Asymptotic analysis for the Generalized Relativistic Langevin Equation

ASYMPTOTIC ANAL YSIS F OR THE GENERALIZED RELA TIVISTIC LANGEVIN EQUA TION ETHAN BAKER 1 , MANH HONG DUONG 1 AND HUNG D ANG NGUYEN 2 Abstract. In this pap er, we study a non-Marko vian generalized relativistic Langevin equation (GRLE). W e show that when the memory kernel is a sum of exponentials, the GRLE is equiv alen t to a Mark ovian system with added v ariables. W e establish the well-posedness and p olynomial ergodicity , obtaining an algebraic rate of con vergence to the unique Gibbs distribution. F rom the Mark ovian GRLE, w e reco v er the relativistic underdamped Langevin dynamics in a small-noise limit, as well as the classical (non-relativistic) generalized Langevin dynamics in the Newtonian limit. 1. Introduction 1.1. The generalized relativistic Langevin equation. In this pap er, w e consider the following gen- er alize d r elativistic L angevin e quation (GRLE) [ 8 , 30 , 36 ]: d q ( t ) = ∇ K ( p ( t )) d t, (1.1a) d p ( t ) = −∇ U ( q ( t )) d t − γ ∇ K ( p ( t )) d t − Z t 0 η ( t − s ) ∇ K ( p ( s )) d s d t + p 2 γ β − 1 d W ( t ) + F ( t ) d t. (1.1b) This stochastic differential equation describ es the motion of a r elativistic particle with p osition q and momen tum p , b oth taking v alues in R d . The first equation ( 1.1a ) is simply a relation b et w een the p osition and the momentum in the special relativistic setting where K is the relativistic kinetic energy K ( p ) = c p m 2 c 2 + | p | 2 . (1.2) Here, c > 0 is the sp eed of ligh t, and m is the particle’s mass at rest. The second equation ( 1.1b ) p osits that the particle mov es under the influence of differen t forces, namely , (i) an external force −∇ U where U : R d 7→ R is an external p oten tial; (ii) a friction − γ ∇ K where γ ≥ 0 is the friction coefficient; (iii) a dela yed drag force from the fluid on the particle giv en by the in tegral R t 0 η ( t − s ) ∇ K ( p ( t )) d s d t where η : [0 , ∞ ) 7→ R d × d is a memory k ernel; (iv) a random p erturbation that consists of tw o components: a standard d -dimensional Brownian motion ( W ( t )) t ≥ 0 and a mean-zero stationary Gaussian pro cess F ( t ) satisfying the fluctuation–dissipation theorem [ 23 , 10 ]: E [ F i ( t ) F j ( s )] = β − 1 ( η ( t − s )) ij δ ij , t ≥ s, (1.3) where δ ij denotes the Kroneck er delta. Finally , the parameter β is the in v erse temperature. The system ( 1.1a )-( 1.1b ) has b een considered in [ 8 , 30 , 36 ] 1 —in particular in [ 30 , 36 ] it is derived from a first-principle particle-bath Lagrangian. It is an extension of the classical (non-relativistic) generalized Langevin dynamics (GLE) [ 28 , 29 ], which is obtained from ( 1.1a )-( 1.1b ) b y replacing the relativistic kinetic energy ( 1.2 ) b y the classical one K ( p ) = | p | 2 / (2 m ), to comply with Einstein’s theory of sp ecial relativit y . Extending the classical Langevin dynamics to the relativistic systems is vital to many areas of physics including relativistic fluids/plasmas, effectiv e field theories of dissipativ e hydrodynamics, and relativistic viscous electron flo w in graphene, just to name a few. The first relativistic Langevin model was in tro duced in [ 10 ]. Since then, it has b ecome an activ e area of research, with a v ast literature dev oting to the study of relativistic systems. W e refer the reader to [ 9 , 12 ] for further information on v arious topics of the relativistic Langevin models. 1 More precisely γ = 0 in these pap ers. 1 Analogous to the non-relativistic GLE, the system ( 1.1a )– ( 1.1b ) exhibits non-Marko vian dynamics ow- ing to the presence of the memory kernel. In conjunction with the nonlinearity induced by the relativistic kinetic energy , this renders the mathematical analysis considerably more in tricate. It is kno wn that, when the memory kernel is a sum of exp onen tials (also known as the Prony series memory kernel), the classical GLE can be equiv alently form ulated as a Mark o vian system b y in troducing additionally auxiliary v ariables [ 24 ]. The Mark ovian system is more computationally tractable, and v arious efficient n umerical sc hemes hav e b een developed for its analysis [ 3 , 16 ]. Under this Mark ovian form ulation, in [ 28 , 13 ], the authors prov e the geometric ergo dicit y and obtain an exp onen tial rate of con v ergence to the equilibrium, whic h is an imp ortan t problem in statistical physics, molecular dynamics and sampling tec hniques. In addition, [ 28 ] also deriv es a white-noise limit of the Marko vian GLE system, that is to show that, when the noise term is appropriately rescaled such that the correlation function b ecomes a Dirac measure, the GLE system conv erges to the underdamp ed Langevin dynamics, th us eliminating the auxiliary v ariables. Concerning relativistic Langevin dynamics, in addition to the questions of ergo dicit y and small mass or small noise limits that arise in classical mo dels, another imp ortan t issue is the Newtonian limit. Sp ecifically , one seeks to show that as the sp eed of ligh t tends to infinit y , the relativistic system conv erges to the corresp onding classical system. Establishing this limit ensures the consistency of the relativistic mo del with its classical counterpart in the non-relativistic regime. These topics for the system ( 1.1 ) in the absence of the memory term ha ve b een studied recen tly b y v arious authors [ 1 , 6 , 5 , 2 , 14 , 15 ]. 1.2. Summary of the main results. The aim of this pap er is to extend the aforemen tioned works to the relativistic generalized Langevin dynamics ( 1.1 ). Under suitable assumptions on the potential U and the k ernel η , our main findings can b e summarized as follo ws. (1) ( Marko vian formulation and w ell-p osedness ) Prop osition 1.2 shows that when the memory k ernel can b e expressed as an exp onen tial form, cf. ( 1.4 ), the GRLE ( 1.1 ) is equiv alen t to a Mark o vian system b y introducing auxiliary v ariables. The w ell-p osedness of this Mark o vian system is pro v ed in Theorem 1.3 . (2) ( Polynomial ergo dicit y ) Theorem 1.4 establishes a polynomial rate of conv ergence to the unique equilibrium measure, which is a Gibbs distribution, for the Marko vian system. (3) ( White-noise limit ) In Theorem 1.6 , w e sho w that under appropriate rescaling, the Marko vian system is w ell approximated by the relativistic underdamp ed Langevin dynamics describing the ev olution of the p osition and momentum v ariables, thus eliminating the auxiliary v ariables and obtaining the friction co efficien t from the parameters of the Mark o vian GRLE system. (4) ( Newtonian limit ) In Theorem 1.8 , we deriv e the Newtonian limit for the Mark o vian GRLE, reco v ering the Mark o vian GLE system when the parameter c , representing the sp eed of ligh t, tends to infinity . The ab o v e results, as w ell as their relationship to existing results, are summarised in Figure 1 . In order to precisely formulate the abov e results, w e make the follo wing assumption on the growth of the external p oten tial. Assumption 1.1. The p otential U ∈ C ∞  R d ; [1 , ∞ )  satisfies the fol lowing: (1) ⟨∇ U ( q ) , q ⟩ ≥ a 1 | q | λ +1 − a 2 for some a 1 , a 2 > 0 and λ ≥ 1 for al l q ∈ R d ; and, (2) 1 a 3 | q | λ +1 − a 3 ≤ U ( q ) ≤ a 3  1 + | q | λ +1  for some a 3 > 0 . 1.3. Mark o vian form ulation. W e no w describ e the results in more detail. W e start b y introducing a Marko vian F ormulation for the GRLE. This is drawn up on the Marko vian framework of the GLE as found in [ 24 , 25 , 29 ]. More sp ecifically , following [ 29 , Definition 1.3], t w o pro cesses X , Y are said to b e equiv alent if X and Y ha v e the same finite dimensional distributions. Under the assumption that the memory k ernel η admits a sp ecial form of exp onen tial functions, we assert that ( q , p ) from ( 1.1 ) is indeed equiv alent to a Mark o v pro cess solving a system of SDEs. This is rigorously v erified through Prop osition 1.2 , whic h can b e regarded as a relativistic analogue of [ 29 , Prop osition 8.1]. 2 Relativistic Langevin Equation Langevin Equation Generalized Langevin Equation c → ∞ c → ∞ ε → 0 ε → 0 Theorem 1.8 [ 14 , Theorem 2.6] Theorem 1.6 [ 28 , Theorem 2.6] Generalized Relativistic Langevin Equation Figure 1. Diagram of the relationships betw een Langevin equations. Prop osition 1.2. L et Λ ∈ R d × k , and A ∈ R k × k b e symmetric p ositive definite. Then, given the memory kernel c an b e written as, η ( t ) = Λ e − A t Λ T , (1.4) ( q , p ) given by ( 1.1 ) is e quivalent to the pr o c ess ( q , p ) solving, d q ( t ) = ∇ K ( p ( t )) d t, (1.5a) d p ( t ) = −∇ U ( q ( t )) d t − γ ∇ K ( p ( t ))d t + Λ z ( t ) d t + p 2 γ β − 1 d W ( t ) , (1.5b) d z ( t ) = − Λ T ∇ K ( p ( t )) d t − A z ( t ) d t + Σ d f W ( t ) , (1.5c) for some k -dimensional Br ownian motion f W , wher e z : [0 , ∞ ) → R k with the initial distribution given by, z (0) ∼ N (0 , β − 1 I k ) , and the matrix Σ ∈ R k × k satisfies ΣΣ T = 2 β − 1 A . The follo wing theorem establishes the well-posedness of the Marko vian system ( 1.5 ). Theorem 1.3. Under Assumption 1.1 , ( 1.5 ) has a unique str ong Markov solution. The pro of of this theorem is rather standard which relies on the Lipsc hitz prop ert y of ∇ K ( p ) and a Ly apuno v condition, see Section 2 for a further discussion of this point. In order to study asypm totic b eha viors of the Marko vian system ( 1.5 ), throughout the rest of the pap er, we will consider a sp ecial case of ( 1.5 ) that is widely explored in the literature [ 13 , 16 , 28 ]. More sp ecifically , w e choose k = M d for some M ∈ N , α = ( α 1 , ..., α M ) ∈ (0 , ∞ ) M and λ = ( λ 1 , ..., λ M ) ∈ (0 , ∞ ) M . W e consider A =      α 1 I d 0 · · · 0 0 α 2 I d · · · 0 . . . . . . . . . . . . 0 0 · · · α M I d ,      and Λ =  λ 1 I d λ 2 I d · · · λ M I d  . 3 In view of ( 1.4 ), we restrict to the class of memory kernels (known as the Pron y series memory kernel) giv en b y η ( t ) = M X i =1 λ 2 i e − α i t I d . (1.6) W e then c ho ose z = ( z 1 , ..., z M ) and f W = ( W 1 , ..., W M ), where z i ∈ R d and W i are indep enden t d - dimensional Bro wnian motions for i = 1 , ..., M . Then, ( 1.5 ) is reduced to d q ( t ) = ∇ K ( p ( t )) d t, (1.7a) d p ( t ) = −∇ U ( q ( t )) d t − γ ∇ K ( p ( t )) d t + M X i =1 λ i z i ( t ) d t + p 2 γ β − 1 d W ( t ) , (1.7b) d z i ( t ) = − λ i ∇ K ( p ( t )) d t − α i z i ( t ) d t + p 2 α i β − 1 d W i ( t ) , i = 1 , . . . , M , (1.7c) q (0) = q 0 , p (0) = p 0 , z i (0) = z i, 0 . 1.4. P olynomial ergo dicit y. W e turn to the asymptotic analysis of ( 1.7 ), the first topic of which is the large-time stabilit y . As a consequence of Theorem 1.3 , w e ma y in tro duce the transition probabilities of ( q , p, z ) solving ( 1.7 ) b y P t ( x, B ) = P [( q ( t ) , p ( t ) , z ( t )) ∈ B | ( q (0) , p (0) , z (0)) = x ] , whic h is w ell-defined for all x ∈ R (2+ M ) d and Borel sets B ⊂ R (2+ M ) d . W e recall that for a measurable space (Ω , F ), the total v ariation norm for tw o probability measures µ and υ is giv en by , ∥ µ − υ ∥ TV = sup A ∈F | µ ( A ) − υ ( A ) | . W e consider the following Gibbs distribution as sociated to ( 1.7 ) d ρ β ( q , p, z ) = 1 Z e − β H ( q ,p,z ) d q d p d z , (1.8) where the Hamiltonian H is giv en b y H ( q , p, z ) = U ( q ) + K ( p ) + 1 2 | z | 2 , (1.9) and Z = Z R (2+ M ) d e − β H ( q ,p,z ) d q d p d z . denotes the normalisation constan t. The next result of the presen t pap er characterizes the long-time b eha viour of ( 1.7 ) and obtains a p olynomial rate of conv ergence to the unique inv ariant probabilit y measure. Theorem 1.4. Under Assumption 1.1 , let ( q , p, z ) denote the solution to ( 1.7 ) . Supp ose that either (i) γ > 0 ; or (ii) γ = 0 and ∇ U is Lipschitz c ontinuous. Then the fol lowing hold: (1) for c > 0 sufficiently lar ge, the Gibbs distribution ρ β define d in ( 1.8 ) is the unique invariant pr ob ability me asur e for the pr o c ess ( q , p, z ) ; and (2) for al l r ∈ N , ther e exists c > 0 sufficiently lar ge and V r : R (2+ M ) d → [1 , ∞ ) such that ∥ P t ( x, · ) − ρ β ∥ TV ≤ C (1 + t ) r V r ( x ) , t ≥ 0 , (1.10) for some C > 0 indep endent of t and x ∈ R (2+ M ) d . T o prov e this theorem, we adopt the framework of [ 19 ] (see [ 4 , 11 , 17 ] for earlier developmen t), whic h consists of three main ingredien ts, namely , a H¨ ormander’s condition, a solv ability of the asso ciated con trol problem, and a suitable Ly apuno v function quan tifying the conv ergent rate. W e will deal with these issues in Section 3 , and recall the framew ork of [ 19 ] in more detail in App endix A . 4 1.5. White-noise limit. Now, w e discuss the white-noise limit of the system ( 1.7 ). F ollowing the frame- w ork of [ 27 , 28 ], w e introduce a diffusiv e rescaling to the memory kernel as follo ws η ( ε ) ( t ) = M X i =1 λ 2 i ε e − α i ε t I d , ε ∈ (0 , 1) . (1.11) This amoun ts to rescaling λ i 7→ λ i √ ε and α i 7→ α i ε in ( 1.7 ). In order to b e consisten t with ( 1.3 ), the rescaled noise is given by F ( ε ) ( t ) := 1 √ ε F  t ε  , By applying this memory k ernel to the Mark o vian form ulation, ( 1.7 ) b ecomes d q ( ε ) ( t ) = ∇ K ( p ( ε ) ( t )) d t, (1.12a) d p ( ε ) ( t ) = −∇ U ( q ( ε ) ( t )) d t − γ ∇ K ( p ( ε ) ( t )) d t + 1 √ ε M X i =1 λ i z ( ε ) i ( t ) d t + p 2 γ β − 1 d W ( t ) , (1.12b) d z ( ε ) i ( t ) = − λ i √ ε ∇ K ( p ( ε ) ( t )) d t − α i ε z ( ε ) i ( t ) d t + r 2 α i ε d W i ( t ) , (1.12c) q ( ε ) (0) = q 0 , p ( ε ) (0) = p 0 , z ( ε ) i (0) = z i, 0 , with initial conditions indep enden t of our c hoice of ε . Such a rescaling is commonly used, not only in the study of white noise limits [ 28 , 27 ], but also in the asymptotic analysis of sto c hastic partial differen tial equations, e.g., the sto c hastic Burgers’ equation [ 7 ]. Note that we ha v e used the sup erscripts in ( 1.12 ) to indicate that the dynamics dep ends on the parameters ε which will b e imp ortan t in the study of the white-noise in the next theorem. As usual, we require the following assumption on the initial conditions: Assumption 1.5. The initial c onditions ( q 0 , p 0 , z 0 ) ∈ R (2+ M ) d have finite moments: for al l n ∈ N , E h | p 0 | n + | q 0 | n + M X i =1 | z i, 0 | n i < ∞ . W e no w state the third main result of the pap er giving the white noise limit for the GRLE. Theorem 1.6 (The Relativistic White Noise Limit) . Under Assumption 1.1 , let  q ( ε ) , p ( ε ) , z ( ε )  b e the solution to ( 1.12 ) with initial c onditions ( q 0 , p 0 , z 0 ) satisfying Assumption 1.5 . F or al l T > 0 and n ∈ N , it holds that E h sup t ∈ [0 ,T ] | q ( ε ) ( t ) − Q ( t ) | n + sup t ∈ [0 ,T ] | p ( ε ) ( t ) − P ( t ) | n i → 0 (1.13) as ε → 0 , wher e ( Q, P ) satisfy the fol lowing under damp e d r elativistic L angevin dynamics d Q ( t ) = ∇ K ( P ( t )) d t, (1.14a) d P ( t ) = −∇ U ( Q ( t )) d t −  γ + M X i =1 λ 2 i α i  ∇ K ( P ( t )) d t + p 2 β − 1 γ dW ( t ) + M X i =1 s 2 β − 1 λ 2 i α i d W i ( t ) , (1.14b) Q (0) = q 0 , P (0) = p 0 . (1.14c) The relativistic Langevin dynamics ( 1.14 ) w as introduce in [ 10 ] and has b een studied in tensiv ely in the literature, see for instance the recent pap er [ 14 ] and references therein. The pro of of Theorem 1.6 relies on the fact that z ( ε ) i is an Ornstein-Uhlen beck pro cess, which can b e solved explicitly . Then b y substituting it to ( 1.12b ), w e obtain a closed system for the evolution of p osition and momen tum. This will be rigorously established in Section 4 . 5 Remark 1.7. As a consequence of Theorem 1.6 , as ε → 0, the pro cess ( q ( ε ) , p ( ε ) ) con v erges in distribution to the solution of the following more familiar-lo oking RLE [ 10 ] d Q ( t ) = ∇ K ( P ( t )) d t, (1.15a) d P ( t ) = −∇ U ( Q ( t )) d t − γ ∗ ∇ K ( P ( t )) d t + p 2 γ ∗ β − 1 d W ( t ) , (1.15b) Q (0) = q 0 , P (0) = p 0 , where the friction co efficien t γ ∗ is giv en by γ ∗ = γ + M X i =1 λ 2 i α i . Note that, comparing to ( 1.14 ), the ab o v e system consists of only one Wiener process and one effectiv e friction coefficient γ ∗ . 1.6. Newtonian limit. W e turn our atten tion to the Newtonian limit of system( 1.7 ) when the sp eed of ligh t c tends to infinit y . Our main finding is that, in this regime, one ma y reco v er the Mark o vian form ulation of the non-relativistic generalised Langevin dynamics from ( 1.7 ). This is summarized b elo w through Theorem 1.8 . Theorem 1.8 (The Newtonian Limit for the GRLE) . Under Assumption 1.1 , let ( q ( c ) , p ( c ) , z ( c ) ) b e the solution to ( 1.7 ) with initial c onditions ( q 0 , p 0 , z 0 ) satisfying Assumption 1.5 . Then, for al l T > 0 and n > 0 , it holds that E h sup t ∈ [0 ,T ]    q ( c ) ( t ) − q ( t )    n + sup t ∈ [0 ,T ]    p ( c ) ( t ) − p ( t )    n + M X i =1 sup t ∈ [0 ,T ]    z ( c ) i ( t ) − z i ( t )    n i → 0 , (1.16) as c → ∞ , wher e the pr o c ess ( q , p, z ) satisfy the classic al (non-r elativistic) gener alize d L angevin e quation [ 28 ] d q ( t ) = p ( t ) m d t, (1.17a) d p ( t ) = −∇ U ( q ( t )) d t − γ p ( t ) m d t + M X i =1 λ i z i ( t ) d t + p 2 γ β − 1 d W ( t ) , (1.17b) d z i ( t ) = − λ i p ( t ) m d t − α i z i ( t ) d t + p 2 α i β − 1 d W i ( t ) , (1.17c) q (0) = q 0 , p (0) = p 0 , z i (0) = z i, 0 . The pro of of this theorem is based on estimating the errors b et w een the tw o systems ( 1.7 ) and ( 1.17 ) while making use of Gron w all’s lemma and suitable moment b ounds. All of this will b e addressed in detail in Section 4 . 1.7. F uture works. In this pap er, we fo cus on the most p opular t yp e of memory kernel, namely the Pron y series memory k ernel as a finite sum of exp onen tials. F or this class, the non-Marko vian GRLE can b e equiv alently written as a Marko vian one by augmenting the original pro cess by a finite n um b er of auxiliary v ariables z . Other class of memory kernels ha ve also been considered in the literature, notably the p o w er-la w ones [ 18 , 27 ]. In this latter case, one needs to use an infinite num ber of auxiliary v ariables and study infinite-dimensional sto c hastic differen tial equations in Hilb ert spaces. It would b e an in teresting problem to extend the results of the prese n t pap er to such settings. Another c hallenging direction for future researc h is to inv estigate systems of interacting particles, in particular when the in teraction p oten tials are singular such as the Coloum b and Lennard-Jones p oten tials. These interacting particle systems pla y a central role in statistical ph ysics and hav e received considerable atten tion in recen t y ears [ 13 , 14 , 15 , 21 , 32 ]. 6 1.8. Organization of the paper. The rest of the pap er is structured as follo ws. In Section 2 , w e deriv e the Marko vian formulation of the GRLE and establish the well-posedness of the resulting Mark o v system ( 1.5 ). Section 3 studies the ergo dicit y of the Marko vian system when the memory k ernel η admits the form as a finite sum of exp onen tials ( 1.6 ). P articularly , in Section 3.2 , we construct Ly apunov functions for the GRLE whereas in Section 3.3 , we complete our pro of of polynomial ergo dicit y b y showing that H¨ ormander’s condition holds and the control problem asso ciated with the Marko vian formulation is solv able. In Section 4 , we prov e the white noise and Newtonian limits of the Marko vian formulation; first when ∇ U is Lipschitz, and then extending to the class of p oten tials satisfying Assumption 1.1 . The pap er concludes with tw o app endices. In App endix A , we recall the framew ork developed in [ 19 ] that w e use in order to prov e the p olynomial ergo dicit y of the GRLE, while in App endix B , w e presen t a technical lemma that is used in the pro ofs of the main results. 2. Well-posdeness of the Marko vian GRLE In this section, we consider the Marko vian system ( 1.5 ) and establish Prop osition 1.2 showing that ( 1.5 ) is indeed equiv alen t to the non-Marko vian GRLE ( 1.1 ) when the memory kernel is given b y the exp onen tial form ( 1.4 ). W e also presen t the pro of of Theorem 1.3 deducing the w ell-p osedness of ( 1.5 ). W e start with the pro of of Prop osition 1.2 , whose argument mainly relies on Duhamel’s form ulas. Pr o of of Pr op osition 1.2 . By the Duhamel’s form ula, we note that the proces s z in ( 1.5 ) can b e recast as z ( t ) = e − A t z (0) + Z t 0 e − A ( t − s ) Σ d W ( s ) − Z t 0 e − A ( t − s ) Λ T ∇ K ( p ( s )) d s. (2.1) Substituting ( 2.1 ) into ( 1.5b ), w e get d p ( t ) = −∇ U ( q ( t )) d t − γ ∇ K ( p ( t )) d t − Z t 0 η ( t − s ) ∇ K ( p ( t )) d s d t + p 2 γ β − 1 d W ( t ) + F ( t ) d t, where η is defined as in ( 1.4 ), and ( F ( t )) t ≥ 0 is an Ornstein-Uhlenbeck pro cess given by F ( t ) = Λ e − A t z (0) + Λ Z t 0 e − A ( t − s ) Σ d W ( s ) . By using a routine calculation similar to the approach found in [ 29 , Prop osition 8.1], F satisfies ( 1.3 ). This completes the pro of. □ Next, we turn to the well-posedness of ( 1.5 ) and establish Theorem 1.3 . In order to do this, w e need the follo wing lemma sho wing that ∇ K is globally Lipsc hitz con tin uous. Lemma 2.1. The function K define d in ( 1.2 ) satisfies that ∇ K is Lipschitz c ontinuous and that ∇ 2 K is b ounde d. In p articular, for al l p 1 , p 2 ∈ R d , |∇ K ( p 1 ) − ∇ K ( p 2 ) | ≤ d m | p 1 − p 2 | , Pr o of. Observe that, for 1 ≤ i, j ≤ M where i  = j ,     δ 2 δ p 2 i K ( p )     =   c 3 m 2 + c | p | 2 − cp 2 i   ( | p | 2 + c 2 m 2 ) 3 2 ≤ 1 m , (2.2) and,     δ 2 δ p i δ p j K ( p )     = c | p i p j | ( | p | 2 + c 2 m 2 ) 3 2 ≤ 1 m , (2.3) T ogether ( 2.2 ) and ( 2.3 ) imply that   ∇ 2 K   F ≤ d m , where ∇ 2 K denotes the Hessian of K and ∥·∥ F denotes the F rob enius norm. □ 7 T o prov e Theorem 1.3 , w e will apply [ 22 , Theorem 3.5] which states that there exists a unique Marko v solution to a time homogeneous SDE, d X ( t ) = a ( X ( t )) d t + σ ( X ( t )) d W ( t ) , if, for all R > 0, there exists C R > 0 such that, whenever | x | , | y | < R , | a ( y ) − a ( x ) | + ∥ σ ( y ) − σ ( x ) ∥ ≤ C R | y − x | ; (2.4) and, there exists V ∈ C 2 ( R ) suc h that lim | x |→∞ V ( x ) = ∞ , and L V ≤ ζ V , (2.5) for some ζ > 0, where L is the generator of the SDE. See also [ 34 ]. Pr o of of The or em 1.3 . W e proceed to v erify conditions ( 2.4 ) and ( 2.5 ) for the Marko vian system ( 1.5 ) whose generator is giv en b y L f ( q , p, z ) = ∇ K ( p ) · ∇ q f − ( ∇ U ( q ) + γ ∇ p K ( p ) − Λ z ) · ∇ p f − (Λ T ∇ K ( p ) + A z ) · ∇ z f + γ β − 1 ∆ p f + 1 2 T r  ΣΣ T ∇ 2 z f  . Condition ( 2.4 ) follows from Lemma 2.1 and the fact that U ∈ C ∞ ( R ). Concerning condition ( 2.5 ), w e recall the formula of the Hamiltonian of the system ( 1.5 ) defined in ( 1.9 ): H ( q , p, z ) = U ( q ) + K ( p ) + 1 2 | z | 2 . Applying L to H , w e ha v e L H = − γ c 2 | p | 2 m 2 c 2 + | p | 2 − z T A z + γ β − 1 ∆ K ( p ) + 1 2 trΣΣ T ≤ C, for some C > 0. The last implication follows from the h yp othesis that A is p ositiv e definite and Lemma 2.1 . It follo ws that condition ( 2.5 ) is satisfied b y c hoosing V = H + 1. □ 3. Ergodicity In this section, w e pro v e Theorem 1.4 on the polynomial ergo dicit y of the Marko vian system ( 1.7 ) when η can b e expressed as a finite sum of exponentials as in ( 1.6 ). As already men tioned in the introduction, w e adopt the framework of [ 19 ], which in turn was built up on the techniques developed earlier in [ 4 , 11 , 17 ]. F or self-consistency , the framework of [ 19 ] will b e summarized in App endix A . F ollowing this framew ork, the pro of of estimate ( 1.10 ) consists of three main steps, namely , v erifying the H¨ ormander’s condition, proving the solv ability of the associated control problem, and constructing a suitable Ly apunov function. The H¨ ormander’s condition, whic h ensures the smo othness of the system’s transition probabilit y , will b e verified b y applying the classical H¨ ormander’s Theorem [ 20 ], which asserts that the state space may b e generated by the collection of vector fields jointly induced by the diffusion and the drifts. Since w e are dealing with finite-dimensional settings, this will follo w from direct computations on Lie brac k ets. The solv ability of the associated con trol problem can b e established by the Supp ort Theorem [ 33 ] sho wing that one can alwa ys find appropriate controls allo wing for driving the dynamics to any bounded ball. The pro of of H¨ ormander theorem and solv abilit y condition are rather standard and will be discussed in Section 3.3 . The third step requires the construction of a Ly apuno v function, which is an energy-lik e function V satisfying an inequality of the form d dt E  V ( q ( t ) , p ( t ) , z ( t ))  ≤ − c 1 E  V ( q ( t ) , p ( t ) , z ( t )) α  + c 2 , t ≥ 0 , (3.1) for a suitable constan t α ∈ (0 , 1]. The construction of such a function V is highly nontrivial due to the lac k of strong dissipation in the p -direction as well as the impact of the nonlinearity of the relativistic 8 kinetic energy . W e are only able to prov e ( 3.1 ) for some α ∈ (0 , 1), th us yielding only an algebraic mixing rate as stated in Theorem 1.4 . This will b e done in Section 3.2 . 3.1. In v arian t measure. W e first prov e that the Gibbs distribution giv en b y ( 1.8 ) is indeed an inv ariant probabilit y measure for ( 1.5 ) (and thus for ( 1.7 )). Lemma 3.1. Supp ose U ∈ C ∞ ( R ) satisfies Assumption 1.1 . Then the Gibbs distribution ρ β define d in ( 1.8 ) is an invariant me asur e of ( 1.5 ) . Pr o of. Note that ( 1.5 ) can b e written in a more compact form as follows d X ( t ) = J ∇ H ( X ( t )) d t − D ∇ H ( X ( t )) d t + p 2 β − 1 D d W ( t ) , (3.2) where X = ( q , p, z ) T and J =   0 1 0 − 1 0 Λ 0 − Λ T 0   , D =   0 0 0 0 γ 0 0 0 A   , W ( t ) =   0 W ( t ) f W ( t )   . (3.3) It is clear that J is anti-symmetric and D is symmetric p ositiv e semi-definite. In fact, to show that ( 3.2 ) is indeed the same as ( 1.5 ), w e compute each term in the RHS of ( 3.2 ) explicitly: J ∇ H =   0 1 0 − 1 0 Λ 0 − Λ T 0     ∇ U ( q ) ∇ K ( p ) z   =   ∇ K ( p ) −∇ U ( q ) + Λ z − Λ T ∇ K ( p )   , D ∇ H = D =   0 0 0 0 γ 0 0 0 A     ∇ U ( q ) ∇ K ( p ) z   =   0 γ ∇ K ( p ) A z   , p 2 β − 1 D =   0 0 0 0 p 2 β − 1 γ 0 0 0 p 2 β − 1 A   =   0 0 0 0 p 2 β − 1 γ 0 0 0 Σ   . Substituting these expressions bac k to ( 3.2 ) we get d   q ( t ) p ( t ) z ( t )   =   ∇ K ( p ) −∇ U ( q ) + Λ z − Λ T ∇ K ( p )   d t −   0 γ ∇ K ( p ) A z   d t +   0 0 0 0 p 2 β − 1 γ 0 0 0 Σ     0 d W ( t ) d f W ( t )   =   ∇ K ( p ) −∇ U ( p ) + Λ z − γ ∇ K ( p ) − Λ T ∇ K ( p ) − A z   d t +   0 p 2 β − 1 γ d W ( t ) Σd f W ( t )   , whic h is precisely ( 1.5 ). The adv an tage of the compact form ( 3.2 ) is that it is very conv enien t for the v erification that ρ β is an inv ariant measure of ( 1.5 ). Indeed, using this form, the adjoin t generator of ( 1.5 ), L ∗ , is given by L ∗ ρ = − div( J ∇ H ρ ) + div ( D ∇ H ρ ) + β − 1 div( D ∇ ρ ) = − div( J ∇ H ρ ) + div [ D ( ∇ H ρ + β − 1 ∇ ρ )] . When ρ = ρ β w e ha v e ∇ ρ β = − β ∇ H ρ . In addition w e ha v e div( J ∇ H ) = J ∇ H · ∇ H = 0 due to the an ti-symmetry of J . It follo ws that β − 1 ∇ ρ + ∇ H ρ = 0 and div( J ∇ H ρ ) = div ( J ∇ H ) ρ + J ∇ H · ∇ ρ = div( J ∇ H ) ρ − β J ∇ H · ∇ H ρ = 0 − 0 = 0 . Therefore, L ∗ ρ β = 0, implying ρ β is an inv arian t measure of ( 1.5 ), as claimed. □ 9 3.2. Construction of Ly apuno v functions. Since the mass constan t m do es not affect the analysis, throughout the rest of the pap er, we assume m = 1. W e also adopt the notation ϵ = 1 c 2 . So, the relativistic kinetic energy K ( p ) from ( 1.2 ) is reduced to K ( p ) = 1 ϵ p 1 + ϵ | p | 2 . Inspired b y [ 13 ], w e now construct ϕ -Lyapuno v functions (see Definition A.1 ) for the GRLE, where ϕ ( t ) = ζ t 1 − 1 2 n for some p ositiv e in teger n and ζ > 0. W e first consider the simpler case when γ > 0. Prop osition 3.2. L et V 1 b e define d by, V 1 ( q , p, z ) = H 1 ( q , p, z ) 2 + ϵ ⟨ q , p ⟩ + κ, (3.4) wher e, H 1 ( q , p, z ) = ϵH ( q , p, z ) = ϵU ( q ) + p 1 + ϵ | p | 2 + 1 2 ϵ ∥ z ∥ 2 , (3.5) and κ is sufficiently lar ge. Then for al l n ∈ N and γ > 0 , L V n 1 ≤ − ζ V n − 1 2 1 + C, (3.6) for al l sufficiently smal l ϵ = ϵ ( n, γ , α, λ, M ) > 0 and for some p ositive c onstants ζ = ζ ( n, γ , α, λ, M ) and C = C ( ϵ, n, γ , α, λ, M , ϵ ) . In the ab ove, L denotes the infinitesimal op er ator for ( 1.7 ) , which is given by L f ( q , p, z ) = J ∇ H · ∇ f − D ∇ H · ∇ f + β − 1 div( D ∇ f ) , (3.7) wher e J and D ar e given in ( 3.3 ) with Λ = diag( λ 1 , . . . , λ M ) and A = diag( α 1 , . . . , α M ) . The form of the Ly apuno v function giv en b y ( 3.4 ) is motiv ated from the Lyapuno v approach in [ 13 ]. W e notice that, instead of the Hamiltonian as often used for the classical Langevin dynamics, we need to emplo y its square due to the w eak dissipation from the relativistic kinetic energy K ( p ). On the other hand, the idea of perturbing a function of Hamiltonian by low er order terms such as the cross term ⟨ q , p ⟩ in ( 3.4 ) is well-kno wn in the literature in the framew ork of h ypo co ercivit y metho d [ 35 ]. It is a p o w erful metho d and has b een employ ed widely when constructing Ly apuno v functions for degenerate systems where the noises act only on some of direction of the phase spaces, see for instance [ 13 , 19 , 26 ] for Langevin-t yp e dynamics. Pr o of of Pr op osition 3.2 . In the b elo w, C , ζ 1 > 0 may c hange from line to line, and ζ 1 is indep enden t of ϵ . W e first consider the case when n = 1 and apply the generator L to H 1 to obtain L H 1 = − γ ϵ | p | 2 1 + ϵ | p | 2 + γ β − 1 dϵ + ( d − 1) ϵ 2 | p | 2 (1 + ϵ | p | 2 ) 3 2 − ϵ M X i =1 α i | z i | 2 + ϵd M X i =1 α i β − 1 (3.8) ≤ − γ ϵ | p | 2 1 + ϵ | p | 2 + γ β − 1 (2 d − 1) ϵ − ϵ M X i =1 α i | z i | 2 + ϵd M X i =1 α i β − 1 . As a consequence, applying L to H 2 1 , w e get L H 2 1 =2 H 1 L H 1 + 2 γ β − 1 ϵ 2 | p | 2 1 + ϵ | p | 2 + ϵ 2 M X k =1 2 α i β − 1 | z i | 2 (3.9) ≤ 2 H 1 L H 1 + 2 γ β − 1 ϵ + ϵ 2 M X i =1 2 α k β − 1 | z i | 2 . Next, considering the cross term ⟨ q , p ⟩ , w e ha v e 10 L⟨ q , p ⟩ = ⟨ p, ∇ K ( p ) ⟩ − ⟨ q , ∇ U ( q ) ⟩ − γ ⟨ q , ∇ K ( p ) ⟩ + M X i =1 λ i ⟨ q , z i ⟩ . By applying Assumption 1.1 and Cauch y-Sc h w arz inequalit y , L⟨ q , p ⟩ ≤ | p | 2 p 1 + ϵ | p | 2 − ζ 1 | q | λ +1 + C − γ ⟨ q , p ⟩ p 1 + ϵ | p | 2 + M X i =1 λ i ⟨ q , z i ⟩ (3.10) ≤ | p | + γ | q | √ ϵ − ζ 1 | q | λ +1 + C + M X i =1 λ i ⟨ q , z i ⟩ . (3.11) So, w e hav e L ϵ ⟨ q , p ⟩ ≤ √ ϵ | p | + √ ϵγ 2 | q | + C − ϵζ 1 | q | λ +1 + M X i =1 λ i | q | 2 + ϵ 2 M X i =1 λ i | z i | 2 , ≤ √ ϵ | p | − ϵ 2 ζ 1 | q | λ +1 + ϵ 2 M X i =1 λ 1 | z i | 2 + C. Putting the ab o v e together, we obtain L  H 2 1 + ϵ ⟨ q , p ⟩  ≤ 2 H 1 − γ ϵ | p | 2 1 + ϵ | p | 2 + (2 d + 1) ϵ − ϵ M X i =1 α i | z i | 2 + ϵd M X i =1 α i β ! + √ ϵ | p | − ϵ 2 ζ 1 | q | λ +1 + ϵ 2 M X i =1 λ 1 | z i | 2 + C + 2 γ β − 1 ϵ + ϵ 2 M X i =1 2 α i β − 1 | z i | 2 . (3.12) W e ha v e the following inequality , ϵ ⟨ p, ∇ K ( p ) ⟩ = ϵ | p | 2 p 1 + ϵ | p | 2 ≥ p 1 + ϵ | p | 2 − 1 . (3.13) Using ( 3.13 ), as well as the inequalit y H 1 ≥ p 1 + ϵ | p | 2 + ϵ 2 M X i =1 | z i | 2 ≥ √ ϵ | p | 2 + 1 2 + ϵ 2 M X i =1 | z i | 2 , (3.14) w e find 2 H 1 − γ ϵ | p | 2 1 + ϵ | p | 2 + (2 d + 1) ϵ − ϵ M X i =1 α i | z i | 2 + ϵd M X i =1 α i β ! ≤ − 2 γ ϵ | p | 2 p 1 + ϵ | p | 2 + 2(2 d − 1) ϵH 1 − 2 ϵH 1 M X i =1 α i | z i | 2 + 2 ϵd M X i =1 α i β H 1 , ≤ − 2 γ p 1 + ϵ | p | 2 + 2(2 d − 1) ϵH 1 + 2 ϵd M X i =1 α i β H 1 − ϵ M X i =1 α i | z i | 2 − ϵ 2 M X i =1 α i | z i | 4 . (3.15) By com bining ( 3.12 ) and ( 3.15 ), w e get L  H 2 1 + ϵ 1 ⟨ q , p ⟩  ≤ − γ p 1 + ϵ | p | 2 + 2(2 d − 1) ϵH 1 + 2 ϵd M X i =1 α i β H 1 − ϵ 2 ζ 1 | q | λ +1 − M X i =1 α i ϵ | z i | 2 + ϵ 2 M X i =1  ( λ i + αβ − 1 ) | z i | 2 − α i | z i | 4  + C. 11 By applying Assumption 1.1 , it holds that − ϵ 2 | q | λ +1 ≤ − ϵζ 1 U ( q ) + C. Th us, c ho osin g ϵ sufficiently small, we can estimate L V 1 ≤ − ζ 1 H 1 + C, (3.16) By our definition of V 1 , w e can c ho ose ϵ > 0 and κ > 0 suc h that, ζ 1 H 2 1 − C ≤ V 1 ≤ C H 2 1 + C. (3.17) Note that V 1 is also b ounded b elo w b y 1. Using this fact, and com bining ( 3.16 ) and ( 3.17 ), L V 1 ≤ − ζ 1 p V 1 + C. (3.18) This finishes the pro of of ( 3.6 ) when n = 1. F or n > 1, we apply L to V n 1 and obtain the iden tit y L V n 1 = nV n − 1 L V 1 + n ( n − 1) V n − 2 1  γ β − 1    2 H 1 ϵp p 1 + ϵ | p | 2 + ϵq    2 + M X i =1 α i β − 1 | 2 H 1 ϵz i | 2  . Since ϵ | p | √ 1+ ϵ | p | 2 ≤ 1 for all ϵ ∈ (0 , 1) and by Assumption 1.1 , we hav e    2 H 1 ϵp p 1 + ϵ | p | 2 + ϵq    2 ≤ C H 2 1 + C. F urthermore, b y definition of H 1 , w e can estimate M X i =1 | 2 H 1 ϵz i | 2 ≤ 4 ϵH 3 1 . By applying ( 3.17 ), we hav e V 3 2 1 ≥ ζ 1 H 3 1 − C. On the other hand, b y applying ( 3.18 ), we get nV n − 1 L V 1 ≤ − ζ 1 V n − 1 2 1 + C V n − 1 1 . Putting the ab o v e together, we obtain L V n 1 ≤ − ζ 1 V n − 1 2 1 + C V n − 1 1 + n ( n − 1) V n − 2 1  C V 1 + ϵζ 1 V 3 2 1 + C  ≤ − ζ 1 − ϵζ 1 n ( n − 1) 2 V n − 1 2 1 + C. Since ζ 1 is indp enden t of ϵ , for ϵ sufficien tly small dep enden t on n ∈ N , ( 3.6 ) holds, thus completing the pro of of this prop osition. □ W e remark that the ab o v e pro of do es not hold for the case γ = 0 since the dissipation term in v olving p on the right hand side of ( 3.15 ) is then cancelled. Therefore, to find the Ly apuno v function for this case, w e will add an additional p erturbation to V 1 , as well as redefine our c hoice of scalar m ultiples for these terms. It is also imp ortan t to note that we hav e to further restrict the construction to Lipschitz ∇ U . 12 Prop osition 3.3. L et γ = 0 , and supp ose that ∇ U is Lipschitz. L et V 2 b e define d by, V 2 ( q , p, z ) = H 1 ( q , p, z ) 2 + ϵ 1 M X i =1 ⟨ p, z i ⟩ + ϵ 2 ⟨ q , p ⟩ + κ, (3.19) wher e H 1 is define d as in ( 3.5 ) , ϵ 1 = Λ 1 ϵ and ϵ 2 = Λ 2 ϵ 1 for Λ 1 , Λ 2 > 0 . Then, for al l n ∈ N , for sufficiently smal l ϵ, Λ 1 , Λ 2 > 0 , with ϵ = ϵ ( n, γ , α, λ, M ) sufficiently smal l, and Λ 1 = Λ 1 ( α, λ ) and Λ 2 = Λ 2 ( α, λ ) sufficiently smal l, indep endent of ϵ , we have L V n 2 ≤ − ζ V n − 1 2 2 + C, (3.20) for some c onstants ζ = ζ ( ϵ, n, Λ 1 , Λ 2 , M ) , C = C ( ϵ, n, Λ 1 , Λ 2 , M ) > 0 . Pr o of. In the b elo w, C, ζ 1 > 0 may c hange from line to line, and ζ 1 is indep enden t of ϵ . When γ = 0, from ( 3.8 ), the generator of ( 1.17 ) acting on H 1 is giv en by L H 1 = − ϵ M X i =1 α i | z i | 2 + ϵd M X i =1 αβ − 1 . (3.21) W e compute L H 2 1 =2 H 1 L H 1 + ϵ 2 M X i =1 2 α i β − 1 | z i | 2 . (3.22) Similar to the pro of of Prop osition 3.2 , w e first consider the case when n = 1. By combining ( 3.21 ) and ( 3.22 ), along with ( 3.14 ), w e can estimate L H 2 1 ≤ − ϵ 3 2 M X i =1 α i | p || z i | 2 − ϵ 2 M X i =1 α i | z i | 4 − ϵ M X i =1 α i | z i | 2 + ϵd M X i =1 2 α i β − 1 H 1 + ϵ 2 M X i =1 2 α i β − 1 | z i | 2 . (3.23) Concerning the fist inner product term of ( 3.19 ), w e compute L⟨ p, z i ⟩ = −⟨ λ i ∇ K ( p ) , p ⟩ − ⟨ α i z i , p ⟩ − ⟨∇ U ( q ) , z i ⟩ + ⟨ z i , M X j =1 λ j z j ⟩ , for i = 1 , ..., M . Note that the Cauch y-Sc h w arz inequalit y and the Lipsc hitz con tin uit y of ∇ U yield, − ϵ ⟨∇ U ( q ) , z i ⟩ ≤ ϵζ 1 (1 + | q | ) | z | . By applying ( 3.13 ), we get ϵ 1 L⟨ p, z i ⟩ ≤ − Λ 1 λ i p 1 + ϵ | p | 2 + Λ 1 ϵα i | z i || p | + ϵζ 1 | q | 2 + ϵa 1 | z i | + ϵ Λ 1 ζ 1 | z i | 2 + Λ 1 ϵ ⟨ z i M X i =1 λ i z i ⟩ + C . Since | z i | + | z i | 2 ≤ 2 | z i | 2 + 1 , w e find ϵ 1 M X i =1 ⟨ p, z i ⟩ ≤ − Λ 1 M X i =1 λ i p 1 + ϵ | p | 2 + Λ 1 ϵ M X i =1 α i | z i || p | + ϵζ 1 | q | 2 + ϵ Λ 1 ζ 1 M X i =1 | z i | 2 + ϵM Λ 1 M X i =1 λ i | z i | 2 + C. (3.24) 13 Applying Assumption 1.1 , and c hoosing Λ 2 ≤ 1 8 λ i for all 1 ≤ i ≤ M , w e hav e ϵ 2 L⟨ q , p ⟩ = ϵ 2 ⟨ p, ∇ K ( p ) ⟩ − ϵ 2 ⟨ q , ∇ U ( q ) ⟩ + ϵ 2 M X i =1 λ i ⟨ q , z i ⟩ ≤ ϵ 2 | p | 2 p 1 + ϵ | p | 2 − ϵ 2 ζ 1 | q | λ +1 + M X i =1 λ i | q | 2 + ϵ 2 2 M X i =1 λ i | z i | 2 + C ≤ Λ 1 8 M X i =1 λ i √ ϵ | p | − Λ 1 Λ 2 ϵζ 1 | q | λ +1 + ϵ 2 M X i =1 λ i | z i | 2 + C. (3.25) Putting together ( 3.23 ), ( 3.24 ), and ( 3.25 ), w e obtain L V 2 ≤ ϵd M X i =1 2 α i β − 1 H 1 + ϵ M X i =1 (Λ 1 M ζ 1 − α i ) | z i | 2 − ϵ 4 ζ 1 | q | λ +1 − 1 2 Λ 1 M X i =1 λ i p 1 + ϵ | p | 2 ! − 1 8 Λ 1 M X i =1 λ i √ ϵ | p | + M X i =1  ϵ Λ 1 α i | z i || p | − 1 4 Λ 1 λ i √ ϵ | p | − ϵ 3 2 α i | p || z i | 2  + ϵ 2 M X i =1  − α i | z i | 4 + λ i | z i | 2 + 2 α i β − 1 | z i | 2  +  ϵζ 2 | q | 2 − ϵ Λ 1 Λ 2 ζ 1 | q | λ +1  + C. Cho osing Λ 1 ≤ λ i α i for all 1 ≤ i ≤ M , w e can estimate ϵ Λ 1 α i | z i | − 1 4 Λ 1 λ i √ ϵ − ϵ 3 2 α i | z i | 2 ≤ 0 . Th us, b y applying Assumption 1.1 , pro ceeding similarly to the pro of of Prop osition 3.2 and choosing Λ 1 sufficien tly small, and then w e find that L V 2 ≤ − ζ H 2 1 + C. By c hoosing Λ 1 ≤ 1 2 , w e can ch o ose κ > 0 suc h that ζ 1 H 2 1 − C ≤ V 1 ≤ C 1 H 2 1 + C. (3.26) By com bining the ab o v e, we find L V 2 ≤ − ζ 1 p V 2 + C, whic h pro v es ( 3.20 ) for the case n = 1. T urning to the case n > 1, note that, L V n = nV n − 1 L V + n ( n − 1) V n − 2  M X i =1 α i β − 1 | 2 H 1 ϵz i + ϵp | 2  . The rest of the proof follows closely the pro of of Prop osition 3.2 when n > 1, and th us is omitted. □ 3.3. H¨ ormander’s and Solv ability Conditions. W e now pro v e H¨ ormander’s condition and the solv- abilit y condition of the GRLE, in order to show the GRLE satisfies the assumptions of Theorem A.4 . T o pro v e H¨ ormander’s condition, w e define the family of vector fields, X 0 = ⟨∇ q , ∇ K ( p ) ⟩ + ⟨∇ p , −∇ U ( q ) − γ ∇ K ( p ) + m X j =1 λ j z j ⟩ + m X j =1 ⟨∇ z j , − λ j ∇ K ( p ) − α j z j ⟩ , (3.27a) X q i = 0 , (3.27b) X p i = p 2 β − 1 γ δ δ p i , (3.27c) X z ki = p 2 β − 1 α k δ δ z ki . (3.27d) 14 Define the family of Lie algebras recursiv ely as A 0 = Lie ( X q i , X p i , X z ki ) i =1 ,...,d , k =1 ,...,m , A n = Lie ([ X 0 , X ]) X ∈A n − 1 , for n ∈ N , where the Lie brack ets denote [ X , Y ] = X ( Y ) − Y ( X ) . F urther w e define A = Lie( A n ) n ∈ N 0 . Then the following Prop osition holds: Prop osition 3.4. The family of ve ctor fields ( 3.27 ) satisfies H¨ ormander’s c ondition: for al l x ∈ R (2+ M ) d , the ve ctor sp ac e, sp an { X ( x ) : X ∈ A} = R (2+ M ) d . Pr o of. Direct computations giv e 1 p 2 β − 1 α k [ X 0 , X z ki ] = − λ k δ δ p i + α k δ δ z ki . Th us, δ δ p i ∈ A , for any choice of γ ≥ 0. F urthermore, w e hav e  X 0 , δ δ p i  = − (1 + ϵ | p | 2 ) δ δ q i − ϵp i P j ≤ d p j δ δ q j (1 + ϵ | p | 2 ) 3 2 ! + γ (1 + ϵ | p | 2 ) δ δ p i − ϵp i P j ≤ d p j δ δ p j (1 + ϵ | p | 2 ) 3 2 ! + m X l =1 λ l (1 + ϵ | p | 2 ) δ δ z li − ϵp i P j ≤ d p j δ δ z l j (1 + ϵ | p | 2 ) 3 2 ! . So that, for i = 1 , .., d , we ha ve 1 p 2 β − 1 α k h X 0 , h X 0 , X z k i ii = λ k (1 + ϵ | p | 2 ) δ δ q i − ϵp i P j ≤ d p j δ δ q j (1 + ϵ | p | 2 ) 3 2 ! − γ λ k (1 + ϵ | p | 2 ) δ δ p i − ϵp i P j ≤ d p j δ δ p j (1 + ϵ | p | 2 ) 3 2 ! − λ k m X l =1 λ l (1 + ϵ | p | 2 ) δ δ z li − ϵp i P j ≤ d p j δ δ z l j (1 + ϵ | p | 2 ) 3 2 ! − α k λ k δ δ p i + α 2 k δ δ z ki . F rom this, it follows that (1 + ϵ | p | 2 ) δ δ q i − ϵp i P j ≤ d p j δ δ q j (1 + ϵ | p | 2 ) 3 2 ∈ A . (3.28) F or all l ∈ 1 , ..., d , there exists a ∈ R d suc h that d X i =1 a i    1 + ϵ | p | 2  δ δ q i − ϵp i X j ≤ d p j δ δ q j   = δ δ q l . (3.29) This statemen t follows from the fact that the matrix: B =      − 1 ϵ − | p | 2 + p 2 1 p 2 p 1 · · · p d p 1 p 1 p 2 − 1 ϵ − | p | 2 + p 2 2 · · · p d p 2 . . . . . . . . . . . . p 1 p d p 2 p d · · · − 1 ϵ − | p | 2 + p 2 d ,      satisfies the conditions of Lemma B.1 , and so is in vertible. Thus we can c ho ose a as, a = − 1 ϵ B − 1 e l . W e deduce from ( 3.28 ) and ( 3.29 ) that δ δ q l ∈ A , which completes the proof. □ 15 The solv abilit y condition is stated as follows: Prop osition 3. 5. The c ontr ol pr oblem asso ciate d with ( 1.7 ) is solvable: for al l ( q 0 , p 0 , z 1 , 0 , ..., z M , 0 ) ∈ R 2+ M , ther e exists a smo oth c ol le ction of p aths U, U 1 , ..., U k , which satisfy the or dinary differ ential e quation d q ( t ) = ∇ K ( p ( t )) d t, (3.30a) d p ( t ) = −∇ U ( q ( t )) d t − γ ∇ K ( p ( t )) d t + M X i =1 λ i z i ( t ) d t + p 2 γ β − 1 d U ( t ) , (3.30b) d z i ( t ) = − λ i ∇ K ( p ( t )) d t − α i z i ( t ) d t + p 2 α i β − 1 d U i ( t ) , (3.30c) subje ct to b oundary c onditions ( q (0) , p (0) , z 1 (0) , ..., z M (0)) = ( q 0 , p 0 , z 1 , 0 , ..., z M , 0 ) , (3.31a) ( q ( T ) , p ( T ) , ..., z 1 ( T ) , ..., z M ( T )) = 0 , (3.31b) for some T ≥ 0 . Pr o of. Case 1 ( γ > 0 ): The result follo ws from Lemma 3.7 from [ 14 ]. Indeed, w e can find q ∈ C ∞ ( R ) suc h that the path of p , given b y p ( s ) = q ′ ( s ) p 1 − ϵ | q ′ ( s ) | 2 , 0 ≤ s ≤ T , (3.32) is w ell defined, that is | q ′ ( s ) | < 1 √ ϵ , 0 ≤ s ≤ T , (3.33) and that q and p satisfy initial conditions. W e then define z k ∈ C ∞ ( R ) b y z k ( s ) =      z k, 0 , 0 ≤ s < ρ, monotonicit y , ρ ≤ s ≤ T − ρ, 0 , T − ρ < s ≤ T , for some ρ > 0. W e then define U ( t ) = 1 p 2 γ β − 1 Z t 0 d p ( s ) d s + ∇ U ( q ( s )) + γ ∇ K ( p ( s )) − M X i =1 λ i z i ( s ) d s, and for i = 1 , . . . , M U i ( t ) = 1 p 2 α i β − 1 Z t 0 d z i ( s ) d s + λ i ∇ K ( p ( s )) + α i z i ( s ) d s. (3.34) Case 2 ( γ = 0 ): The con trol problem reduces to: d q ( t ) = ∇ K ( p ( t )) d t, (3.35a) d p ( t ) = −∇ U ( q ( t )) d t + M X i =1 λ i z i ( t ) d t, (3.35b) d z k ( t ) = − λ k ∇ K ( p ( t )) d t − α k z k ( t ) d t + p 2 α k β − 1 d U k ( t ) . (3.35c) 16 It suffices to c ho ose a path for q , p and z which satisfies ( 3.35a ) and ( 3.35b ). F rom ( 3.35a ) and ( 3.35b ), w e ha v e M X i =1 λ i z i ( s ) = p ′ ( s ) + ∇ U ( q ( s )) , (3.36a) p ′ ( s ) = q ′′ ( s )  1 − ϵ | q ′ ( s ) | 2  + ϵq ′ ( s ) ⟨ q ′ ( s ) , q ′′ ( s ) ⟩ (1 − ϵ | q ′ ( s ) | 2 ) 3 2 , (3.36b) for 0 ≤ s ≤ T . Th us, w e must satisfy the conditions −∇ U (0) = q ′′ ( T ) , (3.37a) q ′ ( T ) = 0 . (3.37b) q ′ (0) = ∇ K ( p (0)) . (3.37c) F or simplicity of notation, let k p := K ( p (0)). In order for p to b e well defined, we must again satisfy ( 3.33 ). There exists q ′′ 0 ∈ R d suc h that q ′′ (0) = q ′′ 0 and satisfies ( 3.36 ). Indeed, q ′′ 0 satisfies the sim ultaneous equation M X i =1 λ i z i, 0 = q ′′ 0 (1 − ϵ | k p | 2 ) + ϵk p ⟨ k p , q ′′ 0 ⟩ (1 − ϵ | k p | 2 ) 3 2 + ∇ U ( q 0 ) . (3.38) In matrix form, ( 3.38 ) is given by B q ′′ (0) = 1 ϵ  1 − ϵ | k p, 0 | 2  3 2 m X i =1 λ i z i, 0 − ∇ U ( q 0 ) ! , where B is giv en by , B =      1 ϵ − | k p | 2 + k 2 p 1 k p 1 k p 2 · · · k p 1 k p d k p 2 k p 1 1 ϵ − | k p | 2 + k 2 p 2 · · · k p 2 k p d . . . . . . . . . . . . k p n k p 1 k p n k p 2 · · · 1 ϵ − | k p | 2 + k 2 p d      . By Lemma B.1 , B is in vertible, th us w e can find initial condition q ′′ (0) for the con trol problem. Next w e seek Q ∈ C ∞ ( R ) so that q ( t ) = q 0 + Z t 0 Q ( s ) d s. W e can construct Q as follo ws: let ψ be a smo oth step function. By appropriately translating and scaling ψ , there exist ψ 0 , ψ 1 ∈ C ∞ ( R ) and t 1 , t 3 > 0 such that, for all n ∈ N , ψ 0 (0) = k p , ψ 0 ( t 1 ) = a, ψ ′ 0 (0) = q ′′ 0 , ψ ( n ) 0 ( t 1 ) = 0 , where 0 < | a | ≤ 1 √ ϵ , and, ψ 1 (0) = sgn( ∇ U (0)) | a | , ψ 1 ( t 3 ) = 0 , ψ ( n ) 1 (0) = 0 , ψ ′ 1 ( t 3 ) = −∇ U (0) . F or an y A ∈ R , and a 1 , a 2 ∈ R , there exist g ∈ C ∞ ( R ) and t 2 > 0 such that, for all n ∈ N , g (0) = a 1 , g ( t 2 ) = a 2 , g ( n ) (0) = g ( n ) ( t 2 ) = 0 , Z t 2 0 g ( s )d s = A. Let T 1 = t 1 , T 2 = t 1 + t 2 and T = t 1 + t 2 + t 3 . Cho osing a 1 = ψ 0 (0), a 2 = ψ 1 (0) and A = − q 0 − R t 1 0 ψ 0 ( s ) d s − R t 3 0 ψ 1 ( s ) d s , w e c hoose, 17 Q ( t ) =      ψ 0 ( t ) , t ∈ [0 , T 1 ] , g ( t − T 1 ) , t ∈ ( T 1 , T 2 ] , ψ 1 ( t − T 2 ) , t ∈ ( T 2 , T ] . By our c hoice of Q , w e hav e found q whic h satisfies ( 3.31 ), ( 3.33 ), and ( 3.37 ), as well as q ′′ (0) = q ′′ 0 . p and z can then b e chosen according to ( 3.32 ) and ( 3.36 ) resp ectiv ely . By choosing U k as defined in ( 3.34 ), w e ha v e found a solution to the control problem. □ 3.4. Pro of of Theorem 1.4 . Ha ving established the crucial prop erties in the previous subsections, we are no w in a position to conclude Theorem 1.4 , b y v erifying the hypothesis of Theorem A.4 . Pr o of of The or em 1.4 . By Prop ositions 3.2 and 3.3 , there exists ζ > 0 suc h that L V ( x ) n ≤ − ζ V ( x ) n − 1 2 . It follows that, for all n ∈ N , for sufficien tly small ϵ > 0, V n is a ϕ - Lyapuno v function, where ϕ ( t ) = ζ t 1 − 1 2 n . F rom Theorem A.4 , as well as Propositions 3.4 and 3.5 , it follo ws that, ∥ P t ( x, · ) − ρ β ∥ TV ≤ ζ 0 V ( x ) n ζ  t 2 nζ + 1  2 n − 1 . The result follows by choosing r = 2 n + 1. □ 4. White Noise and Newtonian Limits In this section, we pro v e Theorem 1.6 and Theorem 1.8 on the white-noise limit and Newtonian limit of the GRLE system ( 1.7 ), resp ectiv ely . This will b e completed in three steps. W e first show that the GRLE has bounded moments. W e then pro ve the limits giv en that ∇ U is Lipschitz contin uous. Finally , w e com bine these results to show that this holds generally for an y U ∈ C ∞ ( R ). This metho d of pro of is adapted from [ 14 , 27 ]. 4.1. Momen ts b ound. W e first show the b oundedness of the momen ts of the GRLE systems. Prop osition 4.1. L et ( q , p, z ) , ( q ( c ) , p ( c ) ) and ( q ( ε ) , p ( ε ) , z ( ε ) ) b e solutions of ( 1.7 ) , ( 1.15 ) , and ( 1.12 ) r esp e ctively. Then under Assumptions 1.1 and 1.5 , for al l n ≥ 1 and T > 0 , the fol lowing estimates hold E " sup t ∈ [0 ,T ] | p ( t ) | n + sup t ∈ [0 ,T ] | q ( t ) | n + M X i =1 sup t ∈ [0 ,T ] | z i ( t ) | n # < C, (4.1) E " sup t ∈ [0 ,T ]    p ( c ) ( t )    n + sup t ∈ [0 ,T ]    q ( c ) ( t )    n # < C ; (4.2) and, E " sup t ∈ [0 ,T ]    p ( ε ) ( t )    n + sup t ∈ [0 ,T ]    q ( ε ) ( t )    n + M X i =1 sup t ∈ [0 ,T ]    z ( ε ) i ( t )    n # < C, (4.3) for some c onstant C = C ( T , n ) > 0 indep endent of ε ∈ (0 , 1] and ϵ = 1 /c 2 ∈ (0 , 1] . Pr o of. W e first pro v e ( 4.3 ). Let ( q ( ε ) , p ( ε ) , z ( ε ) ) denote the solution to ( 1.12 ). In the b elo w, C is indep en- den t of c and ε and may change from line to line. By applying Itˆ o’s formula to e − α ε t z ( ε ) i ( t ), w e find from ( 1.12c ) that z ( ε ) i ( t ) = e − α i ε t z ( ε ) i (0) + Z t 0 e − ( t − s ) α i ε − λ i √ ε ∇ K ( p ( ε ) ( s )) d s + r 2 α i ε d W i ( s ) ! . (4.4) 18 Substituting this expression in to ( 1.12b ) we get p ( ε ) ( t ) = − Z t 0 ∇ U ( q ( ε ) ( s )) d t − γ Z t 0 ∇ K ( p ( ε ) ( s )) d s + p ( ε ) (0) + 1 √ ε M X i =1 λ i  Z t 0 e − α i ε s z ( ε ) i (0) d s + 1 √ ε Z t 0 Z s 0 e − ( s − u ) α i ε  − λ i ∇ K ( p ( ε ) ( u )) d u + √ 2 α i d W i ( u )  d s  + Z t 0 p 2 γ d W ( s ) . (4.5) W e will use the follo wing elemen tary identit y 1 a Z t 0  1 − e − a ( t − s )  d W ( s ) = Z t 0 Z s 0 e − a ( t − u ) d W ( u ) d s. (4.6) By application of ( 4.6 ), as w ell as F ubini’s theorem, we hav e Z t 0 Z s 0 e − ( s − u ) α i ε  − λ i ∇ K ( p ( ε ) ( u )) d u + √ 2 α i d W i ( u )  d s = ε α i Z t 0  1 − e − ( t − u ) α i ε   − λ i ∇ K ( p ( ε ) ( u )) d u + √ 2 α i d W i ( u )  . Substituting this into ( 4.5 ), w e get p ( ε ) ( t ) = − Z t 0 ∇ U ( q ( ε ) ( s )) d t − γ Z t 0 ∇ K ( p ( ε ) ( s )) d s + p ( ε ) (0) + 1 √ ε M X i =1 λ i Z t 0 e − α i ε s z ( ε ) i (0) d s + λ i α i Z t 0  1 − e − ( t − s ) α i ε   − λ i ∇ K ( p ( ε ) ( s )) d s + √ 2 α i d W i ( s )  + Z t 0 p 2 γ d W ( s ) . W e define Γ 1 ( t ) = q 1 + ϵ | p ( ε ) ( t ) | 2 U ( q ( ε ) ( t )) + 1 2 | p ( ε ) ( t ) | 2 + 1 2 ϵU ( q ( ε ) ( t )) 2 . By applying Itˆ o’s form ula, w e can write this term as Γ 1 ( t ) = Γ 1 (0)+ Z t 0 D ϵp ( ε ) ( s ) p 1 + ϵ | p ( ε ) ( s ) | 2 U ( q ( ε ) ( s )) + p ( ε ) ( s ) , γ ∇ K ( p ( ε ) ( s )) + M X i =1 λ i √ ε e − α i ε s z ( ε ) i (0) − λ 2 i α i ( e − ( t − s ) α i ε + 1) ∇ K ( p ( ε ) ( s )) E d s + Z t 0  γ + M X i =1 λ 2 i α i  1 − e − ( t − s ) α i ε  2  ϵdU ( q ( ε ) ( s )) (1 + ϵ | p ( ε ) ( s ) | 2 ) 3 2 + d  d s + Z t 0 p 2 γ D ϵp ( ε ) ( s ) U ( q ( ε ) ( s )) p 1 + ϵ | p ( ε ) ( s ) | 2 + p ( ε ) ( s ) , d W ( s ) E + M X i =1 Z t 0 s 2 λ 2 i α i  e ( t − s ) α i ε + 1  D ϵp ( ε ) ( s ) U ( q ( ε ) ( s )) p 1 + ϵ | p ( ε ) ( s ) | 2 + p ( ε ) ( s ) , d W i ( s ) E . 19 Using the following estimates Z t 0 D ϵp ( ε ) ( s ) p 1 + ϵ | p ( ε ) ( s ) | 2 U ( q ( ε ) ( s )) + p ( ε ) ( s ) , M X i =1 λ i √ ε e − α i ε s z ( ε ) i (0) E d s ≤ Z t 0    ϵp ( ε ) ( s ) p 1 + ϵ | p ( ε ) ( s ) | 2 U ( q ( ε ) ( s )) + p ( ε ) ( s )    2 + M X i =1    λ i √ ε e − α i ε s z ( ε ) i (0)    2 d s ≤ Z t 0    ϵp ( ε ) ( s ) p 1 + ϵ | p ( ε ) ( s ) | 2 U ( q ( ε ) ( s )) + p ( ε ) ( s )    2 d s + M X i =1    z ( ε ) i (0)    2 Z t 0 λ 2 i ε e − 2 α ε d s ≤ Z t 0    ϵp ( ε ) ( s ) p 1 + ϵ | p ( ε ) ( s ) | 2 U ( q ( ε ) ( s )) + p ( ε ) ( s )    2 d s + C M X i =1    z ( ε ) i (0)    2 , w e ha v e Z t 0 D ϵp ( ε ) ( s ) p 1 + ϵ | p ( ε ) ( s ) | 2 U ( q ( ε ) ( s )) + p ( ε ) ( s ) , γ ∇ K ( p ( ε ) ( s )) + M X i =1 λ i √ ε e − α i ε s z ( ε ) i (0) − λ 2 i α i (1 − e − ( t − s ) α i ε ) ∇ K ( p ( ε ) ( s )) E d s ≤ Z t 0 2    ϵp p 1 + ϵ | p ( ε ) | 2 U ( q ( ε ) ( s )) + p ( ε ) ( s )    2 + C |∇ K ( p ( ε ) ( s )) | 2 d s + C M X i =1    z ( ε ) i (0)    2 ≤ Z t 0 C ϵU ( q ( ε ) ( s )) 2 + C    p ( ε ) ( s )    2 d s + C M X i =1    z ( ε ) i (0)    2 . F urthermore, it holds that Z t 0  γ + M X i =1 λ 2 i α i  1 − e − ( t − s ) α i ε  2  ϵdU ( q ( ε ) ( s )) (1 + ϵ | p ( ε ) ( s ) | 2 ) 3 2 + d  d s ≤ Z t 0 C q 1 + ϵ | p ( ε ) ( s ) | 2 U ( q ( s )) d s. Th us, w e obtain E h sup t ∈ [0 ,T ] | Γ( t ) | n i ≤ E [ | Γ(0) | n ] + C M X i =1    z ( ε ) i (0)    2 + Z T 0 C E h sup s ∈ [0 ,t ] | Γ( s ) | n i d t + C E h sup t ∈ [0 ,T ]    Z t 0 D ϵp ( ε ) ( s ) U ( q ( ε ) ( s )) p 1 + ϵ | p ( ε ) ( s ) | 2 + p ( ε ) ( s ) , d W ( s ) E    n + C M X i =1 sup t ∈ [0 ,T ]    Z t 0  1 − e − ( t − s ) α i ε  D ϵp ( ε ) ( s ) U ( q ( ε ) ( s )) p 1 + ϵ | p ( ε ) ( s ) | 2 + p ( ε ) ( s ) , d W i ( s ) E    n i . Through application of the Burkholder-Da vis-Gundy Inequality , for j ∈ { 0 , 1 } , w e hav e the following estimates E h sup t ∈ [0 ,T ]    Z t 0  1 − e − ( t − s ) α i ε  j D ϵp ( ε ) ( s ) U ( q ( ( ε ) s )) p 1 + ϵ | p ( ε ) ( s ) | 2 + p ( ε ) ( s ) , d W ( s ) E    n i ≤ C E h Z T 0  1 − e − ( t − s ) α i ε  2 j ϵ 2 | p ( ε ) ( s ) | 2 U ( q ( ε ) ( s )) 2 1 + ϵ | p ( ε ) ( s ) | 2 + | p ( ε ) ( s ) | 2 d s  n 2 i ≤ C E h Z T 0 ϵ 2 | p ( ε ) ( s ) | 2 U ( q ( ε ) ( s )) 2 1 + ϵ | p ( ε ) ( s ) | 2 + | p ( ε ) ( s ) | 2 d s  n i + C ≤ C Z T 0 E h sup s ∈ [0 ,t ] ϵ n U ( q ( ε ) ( s )) 2 n i d t + C Z T 0 E h sup s ∈ [0 ,t ]    p ( ε ) ( s )    2 n i d t + C . 20 Putting the ab o v e together, we get E h sup t ∈ [0 ,T ] | Γ 1 ( t ) | n i ≤ E [ | Γ 1 (0) | n ] + C M X i =1 E     z ( ε ) i (0)    2 n  + Z T 0 C E h sup s ∈ [0 ,t ] | Γ 1 ( s ) | n i d t + C . By applying Gronw all’s inequality , we get E " sup t ∈ [0 ,T ] | Γ 1 ( t ) | n # ≤ E [ | Γ 1 (0) | n ] + C M X i =1 E     z ( ε ) i (0)    2 n  + C. (4.7) F rom Assumption 1.5 , E [ | Γ(0) | n ] < C . T ogether with ( 4.7 ), we find E " sup t ∈ [0 ,T ] | q ( t ) | n + | p ( t ) | n # ≤ C. It follo ws from ( 4.4 ) that, for 1 ≤ i ≤ M , E " sup t ∈ [0 ,T ] | z ( ε ) i ( t ) | n # ≤ C E h | z ( ε ) i (0) | n i + E " sup t ∈ [0 ,T ] | p ( ε ) ( t ) | n # + C ! . (4.8) The result ( 4.3 ) clearly follo ws. Concerning ( 4.1 )-( 4.2 ), on the one hand, we note that the pro of of ( 4.2 ) is the same as that of [ 13 , Lemma 4.4]. On the other hand, estimate ( 4.1 ) can b e derived using an argument to similar to the ab o v e. Inspired b y of[ 27 , Prop osition 13], w e c hoose Γ 2 ( t ) = 1 2 | p ( ε ) ( t ) | 2 + U ( q ( ε ) ( t )) . By applying It¨ o’s form ula, Γ 2 ( t ) = Γ 2 (0) − Z t 0    p ( ε ) ( s )    2 d s − Z t 0 D p ( ε ) ( s ) , M X i =1 λ i √ ε e − α i ε s z ( ε ) i (0) − λ 2 i α i ( e − ( t − s ) α i ε + 1) p ( ε ) ( s ) E d s + Z t 0 d  γ + M X i =1 λ 2 i α i  1 − e − ( t − s ) α i ε  2  d s + Z t 0 p 2 γ D p ( ε ) ( s ) , d W ( s ) E + M X i =1 Z t 0 s 2 λ 2 i α i  1 − e ( t − s ) α i ε  D p ( ε ) ( s ) , d W i ( s ) E . By modifying the abov e pro of to the classical setting, we obtain E h sup t ∈ [0 ,T ] | Γ 2 ( t ) | n i ≤ E [ | Γ 2 (0) | n ] + C M X i =1 E     z ( ε ) i (0)    2 n  + Z T 0 C E h sup s ∈ [0 ,t ] | Γ 2 ( s ) | n i d t + C . The result then follo ws using the Gronw all-type argument as abov e. □ 4.2. White-noise limit. Owing to the presence of the nonlinearity , w e will not directly establish The- orem 1.6 on the white noise limit of ( 1.12 ). Instead, we will prov e an analogue of Theorem 1.6 while assuming further that ∇ U is Lipsc hitz. Then, we will remov e such a restriction b y leveraging the uniform momen t bounds from Prop osition 4.1 . More sp ecifically , we hav e the follo wing auxiliary result, whose argumen t is similar to [ 28 , Theorem 2.6] adapted to the relativistic setting. 21 Prop osition 4.2. L et  q ( ε ) , p ( ε ) , z ( ε )  b e the solution to ( 1.12 ) . Then, under Assumptions 1.1 and 1.5 , and that ∇ U is Lipschitz c ontinuous, for al l T > 0 and n ∈ N , it holds that E h sup t ∈ [0 ,T ] | q ( ε ) ( t ) − Q ( t ) | n + sup t ∈ [0 ,T ] | p ( ε ) ( t ) − P ( t ) | n i ≤ C ε n 2 , (4.9) for some c onstant C = C ( T , λ, α, n, M ) > 0 . In the ab ove, ( Q, P ) is define d as in ( 1.14 ) . Pr o of. In the b elo w, C is indep enden t of c and ε and ma y change from line to line. Recall that w e assume q ( ε ) and p ( ε ) ha v e the same initial conditions of Q and P resp ectiv ely . Pro ceeding similarly to [ 28 ], from ( 1.12a ), and the Lipsc hitz con tin uit y of ∇ K ( p ), w e ha v e    q ( ε ) ( t ) − Q ( t )    ≤ C Z t 0    p ( ε ) ( s ) − P ( s )    d s. (4.10) F rom ( 1.12c ), w e get 1 √ ε z ( ε ) i ( s )d s = − √ ε α i  z ( ε ) i ( t ) − z ( ε ) i (0)  − λ i α i Z t 0 ∇ K ( p ( ε ) ( s ))d s + r 2 α i d W i ( t ) . (4.11) Next from ( 1.12b ) and ( 1.14b ), we hav e p ( ε ) ( t ) − P ( t ) = Z t 0  ∇ U ( Q ( s )) − ∇ U ( q ( ε ) ( s ))  d s − M X i =1 √ ε  λ i α i  z ( ε ) i ( t ) − z ( ε, i (0)   +  γ + M X i =1 λ 2 i α i  Z t 0  ∇ K ( P ( s )) − ∇ K ( p ( ε ) ( s ))  d s. F or notational con v enience, we define χ 1 ( T ) = E h sup t ∈ [0 ,T ]     q ( ε ) ( t ) − Q ( t )    n +    p ( ε ) ( t ) − P ( t )    n  i . By the Lipschitz contin uit y of ∇ U and H¨ older’s inequality , w e hav e χ ‘ ( T ) ≤ C T n − 1 Z T 0 E " sup s ∈ [0 ,t ] | q ( ε ) ( s ) − Q ( s ) | n # d t + C γ + M X i =1 λ 2 i α i ! T n − 1 Z T 0 E " sup s ∈ [0 ,t ] | p ( ε ) ( s ) − P ( s ) | n # d t + C ε n 2 M X i =1  λ i α i  n E " sup t ∈ [0 ,T ] | z ( ε ) i ( t ) − z ( ε ) i (0) | n # . F rom this w e deduce χ 1 ( T ) ≤ C Z T 0 χ 1 ( t )d t + C ε n 2 M X i =1 E " sup t ∈ [0 ,T ] | z ( ε ) i ( t ) − z ( ε ) i (0) | n # . By applying Gronw all’s lemma, w e obtain χ 1 ( T ) ≤ C ε n 2 M X i =1 E h sup t ∈ [0 ,T ] | z ( ε ) i ( t ) − z ( ε ) i (0) | n i + C ε n 2 Z T 0 M X i =1 E h sup s ∈ [0 ,t ] | z ( ε ) i ( s ) − z ( ε ) i (0) | n i d t, whence χ 1 ( T ) ≤ C ε n 2 M X i =1 E h sup t ∈ [0 ,T ] | z ( ε ) i ( t ) − z ( ε ) i (0) | n i . 22 In ligh t of Prop osition 4.1 together with 1.5 , the supremum on the ab o v e right hand side is uniformly b ounded indep enden t of ε . This establishes ( 4.9 ), as claimed. □ W e now presen t the pro of of Theorem 1.6 , whose argumen t will employ the results from Prop ositions 4.1 and 4.2 . Pr o of of The or em 1.6 . F or R > 0, w e define the follo wing stopping times τ R = inf n T : sup t ∈ [0 ,T ] | q ( ε ) ( t ) | > R o ; and τ ε R = inf n T : sup t ∈ [0 ,T ] | Q ( t ) | > R o . W e also define a cut-off function by θ R ( x ) = ( 1 , | x | ≤ R, 0 , | x | ≥ R + 1 . Using this cut-off function, w e next introduce the cut-off system for the GRLE dq ( R,ε ) ( t ) = ∇ K ( p ( R,ε ) ( t )) d t, dp ( R,ε ) ( t ) = −∇ θ R ( q ( t )) U ( q ( R,ε ) ( t )) d t − γ ∇ K ( p ( R,ε ) ( t )) d t + 1 √ ε M X i =1 λ i z ( R,ε ) i ( t ) d t + p 2 γ d W ( t ) , dz ( R,ε ) i ( t ) = − λ i ε ∇ K ( p ( R,ε ) ( t )) d t − α i ε z ( R,ε ) i ( t ) d t + r 2 α i ε d W i ( t ) , q ( R,ε ) (0) = q 0 , p ( R,ε ) (0) = p 0 , z ( R,ε ) i (0) = z i, 0 , and the corresp onding cut-off system for the relativistic underdamp ed Langevin dynamics: d Q ( R ) ( t ) = ∇ K ( P ( R ) ( t )) d t, d P ( R ) ( t ) = −∇ U ( Q ( R ) ( t )) d t − γ + M X i =1 λ 2 i α i ! ∇ K ( P ( R ) ( t )) d t + p 2 β − 1 γ dW ( t ) + M X i =1 s 2 β − 1 λ 2 i α i d W i ( t ) , Q (0) = q 0 , P (0) = p 0 . By conditioning on the ev en ts { τ R < T } and { τ ε R < T } w e ha v e E h sup t ∈ [0 ,T ]    q ( ε ) ( t ) − Q ( t )    n + sup t ∈ [0 ,T ]    p ( ε ) ( t ) − P ( t )    n i ≤ E " sup t ∈ [0 ,T ]    q ( R,ε ) ( t ) − Q ( R ) ( t )    n + sup t ∈ [0 ,T ]    p ( R,ε ) ( t ) − P ( R ) ( t )    n # + E " sup t ∈ [0 ,T ]    q ( ε ) ( t ) − Q ( t )    n + sup t ∈ [0 ,T ]    p ( ε ) ( t ) − P ( t )    n ! 1 { τ R > T } # + E " sup t ∈ [0 ,T ]    q ( ε ) ( t ) − Q ( t )    n + sup t ∈ [0 ,T ]    p ( ε ) ( t ) − P ( t )    n ! 1 { τ ϵ R > T } # . By Proposition 4.2 , w e hav e E " sup t ∈ [0 ,T ]    q ( R,ε ) ( t ) − Q ( R ) ( t )    n + sup t ∈ [0 ,T ]    p ( R,ε ) ( t ) − P ( R ) ( t )    n # ≤ C ε n 2 . 23 W e emphasize that the ab o v e constant C = C ( R ) does not dep end on ε . F rom H¨ older’s inequalit y , we can estimate E " sup t ∈ [0 ,T ]    q ( ε ) ( t ) − Q ( t )    n + sup t ∈ [0 ,T ]    p ( ε ) ( t ) − P ( t )    n ! 1 { τ R > T } # ≤ E " sup t ∈ [0 ,T ]    q ( ε ) ( t ) − Q ( t )    2 n + sup t ∈ [0 ,T ]    p ( ε ) ( t ) − P ( t )    2 n # 1 2 P [ τ R > T ] 1 2 . By Mark o v’s Inequality , and Prop osition 4.1 , we hav e P [ τ R > T ] ≤ E h sup t ∈ [0 ,T ] | Q ( t ) | i R ≤ C R → 0 , R → ∞ . By Proposition 4.1 , w e hav e E " sup t ∈ [0 ,T ]    q ( ε ) ( t ) − Q ( t )    2 n + sup t ∈ [0 ,T ]    p ( ε ) ( t ) − P ( t )    2 n # 1 2 ≤ E " sup t ∈ [0 ,T ]    q ( ε ) ( t )    2 n + sup t ∈ [0 ,T ] | Q ( t ) | 2 n + sup t ∈ [0 ,T ]    p ( ε ) ( t )    2 n + sup t ∈ [0 ,T ] | P ( t ) | 2 n # < C. Th us, it holds that E h sup t ∈ [0 ,T ]    q ( ε ) ( t ) − Q ( t )    n + sup t ∈ [0 ,T ]    p ( ε ) ( t ) − P ( t )    n  1 { τ R < T } i ≤ ˜ C √ R , where the p ositiv e constan t ˜ C is independent of b oth ε and R . By a similar argument, we also obtain E h sup t ∈ [0 ,T ]    q ( ε ) ( t ) − q ( t )    n + sup t ∈ [0 ,T ]    p ( ε ) ( t ) − p ( t )    n  1 { τ ϵ R < T } i ≤ ˜ C √ R . Altogether, w e get E h sup t ∈ [0 ,T ]    q ( ε ) ( t ) − Q ( t )    n + sup t ∈ [0 ,T ]    p ( ε ) ( t ) − P ( t )    n i ≤ C ( R ) ε n 2 + ˜ C √ R . The result follo ws by sending R > 0 to infinity and then shrinking ε to zero. The pro of is th us finished. □ 4.3. Newtonian limit. In this section, w e prov e Theorem 1.8 giving the v alidity of the appro ximation of the GRLE ( 1.7 ) by the GLE ( 1.17 ) in the Newtonian limit as the light speed parameter c tends to infinit y . Thanks to Proposition 4.1 , the approach of Theorem 1.8 is similar to that of the white noise regime presented in Section 4.2 . More sp ecifically , w e pro ceed to establish limit ( 1.16 ) under the extra condition that ∇ U is Lipschitz. This is demonstrated through Prop osition 4.3 b elow. Then, the uniformit y with resp ect to c on the energy estimates from Prop osition 4.1 will allo w us to ov ercome the Lipschitz restriction, thereb y recov ering limit ( 1.16 ) for the general class of p oten tials U as in Assumption 1.1 . Prop osition 4.3. L et ( q ( c ) , p ( c ) , z ( c ) ) b e the solution to ( 1.7 ) . Then, under Assumptions 1.1 and 1.5 , and that ∇ U is Lipschitz c ontinuous, for al l T > 0 and n ∈ N , the fol lowing holds E " sup t ∈ [0 ,T ]    q ( c ) ( t ) − q ( t )    n + sup t ∈ [0 ,T ]    p ( c ) ( t ) − p ( t )    n + M X i =1 sup t ∈ [0 ,T ]    z ( c ) i ( t ) − z i ( t )    n # ≤ C ϵ n 2 , (4.12) for some c onstant C = C ( U, T , λ, α, n, M ) > 0 indep endent of ϵ = 1 /c 2 . In the ab ove, ( q , p, z ) is the pr o c ess solving the non-r elativistic gener alize d L angevin dynamics ( 1.17 ) . 24 Pr o of. In the b elo w, C is indep enden t of c and may change from line to line. Recall that w e assume q ( c ) , p ( c ) and z ( c ) ha v e the same initial conditions of q , p and z resp ectively . Note that for all p ∈ R d , we hav e      p p 1 + ϵ | p | 2 − p      = ϵ | p | 3  1 + p 1 + ϵ | p | 2  p 1 + ϵ | p | 2 ≤ √ ϵ | p | 2 . By the triangle inequalit y , it follo ws that |∇ K ( p ( c ) ( t )) − p ( t ) | =    p ( c ) ( t ) p 1 + ϵ | p ( c ) ( t ) | 2 − p ( t )    ≤    p ( c ) ( t ) p 1 + ϵ | p ( c ) ( t ) | 2 − p ( c ) ( t )    + | p ( c ) ( t ) − p ( t ) | ≤ | p ( c ) ( t ) − p ( t ) | + √ ϵ | p ( c ) ( t ) | 2 . (4.13) By ( 4.13 ), ( 1.7a ) and ( 1.17a ), w e hav e | q ( c ) ( t ) − q ( t ) | =     Z t 0 ∇ K ( p ( c ) ( s )) − p ( s ) d s     ≤ C Z t 0 ( √ ϵ | p ( c ) ( s ) | 2 + | p ( c ) ( s ) − p ( s ) | ) d s. (4.14) F rom ( 1.17c ) and ( 1.7c ), w e get Z t 0 z ( c ) k ( s ) − z k ( s )d s = 1 α i ( z k ( t ) − z k (0)) − 1 α i ( z K ( t ) − z K (0)) − λ i α i Z t 0  ∇ K ( p ( c ) ( s )) − p ( s )  d s. Also, from ( 1.17b ) and ( 1.7b ), we hav e p ( c ) ( t ) − p ( t ) = Z t 0 ∇ U ( q ( s )) − ∇ U ( q ( c ) ( s )) d s + Z t 0 p ( s ) − p ( c ) ( s ) p 1 + ϵ | p ( c ) ( s ) | 2 d t + M X i =1 λ i Z t 0 z ( c ) i ( s ) − z i ( s ) d s. (4.15) W e define χ 2 ( T ) = E h sup t ∈ [0 ,T ]     q ( c ) ( t ) − q ( t )    n +    p ( c ) ( t ) − p ( t )    n + M X i =1    z ( c ) ( t ) − z ( t )    n i . Applying the Lipschitz contin uit y of ∇ K and ∇ U , as w ell as ( 4.13 ), w e can estimate χ 2 ( T ) as follo ws: χ 2 ( T ) ≤ Z T 0 E h sup s ∈ [0 ,t ] | q ( c ) ( s ) − q ( s ) | n i d t + C  1 + M X i =1 λ 2 i α i  T n − 1 Z T 0 E h sup s ∈ [0 ,t ] | p ( c ) ( s ) − p ( s ) | n i d t + C ϵ n 2  1 + M X i =1 λ 2 i α i  T n − 1 Z T 0 E h sup s ∈ [0 ,t ] | p ( c ) ( s ) | 2 n i d t + C ϵ n 2 M X i =1  λ i α i  n E h sup t ∈ [0 ,T ] | z ( c ) i ( t ) − z ( c ) i (0) | n i + C ϵ n 2 M X i =1  λ i α i  n E h sup t ∈ [0 ,T ] | z i ( t ) − z i (0) | n i + M X i =1 α i T n − 1 Z T 0 E h sup s ∈ [0 ,t ] | z ( c ) i ( s ) − z i ( s ) | n i d s. 25 In view of Prop osition 4.1 , w e deduce that χ 2 ( T ) ≤ C Z T 0 χ 2 ( t ) d t + C ϵ n 2  M X i =1 E h sup t ∈ [0 ,T ] | z ( c ) i ( t ) − z ( c ) i (0) | n i + E h sup t ∈ [0 ,T ] | z i ( t ) − z i (0) | n i + 1  . An application of Gron w all’s lemma pro duces the b ound χ 2 ( T ) ≤ C ϵ n 2  M X i =1 E h sup t ∈ [0 ,T ] | z ( c ) i ( t ) − z ( c ) i (0) | n i + M X i =1 E h sup t ∈ [0 ,T ] | z i ( t ) − z i (0) | n i + 1  + C ϵ n 2 Z T 0  M X i =1 E h sup s ∈ [0 ,t ] | z ( c ) i ( s ) − z ( c ) i (0) | n i + M X i =1 E h sup s ∈ [0 ,t ] | z i ( s ) − z i (0) | n i + 1  d t, whence χ 2 ( T ) ≤ C ϵ n 2  M X i =1 E h sup t ∈ [0 ,T ] | z ( c ) i ( t ) − z ( c ) i (0) | n i + M X i =1 E h sup t ∈ [0 ,T ] | z i ( t ) − z i (0) | n i + 1  , where C = C ( T ) is indep enden t of ϵ . W e inv ok e Prop osition 4.1 once again to deduce estimate ( 4.12 ), as claimed. □ Lastly , let us conclude Theorem 1.8 by combining the auxiliary results from Prop ositions 4.1 and 4.3 . Pr o of of The or em 1.8 . Through applications of Prop ositions 4.1 , 4.3 , the pro of of Theorem 1.8 is can b e carried out by making use of suitable stopping times of exiting b ounded sets. Since the argument is similar to that of Theorem 1.6 , its detail is th us omitted. □ Appendix A. Ergodicity for a general SDE In this appendix, we briefly review the framework of [ 19 ] for obtaining rate of conv ergence to the equilibrium for a general sto c hastic differen tial equation. W e ha ve employ ed this approach to establish Theorem 1.4 giving p olynomial mixing for the GRLE ( 1.7 ). W e denote X to b e the solution of the Itˆ o SDE: d X ( t ) = a ( X ( t )) d t + σ ( X ( t )) d W ( t ) , X (0) = x 0 ∈ R d . (A.1) Let L denote the infinitesimal generator for X . F ollo wing [ 19 , Section 3.2], w e recall the notion of ϕ -Ly apuno v functions: Definition A.1. Giv en ϕ : R + → R + , a function V ∈ C 2 ( R d ; R + ) is called ϕ -Lyapunov if L V ( x ) ≤ − ϕ ( V ( x )) , outside some compact set, and lim | x |→∞ V ( x ) = ∞ . W e in tro duce the notation: X 0 = d X i =1 a i ( x ) δ δ x i , X k = k X i =1 σ ik ( x ) δ δ x i . W e also define the v ector fields recursively as: A 0 = Lie ( X i ) i =1 ,...,d , A n = Lie ([ X 0 , X ]) X ∈A n − 1 , and define A = Lie( A n ) n ∈ N 0 . Assumption A.2 ([ 29 ], Assumption 6.3) . F or al l x ∈ R d , the fol lowing identity holds sp an { X ( x ) : X ∈ A} = R d . 26 In the literature, Assumption A.2 is referred to as H¨ ormander’s condition, which guaran tees that the solution to ( A.1 ) admits a smo oth probability density function. Assumption A.3. F or any initial c ondition X (0) ∈ R d , ther e exists a p ath U , which satisfies the or dinary differ ential e quation, d X ( t ) = a ( X ( t )) d t + σ ( X ( t )) d U ( t ) . Assumption A.3 ensures that the origin is reac hable given any starting p oin t. Theorem A.4. [ 19 , The or em 3.5] Supp ose ther e exists an incr e asing, smo oth, strictly subline ar ϕ - Lyapunov function V , and that Assumptions A.2 and A.3 hold. L et ( P t ( x, · )) t ≥ 0 denote the tr ansition pr ob ability of X given in ( A.1 ) . Then ther e exist a unique invariant me asur e π and c onstant ζ > 0 such that ∥ P t ( x, · ) − π ∥ TV ≤ ζ 0 V ( x ) ψ ( t ) , wher e ψ ( t ) = 1 ϕ ◦ H − 1 ϕ ( t ) , and H ϕ ( t ) = Z t 1 1 ϕ ( s ) d s. Appendix B. A Non-Singular Ma trix In this app endix, through Lemma B.1 b elo w, w e pro vide sufficien t conditions guaranteeing the non- singularit y of certain matrices. The result of whic h is used in the proof of Prop ositions 3.4 and 3.5 that are emplo y ed to construct Ly apuno v functions for ( 1.7 ). Lemma B.1. L et B = ( a ij ) 1 ≤ i,j ≤ n b e a n × n matrix b e given in the form, A =      λ + b 2 1 b 1 b 2 · · · b 1 b n b 1 b 2 λ + b 2 2 · · · b 2 b n . . . . . . . . . . . . b 1 b n b 2 b n · · · λ + b 2 n      wher e b = ( b 1 , . . . , b n ) ∈ R n and λ ∈ R \ { 0 } . Supp ose further that λ + n X i =1 b 2 i  = 0 . (B.1) Then B is non-singular. Pr o of. W e decompose B as follows: B = λ I n +      b 1 0 · · · 0 0 b 2 · · · 0 . . . . . . . . . . . . 0 0 · · · b n           b 1 b 2 · · · b n b 1 b 2 · · · b n . . . . . . . . . . . . b 1 b 2 · · · b n      . By applying the W einstein-Aronsza jn identit y (see [ 31 , p.271]), we hav e det B = det B 1 where B 1 =      λ + b 2 1 b 2 2 · · · b 2 n b 2 1 λ + b 2 2 · · · b 2 n . . . . . . . . . . . . b 2 1 b 2 2 · · · λ + b 2 n      . By subtracting the first ro w from the other rows, we also ha v e det B = det B 2 27 where B 2 =      λ + b 2 1 b 2 2 · · · b 2 n − λ λ · · · 0 . . . . . . . . . . . . − λ 0 · · · λ      . (B.2) W e claim that by virtue of ( B.1 ), B 2 is non-singular. Indeed, observe that if a ∈ R n satisfies B 2 a = 0 , then, from the the second row to the n -th row of B 2 , we get a 1 = ... = a n . It follows that the first row, together with ( B.1 ), implies that a 1 = ... = a n = 0. 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