g − ( g, H , then the constants κ and κ are given by κ = s ε − ε , κ = κ ε . In order to prove (H4), we will make use of the following result:
Lemma 2.3.
Assume (H3). Let ( T, D ( T )) be a linear operator with D ⊂ D ( T ) andassume AP ( D ) ⊂ D ( T ∗ ) . Then ( I − G )( D ) ⊂ D (( BT ) ∗ ) with ( BT ) ∗ ( I − G ) f = T ∗ AP f, f ∈ D. If there exists some
C < ∞ such that k ( BT ) ∗ g k ≤ C k g k for all g = ( I − G ) f, f ∈ D, (2.2) then ( BT, D ( T )) is bounded and its closure ( BT ) is a bounded operator on H with k BT k = k ( BT ) ∗ k .In particular, if ( S, D ( S )) and ( A, D ( A )) satisfy these assumptions, the correspondinginequalities in (H4) are satisfied with c = k ( BS ) ∗ k and c = k ( BA ) ∗ k . roof: Let h ∈ D (( AP ) ∗ ) and f ∈ D . Set g = ( I − G ) f . By the representation of B on D (( AP ) ∗ ) together with self-adjointness of ( I + ( AP ) ∗ AP ) − and D ⊂ D ( AP ), we get( h, B ∗ g ) H = ( Bh, ( I − G ) f ) H = (( AP ) ∗ h, f ) H = ( h, AP f ) H . So B ∗ g = AP f ∈ D ( T ∗ ). By Lemma 2.1 (v), (( BT ) ∗ , D (( BT ) ∗ )) = ( T ∗ B ∗ , D ( T ∗ B ∗ )),which implies ( BT ) ∗ g = T ∗ B ∗ g = T ∗ AP f .By essential self-adjointness and hence essential m-dissipativity of G , ( I − G )( D ) is densein H . Therefore by (2.2), the closed operator (( BT ) ∗ , D (( BT ) ∗ )) is a bounded operatoron H . Since ( BT, D ( T )) is densely defined, by Lemma 2.1 (i) and (ii), it is closable with BT = ( BT ) ∗∗ , which is a bounded operator on H with the stated norm.The last part follows directly by Sf = S ( I − P ) f for f ∈ D . (cid:3) As stated in the introduction, the aim of this section is to prove exponential convergenceto equilibrium of the semigroup solving the abstract Kolmogorov equation correspondingto the Langevin equation with multiplicative noise (1.1).We remark that most of the conditions are verified analogously to [GS16], the maindifference being the proof of essential m-dissipativity for the operator (
L, D ) as wellas the first inequality in (H4). Nevertheless, some care has to be taken whenever S isinvolved, as it doesn’t preserve regularity to the same extent as in the given reference. We start by introducing the setting and verifying the data conditions (D) . The notationsintroduced in this part will be used for the remainder of the section without furthermention.Let d ∈ N and set the state space as E = R d , F = B ( R d ). In the following, the first d components of E will be written as x , the latter d components as v . Let ν be thenormalised Gaussian measure on R d with mean zero and covariance matrix I , i.e. ν ( A ) = Z A (2 π ) − d e − x d x. Assumption (P).
The potential Φ : R d → R is assumed to depend only on the positionvariable x and to be locally Lipschitz-continuous. We further assume e − Φ( x ) d x to be aprobability measure on ( R d , B ( R d )). 8ote that the first part implies Φ ∈ H , ∞ loc ( R d ). Moreover, Φ is differentiable d x -a.e. on R d , such that the weak gradient and the derivative of Φ coincide d x -a.e. on R d . In thefollowing, we fix a version of ∇ Φ.The probability measure µ on ( E, F ) is then given by µ = e − Φ( x ) d x ⊗ ν , and we set H .. = L ( E, µ ), which satisfies condition (D1). Next we assume the following aboutΣ = ( a ij ) ≤ i,j ≤ d with a ij : R d → R : Assumption (Σ1).
Σ is symmetric and uniformly strictly elliptic, i.e. there is some c Σ > v, Σ( v ) v ) ≥ c Σ · | v | for all v ∈ R d . Assumption (Σ2).
There is some p > d such that for all 1 ≤ i, j ≤ d , it holds that a ij ∈ H ,p loc ( R d , ν ) ∩ L ∞ ( R d ). Additionally, a ij is locally Lipschitz-continuous for all1 ≤ i, j ≤ d .Additionally, we will consider one of the following conditions on the growth of the partialderivatives: Assumption (Σ3).
There are constants 0 ≤ M < ∞ , −∞ < β ≤ ≤ i, j ≤ d | ∂ j a ij ( v ) | ≤ M | v | β for ν -almost all v ∈ R d . Assumption (Σ3 ′ ). There are constants 0 ≤ M < ∞ , 0 < β < ≤ i, j ≤ d | ∂ j a ij ( v ) | ≤ M ( B (0) ( v ) + | v | β ) for ν -almost all v ∈ R d . We note that any of these together with (Σ2) implies ∂ j a ij ∈ L ( R d , ν ) for all 1 ≤ i, j ≤ d . Definition 3.1.
Let Σ satisfy (Σ2). Then we set M Σ .. = max {k a ij k ∞ : 1 ≤ i, j ≤ d } and B Σ .. = max {| ∂ j a ij ( v ) | : v ∈ B (0) , ≤ i, j ≤ d } . If Σ additionally satisfies (Σ3), then we define N Σ .. = q M + ( B Σ ∨ M ) . If instead (Σ3 ′ ) is fulfilled, then we consider instead N Σ .. = q M + B + dM . efinition 3.2. Let D = C ∞ c ( E ) be the space of compactly supported smooth functionson E . We define the linear operators S, A and L on D via Sf = d X i,j =1 a ij ∂ v j ∂ v i f + d X i =1 b i ∂ v i f, where b i ( v ) = d X j =1 ( ∂ j a ij ( v ) − a ij ( v ) v j ) ,Af = ∇ Φ · ∇ v f − v · ∇ x f,Lf = ( S − A ) f, for f ∈ D. Integration by parts shows that (
S, D ) is symmetric and non-positive definite on H , and( A, D ) is antisymmetric on H . Hence, all three operators with domain D are dissipativeand therefore closable. We denote their closure respectively by ( S, D ( S )) , ( A, D ( A )) and( L, D ( L )).For f ∈ D and g ∈ H , ( E, µ ), integration by parts yields(
Lf, g ) H = − Z E ∇ f, − II Σ ! ∇ g ! euc d µ. In particular, (D6) is obviously fulfilled. Next we provide an estimate which will beneeded later:
Proposition 3.3.
Let ( Σ and either ( Σ or ( Σ ′ ) hold respectively and recall Defi-nition 3.1. Then for all ≤ i, j ≤ d , it holds that k ∂ j a ij − a ij v j k L ( ν ) ≤ N Σ . Proof:
Due to integration by parts, it holds that Z R d a ij v j d ν = Z R d a ij + 2 a ij v j ∂ j a ij d ν. Hence we obtain in the case ( Σ ′ ) Z R d ( ∂ j a ij − a ij v j ) d ν = Z R d ( ∂ j a ij ) + a ij d ν ≤ Z B (0) ( ∂ j a ij ) d ν + Z R d \ B (0) ( ∂ j a ij ) d ν + M ≤ B + Z R d \ B (0) ( M | v | β ) d ν + M ≤ B + M + d X k =1 M Z R d v k d ν = B + M + M d. The case ( Σ follows from ( ∂ j a ij ) ≤ ( B Σ ∨ M ) . (cid:3)
10e now state the essential m-dissipativity result, which will be proven in the nextsection.
Theorem 3.4.
Let ( Σ , ( Σ and either ( Σ or ( Σ ′ ) be fulfilled, and let Φ be as in (P) . Assume further that Φ is bounded from below and that |∇ Φ | ∈ L ( R d , e − Φ( x ) d x ) .If β is larger than − , then assume additionally that there is some N < ∞ such that |∇ Φ( x ) | ≤ N (1 + | x | γ ) , where γ <
21 + β .
Then the linear operator ( L, C ∞ c ( R d )) is essentially m-dissipative and hence its closure ( L, D ( L )) generates a strongly continuous contraction semigroup on H . In particular,the conditions (D2)-(D4) are satisfied. Let us now introduce the orthogonal projections P S and P : Definition 3.5.
Define P S : H → H as P S f = Z R d f d ν ( v ) , f ∈ H, where integration is understood w.r.t the velocity variable v . By Fubini’s theorem andthe fact that ν is a probability measure on ( E, F ), it follows that P S is a well-definedorthogonal projection on H with P S f ∈ L ( R d , e − Φ( x ) d x ) , k P S f k L ( R d , e − Φ( x ) d x ) = k P S f k H , f ∈ H, where L ( R d , e − Φ( x ) d x ) is interpreted as embedded in H .Then define P : H → H via P f = P S f − ( f, H for f ∈ H . Again, P is an orthogonalprojection on H with P f ∈ L ( R d , e − Φ( x ) d x ) , k P f k L ( R d , e − Φ( x ) d x ) = k P f k H , f ∈ H. Additionally, for each f ∈ D , P S f admits a unique representation in C ∞ c ( R d ), which wewill denote by f S ∈ C ∞ c ( R d ).In order to show the last remaining conditions (D5) and (D7), we will make use of astandard sequence of cutoff functions as specified below: Definition 3.6.
Let ϕ ∈ C ∞ c ( R d ) such that 0 ≤ ϕ ≤ ϕ = 1 on B (0) and ϕ = 0outside of B (0). Define ϕ n ( z ) .. = ϕ ( zn ) for each z ∈ R d , n ∈ N . Then there exists aconstant C < ∞ independent of n ∈ N such that | ∂ i ϕ n ( z ) | ≤ Cn , | ∂ ij ϕ n ( z ) | ≤ Cn for all z ∈ R d , ≤ i, j ≤ d. Moreover 0 ≤ ϕ n ≤ n ∈ N and ϕ n → R d as n → ∞ .11 emma 3.7. Let ( Σ and either ( Σ or ( Σ ′ ) be fulfilled, and let Φ be as in (P) .Then the operator L satisfies the following:(i) P ( H ) ⊂ D ( S ) with SP f = 0 for all f ∈ H ,(ii) P ( D ) ⊂ D ( A ) and AP f = − v · ∇ x ( P S f ) ,(iii) AP ( D ) ⊂ D ( A ) with A P f = h v, ∇ x ( P S f ) v i − ∇ Φ · ∇ x ( P S f ) .(iv) It holds ∈ D ( L ) and L .In particular, (D5) and (D7) are fulfilled. Proof:
We only show (i), as the other parts can be shown exactly as in [GS16]. First, let f ∈ C ∞ c ( R d ) and define f n ∈ D via f n ( x, y ) .. = f ( x ) ϕ n ( v ). Then by Lebesgue’s dominatedconvergence theorem and the inequalities in the previous definition, Sf n = f · d X i,j =1 a ij ∂ ij ϕ n + d X i =1 b i ∂ i ϕ n → H as n → ∞ , since a ij ∈ L ∞ ( R d ) ⊂ L ( R d , ν ), | v | ∈ L ( R d , ν ) and ∂ j a ij ∈ L ( R d , ν ) for all 1 ≤ i, j ≤ d .Since f n → f in H and by closedness of ( S, D ( S )), this implies f ∈ D ( S ) with Sf = 0,where f is interpreted as an element of H .Now let g ∈ P ( H ) and identify g as an element of L ( R d , e − Φ( x ) d x ). Then there exist g n ∈ C ∞ c ( R d ) with g n → g in L ( R d , e − Φ( x ) d x ) as n → ∞ . Identifying all g n and g withelements in H then yields g n → g in H as n → ∞ and g n ∈ D ( S ), Sg n = 0 for all n ∈ N .Therefore, again by closedness of ( S, D ( S )), g ∈ D ( S ) and Sg = 0. (cid:3) Now we verify the hypocoercivity conditions (H1) - (H4) for the operator L . From hereon, we will assume Σ to satisfy ( Σ , ( Σ and either ( Σ or ( Σ ′ ) , with N Σ referringto the appropriate constant as in Definition 3.1. Analogously to [GS16] we introducethe following conditions: Hypocoercivity assumptions (C1)-(C3).
We require the following assumptions onΦ : R d → R :(C1) The potential Φ is bounded from below, is an element of C ( R d ) and e − Φ( x ) d x isa probability measure on ( R d , B ( R d )).(C2) The probability measure e − Φ( x ) d x satisfies a Poincaré inequality of the form k∇ f k L (e − Φ( x ) d x ) ≥ Λ k f − ( f, L (e − Φ( x ) d x ) k L (e − Φ( x ) d x ) for some Λ ∈ (0 , ∞ ) and all f ∈ C ∞ c ( R d ).12C3) There exists a constant c < ∞ such that |∇ Φ( x ) | ≤ c (1 + |∇ Φ( x ) | ) for all x ∈ R d . Note that in particular, (C1) implies (P) . As shown in [Vil06, Lemma A.24], conditions(C3) and (C1) imply ∇ Φ ∈ L (e − Φ( x ) d x ).Since we only change the operator ( S, D ( S )) in comparison to the framework of [GS16],the results stated there involving only ( A, D ( A )) and the projections also hold here andare collected as follows: Proposition 3.8.
Let Φ satisfy (P) . Then the following hold:(i) Assume additionally ∇ Φ ∈ L (e − Φ( x ) d x ) . Then (H1) is fulfilled.(ii) Assume that Φ satisfies (C1) and that ∇ Φ ∈ L (e − Φ( x ) d x ) . Then the operator ( G, D ) defined by G .. = P A P is essentially self-adjoint, equivalently essentiallym-dissipative. For f ∈ D , it holds Gf = P AAP f = ∆ f S − ∇ Φ · ∇ f S . (iii) Assume that Φ satisfies (C1) and (C2) as well as ∇ Φ ∈ L (e − Φ( x ) d x ) . Then (H3) holds with Λ M = Λ .(iv) Assume that Φ satisfies (C1)-(C3). Then the second estimate in (H4) is satisfied,and the constant there is given as c = c Φ ∈ [0 , ∞ ) , which only depends on thechoice of Φ . It remains to show (H2) and the first half of (H4) : Proposition 3.9.
Let Φ be as in (P) . Then Condition (H2) is satisfied with Λ m = c Σ . Proof:
Let g ∈ C ∞ c ( R d ). The Poincaré inequality for Gaussian measures, see for example[Bec89], states k∇ g k L ( ν ) ≥ (cid:13)(cid:13)(cid:13)(cid:13) g − Z R d g ( v ) d ν ( v ) (cid:13)(cid:13)(cid:13)(cid:13) L ( ν ) . Therefore, integration by parts yields for all f ∈ D :( − Sf, f ) H = Z E h∇ v f, Σ ∇ v f i d µ ≥ Z R d Z R d c Σ |∇ v f ( x, v ) | d ν e − Φ( x ) d x ≥ c Σ Z R d Z R d ( f − P S f ) d ν e − Φ( x ) d x = c Σ k ( I − P S ) f k H (cid:3) Finally, we verify the first part of (H4) : 13 roposition 3.10.
Assume that Φ satisfies (C1) and (C2) as well as ∇ Φ ∈ L (e − Φ d x ) .Then the first inequality of (H4) is also satisfied with c = d Σ .. = √ d N Σ . Proof:
For f ∈ D , define T f ∈ H by T f .. = d X i =1 b i ∂ i ( f S ) = d X i,j =1 ( ∂ j a ij − a ij v j ) ∂ x i ( P S f ) . We want to apply Lemma 2.3 to the operator (
S, D ( S )). Let f ∈ D , h ∈ D ( S ) and h n ∈ D such that h n → h and Sh n → Sh in H as n → ∞ . Then, by integration byparts, ( Sh, AP f ) H = lim n →∞ ( Sh n , − v · ∇ x ( P S f )) H = lim n →∞ ( h n , T f ) H = ( h, T f ) H . This shows
AP f ∈ D ( S ∗ ) and by the first part of Lemma 2.3, ( I − G ) f ∈ D (( BS ) ∗ ) and( BS ) ∗ ( I − G ) f = S ∗ AP f = T f . Now set g = ( I − G ) f , then, via Proposition 3.3, k ( BT ) ∗ g k H = k T f k H = Z E d X i =1 b i ∂ i f S ! d µ ≤ d d X i,j =1 Z R d Z R d ( ∂ j a ij ( v ) − a ij ( v ) v j ) d ν ( v ) ( ∂ x i ( P S f )( x )) e − Φ( x ) d x ≤ d N d X i =1 Z R d ∂ x i ( P f ) · ∂ x i ( P S f ) e − Φ( x ) d x. A final integration by parts then yields k ( BT ) ∗ g k H ≤ − d N Z R d P f · (∆ x P S f − ∇ Φ ∇ x ( P S f )) e − Φ( x ) d x = − d N Z R d P f · Gf e − Φ( x ) d x ≤ d N k P f k L (e − Φ( x ) d x ) · k Gf k L (e − Φ( x ) d x ) ≤ d N k P f k H ( k ( I − G ) f k H + k f k H ) ≤ d N k f k H , where the last inequality is due to dissipativity of ( G, D ). (cid:3) Proof (of Theorem 1.1):
Under the given assumptions, all conditions (C1)-(C3) , ( Σ , ( Σ and either ( Σ or ( Σ ′ ) are satisfied. Therefore hypocoercivity follows by the previous propositions andTheorem 2.2. It remains to show the stated convergence rate, which will be done as in[GS16] or [DKMS13] using the determined values for c , c , Λ M and Λ m . Fix δ .. = Λ1 + Λ 11 + c Φ + d Σ . c Σ − εr Φ ( N Σ ) and εs Φ respectively, where r Φ ( N Σ ) .. = (1 + c Φ + √ d N Σ ) (cid:18) c Φ + √ d N Σ ) (cid:19) and s Φ .. = 12 Λ1 + Λ . and ε = ε Φ (Σ) ∈ (0 ,
1) still needs to be determined. Write r Φ ( N Σ )+ s Φ as the polynomial r Φ ( N Σ ) + s Φ = a + a N Σ + a N , where all a i ∈ (0 , ∞ ), i = 1 , . . . , ε Φ ( N Σ ) .. = N Σ r Φ ( N Σ ) + s Φ = N Σ a + a N Σ + a N . Some rough estimates show ˜ ε Φ ( N Σ ) ∈ (0 , v > ε .. = v v c Σ N Σ ˜ ε Φ ( N Σ ) ∈ (0 , . Then εr Φ ( N Σ ) + εs Φ = v v c Σ < c Σ , hence we get the estimate c Σ − εr Φ ( N Σ ) > εs Φ = v v c Σ n + n N Σ + n N = .. κ, where all n i ∈ (0 , ∞ ) depend on Φ and are given by n i .. = 2 s Φ a i , for each i = 1 , . . . , . Clearly, κ , ε and δ now solve (2.1) and the convergence rate coefficients are given viaTheorem 2.2 by κ = s ε − ε = vuut v + c Σ N Σ ˜ ε Φ ( N Σ ) v v − c Σ N Σ ˜ ε Φ ( N Σ ) v ≤ p v + v = 1 + v and κ = κ ε > κ Hence, by choosing θ = 1+ v and θ = κ = θ − θ c Σ n + n N Σ + n N , the rate of convergenceclaimed in the theorem is shown. (cid:3) Remark 3.11.
We remark here that all previous considerations up to the explicit rateof convergence can also be applied to the formal adjoint operator ( L ∗ , D ) with L ∗ = S + A , the closure of which generates the adjoint semigroup ( T ∗ t ) t ≥ on H . For example,the perturbation procedure to prove essential m-dissipativity is exactly the same asfor L , since the sign of A does not matter due to antisymmetry. We can use thisto construct solutions to the corresponding Fokker-Planck PDE associated with ourLangevin dynamics, see Section 5.3. 15 Essential m-dissipativity of the Langevin operator
The goal of this section is to prove Theorem 3.4. We start by giving some basics onperturbation of semigroup generators.
Definition 4.1.
Let (
A, D ( A )) and ( B, D ( B )) be linear operators on H . Then B issaid to be A -bounded if D ( A ) ⊂ D ( B ) and there exist constants a, b < ∞ such that k Bf k H ≤ a k Af k H + b k f k H (4.1)holds for all f ∈ D ( A ). The number inf { a ∈ R | (4.1) holds for some b < ∞} is calledthe A -bound of B . Theorem 4.2.
Let D ⊂ H be a dense linear subspace. Let ( A, D ) be an essentiallym-dissipative linear operator on H and let ( B, D ) be dissipative and A -bounded with A -bound strictly less than . Then ( A + B, D ) is essentially m-dissipative and its closureis given by ( A + B, D ( A )) . A useful criterion for verifying A -boundedness is given by: Lemma 4.3.
Let D ⊂ H be a dense linear subspace, ( A, D ) be essentially m-dissipativeand ( B, D ) be dissipative. Assume that there exist constants c, d < ∞ such that k Bf k H ≤ c ( Af, f ) H + d k f k H holds for all f ∈ D . Then B is A -bounded with A -bound . We also require the following generalization of the perturbation method:
Lemma 4.4.
Let D ⊂ H be a dense linear subspace, ( A, D ) be essentially m-dissipativeand ( B, D ) be dissipative on H . Assume that there exists a complete orthogonal fam-ily ( P n ) n ∈ N , i.e. each P n is an orthogonal projection, P n P m = 0 for all n = m and P n ∈ N P n = I strongly, such that P n ( D ) ⊂ D, P n A = AP n , and P n B = BP n for all n ∈ N . Set A n .. = AP n , B n .. = BP n , both with domain D n .. = P n ( D ) , as operatorson P n ( H ) . Assume that each B n is A n -bounded with A n -bound strictly less than . Then ( A + B, D ) is essentially m-dissipative. .2 The symmetric part We first prove essential self-adjointness, equivalently essential m-dissipativity, for a cer-tain class of symmetric differential operators on specific Hilbert spaces. This is essentiallya combination of two results by Bogachev, Krylov, and Röckner, namely [BKR01, Corol-lary 2.10] and [BKR97, Theorem 7], however, the combined statement does not seemto be well known and might hold interest as the basis for similar m-dissipativity proofs.We use the slightly more general statement from [BGS13, Theorem 5.1] in order to relaxthe assumptions.
Theorem 4.5.
Let d ≥ and consider H = L ( R d , µ ) where µ = ρ d x , ρ = ϕ for some ϕ ∈ H , ( R d ) such that ρ ∈ L ∞ loc ( R d ) . Let A = ( a ij ) ≤ i,j ≤ d : R d → R d × d be symmetricand locally strictly elliptic with a ij ∈ L ∞ ( R d ) for all ≤ i, j ≤ d . Assume there is some p > d such that a ij ∈ H ,p loc ( R d ) for all ≤ i, j ≤ d and that |∇ ρ | ∈ L p loc ( R d ) . Considerthe bilinear form ( B, D ) given by D = C ∞ c ( R d ) and B ( f, g ) .. = ( ∇ f, A ∇ g ) H = Z R d ( ∇ f ( x ) , A ( x ) ∇ g ( x )) euc ρ ( x ) d x, f, g ∈ D. Define further the linear operator ( S, D ) via Sf .. = d X i,j =1 a ij ∂ j ∂ i f + d X i =1 b i ∂ i f, f ∈ D, where b i = P dj =1 ( ∂ j a ij + a ij ∂ j ρρ ) ∈ L p loc ( R d ) , so that B ( f, g ) = ( − Sf, g ) H . Then ( S, D ) is essentially self-adjoint on H . Proof:
Analogously to the proof of [BKR97, Theorem 7], it can be shown that ρ is continuous,hence locally bounded. Assume that there is some g ∈ H such that Z R d ( S − I ) f ( x ) · g ( x ) · ρ ( x ) d x = 0 for all f ∈ D. (4.2)Define the locally finite signed Borel measure ν via ν = gρ d x , which is then absolutelycontinuous with respect to the Lebesgue measure. By definition it holds that Z R d d X i,j =1 a ij ∂ j ∂ i f + d X i =1 b i ∂ i f − f d ν = 0 for all f ∈ D, so by [BGS13, Theorem 5.1], the density g · ρ of ν is in H ,p loc ( R d ) and locally Höldercontinuous, hence locally bounded. This implies g = gρ · ρ ∈ L p loc ( R d ) ∩ L ∞ loc ( R d ) and ∇ g = ∇ ( gρ ) · ρ − ( gρ ) ∇ ρρ ∈ L p loc ( R d ). Hence g ∈ H ,p loc ( R d ), is locally bounded, and g · b i ∈ L p loc ( R d ) for all 1 ≤ i ≤ d . Therefore, we can apply integration by parts to (4.2)17nd get for every f ∈ D :0 = − d X i,j =1 ( a ij ∂ i f, ∂ j g ) H − d X i =1 ( ∂ i f, b i g ) H + d X i =1 ( ∂ i f, b i g ) H − ( f, g ) H = − Z R d ( ∇ f, A ∇ g ) euc d µ − ( f, g ) H . (4.3)Note that this equation can then be extended to all f ∈ H , ( R d ) with compact support,since p > ψ ∈ C ∞ c ( R d ) and set η = ψg ∈ H , ( R d ), which hascompact support. The same then holds for f .. = ψη ∈ H , ( R d ). Elementary applicationof the product rule yields( ∇ η, A ∇ ( ψg )) euc = ( ∇ f, A ∇ g ) euc − η ( ∇ ψ, A ∇ g ) euc + g ( ∇ η, A ∇ ψ ) euc . (4.4)From now on, for a, b : R d → R d , let ( a, b ) always denote the evaluation of the Euclideaninner product ( a, b ) euc . By using (4.4) and applying (4.3) to f , we get Z R d ( ∇ ( ψg ) , A ∇ ( ψg )) d µ + Z R d ( ψg ) d µ = Z R d ( ∇ η, A ∇ ( ψg )) d µ + Z R d ηψg d µ = Z R d ( ∇ f, A ∇ g ) d µ − Z R d η ( ∇ ψ, A ∇ g ) d µ + Z R d g ( ∇ η, A ∇ ψ ) d µ + Z R d f g d µ = − Z R d ψg ( ∇ ψ, A ∇ g ) d µ + Z R d g ( ∇ ( ψg ) , A ∇ ψ ) d µ = Z R d g ( ∇ ψ, A ∇ ψ ) d µ, where the last step follows from the product rule and symmetry of A . Since A is locallystrictly elliptic, there is some c > ≤ Z R d c ( ∇ ( ψg ) , ∇ ( ψg )) d µ ≤ Z R d ( ∇ ( ψg ) , A ∇ ( ψg )) d µ and therefore it follows that Z R d ( ψg ) d µ ≤ Z R d g ( ∇ ψ, A ∇ ψ ) d µ. (4.5)Let ( ψ n ) n ∈ N be as in Definition 3.6. Then (4.5) holds for all ψ = ψ n . By dominatedconvergence, the left part converges to k g k H as n → ∞ . The integrand of the righthand side term is dominated by d C M · g ∈ L ( µ ), where C is from Definition 3.6 and M .. = max ≤ i,j ≤ d k a ij k ∞ . By definition of the ψ n , that integrand converges pointwiselyto zero as n → ∞ , so again by dominated convergence it follows that g = 0 in H .This implies that ( S − I )( D ) is dense in H and therefore that ( S, D ) is essentially self-adjoint. (cid:3)
Remark 4.6.
The above theorem also holds for d = 1, as long as p ≥
2. Indeed,continuity of ρ follows from similar regularity estimates, see [BKR97, Remark 2]. The18roof of [BGS13, Theorem 5.1] mirrors the proof of [BKR01, Theorem 2.8], where d ≥ p ′ < q always holds). Finally, the extension of (4.3)requires p ≥ S, D ) of ouroperator L : Theorem 4.7.
Let
H, D and the operator S be defined as in Section 3.1. Then ( S, D ) isessentially m-dissipative on H . Its closure ( S, D ( S )) generates a sub-Markovian stronglycontinuous contraction semigroup on H . Proof:
Define the operator ( ˜
S, C ∞ c ( R d )) on L ( R d , ν ) by˜ Sf .. = d X i,j =1 a ij ∂ j ∂ i f + d X i =1 b i ∂ i f, f ∈ C ∞ c ( R d ) . The density ρ of ν wrt. the Lebesgue measure is given by ρ ( v ) = e − v / = (e − v / ) . Dueto the conditions ( Σ , ( Σ and either ( Σ or ( Σ ′ ) , all assumptions from Theorem 4.5are fulfilled and therefore, ( ˜ S, C ∞ c ( R d )) is essentially m-dissipative in L ( ν ). Let g = g ⊗ g ∈ C ∞ c ( R d ) ⊗ C ∞ c ( R d ) be a pure tensor. Then there is a sequence ( ˜ f n ) n ∈ N in C ∞ c ( R d ) such that ( I − ˜ S ) ˜ f n → g in L ( ν ) as n → ∞ . Define f n ∈ D for each n ∈ N by f n ( x, v ) .. = g ( x ) ˜ f n ( v ) . Then k ( I − S ) f n − g k H = k g ⊗ (( I − ˜ S ) ˜ f n − g ) k H = k g k L (e − Φ( x ) d x ) · k ( I − ˜ S ) ˜ f n − g k L ( ν ) , which converges to zero as n → ∞ . By taking linear combinations, this shows that( I − S )( D ) is dense in C ∞ c ( R d ) ⊗ C ∞ c ( R d ) wrt. the H -norm. Since C ∞ c ( R d ) ⊗ C ∞ c ( R d )is dense in H , ( S, D ) is essentially m-dissipative and its closure (
S, D ( S )) generates astrongly continuous contraction semigroup.It can easily be shown that ( Sf, f + ) H ≤ f ∈ D . Parallelly to the proof of (D7),it holds that 1 ∈ D ( S ) and S S, D ( S )) is a Dirichletoperator and the generated semigroup is sub-Markovian. (cid:3) Now we extend the essential m-dissipativity stepwise to the non-symmetric operator L by perturbation. This follows and is mostly based on the method seen in the proof of[Con11, Theorem 6.3.1], which proved that result for Σ = I .19ince S is dissipative on D = L (e − Φ d x ) ⊗ C ∞ c ( R d ) ⊃ D , the operator ( S, D ) isessentially m-dissipative as well. The unitary transformation T : L ( R d , d( x, v )) → H given by T f ( x, v ) = e v + Φ( x )2 f ( x, v ) leaves D invariant. This implies that the operator( S , D ) on L ( R d , d( x, v )) , where S = T − ST , is again essentially m-dissipative. Notethat S is explicitly given by S f = d X i,j =1 a ij ∂ v j ∂ v i f −
14 ( v, Σ v ) f + 12 tr(Σ) f + d X i,j =1 ∂ j a ij ( v i f + ∂ v i f )Now consider the operator ( ivxI, D ), which is dissipative as Re( ivxf, f ) L ( R d , d( x,v )) = 0for f ∈ D . We show the following perturbation result: Proposition 4.8.
Let Σ satisfy ( Σ with β ≤ − . Then the operator ( S + ivxI, D ) is essentially m-dissipative on L ( R d , d( x, v )) . Proof:
Define the orthogonal projections P n via P n f ( x, v ) .. = ξ n ( x ) f ( x, v ), where ξ n is given by ξ n = [ n − ,n ) ( | x | ), which leave D invariant. Then the conditions for Lemma 4.4 arefulfilled, and we are left to show the A n -bounds. Note that due to the restriction on β ,there is some constant C < ∞ such that ∂ j a ij ( v ) v i ≤ C for all 1 ≤ i, j ≤ d , v ∈ R d . Foreach fixed n ∈ N it holds for all f ∈ P n D : k ivxf k L ≤ n Z R d | v | f d( x, v ) ≤ c Σ n Z R d ( v, Σ v )4 f d( x, v ) ≤ c Σ n Z R d ( v, Σ v )4 f + ( ∇ v f, Σ ∇ v f ) d( x, v )= 4 c Σ n Z R d − d X i,j =1 a ij ∂ v j ∂ v i f − d X i,j =1 ∂ j a ij ∂ v i f + ( v, Σ v )4 f f d( x, v )= 4 c Σ n ( − P n S f, f ) + Z R d
12 tr(Σ) f + d X i,j =1 ∂ j a ij v i f d( x, v ) ≤ c Σ n (cid:18) ( − S f, f ) + ( d C + dM Σ k f k L (cid:19) . Hence by Lemma 4.3, ( ivxIP n , P n D ) is S P n -bounded with Kato-bound zero. Appli-cation of Lemma 4.4 yields the statement. (cid:3) Since C ∞ c ( R d ) ⊗ C ∞ c ( R d ) is dense in D wrt. the graph norm of S + ivxI , we obtainessential m-dissipativity of ( S + ivxI, C ∞ c ( R d ) ⊗ C ∞ c ( R d )) and therefore also of its dis-sipative extension ( S + ivxI, D ) with D = S ( R d ) ⊗ C ∞ c ( R d )), where S ( R d ) denotesthe set of smooth functions of rapid decrease on R d . Applying Fourier transform in the x -component leaves D invariant and shows that ( L , D ) is essentially m-dissipative,where L = S + v ∇ x . Now we add the part depending on the potential Φ.20 roposition 4.9. Let Σ satisfy ( Σ with β ≤ − and Φ be Lipschitz-continuous. Thenthe operator ( L ′ , D ) with L ′ = L −∇ Φ ∇ v is essentially m-dissipative on L ( R d , d( x, v )) . Proof:
It holds due to antisymmetry of v ∇ x that k∇ Φ ∇ v f k L ≤ k|∇ Φ |k ∞ c Σ (cid:18) ( ∇ v f, Σ ∇ v f ) L + (cid:18) ( v, Σ v )4 f, f (cid:19) L − ( v ∇ x f, f ) L (cid:19) ≤ k|∇ Φ |k ∞ c Σ (cid:18) ( − L f, f ) L + ( d C + dM Σ k f k L (cid:19) , analogously to the proof of Proposition 4.8, which again implies that the antisymmetric,hence dissipative operator ( ∇ Φ ∇ v , D ) is L -bounded with bound zero. This shows theclaim. (cid:3) Denote by H , ∞ c ( R d ) the space of functions in H , ∞ ( R d ) with compact support and set D ′ .. = H , ∞ c ( R d ) ⊗ C ∞ c ( R d ). As ( L ′ , D ′ ) is dissipative and its closure extends ( L ′ , D ), itis itself essentially m-dissipative. The unitary transformation T from the beginning ofthis section leaves D ′ invariant, and it holds that T L ′ T − = L on D ′ . This brings us tothe first m-dissipativity result for the complete Langevin operator: Theorem 4.10.
Let Σ satisfy ( Σ with β ≤ − and Φ be Lipschitz-continuous. Then ( L, D ) with is essentially m-dissipative on H . Proof:
By the previous considerations, (
L, D ′ ) is essentially m-dissipative on H . Let f ∈ D ′ with f = g ⊗ h . It holds g ∈ H , ∞ c ( R d ) ⊂ H , ( R d ). Choose a sequence ( g n ) n ∈ N with g n ∈ C ∞ c ( R d ), such that g n → g in H , ( R d ) as n → ∞ . Due to boundedness of e − Φ and v j e − v / for all 1 ≤ j ≤ d , it follows immediately that g n ⊗ h → f and L ( g n ⊗ h ) → Lf in H as n → ∞ . This extends to arbitrary f ∈ D ′ via linear combinations and thereforeshows that C ∞ c ( R d ) ⊗ C ∞ c ( R d ) and hence also D , is a core for ( L, D ( L )). (cid:3) It is now left to relax the assumptions on Σ and Φ by approximation. Let the assumptionsof Theorem 3.4 hold and wlog Φ ≥
0. For n ∈ N we define Σ n viaΣ n = ( a ij,n ) ≤ i,j ≤ d , a ij,n ( v ) .. = a ij (cid:18)(cid:18) n | v | ∧ (cid:19) v (cid:19) . Then each Σ n also satisfies ( Σ - ( Σ with β = −
1, since ∂ j a ij,n = ∂ j a ij on B n (0)and | ∂ j a ij,n | ≤ nL Σ ,n | v | outside of B n (0), where L Σ ,n denotes the Lipschitz constant of a ij on B n (0). Let further η m ∈ C ∞ c ( R d ) for each m ∈ N with η = 1 on B m (0) and set21 m = η m Φ, which is Lipschitz-continuous. Define H m as L ( R d , e − v − Φ m ( x ) d( x, v ))and ( L n,m , D ) via L n,m f = d X i,j =1 a ij,n ∂ v j ∂ v i f + d X i =1 d X j =1 ( ∂ j a ij,n ( v ) − a ij,n ( v ) v j ) ∂ v i f + v · ∇ x f − ∇ Φ m · ∇ v f. Then Theorem 4.10 shows that for each n, m ∈ N , ( L n,m , D ) is essentially m-dissipativeon H m , and it holds that L n,m f = Lf for all f ∈ D on B m (0) × B n (0). Note furtherthat k · k H ≤ k · k H m .We need the following estimates: Lemma 4.11.
Let n, m ∈ N and Σ n , Φ m as defined above. Then there is a constant D < ∞ independent of n, m such that for each ≤ j ≤ d , the following hold for all f ∈ D : k v j f k H m ≤ D n β k ( I − L n,m ) f k H m , k ∂ v j f k H m ≤ D n β k ( I − L n,m ) f k H m . Proof:
Recall the unitary transformations T m : L ( R d , d( x, v )) → H m defined by T m f =e v + Φ m ( x )2 f , as well as the operator L ′ n,m = T − m L n,m T m , and let f ∈ T − m D . Then L ′ n,m f = d X i,j =1 a ij,n ∂ v j ∂ v i f −
14 ( v, Σ n v ) f + 12 tr(Σ n ) f + d X i,j =1 ∂ j a ij,n ( v i f + ∂ v i f ) − v ∇ x f + ∇ Φ m ∇ v f. Analogously to the proof of Proposition 4.8 and due to antisymmetry of v ∇ x and ∇ Φ m ∇ v on L (d( x, v )), it holds that k v j T m f k H m = k v j f k L (d( x,v )) ≤ c Σ Z R d
14 ( v, Σ n v ) f d( x, v ) ≤ c Σ ( − L ′ n,m f, f ) L (d( x,v )) + Z R d f tr(Σ n ) + d X i,j =1 ∂ j a ij,n v i d( x, v ) . Since | tr(Σ n ) | ≤ | tr(Σ) | ≤ d · M Σ and | ∂ j a ij,n ( v ) v i | ≤ | ∂ j a ij ( v ) | · | v i | ≤ max { B Σ , M · n β +1 } for all v ∈ B n (0) , as well as | ∂ j a ij,n ( v ) v i | ≤ nL Σ ,n | v i || v | ≤ M n β +1 for all v / ∈ B n (0) , and wlog B Σ ≤ M · n β +1 , it follows that k v j T m f k H m ≤ c Σ ( − L ′ n,m f, f ) L (d( x,v )) + 2 c Σ ( dM Σ + 2 d M n β +1 ) k f k L (d( x,v )) . − L ′ n,m f, f ) L (d( x,v )) ≤ (cid:16) k L ′ n,m f k L (d( x,v )) + k f k L (d( x,v )) (cid:17) and k f k L (d( x,v )) ≤ (cid:16) k L ′ n,m f k L (d( x,v )) + k f k L (d( x,v )) (cid:17) . Dissipativity of ( L ′ n,m , T − m D ) on L (d( x, v )) implies k L ′ n,m f k L (d( x,v )) + k f k L (d( x,v )) ≤ k ( I − L ′ n,m ) f k L (d( x,v )) + 2 k ( I − L ′ n,m ) f k L (d( x,v )) . Overall, we get k v j T m f k H m ≤ c Σ (1 + 2( dM Σ + 2 d M n β +1 )) k ( I − L ′ n,m ) f k L (d( x,v )) ≤ c Σ d n β +1 max { M Σ , M }k ( I − L ′ n,m ) f k L (d( x,v )) . Since k ( I − L ′ n,m ) f k L (d( x,v )) = k T − m ( I − L n,m ) T m f k L (d( x,v )) = k ( I − L n,m ) T m f k H m , this proves the first statement with D = d p c Σ max { M Σ , M } .For the second part, note that ∂ v j T m f = T m ∂ v j f + v j T m f and that k T m ∂ v j f k H m = ( ∂ v j f, ∂ v j f ) L (d( x,v )) ≤ c Σ Z R d ( ∇ v f, Σ n ∇ v f ) euc d( x, v ) ≤ c Σ ( − L ′ n,m f, f ) L + Z R d
12 tr(Σ n ) f + d X i,j =1 ∂ j a ij,n v i f d( x, v ) . Repeating all calculations of the first part yields k ∂ v j T m f k H m ≤ (cid:18) D D (cid:19) n β k ( I − L n,m ) T m f k H m . (cid:3) Fix some pure tensor g ∈ C ∞ c ( R d ) ⊗ C ∞ c ( R d ). We prove that for every ε >
0, we canfind some f ∈ D such that k ( I − L ) f − g k H < ε . This then extends to arbitrary g ∈ C ∞ c ( R d ) ⊗ C ∞ c ( R d ) via linear combinations and therefore implies essential m-dissipativityof ( L, D ) on H , since C ∞ c ( R d ) ⊗ C ∞ c ( R d ) is dense in H . If β ≤ −
1, then the proof iseasier and follows analogously to the proof of of [Con11, Theorem 6.3.1]. Therefore wewill assume β > −
1. Recall that in this case, we have |∇ Φ( x ) | ≤ N (1 + | x | γ ) for all x ∈ R d , where γ < β , see the assumptions of Theorem 3.4.Denote the support of g by K x × K v , where K x and K v are compact sets in R d . Bya standard construction, for each δ x , δ v >
0, there are smooth cutoff functions 0 ≤ φ δ x , ψ δ v ≤ ∈ C ∞ c ( R d ) with supp( φ δ x ) ⊂ B δ x ( K x ), supp( ψ δ v ) ⊂ B δ v ( K v ), φ δ x = 1 on23 x , ψ δ v = 1 on K v . Moreover, there are constants C φ , C ψ independent of δ x and δ v suchthat k ∂ s φ δ x k ∞ ≤ C φ δ −| s | x and k ∂ s ψ δ v k ∞ ≤ C ψ δ −| s | v for all multi-indices s ∈ N d . Fix α such that β < α < γ . For any δ >
0, we set δ x .. = δ α and δ v .. = δ , and then define χ δ ( x, v ) .. = φ δ x ( x ) ψ δ v ( v ) = φ δ α ( x ) ψ δ ( v ).For f ∈ D , δ >
0, consider f δ .. = χ δ f , which is an element of D , as χ δ ∈ D . Without lossof generality, we consider δ and hence δ α sufficiently large such that supp( φ δ α ) ⊂ B δ α (0),supp( ψ δ ) ⊂ B δ (0) and that there are n, m ∈ N that satisfysupp( φ δ α ) × supp( ψ δ ) ⊂ B m (0) × B n (0) ⊂ B δ α (0) × B δ (0) . (4.6)The following then holds: Lemma 4.12.
Let g ∈ C ∞ c ( R d ) ⊗ C ∞ c ( R d ) and φ, ψ as above. Then there is a constant C < ∞ and a function ρ : R → R satisfying ρ ( s ) → as s → ∞ , such that for any δ , n and m satisfying (4.6) , k ( I − L ) f δ − g k H ≤ k ( I − L n,m ) f − g k H m + D · ρ ( δ ) k ( I − L n,m ) f k H m holds for all f ∈ D . Proof:
By the product rule, k ( I − L ) f δ − g k H ≤ k χ δ (( I − L ) f − g ) k H + d X i,j =1 k a ij φ δ α ( x ) ∂ j ∂ i ψ δ ( v ) f k H + 2 d X i,j =1 k a ij φ δ α ( x ) ∂ i ψ δ ( v ) ∂ v j f k H + d X i,j =1 k ∂ j a ij φ δ α ( x ) ∂ i ψ δ ( v ) f k H + d X i,j =1 k a ij v j φ δ α ( x ) ∂ i ψ δ ( v ) f k H + d X i =1 k v i ∂ i φ δ α ( x ) ψ δ ( v ) f k H + d X i =1 k ∂ i Φ φ δ α ( x ) ∂ i ψ δ ( v ) f k H . Due to the choice of n and m , every k · k H on the right hand side can be replaced with k · k H m , a ij by a ij,n , and Φ by Φ m , hence L by L n,m .We now give estimates for each summand of the right hand side, in their order ofappearance:(1) k χ δ (( I − L ) f − g ) k H ≤ k ( I − L n,m ) f − g k H m ,(2) k a ij φ δ α ( x ) ∂ j ∂ i ψ δ ( v ) f k H ≤ M Σ C ψ δ − k f k H m ,(3) k a ij φ δ α ( x ) ∂ i ψ δ ( v ) ∂ v j f k H ≤ M Σ C ψ δ − k ∂ v j f k H m ,244) k ∂ j a ij φ δ α ( x ) ∂ i ψ δ ( v ) f k H ≤ max { B Σ , M · (2 δ ) β ∨ } C ψ δ − k f k H m ,(5) k a ij v j φ δ α ( x ) ∂ i ψ δ ( v ) f k H ≤ M Σ C ψ δ − k v j f k H m ,(6) k v i ∂ i φ δ α ( x ) ψ δ ( v ) f k H ≤ C φ δ − α k v i f k H m ,(7) k ∂ i Φ φ δ α ( x ) ∂ i ψ δ ( v ) f k H ≤ N (1 + (2 δ α ) γ ) C ψ δ − k f k H m ,where the last inequality is due to | ∂ i Φ( x ) | ≤ N (1+ | x | γ ) for all x ∈ R d and the support ofthe cutoff as in (4.6). Application of Lemma 4.11 shows the existence of D independentof n, m , such that k ( I − L ) f δ − g k H ≤ k ( I − L n,m ) f − g k H m + D · ρ ( δ ) k ( I − L n,m ) f k H m where ρ ( δ ) .. = δ − + 2 β δ β − + 2 β ∨ δ ( β ∨ − + 2 β δ β − α + δ − + 2 γ δ αγ − . Clearly ρ ( δ ) → δ → ∞ due to β < α . (cid:3) Now finally we show that for each ε >
0, we can find some f δ ∈ D such that k ( I − L ) f δ − g k H < ε. Choose δ > ρ ( δ ) < ε D k g k H (where ρ, D are provided byLemma 4.12) and that there exist n, m satisfying (4.6).Then choose f ∈ D via Theorem 4.10 such that k ( I − L n,m ) f − g k H m < min { ε , k g k H } anddefine f δ as before. Note that due to the choice of the cutoffs, it holds k g k H = k g k H m ,therefore k ( I − L ) f δ − g k H < ε ε k g k H m ( k ( I − L n,m ) f − g k H m + k g k H m ) < ε. As mentioned earlier, this shows essential m-dissipativity of the operator (
L, D ) on H and therefore concludes the proof of Theorem 3.4. We consider the abstract Cauchy problem associated with the operator L . Given theinitial condition u ∈ H , u : [0 , ∞ ) → H should satisfy ∂ t u ( t ) = (tr (Σ H v ) + b · ∇ v + v · ∇ x − ∇ Φ · ∇ v ) u ( t ) and u (0) = u . (5.1)If we set u ( t ) .. = T t u , where ( T t ) t ≥ is the semigroup on H generated by the closure( L, D ( L )) of ( L, D ), then the map t u ( t ) is continuous in H . For all t ≥
0, it holds25hat R t u ( s ) d s ∈ D ( L ) with L R t u ( s ) d s = T t u − u = u ( t ) − u , hence u is the uniquemild solution to the abstract Cauchy problem.If u ∈ D ( L ), then u ( t ) ∈ D ( L ) for all t ≥
0, and ∂ t u ( t ) = LT t u = Lu ( t ), so u is evena classical solution to the abstract Cauchy problem associated to L . In particular, thisholds for all u ∈ C c ( R d × d ), since L is dissipative there and it extends D , which implies C c ( R d × d ) ⊂ D ( L ).In this context, Theorem 1.1 shows exponential convergence of the unique solution u ( t )to a constant as t → ∞ . More precisely, for each θ > θ ∈ (0 , ∞ )depending on the choice of Σ and Φ such that for all t ≥ (cid:13)(cid:13)(cid:13)(cid:13) u ( t ) − Z E u d µ (cid:13)(cid:13)(cid:13)(cid:13) H ≤ θ e − θ t (cid:13)(cid:13)(cid:13)(cid:13) u − Z E u d µ (cid:13)(cid:13)(cid:13)(cid:13) H . So far, our considerations have been purely analytical, giving results about the coreproperty of D for L and rate of convergence for the generated semigroup ( T t ) t ≥ in H .However, this approach is still quite natural in the context of the Langevin SDE (1.1),as the semigroup has a meaningful stochastic representation. The connection is achievedvia the powerful theory of generalized Dirichlet forms as developed by Stannat in [Sta99],which gives the following:Assume the context of Theorem 3.4. There exists a hunt process M = (cid:16) Ω , F , ( F t ) t ≥ , ( X t , V t ) , ( P ( x,v ) ) ( x,v ) ∈ R d × R d (cid:17) with state space E = R d × R d , infinite lifetime and continuous sample paths ( P ( x,v ) -a.s.for all ( x, v ) ∈ E ), which is properly associated in the resolvent sense with ( T t ) t ≥ . Inparticular (see [Con11, Lemma 2.2.8]), this means that for each bounded measurable f which is also square-integrable with respect to the invariant measure µ and all t > T t f is a µ -version of p t f , where ( p t ) t ≥ is the transition semigroup of M with p t f : R d × R d → R , ( x, v ) E ( x,v ) [ f ( X t , V t )] . This representation can be further extended to all f ∈ H , see for example [MR92,Exercise IV.2.9]. Moreover, if µ -versions of Σ and Φ are fixed, then P ( x,v ) solves themartingale problem for L on C c ( E ) for L -quasi all ( x, v ) ∈ E , i.e. for each f ∈ C c ( E ),the stochastic process ( M [ f ] t ) t ≥ defined by M [ f ] t .. = f ( X t , V t ) − f ( X , V ) − Z t Lf ( X s , V s ) d s, is a martingale with respect to P ( x,v ) . If h ∈ L ( µ ) is a probability density with respectto µ , then the law P h .. = R E P ( x,v ) h ( x, v ) d µ solves the martingale problem for ( L, D ( L )),26ithout the need to fix specific versions of Σ and Φ. In particular, this holds for h = 1.As in [Con11, Lemma 2.1.8], for f ∈ D ( L ) with f ∈ D ( L ) and Lf ∈ L ( µ ), a martingaleis also defined via N [ f ] t .. = ( M [ f ] t ) − Z t L ( f )( X s , V s ) − (2 f Lf )( X s , V s ) d s, t ≥ , which may serve as a way to verify that M is already a weak solution of (1.1), as it allowsa representation of the quadratic variation process. Indeed, if we set f in ( x, v ) .. = ϕ n ( x i ) x i for a suitable sequence ( ϕ n ) n ∈ N of cutoff functions as in Definition 3.6, evaluation of N [ f in ] t shows that the quadratic variation [ M [ f in ] ] t of M [ f in ] t is constantly zero, which impliesthe same for M [ f in ] t . Hence, by introducing appropriate stopping times, it follows that X it − X i = R t V is d s , so the first line of the SDE (1.1) is satisfied.In an analogous procedure, using g in ( x, v ) .. = ϕ n ( v i ) v i , we can see that the quadratic co-variation [ V i , V j ] t is given by 2 R t a ij ( V s ) d s . Since Σ is strictly elliptic, the diffusion ma-trix σ is invertible and by Lévy’s characterization, the process B t .. = R t √ σ − ( V s ) d M s is a standard d -dimensional Brownian motion, where M t .. = ( M [ v ] t , . . . , ( M [ v d ] t ), which isa local martingale. Moreover, it holds thatd V t = d M t + b ( V t ) − ∇ Φ( X t ) d t = √ σ ( V t )d B t + b ( V t ) − ∇ Φ( X t ) d t, so ( X t , V t ) is a weak solution to the SDE (1.1) with initial distribution hµ under P h .Finally, in this context, the statement on hypocoercivity (Theorem 1.1) shows that forevery θ >
1, there is an explicitly computable θ ∈ (0 , ∞ ) depending on the choice ofΣ and Φ, such that the transition semigroup ( p t ) t ≥ satisfies k p t g − Z E g d µ k L ( µ ) ≤ θ e − θ t k g − Z E g d µ k L ( µ ) (5.2)for all g ∈ L ( µ ) and t ≥
0. In particular, this implies that the probability law P µ onthe space of continuous paths on E with initial distribution (and invariant measure) µ has the strong mixing property, i.e. for any Borel sets A , A on the path space, it holdsthat P µ ( ϕ t A ∩ A ) → P µ ( A ) P µ ( A ) as t → ∞ , where ϕ t A = { ( Z s ) s ≥ ∈ C ([0 , ∞ ) , E ) | ( Z s + t ) s ≥ ∈ A } . This follows from (5.2)and associatedness of the semigroups to the probability law P µ , see for example [Con11,Remark 2.1.13]. Given a Kolmogorov backwards equation of the form − ∂ t u ( x, t ) = L K u ( x, t ), the corre-sponding Fokker-Planck equation is given by ∂ t f ( x, t ) = L FP f ( x, t ), where L FP = ( L K ) ′
27s the adjoint operator of L K in L ( R d , d x ), restricted to smooth functions. In oursetting, we have L K = L , which therefore yields via integration by parts for f ∈ D : L FP f = d X i,j =1 ∂ v i ( a ij ∂ v j f + v j a ij f ) − v · ∇ x f + ∇ Φ ∇ v f. (5.3)Consider the Fokker-Planck Hilbert space ˜ H .. = L ( E, ˜ µ ), where˜ µ .. = (2 π ) − d e Φ( x )+ v d x ⊗ d v. Then a unitary Hilbert space transformation between H and ˜ H is given by T : H → ˜ H, T g = ρg with ρ ( x, v ) .. = e − Φ( x ) − v . Let ( T t ) t ≥ be the semigroup on H generated by ( L, D ( L )) and denote by ( T ∗ t ) t ≥ and L ∗ the adjoint semigroup on H and its generator, respectively. It is evident that for f ∈ D , L ∗ is given as L ∗ f = ( S + A ) f , where S and A refer to the symmetric andantisymmetric components of L respectively, as defined in Definition 3.2. As mentionedin Remark 3.11, we achieve the exact same results for the equation corresponding to L ∗ as for the one corresponding to L , which we considered in Section 3. In particular,( L ∗ , D ) is essentially m-dissipative and its closure ( L ∗ , D ( L ∗ )) generates ( T ∗ t ) t ≥ , whichconverges exponentially to equilibrium with the same rate as ( T t ) t ≥ .Let e T t g .. = T ( T ∗ t ) T − g for t ≥ g ∈ ˜ H . Then ( e T t ) t ≥ is a strongly continuous contractionsemigroup on ˜ H with the generator ( T L ∗ T − , T ( D ( L ∗ ))). It is easy to see that L FP = T L ∗ T − , so for each initial condition u ∈ ˜ H , u ( t ) .. = e T t u is a mild solution to theFokker-Planck Cauchy problem. Note that for Φ ∈ C ∞ ( R d ), the transformation T leaves D invariant, which implies D ⊂ T ( D ( L ∗ )) and essential m-dissipativity of ( L FP , D ) on˜ H .If u ∈ T ( D ( L ∗ )), then ∂ t e T t u = T ( L ∗ T ∗ t ) T − u , and therefore Z E ∂ t u ( t ) f d( x, v ) = Z E L ∗ T ∗ t T − u f d µ = Z E T ∗ t T − u Lf d µ = Z E T T ∗ t T − u Lf d( x, v ) = Z E L FP u ( t ) f d( x, v ) , so u ( t ) is also a classical solution. Due to the invariance of µ for L , a stationary solutionis given by ρ and by Theorem 1.1, for every θ > θ it holds that (cid:13)(cid:13) u ( t ) − ρ ( u , ρ ) ˜ H (cid:13)(cid:13) ˜ H = (cid:13)(cid:13)(cid:13) T ∗ t T − u − ( T − u , H (cid:13)(cid:13)(cid:13) H ≤ θ e − θ t (cid:13)(cid:13)(cid:13) T − u − ( T − u , H (cid:13)(cid:13)(cid:13) H = θ e − θ t (cid:13)(cid:13) u − ρ ( u , ρ ) ˜ H (cid:13)(cid:13) ˜ H . This shows exponential convergence to a stationary state for solutions to the Fokker-Planck equation. 28 eferences [Bec89] William Beckner. A Generalized Poincaré Inequality for Gaussian Measures.
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