A CLT for degenerate diffusions with periodic coefficients, and application to homogenisation of linear PDEs
AA CLT FOR DEGENERATE DIFFUSIONS WITH PERIODIC COEFFICIENTS,AND APPLICATION TO HOMOGENIZATION OF LINEAR PDEs
NIKOLA SANDRIĆ AND IVANA VALENTIĆA
BSTRACT . In this article, we obtain a functional CLT for a class of degenerate diffusion pro-cesses with periodic coefficients, thus generalizing the already classical results in the context ofuniformly elliptic diffusions. As an application, we also discuss periodic homogenization of aclass of linear degenerate elliptic and parabolic PDEs.
1. I
NTRODUCTION
Let 𝜀 , 𝜀 > , be a second-order elliptic differential operator of the form(1.1) 𝜀 = 2 −1 Tr ( 𝑎 ( ⋅ ∕ 𝜀 ) ∇∇ T ) + ( 𝜀 −1 𝑏 ( ⋅ ∕ 𝜀 ) + 𝑐 ( ⋅ ∕ 𝜀 ) ) T ∇ . The main goal of this article is to discuss periodic homogenization (that is, asymptotic behaviorof the solution as 𝜀 → ) of the associated elliptic boundary-value problem(1.2) 𝜀 𝑢 𝜀 ( 𝑥 ) + 𝑒 ( 𝑥 ∕ 𝜀 ) 𝑢 𝜀 ( 𝑥 ) + 𝑓 ( 𝑥 ) = 0 , 𝑥 ∈ 𝒟 ,𝑢 𝜀 ( 𝑥 ) = 𝑔 ( 𝑥 ) , 𝑥 ∈ 𝜕 𝒟 , as well as the parabolic initial-value problem(1.3) 𝜕 𝑡 𝑢 𝜀 ( 𝑥, 𝑡 ) = 𝜀 𝑢 𝜀 ( 𝑥, 𝑡 ) + ( 𝜀 −1 𝑑 ( 𝑥 ∕ 𝜀 ) + 𝑒 ( 𝑥 ∕ 𝜀 ) ) 𝑢 𝜀 ( 𝑥, 𝑡 ) + 𝑓 ( 𝑥 ) 𝑢 𝜀 ( 𝑥,
0) = 𝑔 ( 𝑥 ) , 𝑥 ∈ (cid:82) n , in the case of degenerate (possibly vanishing on a set of positive Lebesgue measure) diffusioncoefficient 𝑎 ( 𝑥 ) . Our approach to this problem relies on probabilistic techniques: we first showthat the (appropriately centered) diffusion process associated to 𝜀 satisfies a functional CLTwith Brownian limit as 𝜀 → (see Theorems 3.1 and 3.5), and then by employing probabilis-tic representation (the Feynman-Kac formula) of the (viscosity) solutions to the problems ineqs. (1.2) and (1.3) obtained in [32, Chapter 3] we conclude the homogenization result (seeTheorems 4.2 and 4.3). This idea goes back to M. I. Fre˘ıdlin [15] (see also [3, Chapter 3]). Inthe non-degenerate (uniformly elliptic) case these steps can be carried out by combining classi-cal PDE results (existence of a smooth solution to the corresponding Poisson equation) and thefact that the underlying diffusion process does not show a singular behavior in its motion, that is,it is irreducible (see [3, Chapter 3] and [5] for a detailed exposition). In the case of a degeneratediffusion part, this deficiency is compensated by the assumption that the underlying diffusionprocess with positive probability reaches the part of the state space where the diffusion term isnon-degenerate (see assumption (A3)). In oder words, this condition ensures irreducibility ofthe process (see Section 2 for details). Also, in this case it is not clear that we can rely on PDEtechniques therefore the analysis of a solution to the corresponding Poisson equation is com-pletely based on stochastic analysis tools, in particular Motoo’s theorem [8, Proposition 3.56](see Section 3). Mathematics Subject Classification.
Key words and phrases. characteristics of semimartingale, degenerate diffusion process, Feynman-Kac formula,periodic homogenization. a r X i v : . [ m a t h . P R ] F e b ERIODIC HOMOGENIZATION OF LINEAR DEGENERATE PDEs 2
Literature review.
Our work contributes to the classical theory of periodic homogeniza-tion. Most of the existing literature on this subject focuses on the problem of homogenization ofnon-degenerate PDEs; for instance, see the classical monographs [1], [3], [21] and [39]. How-ever, in the recent years there have been developments in understanding the homogenization ofdegenerate PDEs. We refer the readers to [10], [34], [35] and [36] for a PDE approach to thisproblem, and [9], [11], [37] and [38] for a probabilistic approach. However, in all these worksthe major limitation is that the diffusion term can fully degenerate (vanish) on a “small” part ofthe domain only. In the first five references it is allowed that it vanishes on a set of Lebesguemeasure zero only and in the rest of the domain it must have a full rank. While in [11], [37] and[38] it is allowed that it degenerates everywhere, but its rank must be greater than or equal toone except maybe on a set of Lebesgue measure zero. In the present work we partly fill this gapand focus on the case when the diffusion part vanishes on a set of positive Lebesgue measure.In the closely related article [18] (see also [31] and [33] in the context of semilinear elliptic andparabolic PDEs), by also employing probabilistic methods, the authors are concerned with thesame questions we discuss in this article. However, unfortunately, there seems to be a doubtabout their proof of the functional CLT in [18, Theorem 3.1] (see Section 3.2 for details). In thisarticle, under slightly weaker assumptions (and by employing different techniques) we resolvethis issue, or at least suggest an alternative approach to the problem. We distinguish two cases:(i) 𝑐 ( 𝑥 ) ≡ , and (ii) 𝑐 ( 𝑥 ) ≢ . In the first case, in Theorem 3.1 we obtain the functional CLTunder the assumptions in (A1)-(A3). Here, (A1) should be compared to (H.1) from [18], and(A2)-(A3) to (H.2). Note that the condition in (H.2) assumes existence of a fixed time 𝑡 > suchthat for every 𝜀 ∈ [0 , 𝜀 ] , for some 𝜀 > , the corresponding diffusion process observed at 𝑡 iswith positive probability in a part of the state space where the diffusion term is non-degenerate,which is a slightly stronger assumption than (A3). Observe also that in addition to (H.1)-(H.2)the authors in [18] assume a regularizing condition (H.3), which we do not require in this case.On the other hand, the case 𝑐 ( 𝑥 ) ≢ is technically more delicate. Namely, an analogous analysisas in Theorem 3.1 cannot be performed in this situation. To overcome this difficulty we adoptassumption (H.3) from [18] (see (A4)), which allows us to conclude an Itô-type formula for thediffusion process associated to the operator −1 Tr( 𝑎 ( ⋅ ) ∇∇ T ) + ( 𝑏 ( ⋅ ) + 𝜀𝑐 ( ⋅ )) T ∇ (see Lemma 3.4).With this in hand, and basing on the ideas from [15], we are then able to obtain the requiredfunctional CLT (see Theorem 3.5) and conclude the homogenization results in Theorems 4.2and 4.3.The results of this article can also be found as a part of the second-named author’s doctoraldissertation [41], where the problem of periodic homogenization of a class of Lévy-type opera-tors has been discussed.1.2. Notation.
We summarize some notation used throughout the article. We use (cid:82) n , n ∈ (cid:78) , todenote real-valued 𝑛 -dimensional vectors, and write (cid:82) for n = 1 . All vectors will be column vec-tors. cis denoted by | ⋅ | . By 𝑀 T and ‖ 𝑀 ‖ HS ∶= (Tr 𝑀 𝑀 T ) we denote the transpose and theHilbert-Schmidt norm of a 𝑛 × 𝑚 -matrix 𝑀 , respectively. For a square matrix 𝑀 , Tr 𝑀 standsfor its trace. For a set 𝐴 ⊆ (cid:82) n , the symbols 𝐴 𝑐 , (cid:49) 𝐴 , 𝐴 and 𝜕𝐴 stand for the complement, indica-tor function, (topological) closure and (topological) boundary of 𝐴 , respectively. We let 𝔅 ( (cid:82) n ) and ( (cid:82) n , (cid:82) m ) denote the Borel 𝜎 -algebra on (cid:82) n and the space of 𝔅 ( (cid:82) n )∕ 𝔅 ( (cid:82) m ) -measurablefunctions, respectively. Also, for 𝐴 ⊆ (cid:82) n , 𝔅 ( 𝐴 ) stands for { 𝐴 ∩ 𝐵 ∶ 𝐵 ∈ 𝔅 ( (cid:82) 𝑛 )} . For a Borelmeasure µ (d 𝑥 ) on 𝔅 ( (cid:82) n ) and 𝑓 = ( 𝑓 , … , 𝑓 m ) T ∈ ( (cid:82) n , (cid:82) m ) , we often use the convenient no-tation µ ( 𝑓 ) = ∫ (cid:82) n 𝑓 ( 𝑥 ) µ (d 𝑥 ) ∶= ( ∫ (cid:82) n 𝑓 ( 𝑥 ) µ (d 𝑥 ) , … , ∫ (cid:82) n 𝑓 m ( 𝑥 ) µ (d 𝑥 )) T . For 𝑓 ∈ ( (cid:82) n , (cid:82) m ) we let ‖ 𝑓 ‖ ∞ ∶= sup 𝑥 ∈ (cid:82) n | 𝑓 ( 𝑥 ) | denote its supremum norm, and 𝑏 ( (cid:82) n , (cid:82) m ) stands for { 𝑓 ∈ ( (cid:82) n , (cid:82) m ) ∶ ‖ 𝑓 ‖ ∞ < ∞} . We use 𝑘𝑏 ( (cid:82) n , (cid:82) m ) , 𝑘𝑢,𝑏 ( (cid:82) n , (cid:82) m ) , 𝑘 ∞ ( (cid:82) n , (cid:82) m ) and 𝑘𝑐 ( (cid:82) n , (cid:82) m ) , 𝑘 ∈ (cid:78) ∪ {∞} , to denote the subspaces of 𝑏 ( (cid:82) n , (cid:82) m ) ∩ 𝑘 ( (cid:82) n , (cid:82) m ) of all 𝑘 times differentiablefunctions such that all derivatives up to order 𝑘 are bounded, uniformly continuous and bounded, ERIODIC HOMOGENIZATION OF LINEAR DEGENERATE PDEs 3 vanish at infinity, and have compact support, respectively. Gradient of 𝑓 ∈ ( (cid:82) n , (cid:82) ) is denotedby ∇ 𝑓 ( 𝑥 ) = ( 𝜕 𝑓 ( 𝑥 ) , … , 𝜕 n 𝑓 ( 𝑥 )) T , and for 𝑓 = ( 𝑓 , … , 𝑓 m ) T ∈ ( (cid:82) n , (cid:82) m ) we write D 𝑓 ( 𝑥 ) =(∇ 𝑓 ( 𝑥 ) , … , ∇ 𝑓 m ( 𝑥 )) T for the corresponding Jacobian. For 𝜏 = ( 𝜏 , … , 𝜏 n ) T ∈ (0 , ∞) n , we let (cid:90) n 𝜏 ∶= {( 𝜏 𝑘 , … , 𝜏 n 𝑘 n ) T ∶( 𝑘 , … , 𝑘 n ) T ∈ (cid:90) n } , and, for 𝑥 ∈ (cid:82) n , [ 𝑥 ] 𝜏 ∶= { 𝑦 ∈ (cid:82) n ∶ 𝑥 − 𝑦 ∈ (cid:90) n 𝜏 } , and (cid:84) n 𝜏 ∶= { [ 𝑥 ] 𝜏 ∶ 𝑥 ∈ (cid:82) n } . Clearly, (cid:84) n 𝜏 is obtained by identifying the opposite faces of [0 , 𝜏 ] ∶= [0 , 𝜏 ] × ⋯ × [0 , 𝜏 n ] . Thecorresponding Borel 𝜎 -algebra is denoted by 𝔅 ( (cid:84) n 𝜏 ) , which can be identified with the sub- 𝜎 -algebra of 𝔅 ( (cid:82) n ) of sets of the form ⋃ 𝑘 𝜏 ∈ (cid:90) n 𝜏 { 𝑥 + 𝑘 𝜏 ∶ 𝑥 ∈ 𝐵 𝜏 } , 𝐵 𝜏 ∈ 𝔅 ([0 , 𝜏 ]) . The coveringmap (cid:82) n ∋ 𝑥 ↦ [ 𝑥 ] 𝜏 ∈ (cid:84) n 𝜏 is denoted by Π 𝜏 ( 𝑥 ) . A function 𝑓 ∶ (cid:82) n → (cid:82) m is called 𝜏 -periodic if 𝑓 ( 𝑥 + 𝑘 𝜏 ) = 𝑓 ( 𝑥 ) ∀ ( 𝑥, 𝑘 𝜏 ) ∈ (cid:82) n × (cid:90) n 𝜏 . Clearly, every 𝜏 -periodic function 𝑓 ( 𝑥 ) is completely and uniquely determined by its restriction 𝑓 | [0 ,𝜏 ] ( 𝑥 ) to [0 , 𝜏 ] , and since 𝑓 | [0 ,𝜏 ] ( 𝑥 ) assumes the same value on opposite faces of [0 , 𝜏 ] it canbe identified by a function 𝑓 𝜏 ∶ (cid:84) d 𝜏 → (cid:82) m given with 𝑓 𝜏 ([ 𝑥 ] 𝜏 ) ∶= 𝑓 ( 𝑥 ) . Using this identification,in an analogous way as above we define ( (cid:84) n 𝜏 , (cid:82) m ) , 𝑏 ( (cid:84) n 𝜏 , (cid:82) m ) and 𝑘𝑏 ( (cid:84) n 𝜏 , (cid:82) m ) = 𝑘 ( (cid:84) n 𝜏 , (cid:82) m ) , 𝑘 ∈ (cid:78) ∪{∞} . For notational convenience, we write 𝑥 instead of [ 𝑥 ] 𝜏 , and 𝑓 ( 𝑥 ) instead of 𝑓 𝜏 ( 𝑥 ) .1.3. Organization of the article.
In the next section, we first discuss certain structural andergodic properties of a diffusion process associated to the operator 𝜀 . Then, in Section 3, weprove that under an appropriate centering this process satisfies a functional CLT, as 𝜀 → , witha Browninan limit, which is the key probabilistic argument in discussing the homogenization ofthe problems in eqs. (1.1) and (1.2). Finally, in Section 4, we prove the homogenization results.2. S TRUCTURAL PROPERTIES OF THE ASSOCIATED DIFFUSION PROCESS
Throughout the article we impose the following assumptions on the coefficients 𝑎 ( 𝑥 ) , 𝑏 ( 𝑥 ) and 𝑐 ( 𝑥 ) : (A1): (i) there is σ ∈ ( (cid:82) n , (cid:82) n×m ) such that 𝑎 ( 𝑥 ) = σ ( 𝑥 ) σ ( 𝑥 ) T for all 𝑥 ∈ (cid:82) n ;(ii) σ ( 𝑥 ) , 𝑏 ( 𝑥 ) and 𝑐 ( 𝑥 ) are continuous and 𝜏 -periodic;(iii) there is Θ > and a non-decreasing concave function θ ∶ (0 , ∞) → (0 , ∞) satisfy-ing ∫ d 𝑣 θ ( 𝑣 ) = ∞ , such that for all 𝑥, 𝑦 ∈ [0 , 𝜏 ] ,(2.1) max {‖ σ ( 𝑥 )− σ ( 𝑦 ) ‖ , ( 𝑏 ( 𝑥 )− 𝑏 ( 𝑦 ))( 𝑥 − 𝑦 ) T , ( 𝑐 ( 𝑥 )− 𝑐 ( 𝑦 ))( 𝑥 − 𝑦 ) T } ≤ Θ | 𝑥 − 𝑦 | θ ( | 𝑥 − 𝑦 | ) . According to [42, Theorems 2.2 and 2.4], (A1) implies that for any 𝜀 > , 𝑥 ∈ (cid:82) n and a givenstandard 𝑚 -dimensional Brownian motion { 𝐵 ( 𝑡 )} 𝑡 ≥ (defined on a stochastic basis (Ω , , { 𝑡 } 𝑡 ≥ , (cid:80) ) satisfying the usual conditions), the following stochastic differential equation (SDE): d 𝑋 𝜀 ( 𝑥, 𝑡 ) = ( 𝜀 −1 𝑏 ( 𝑋 𝜀 ( 𝑥, 𝑡 )∕ 𝜀 ) + 𝑐 ( 𝑋 𝜀 ( 𝑥, 𝑡 )∕ 𝜀 )) d 𝑡 + σ ( 𝑋 𝜀 ( 𝑥, 𝑡 )∕ 𝜀 ) d 𝐵 ( 𝑡 ) 𝑋 𝜀 ( 𝑥,
0) = 𝑥 ∈ (cid:82) n , admits a unique strong solution { 𝑋 𝜀 ( 𝑥, 𝑡 )} 𝑡 ≥ which is a conservative (non-explosive) strongMarkov process with continuous sample paths, and transition kernel 𝑝 𝜀 ( 𝑡, 𝑥, d 𝑦 ) = (cid:80) ( 𝑋 𝜀 ( 𝑥, 𝑡 ) ∈ ERIODIC HOMOGENIZATION OF LINEAR DEGENERATE PDEs 4 d 𝑦 ) , 𝑡 ≥ , 𝑥 ∈ (cid:82) n . Furthermore, due to [42, Proposition 4.2] the process { 𝑋 𝜀 ( 𝑥, 𝑡 )} 𝑡 ≥ possessesthe 𝑏 -Feller property, that is, 𝜀𝑡 𝑓 ∈ 𝑏 ( (cid:82) n , (cid:82) ) for any 𝑡 ≥ and 𝑓 ∈ 𝑏 ( (cid:82) n , (cid:82) ) , where 𝜀𝑡 𝑓 ( ⋅ ) ∶= ∫ (cid:82) d 𝑓 ( 𝑦 ) 𝑝 𝜀 ( 𝑡, ⋅ , d 𝑦 ) , 𝑡 ≥ , 𝑓 ∈ 𝑏 ( (cid:82) n , (cid:82) ) , stands for the corresponding operator semigroup defined on the Banach space ( 𝑏 ( (cid:82) n , (cid:82) ) , ‖ ⋅ ‖ ∞ ) . The 𝑏 -infinitesimal generator ( 𝜀 , 𝜀 ) of { 𝜀𝑡 } 𝑡 ≥ (or of { 𝑋 𝜀 ( 𝑥, 𝑡 )} 𝑡 ≥ )is a linear operator 𝜀 ∶ 𝜀 → 𝑏 ( (cid:82) n , (cid:82) ) defined by 𝜀 𝑓 ∶= lim 𝑡 → 𝜀𝑡 𝑓 − 𝑓𝑡 , 𝑓 ∈ 𝜀 ∶= { 𝑓 ∈ 𝑏 ( (cid:82) n , (cid:82) ) ∶ lim 𝑡 → 𝜀𝑡 𝑓 − 𝑓𝑡 exists in ‖ ⋅ ‖ ∞ } . By employing Itô’s formula we easily see that lim 𝑡 → ‖‖‖( 𝜀𝑡 𝑓 − 𝑓 ) ∕ 𝑡 − 𝜀 𝑓 ‖‖‖ ∞ = 0 ∀ 𝑓 ∈ 𝑢,𝑏 ( (cid:82) n , (cid:82) ) , that is, 𝑢,𝑏 ( (cid:82) n , (cid:82) ) ⊆ 𝜀 and 𝜀 | 𝑢,𝑏 ( (cid:82) n , (cid:82) ) = 𝜀 . Following [15] (see also [3, Lemma 3.4.1]), for 𝜀 > let ̄𝑋 𝜀 ( 𝑥, 𝑡 ) ∶= 𝜀 −1 𝑋 𝜀 ( 𝜀𝑥, 𝜀 𝑡 ) , 𝑡 ≥ .Clearly, { ̄𝑋 𝜀 ( 𝑥, 𝑡 )} 𝑡 ≥ satisfies(2.2) d ̄𝑋 𝜀 ( 𝑥, 𝑡 ) = ( 𝑏 ( ̄𝑋 𝜀 ( 𝑥, 𝑡 ) ) + 𝜀𝑐 ( ̄𝑋 𝜀 ( 𝑥, 𝑡 ) )) d 𝑡 + σ ( ̄𝑋 𝜀 ( 𝑥, 𝑡 ) ) d 𝐵 𝜀 ( 𝑡 ) ̄𝑋 𝜀 ( 𝑥,
0) = 𝑥 ∈ (cid:82) n , where 𝐵 𝜀 ( 𝑡 ) ∶= 𝜀 −1 𝐵 ( 𝜀 𝑡 ) , 𝑡 ≥ . Observe that { 𝐵 𝜀 ( 𝑡 )} 𝑡 ≥ = { 𝐵 ( 𝑡 )} 𝑡 ≥ , although it is nota martingale with respect to { 𝑡 } 𝑡 ≥ . Here, (d) = denotes the equality in distribution. Let also { ̄𝑋 ( 𝑥, 𝑡 )} 𝑡 ≥ be a solution to(2.3) d ̄𝑋 ( 𝑥, 𝑡 ) = 𝑏 ( ̄𝑋 ( 𝑥, 𝑡 ) ) d 𝑡 + σ ( ̄𝑋 ( 𝑥, 𝑡 ) ) d 𝐵 ( 𝑡 ) ̄𝑋 ( 𝑥,
0) = 𝑥 ∈ (cid:82) n . Clearly, the processes { ̄𝑋 𝜀 ( 𝑥, 𝑡 )} 𝑡 ≥ , 𝜀 ≥ , share the same structural properties as { 𝑋 𝜀 ( 𝑥, 𝑡 )} 𝑡 ≥ , 𝜀 > , mentioned above. Denote by ̄𝑝 𝜀 ( 𝑡, 𝑥, d 𝑦 ) = (cid:80) ( ̄𝑋 𝜀 ( 𝑥, 𝑡 ) ∈ d 𝑦 ) , 𝑡 ≥ , 𝑥 ∈ (cid:82) n , { ̄ 𝜀𝑡 } 𝑡 ≥ and ( ̄ 𝜀 , ̄ 𝜀 ) the corresponding transition kernel, operator semigroup and 𝑏 -infinitesimal gener-ator, respectively. From [20, Theorem IX.4.8] it follows that(2.4) { ̄𝑋 𝜀 ( 𝑥, 𝑡 )} 𝑡 ≥ ⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇒ 𝜀 → { ̄𝑋 ( 𝑥, 𝑡 )} 𝑡 ≥ . In particular, for any 𝑡 ≥ and 𝑥 ∈ (cid:82) n , ̄𝑝 𝜀 ( 𝑡, 𝑥, d 𝑦 ) (w) ⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇒ 𝜀 → ̄𝑝 ( 𝑡, 𝑥, d 𝑦 ) , that is, lim 𝜀 → ̄ 𝜀𝑡 𝑓 ( 𝑥 ) = ̄ 𝑡 𝑓 ( 𝑥 ) for any 𝑡 ≥ , 𝑥 ∈ (cid:82) n and 𝑓 ∈ 𝑏 ( (cid:82) n , (cid:82) ) . Here, (d) ⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇒ denotes the convergence in the spaceof continuous functions endowed with the locally uniform topology (see [20, Chapter VI] fordetails), and (w) ⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇒ stands for the weak convergence of probability measures.Next, observe that due to 𝜏 -periodicity of the coefficients { ̄𝑋 𝜀 ( 𝑥 + 𝑘 𝜏 , 𝑡 )} 𝑡 ≥ and { ̄𝑋 𝜀 ( 𝑥, 𝑡 ) + 𝑘 𝜏 } 𝑡 ≥ , 𝜀 ≥ , 𝑥 ∈ (cid:82) n , 𝑘 𝜏 ∈ (cid:90) n 𝜏 , are indistinguishable. In particular, ̄𝑝 𝜀 ( 𝑡, 𝑥 + 𝑘 𝜏 , 𝐵 ) = ̄𝑝 𝜀 ( 𝑡, 𝑥, 𝐵 − 𝑘 𝜏 ) for all 𝜀 ≥ , 𝑡 ≥ , 𝑥 ∈ (cid:82) n , 𝑘 𝜏 ∈ (cid:90) n 𝜏 and 𝐵 ∈ 𝔅 ( (cid:82) n ) , which implies that { ̄ 𝜀𝑡 } 𝑡 ≥ preservesthe class of 𝜏 -periodic functions in 𝑏 ( (cid:82) n , (cid:82) ) . Thus, according to [23, Proposition 3.8.3] the ERIODIC HOMOGENIZATION OF LINEAR DEGENERATE PDEs 5 projection of { ̄𝑋 𝜀 ( 𝑥, 𝑡 )} 𝑡 ≥ with respect to Π 𝜏 ( 𝑥 ) on the torus (cid:84) n 𝜏 , denoted by { ̄𝑋 𝜀,𝜏 ( 𝑥, 𝑡 )} 𝑡 ≥ , is aMarkov process on ( (cid:84) n 𝜏 , 𝔅 ( (cid:84) n 𝜏 )) with transition kernel given by(2.5) ̄𝑝 𝜀,𝜏 ( 𝑡, 𝑥, 𝐵 ) = ̄𝑝 𝜀 ( 𝑡, 𝑧 𝑥 , Π −1 𝜏 ( 𝐵 ) ) for 𝜀 ≥ , 𝑡 ≥ , 𝑥 ∈ (cid:84) n 𝜏 , 𝐵 ∈ 𝔅 ( (cid:84) n 𝜏 ) and 𝑧 𝑥 ∈ Π −1 𝜏 ({ 𝑥 }) . In particular, { ̄𝑋 𝜀,𝜏 ( 𝑥, 𝑡 )} 𝑡 ≥ is a 𝑏 -Feller process. Proposition 2.1.
Under ( A1 ) , for any 𝑡 ≥ and 𝜏 -periodic 𝑓 ∈ 𝑏 ( (cid:82) n , (cid:82) ) it holds that lim 𝜀 → ‖ ̄ 𝜀𝑡 𝑓 − ̄ 𝑡 𝑓 ‖ ∞ = 0 . Proof.
From eqs. (2.4) and (2.5) we see that { ̄𝑋 𝜀,𝜏 ( 𝑥, 𝑡 )} 𝑡 ≥ ⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇒ 𝜀 → { ̄𝑋 ,𝜏 ( 𝑥, 𝑡 )} 𝑡 ≥ . Now, since (cid:84) d 𝜏 is compact, the assertion follows from [22, Theorem 17.25]. (cid:3) Clearly, for any 𝑥 ∈ (cid:82) n the matrix 𝑎 ( 𝑥 ) is symmetric and non-negative definite. We furtherassume (A2): (i) there is an open connected set 𝒪 ⊂ [0 , 𝜏 ] such that the matrix 𝑎 ( 𝑥 ) is positivedefinite on 𝒪 , that is, 𝜉 T 𝑎 ( 𝑥 ) 𝜉 > 𝑥, 𝜉 ) ∈ 𝒪 × (cid:82) n ⧵ {0} ; (ii) 𝑎 ( 𝑥 ) , 𝑏 ( 𝑥 ) and 𝑐 ( 𝑥 ) are 𝛾 -Hölder continuous for some < 𝛾 ≤ , that is, there is Γ > such that(2.6) ‖ 𝑎 ( 𝑥 ) − 𝑎 ( 𝑦 ) ‖ HS + | 𝑏 ( 𝑥 ) − 𝑏 ( 𝑦 ) | + | 𝑐 ( 𝑥 ) − 𝑐 ( 𝑦 ) | ≤ Γ | 𝑥 − 𝑦 | 𝛾 ∀ 𝑥, 𝑦 ∈ (cid:82) n . Remark 2.2. (i) For a given symmetric, non-negative definite and Borel measurable 𝑛 × 𝑛 -matrix-valued function 𝑎 ( 𝑥 ) there is a unique non-negative definite and Borel measurable 𝑛 × 𝑛 -matrix-valued function ̄ σ ( 𝑥 ) such that 𝑎 ( 𝑥 ) = ̄ σ ( 𝑥 ) ̄ σ ( 𝑥 ) T for all 𝑥 ∈ (cid:82) n . In general,it is not clear that smoothness (Hölder continuity or differentiability) of 𝑎 ( 𝑥 ) impliessmoothness of ̄ σ ( 𝑥 ) . However, if 𝑎 ( 𝑥 ) is additionally positive definite or twice continu-ously differentiable this will be the case (see [16, Lemma 6.1.1 and Theorem 6.1.2]). Inparticular, under (A2), ̄ σ ( 𝑥 ) will be 𝛾 -Hölder continuous on 𝒪 .(ii) Equation (2.1) holds true if for all 𝑥, 𝑦 ∈ (cid:82) n ,(2.7) ‖ σ ( 𝑥 ) − σ ( 𝑦 ) ‖ + | 𝑥 − 𝑦 |(| 𝑏 ( 𝑥 ) − 𝑏 ( 𝑦 ) | + | 𝑐 ( 𝑥 ) − 𝑐 ( 𝑦 ) |) ≤ Θ | 𝑥 − 𝑦 | 𝜃 ( | 𝑥 − 𝑦 | ) . Clearly, eq. (2.7), together with periodicity of σ ( 𝑥 ) , automatically implies -Höldercontinuity of σ ( 𝑥 ) . Moreover, since ‖ 𝑎 ( 𝑥 ) − 𝑎 ( 𝑦 ) ‖ HS = ‖ σ ( 𝑥 ) σ ( 𝑥 ) T − σ ( 𝑦 ) σ ( 𝑦 ) T ‖ HS ≤ ‖ ( σ ( 𝑥 ) − σ ( 𝑦 )) σ ( 𝑥 ) T ‖ HS + ‖ σ ( 𝑦 ) ( σ ( 𝑥 ) T − σ ( 𝑦 ) T ) ‖ HS ≤ ‖ σ ‖ ∞ ‖ σ ( 𝑥 ) − σ ( 𝑦 ) ‖ HS it also implies -Hölder continuity of 𝑎 ( 𝑥 ) . In addition, if lim sup 𝑣 → θ ( 𝑣 )∕ 𝑣 𝛾 < ∞ forsome 𝛾 ∈ (0 , , it is easy to see that eq. (2.7) implies 𝛾 -Hölder continuity of 𝑏 ( 𝑥 ) and 𝑐 ( 𝑥 ) , and (1 + 𝛾 )∕2 -Hölder continuity of σ ( 𝑥 ) and 𝑎 ( 𝑥 ) . ERIODIC HOMOGENIZATION OF LINEAR DEGENERATE PDEs 6 (iii) Assumptions (A1)-(A2) imply that 𝑎 ( 𝑥 ) is uniformly elliptic on 𝒪 , that is, there is 𝛼 > such that 𝜉 T 𝑎 ( 𝑥 ) 𝜉 ≥ 𝛼 | 𝜉 | ∀ ( 𝑥, 𝜉 ) ∈ 𝒪 × (cid:82) n . Indeed, since for every 𝑥 ∈ 𝒪 the matrix 𝑎 ( 𝑥 ) is symmetric and positive definite, the cor-responding eigenvalues λ ( 𝑥 ) , … , λ n ( 𝑥 ) are real and positive. Also, since λ ( 𝑥 ) , … , λ n ( 𝑥 ) are roots of the polynomial 𝜆 ↦ det( 𝑎 ( 𝑥 ) − 𝜆 (cid:73) n ) we see that each λ 𝑖 ∶ 𝒪 → (0 , ∞) iscontinuous. Here, (cid:73) n stands for the 𝑛 × 𝑛 -identity matrix. Hence, due to compactness of 𝒪 , we conclude that there is 𝛼 > such that 𝑎 ( 𝑥 ) − 𝛼 (cid:73) n is positive definite on 𝒪 , whichproves the assertion.For 𝜀 ≥ , 𝑥 ∈ (cid:82) n and 𝐵 ∈ 𝔅 ( (cid:82) n ) , let ̄ τ 𝜀,𝑥𝐵 ∶= inf { 𝑡 ≥ ̄𝑋 𝜀 ( 𝑥, 𝑡 ) ∈ 𝐵 } be the first entrytime of 𝐵 by { ̄𝑋 𝜀 ( 𝑥, 𝑡 )} 𝑡 ≥ . Assume (A3): there is 𝜀 > such that (cid:80) ( ̄ τ 𝜀,𝑥 𝒪 + 𝜏 < ∞ ) > 𝜀, 𝑥 ) ∈ [0 , 𝜀 ] × (cid:82) n , where 𝒪 + 𝜏 ∶= { 𝑥 + 𝑘 𝜏 ∶ 𝑥 ∈ 𝒪 , 𝑘 𝜏 ∈ (cid:90) n 𝜏 } .Under (A1), [27, Theorem 3.1] shows that { ̄𝑋 𝜀,𝜏 ( 𝑥, 𝑡 )} 𝑡 ≥ , 𝜀 ≥ , admits at least one invariantprobability measure. Assuming additionally (A2)-(A3), in what follows we show that { ̄𝑋 𝜀,𝜏 ( 𝑥, 𝑡 )} 𝑡 ≥ , 𝜀 ∈ [0 , 𝜀 ] , admits one, and only one, invariant measure, and the correspond-ing marginals converge as 𝑡 → ∞ to the invariant measure in the total variation norm (withexponential rate). Proposition 2.3.
Under ( A1 ) - ( A3 ) , there exists a measure ψ (d 𝑥 ) on (cid:84) n 𝜏 such that (i) supp( ψ ) has nonempty interior; (ii) for every 𝑥 ∈ (cid:84) n 𝜏 and 𝜀 ∈ [0 , 𝜀 ] there is 𝑡 𝑥,𝜀 ≥ such that ψ ( 𝐵 ) > ⟹ ̄𝑝 𝜀,𝜏 ( 𝑡, 𝑥, 𝐵 ) > 𝑡 ∈ [ 𝑡 𝑥,𝜀 , ∞) . Proof.
According to [13, Theorems 7.3.6 and 7.3.7] there is a strictly positive function 𝑞 𝜀 ( 𝑡, 𝑥, 𝑦 ) on (0 , ∞) × 𝒪 + 𝜏 × 𝒪 + 𝜏 , jointly continuous in 𝑡 , 𝑥 and 𝑦 , satisfying (cid:69) [ 𝑓 ( ̄𝑋 𝜀 ( 𝑥, 𝑡 )) (cid:49) ( 𝑡, ∞] ( ̄ τ 𝜀,𝑥 ( 𝒪 + 𝜏 ) 𝑐 )] = ∫ 𝒪 + 𝜏 𝑓 ( 𝑦 ) 𝑞 𝜀 ( 𝑡, 𝑥, 𝑦 ) d 𝑦 for all 𝑡 > , 𝑥 ∈ 𝒪 + 𝜏 and 𝑓 ∈ 𝑏 ( (cid:82) n , (cid:82) ) . By employing dominated convergence theorem it isstraightforward to check that the above relation holds also for (cid:49) 𝒰 + 𝜏 ( 𝑥 ) for any open set 𝒰 ⊆ 𝒪 .Denote by the class of all 𝐵 ∈ 𝔅 ( 𝒪 ) such that (cid:80) ( ̄𝑋 𝜀 ( 𝑥, 𝑡 ) ∈ 𝐵 + 𝜏 , ̄ τ 𝜀,𝑥 ( 𝒪 + 𝜏 ) 𝑐 > 𝑡 ) = ∫ 𝐵 + 𝜏 𝑞 𝜀 ( 𝑡, 𝑥, 𝑦 ) d 𝑦 . Obviously, contains the 𝜋 -system of open rectangles in 𝔅 ( 𝒪 ) , and forms a 𝜆 -system. Thus,Dynkin’s 𝜋 - 𝜆 theorem implies that = 𝔅 ( 𝒪 ) , and for all 𝑡 > , 𝑥 ∈ 𝒪 + 𝜏 and 𝐵 ∈ 𝔅 ([0 , 𝜏 ]) we have ̄𝑝 𝜀 ( 𝑡, 𝑥, 𝐵 + 𝜏 ) ≥ ∫ ( 𝐵 ∩ 𝒪 )+ 𝜏 𝑞 𝜀 ( 𝑡, 𝑥, 𝑦 ) d 𝑦 . Set now ψ ( 𝐵 + 𝜏 ) ∶= λ (( 𝐵 ∩ 𝒪 ) + 𝜏 ) , 𝐵 ∈ 𝔅 ([0 , 𝜏 ]) , where λ (d 𝑥 ) stands for the Lebesguemeasure on (cid:82) n . Clearly, by construction, ψ (d 𝑥 ) is a measure on 𝜎 -algebra 𝔅 ([0 , 𝜏 ]) + 𝜏 ∶={ 𝐵 + 𝜏 ∶ 𝐵 ∈ 𝔅 ([0 , 𝜏 ])} , supp( ψ ) has non-empty interior, and for 𝐵 ∈ 𝔅 ([0 , 𝜏 ]) it holds that(2.8) ψ ( 𝐵 + 𝜏 ) > ⟹ ̄𝑝 𝜀 ( 𝑡, 𝑥, 𝐵 + 𝜏 ) > 𝑡, 𝑥 ) ∈ (0 , ∞) × 𝒪 + 𝜏 . ERIODIC HOMOGENIZATION OF LINEAR DEGENERATE PDEs 7
To this end, it remains to show that for each 𝑥 ∈ ([0 , 𝜏 ] ⧵ 𝒪 ) + 𝜏 there is 𝑡 𝑥,𝜀 ≥ such that theimplication in eq. (2.8) holds for all 𝑡 ≥ 𝑡 𝑥,𝜀 . Since { ̄𝑋 𝜀 ( 𝑥, 𝑡 )} 𝑡 ≥ has continuous sample pathsand 𝒪 is an open set, we have that ∑ 𝑡 ∈ (cid:81) + ̄𝑝 𝜀 ( 𝑡, 𝑥, 𝒪 + 𝜏 ) ≥ (cid:80) ( ∃ 𝑡 ∈ (cid:81) + such that ̄𝑋 𝜀 ( 𝑥, 𝑡 ) ∈ 𝒪 + 𝜏 ) = (cid:80) ( ∃ 𝑡 ≥ such that ̄𝑋 𝜀 ( 𝑥, 𝑡 ) ∈ 𝒪 + 𝜏 ) = (cid:80) ( ̄ τ 𝜀,𝑥 𝒪 + 𝜏 < ∞ ) . From (A3) we see that there is 𝑡 𝑥,𝜀 ∈ (cid:81) + such that ̄𝑝 𝜀 ( 𝑡 𝑥,𝜀 , 𝑥, 𝒪 + 𝜏 ) > . Let 𝐵 ∈ 𝔅 ([0 , 𝜏 ]) besuch that ψ ( 𝐵 + 𝜏 ) > . For any 𝑡 > 𝑡 𝑥,𝜀 we then have ̄𝑝 𝜀 ( 𝑡, 𝑥, 𝐵 + 𝜏 ) ≥ (cid:80) ( ̄𝑋 𝜀 ( 𝑥, 𝑡 ) ∈ 𝐵 + 𝜏 , ̄𝑋 𝜀 ( 𝑥, 𝑡 𝑥,𝜀 ) ∈ 𝒪 + 𝜏 ) = ∫ 𝒪 + 𝜏 ̄𝑝 𝜀 ( 𝑡 − 𝑡 𝑥,𝜀 , 𝑦, 𝐵 + 𝜏 ) ̄𝑝 𝜀 ( 𝑡 𝑥,𝜀 , 𝑥, d 𝑦 ) , which is strictly positive because of eq. (2.8). The result now follows by setting ψ ( 𝐵 ) ∶= ψ (Π −1 𝜏 ( 𝐵 )) , 𝐵 ∈ 𝔅 ( (cid:84) n 𝜏 ) , and using eq. (2.5). (cid:3) From Proposition 2.3 it immediately follows that { ̄𝑋 𝜀,𝜏 ( 𝑥, 𝑡 )} 𝑡 ≥ is irreducible in the senseof [12], that is, ψ ( 𝐵 ) > implies that ∫ ∞0 ̄𝑝 𝜀,𝜏 ( 𝑡, 𝑥, 𝐵 ) d 𝑡 > for all 𝑥 ∈ (cid:84) n 𝜏 . This automati-cally entails that { ̄𝑋 𝜀,𝜏 ( 𝑥, 𝑡 )} 𝑡 ≥ admits one, and only one, invariant probability measure π 𝜀 (d 𝑥 ) .Namely, according to [40, Theorem 2.3] every irreducible Markov process is either transient orrecurrent. Due to the fact that { ̄𝑋 𝜀,𝜏 ( 𝑥, 𝑡 )} 𝑡 ≥ admits at least one invariant probability measure itclearly cannot be transient. The assertion now follows from [40, Theorem 2.6] which states thatevery recurrent Markov process admits a unique (up to constant multiplies) invariant measure. Proposition 2.4.
Under ( A1 ) - ( A3 ) , there are 𝛾 > and 𝛤 > , such that sup 𝑥 ∈ (cid:84) n 𝜏 ‖ ̄𝑝 𝜀,𝜏 ( 𝑡, 𝑥, d 𝑦 ) − π 𝜀 (d 𝑦 ) ‖ TV ≤ 𝛤 e − 𝛾𝑡 ∀ ( 𝜀, 𝑡 ) ∈ [0 , 𝜀 ] × [0 , ∞) , where ‖ ⋅ ‖ TV denotes the total variation norm on the space of signed measures on 𝔅 ( (cid:84) n 𝜏 ) .Proof. First, [40, Theorems 5.1 and 7.1] together with the 𝑏 -Feller property of { ̄𝑋 𝜀,𝜏 ( 𝑥, 𝑡 )} 𝑡 ≥ and Proposition 2.3 imply that (cid:84) n 𝜏 is a petite set for { ̄𝑋 𝜀,𝜏 ( 𝑥, 𝑡 )} 𝑡 ≥ (see [40] for the definition ofpetite sets). Next, from [28, Theorem 4.2] (with 𝑐 = 𝑑 = 1 , 𝐶 = (cid:84) n 𝜏 , 𝑓 ( 𝑥 ) = 𝑉 ( 𝑥 ) ≡ , and 𝑉 ( 𝑥 ) ≡ ), Proposition 2.3 (which implies that ∑ ∞ 𝑖 =1 ̄𝑝 𝜀,𝜏 ( 𝑖, 𝑥, 𝐵 ) > for all 𝑥 ∈ (cid:84) n 𝜏 whenever ψ ( 𝐵 ) > ), and [27, Proposition 6.1] we see that { ̄𝑋 𝜀,𝜏 ( 𝑥, 𝑡 )} 𝑡 ≥ is aperiodic in the sense of [12].The desired result now follows from [12, Theorem 5.2] by taking 𝑐 = 𝑏 = 1 , 𝐶 = (cid:84) n 𝜏 , ̃𝑉 ( 𝑥 ) ≡ ,and ̃ ̃𝑉 ( 𝑥 ) ≡ . (cid:3) From Proposition 2.4 we see that for any 𝑓 ∈ 𝑏 ( (cid:84) n 𝜏 , (cid:82) ) satisfying π 𝜀 ( 𝑓 ) = 0 , 𝜀 ∈ [0 , 𝜀 ] , itholds that(2.9) ‖ ̄ 𝜀,𝜏𝑡 𝑓 ‖ ∞ ≤ 𝛤 ‖ 𝑓 ‖ ∞ e − 𝛾𝑡 ∀ 𝑡 ≥ . Proposition 2.5.
Under ( A1 ) - ( A3 ) , π 𝜀 (d 𝑥 ) (w) ⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇒ 𝜀 → π (d 𝑥 ) . ERIODIC HOMOGENIZATION OF LINEAR DEGENERATE PDEs 8
Proof.
Since (cid:84) n 𝜏 is compact the family of probability measures { π 𝜀 (d 𝑥 )} 𝜀 ≥ is tight. Hence, forany sequence { 𝜀 𝑖 } 𝑖 ∈ (cid:78) ⊂ [0 , 𝜀 ] converging to there is a further subsequence { 𝜀 𝑖 𝑗 } 𝑗 ∈ (cid:78) such that { π 𝜀 𝑖𝑗 (d 𝑥 )} 𝑗 ∈ (cid:78) converges weakly to some probability measure ̄ π (d 𝑥 ) . Take 𝑓 ∈ ( (cid:84) n 𝜏 , (cid:82) ) , andfix 𝑡 ≥ and 𝜖 > . From Proposition 2.1 we have that there is < 𝜀 ≤ 𝜀 such that ‖ ̄ 𝜀,𝜏𝑡 𝑓 − ̄ ,𝜏𝑡 𝑓 ‖ ∞ ≤ 𝜖 ∀ 𝜀 ∈ [0 , 𝜀 ] . We now have that | ̄ π ( 𝑓 ) − ̄ π ( ̄ ,𝜏𝑡 𝑓 )| = lim 𝑗 → ∞ | π 𝜀 𝑖𝑗 ( 𝑓 ) − ̄ π ( ̄ ,𝜏𝑡 𝑓 )| = lim 𝑗 → ∞ | ̄ π 𝜀 𝑖𝑗 ( ̄ 𝜀 𝑖𝑗 ,𝜏𝑡 𝑓 ) − ̄ π ( ̄ ,𝜏𝑡 𝑓 )| ≤ lim sup 𝑗 → ∞ | π 𝜀 𝑖𝑗 ( ̄ 𝜀 𝑖𝑗 ,𝜏𝑡 𝑓 ) − ̄ π 𝜀 𝑖𝑗 ( ̄ ,𝜏𝑡 𝑓 )| + lim 𝑗 → ∞ | π 𝜀 𝑖𝑗 ( ̄ ,𝜏𝑡 𝑓 ) − ̄ π ( ̄ ,𝜏𝑡 𝑓 )| ≤ 𝜖 , which implies that ̄ π (d 𝑥 ) is an invariant probability measure for { 𝑋 ,𝜏 ( 𝑥, 𝑡 )} 𝑡 ≥ . Thus, ̄ π (d 𝑥 ) = π (d 𝑥 ) , which proves the assertion. (cid:3)
3. CLT
FOR THE PROCESS { 𝑋 𝜀 ( 𝑥, 𝑡 )} 𝑡 ≥ Under the assumption that 𝑎 ( 𝑥 ) is uniformly elliptic (that is, 𝒪 = (0 , 𝜏 )× ⋯ ×(0 , 𝜏 d ) ) and twicecontinuously differentiable, and that 𝑏, 𝑐 ∈ ( (cid:82) n , (cid:82) 𝑛 ) (in particular (A1)-(A3) are automaticallysatisfied with 𝜃 ( 𝑣 ) = 𝑣 , 𝛾 = 1 and 𝒪 = (0 , 𝜏 ) × ⋯ × (0 , 𝜏 d ) ), in [3, Theorem 3.4.4] it hasbeen shown that: (i) the equation ̄ 𝛽 ( 𝑥 ) = 𝑏 ( 𝑥 ) − π ( 𝑏 ) admits a unique 𝜏 -periodic solution 𝛽 ∈ 𝐶 ( (cid:82) n , (cid:82) n ) , and (ii) it holds that(3.1) { 𝑋 𝜀 ( 𝑥, 𝑡 ) − 𝜀 −1 π ( 𝑏 ) 𝑡 } 𝑡 ≥ ⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇒ 𝜀 → { 𝑊 𝖺 , 𝖻 ( 𝑥, 𝑡 )} 𝑡 ≥ , where { 𝑊 𝖺 , 𝖻 ( 𝑥, 𝑡 )} 𝑡 ≥ is a 𝑛 -dimensional Brownian motion determined by covariance matrix anddrift vector(3.2) 𝖺 = π ( ( (cid:73) n − D 𝛽 ) 𝑎 ( (cid:73) n − D 𝛽 ) T ) and 𝖻 = π ( ( (cid:73) n − D 𝛽 ) 𝑐 ) , respectively. In this section, we derive an analogous results for the case when 𝑎 ( 𝑥 ) is not neces-sarily uniformly elliptic. From (A1)-(A2), that is, from eq. (2.9), we see that the function 𝑥 ↦ − ∫ ∞0 ̄ 𝑡 ( 𝑏 − π ( 𝑏 ) ) ( 𝑥 ) d 𝑡 , 𝑥 ∈ (cid:82) n , (which we again denote by 𝛽 ( 𝑥 ) ) is well defined, 𝜏 -periodic, continuous, and satisfies 𝛽 ∈ ̄ and ̄ 𝛽 ( 𝑥 ) = 𝑏 ( 𝑥 ) − π ( 𝑏 ) . Note that under the uniform ellipticity (and smoothness) as-sumption this function coincides with the function 𝛽 ( 𝑥 ) discussed above. A crucial step in theproof of eq. (3.1) in the uniformly elliptic case is an application of Itô’s formula to the process { 𝛽 ( 𝑋 𝜀 ( 𝑥, 𝑡 ))} 𝑡 ≥ (recall that in this case 𝛽 ∈ 𝐶 ( (cid:82) n , (cid:82) n ) ). On the other hand, in the case whenthe coefficient 𝑎 ( 𝑥 ) can be degenerate it is not clear how to conclude necessary smoothness of 𝛽 ( 𝑥 ) .3.1. The case 𝑐 ( 𝑥 ) ≡ . Observe first that(3.3) { 𝑋 𝜀 ( 𝑥, 𝑡 )} 𝑡 ≥ = { 𝜀 ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 )} 𝑡 ≥ = { 𝜀 ̄𝑋 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 )} 𝑡 ≥ . We now conclude the following.
ERIODIC HOMOGENIZATION OF LINEAR DEGENERATE PDEs 9
Theorem 3.1.
Under ( A1 ) - ( A3 ) , the relation in eq. (3.1) holds with 𝖺 = π ( 𝑎 − ̄𝑎 − ̄𝑎 T − 𝛽 ( ̄ ,𝜏 𝛽 ) T − ( ̄ ,𝜏 𝛽 ) 𝛽 T ) and 𝖻 = 0 , where ̄𝑎 ∈ ( (cid:82) n , (cid:82) n×n ) is 𝜏 -periodic and such that π ( ‖ ̄𝑎 ‖ HS ) < ∞ .Proof. We have(3.4) 𝑋 𝜀 ( 𝑥, 𝑡 ) − 𝑥 − 𝜀 −1 π ( 𝑏 ) 𝑡 − 𝜀𝛽 ( 𝑋 𝜀 ( 𝑥, 𝑡 )∕ 𝜀 ) + 𝜀𝛽 ( 𝑥 ∕ 𝜀 ) (d) = 𝜀 ̄𝑋 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 ) − 𝑥 − 𝜀 −1 π ( 𝑏 ) 𝑡 − 𝜀𝛽 ( ̄𝑋 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 ) ) + 𝜀𝛽 ( 𝑥 ∕ 𝜀 )= − 𝜀𝛽 ( ̄𝑋 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 ) ) + 𝜀𝛽 ( 𝑥 ∕ 𝜀 ) + 𝜀 ∫ 𝜀 −2 𝑡 ( 𝑏 ( ̄𝑋 ( 𝑥 ∕ 𝜀, 𝑠 ) ) − π ( 𝑏 ) ) d 𝑠 + 𝜀 ∫ 𝜀 −2 𝑡 σ ( ̄𝑋 ( 𝑥 ∕ 𝜀, 𝑠 ) ) d 𝐵 ( 𝑠 ) ∀ 𝑡 ≥ . Due to boundedness of 𝛽 ( 𝑥 ) , { 𝑋 𝜀 ( 𝑥, 𝑡 ) − 𝑥 − 𝜀 −1 π ( 𝑏 ) 𝑡 } 𝑡 ≥ converges in law if, and only if, { 𝑋 𝜀 ( 𝑥, 𝑡 ) − 𝑥 − 𝜀 −1 π ( 𝑏 ) 𝑡 − 𝜀𝛽 ( 𝑋 𝜀 ( 𝑥, 𝑡 )∕ 𝜀 ) + 𝜀𝛽 ( 𝑥 ∕ 𝜀 )} 𝑡 ≥ converges, and if this is the case thelimit is the same. Denote 𝑀 ( 𝑥, 𝑡 ) ∶= 𝛽 ( ̄𝑋 ( 𝑥, 𝑡 ) ) − 𝛽 ( 𝑥 ) − ∫ 𝑡 ( 𝑏 ( ̄𝑋 ( 𝑥, 𝑠 ) ) − π ( 𝑏 ) ) d 𝑠 , 𝑡 ≥ ,𝑀 ( 𝑥, 𝑡 ) ∶= ∫ 𝑡 σ ( ̄𝑋 ( 𝑥, 𝑠 ) ) d 𝐵 ( 𝑠 ) , 𝑡 ≥ . According to [14, Proposition 4.1.7] the processes { 𝑀 𝑘 ( 𝑥, 𝑡 )} 𝑡 ≥ , 𝑘 = 1 , , are { 𝑡 } 𝑡 ≥ -martingales.Hence, { 𝑋 𝜀 ( 𝑥, 𝑡 ) − 𝑥 − 𝜀 −1 π ( 𝑏 ) 𝑡 − 𝜀𝛽 ( 𝑋 𝜀 ( 𝑥, 𝑡 )∕ 𝜀 ) + 𝜀𝛽 ( 𝑥 )} 𝑡 ≥ is also a { 𝑡 } 𝑡 ≥ -martingale. Next,let { 𝜃 𝑡 } 𝑡 ≥ be the family of shift operators on (Ω , , { 𝑡 } 𝑡 ≥ ) satisfying 𝜃 𝑠 ◦ ̄𝑋 ( 𝑥, 𝑡 ) = ̄𝑋 ( 𝑥, 𝑠 + 𝑡 ) for all 𝑠, 𝑡 ≥ (see [29, p. 119]). We have that 𝑀 𝑘 ( 𝑥, 𝑠 + 𝑡 ) = 𝑀 𝑘 ( 𝑥, 𝑡 ) + 𝜃 𝑡 ◦ 𝑀 𝑘 ( 𝑥, 𝑠 ) , 𝑠, 𝑡 ≥ , 𝑘 = 1 , . In other words, the processes { 𝑀 𝑘 ( 𝑥, 𝑡 )} 𝑡 ≥ , 𝑘 = 1 , , are continuous additive martingales withrespect to { ̄𝑋 ( 𝑥, 𝑡 )} 𝑡 ≥ , in the sense of [7]. Observe that 𝑀 ( 𝑥, 𝑡 ) = ̄𝑋 ( 𝑥, 𝑡 ) − 𝑥 − ∫ 𝑡 𝑏 ( ̄𝑋 ( 𝑥, 𝑠 ) ) d 𝑠 , 𝑡 ≥ . According to [20, Theorem VIII.2.17], in order to conclude eq. (3.1) it suffices to show that(3.5) 𝜀 ⟨ − 𝑀 ( 𝑥 ∕ 𝜀, ⋅ ) + 𝑀 ( 𝑥 ∕ 𝜀, ⋅ ) , − 𝑀 ( 𝑥 ∕ 𝜀, ⋅ ) + 𝑀 ( 𝑥 ∕ 𝜀, ⋅ ) ⟩ 𝑡 ∕ 𝜀 (cid:80) ←←←←←←←←←←←←←←←→ 𝜀 → 𝖺 𝑡 , 𝑡 ≥ . Here, (cid:80) ←←←←←←→ stands for the convergence in probability, and for two locally square-integrable martin-gales { 𝑀 ( 𝑡 )} 𝑡 ≥ and { 𝑁 ( 𝑡 )} 𝑡 ≥ , { ⟨ 𝑀 ( ⋅ ) , 𝑁 ( ⋅ ) ⟩ 𝑡 } 𝑡 ≥ denotes the corresponding predictable qua-dratic covariation process and we write { ⟨ 𝑀 ( ⋅ ) ⟩ 𝑡 } 𝑡 ≥ instead of { ⟨ 𝑀 ( ⋅ ) , 𝑀 ( ⋅ ) ⟩ 𝑡 } 𝑡 ≥ .For 𝑖, 𝑗 = 1 , … , d and 𝑘, 𝑙 = 1 , , we have that ⟨ 𝑀 𝑖𝑘 ( 𝑥, ⋅ ) , 𝑀 𝑗𝑙 ( 𝑥, ⋅ ) ⟩ 𝑡 = 4 −1 (⟨ 𝑀 𝑖𝑘 ( 𝑥, ⋅ ) + 𝑀 𝑗𝑙 ( 𝑥, ⋅ ) ⟩ 𝑡 − ⟨ 𝑀 𝑖𝑘 ( 𝑥, ⋅ ) − 𝑀 𝑗𝑙 ( 𝑥, ⋅ ) ⟩ 𝑡 ) , 𝑡 ≥ . Next, form the martingale representation theorem we see that for each 𝑖, 𝑗 = 1 , … d and 𝑘, 𝑙 =1 , it holds that d ⟨ 𝑀 𝑖𝑘 ( 𝑥, ⋅ ) ± 𝑀 𝑗𝑙 ( 𝑥, ⋅ ) ⟩ 𝑡 ≪ d 𝑡 . Thus, due to 𝜏 -periodicity of the coefficientsand the fact that { ̄𝑋 ( 𝑥 + 𝑘 𝜏 , 𝑡 )} 𝑡 ≥ and { ̄𝑋 ( 𝑥, 𝑡 ) + 𝑘 𝜏 } 𝑡 ≥ are indistinguishable for all 𝑥 ∈ (cid:82) d and 𝑘 𝜏 ∈ (cid:90) d 𝜏 , [8, Proposition 3.56] implies that for each 𝑖, 𝑗 = 1 , … , d and 𝑘, 𝑙 = 1 , there is anon-negative and 𝜏 -periodic ̃𝑎 𝑘,𝑙 ± 𝑖𝑗 ∈ ( (cid:82) d , (cid:82) ) such that ⟨ 𝑀 𝑖𝑘 ( 𝑥, ⋅ ) ± 𝑀 𝑗𝑙 ( 𝑥, ⋅ ) ⟩ 𝑡 = ∫ 𝑡 ̃𝑎 𝑘𝑙 ± 𝑖𝑗 ( ̄𝑋 ( 𝑥, 𝑠 ) ) d 𝑠 , 𝑡 ≥ . ERIODIC HOMOGENIZATION OF LINEAR DEGENERATE PDEs 10
Due to boundedness of 𝑏 ( 𝑥 ) , 𝛽 ( 𝑥 ) and σ ( 𝑥 ) we have π ( ̃𝑎 𝑘𝑙 ± 𝑖𝑗 ) = ∫ (cid:84) d 𝜏 (cid:69) [ ∫ ̃𝑎 𝑘𝑙 ± 𝑖𝑗 ( ̄𝑋 ,𝜏 ( 𝑥, 𝑠 ) ) d 𝑠 ] π (d 𝑥 )= ∫ (cid:84) d 𝜏 (cid:69) [⟨ 𝑀 𝑖𝑘 ( 𝑧 𝑥 , ⋅ ) ± 𝑀 𝑗𝑙 ( 𝑧 𝑥 , ⋅ ) ⟩ ] π (d 𝑥 )= ∫ (cid:84) d 𝜏 (cid:69) [( 𝑀 𝑖𝑘 ( 𝑧 𝑥 ,
1) ± 𝑀 𝑗𝑙 ( 𝑧 𝑥 , ) ] π (d 𝑥 ) ≤ ∫ (cid:84) d 𝜏 (cid:69) [( 𝑀 𝑖𝑘 ( 𝑧 𝑥 , ) ] π (d 𝑥 ) + 2 ∫ (cid:84) d 𝜏 (cid:69) [( 𝑀 𝑗𝑙 ( 𝑧 𝑥 , ) ] π (d 𝑥 ) < ∞ , where 𝑧 𝑥 ∈ Π −1 𝜏 ({ 𝑥 }) is arbitrary. Set now ̄𝑎 𝑘𝑙𝑖𝑗 ( 𝑥 ) ∶= ( ̃𝑎 𝑘𝑙 + 𝑖𝑗 ( 𝑥 ) − ̃𝑎 𝑘𝑙 − 𝑖𝑗 ( 𝑥 ))∕4 , and ̄𝑎 𝑘𝑙 ( 𝑥 ) ∶=( ̄𝑎 𝑘𝑙𝑖𝑗 ( 𝑥 )) 𝑖,𝑗 =1 , … , d . Clearly, for all 𝑘, 𝑙 = 1 , , ̄𝑎 𝑘𝑙 ∈ ( (cid:82) d , (cid:82) d×d ) is 𝜏 -periodic, and satisfies π ( ‖ ̄𝑎 𝑘𝑙 ‖ HS ) < ∞ and ⟨ 𝑀 𝑘 ( 𝑥, ⋅ ) , 𝑀 𝑙 ( 𝑥, ⋅ ) ⟩ 𝑡 = ∫ 𝑡 ̄𝑎 𝑘𝑙 ( ̄𝑋 ( 𝑥, 𝑠 ) ) d 𝑠 , 𝑡 ≥ . Furthermore, ̄𝑎 ( 𝑥 ) and ̄𝑎 ( 𝑥 ) are symmetric and non-negative definite. Directly from Propo-sition 2.4 and Birkhoff ergodic theorem it follows that for all 𝑡 ≥ (3.6) 𝜀 ⟨ 𝑀 𝑘 ( 𝑥 ∕ 𝜀, ⋅ ) , 𝑀 𝑙 ( 𝑥 ∕ 𝜀, ⋅ ) ⟩ 𝑡 ∕ 𝜀 = 𝜀 ∫ 𝑡 ∕ 𝜀 ̄𝑎 𝑘𝑙 ( ̄𝑋 ,𝜏 (Π 𝜏 ( 𝑥 ∕ 𝜀 ) , 𝑠 ) ) d 𝑠 (cid:80) -a.s. ←←←←←←←←←←←←←←←←←←→ 𝜀 → π ( ̄𝑎 𝑘𝑙 ) 𝑡 . It remains to determine π ( ̄𝑎 𝑘𝑙 ) . For 𝑘 = 𝑙 = 1 dominated convergence theorem implies that π ( ̄𝑎 ) = lim 𝜀 → 𝜀 ∫ (cid:84) n 𝜏 (cid:69) [⟨ 𝑀 ( 𝑧 𝑥 , ⋅ ) , 𝑀 ( 𝑧 𝑥 , ⋅ ) ⟩ 𝜀 ] π (d 𝑥 )= lim 𝜀 → 𝜀 ∫ (cid:84) n 𝜏 (cid:69) [ ( ∫ 𝜀 −2 ̄ 𝛽 ( ̄𝑋 ( 𝑧 𝑥 , 𝑠 ) ) d 𝑠 ) ( ∫ 𝜀 −2 ̄ 𝛽 ( ̄𝑋 ( 𝑧 𝑥 , 𝑠 ) ) d 𝑠 ) T − ( ∫ 𝜀 −2 ̄ 𝛽 ( ̄𝑋 ( 𝑧 𝑥 , 𝑠 ) ) d 𝑠 ) ( 𝛽 ( ̄𝑋 ( 𝑧 𝑥 , 𝜀 ) ) − 𝛽 ( 𝑧 𝑥 ) ) T − ( 𝛽 ( ̄𝑋 ( 𝑧 𝑥 , 𝜀 ) ) − 𝛽 ( 𝑧 𝑥 ) ) ( ∫ 𝜀 −2 ̄ 𝛽 ( ̄𝑋 ( 𝑧 𝑥 , 𝑠 ) ) d 𝑠 ) T ] π (d 𝑥 )= lim 𝜀 → 𝜀 ∫ (cid:84) n 𝜏 (cid:69) [ ( ∫ 𝜀 −2 ̄ ,𝜏 𝛽 ( ̄𝑋 ,𝜏 ( 𝑥, 𝑠 ) ) d 𝑠 ) ( ∫ 𝜀 −2 ̄ ,𝜏 𝛽 ( ̄𝑋 ,𝜏 ( 𝑥, 𝑠 ) ) d 𝑠 ) T − ( ∫ 𝜀 −2 ̄ ,𝜏 𝛽 ( ̄𝑋 ,𝜏 ( 𝑥, 𝑠 ) ) d 𝑠 ) ( 𝛽 ( ̄𝑋 ,𝜏 ( 𝑥, 𝜀 ) )) T − ( 𝛽 ( ̄𝑋 ,𝜏 ( 𝑥, 𝜀 ) )) ( ∫ 𝜀 −2 ̄ ,𝜏 𝛽 ( ̄𝑋 ,𝜏 ( 𝑥, 𝑠 ) ) d 𝑠 ) T ] π (d 𝑥 )+ ∫ (cid:84) n 𝜏 ( lim 𝜀 → 𝜀 ∫ 𝜀 −2 ̄ ,𝜏𝑠 ̄ ,𝜏 𝛽 ( 𝑥 ) d 𝑠 ) ( 𝛽 ( 𝑥 ) ) T π (d 𝑥 ) ERIODIC HOMOGENIZATION OF LINEAR DEGENERATE PDEs 11 + ∫ (cid:84) n 𝜏 𝛽 ( 𝑥 ) ( lim 𝜀 → 𝜀 ∫ 𝜀 −2 ̄ ,𝜏𝑠 ̄ ,𝜏 𝛽 ( 𝑥 ) d 𝑠 ) T π (d 𝑥 )= lim 𝜀 → 𝜀 ∫ (cid:84) n 𝜏 (cid:69) [ ∫ 𝜀 −2 ∫ 𝜀 −2 𝑠 ̄ ,𝜏 𝛽 ( ̄𝑋 ,𝜏 ( 𝑥, 𝑣 ) )( ̄ ,𝜏 𝛽 ( ̄𝑋 ,𝜏 ( 𝑥, 𝑠 ) )) T d 𝑣 d 𝑠 − ( ∫ 𝜀 −2 ̄ ,𝜏 𝛽 ( ̄𝑋 ,𝜏 ( 𝑥, 𝑠 ) ) d 𝑠 ) ( 𝛽 ( ̄𝑋 ,𝜏 ( 𝑥, 𝜀 ) )) T − ( 𝛽 ( ̃𝑋 ,𝜏 ( 𝑥, 𝜀 ) )) ( ∫ 𝜀 −2 ̄ ,𝜏 𝛽 ( ̄𝑋 ,𝜏 ( 𝑥, 𝑠 ) ) d 𝑠 ) T ] π (d 𝑥 )+ π ( ̄ ,𝜏 𝛽 ) π ( 𝛽 T ) + π ( ( ̄ ,𝜏 𝛽 ) T ) π ( 𝛽 )= lim 𝜀 → 𝜀 ∫ (cid:84) n 𝜏 (cid:69) [ ∫ 𝜀 −2 ( ∫ 𝜀 −2 𝑠 ̄ ,𝜏 𝛽 ( ̄𝑋 ,𝜏 ( 𝑥, 𝑣 ) ) d 𝑣 − 𝛽 ( ̄𝑋 ,𝜏 ( 𝑥, 𝜀 ) ))( ̄ ,𝜏 𝛽 ( ̄𝑋 ,𝜏 ( 𝑥, 𝑠 ) )) T d 𝑠 + ∫ 𝜀 −2 ̄ ,𝜏 𝛽 ( ̄𝑋 ,𝜏 ( 𝑥, 𝑠 ) )( ∫ 𝜀 −2 𝑠 ̄ ,𝜏 𝛽 ( ̃𝑋 ,𝜏 ( 𝑥, 𝑣 ) ) d 𝑣 − 𝛽 ( ̄𝑋 ,𝜏 ( 𝑥, 𝜀 ) )) T d 𝑠 ] π (d 𝑥 ) . Set 𝑀 𝜏 ( 𝑥, 𝑡 ) ∶= 𝛽 ( ̄𝑋 ,𝜏 ( 𝑥, 𝑡 ) ) − 𝛽 ( 𝑥 ) − ∫ 𝑡 ̄ ,𝜏 𝛽 ( ̄𝑋 ,𝜏 ( 𝑥, 𝑠 ) ) d 𝑠 , 𝑡 ≥ . Clearly, { 𝑀 𝜏 ( 𝑥, 𝑡 )} 𝑡 ≥ is a { 𝑡 } 𝑡 ≥ -martingale. We now have π ( ̄𝑎 ) = lim 𝜀 → 𝜀 ∫ (cid:84) n 𝜏 (cid:69) [ ∫ 𝜀 −2 ( 𝑀 𝜏 ( 𝑥, 𝑠 ) − 𝑀 𝜏 ( 𝑥, 𝜀 ) − 𝛽 ( ̄𝑋 ,𝜏 ( 𝑥, 𝑠 ) ))( ̄ ,𝜏 𝛽 ( ̄𝑋 ,𝜏 ( 𝑥, 𝑠 ) )) T d 𝑠 + ∫ 𝜀 −2 ̄ ,𝜏 𝛽 ( ̄𝑋 ,𝜏 ( 𝑥, 𝑠 ) ) ( 𝑀 𝜏 ( 𝑥, 𝑠 ) − 𝑀 𝜏 ( 𝑥, 𝜀 ) − 𝛽 ( ̄𝑋 ,𝜏 ( 𝑥, 𝑠 ) )) T d 𝑠 ] π (d 𝑥 )= − lim 𝜀 → 𝜀 ∫ (cid:84) n 𝜏 (cid:69) [ ∫ 𝜀 −2 𝛽 ( ̄𝑋 ,𝜏 ( 𝑥, 𝑠 ) ) ( ̄ ,𝜏 𝛽 ( ̄𝑋 ,𝜏 ( 𝑥, 𝑠 ) )) T d 𝑠 + ∫ 𝜀 −2 ̄ ,𝜏 𝛽 ( ̄𝑋 ,𝜏 ( 𝑥, 𝑠 ) ) ( 𝛽 ( ̄𝑋 ,𝜏 ( 𝑥, 𝑠 ) )) T d 𝑠 ] π (d 𝑥 )= − ∫ (cid:84) n 𝜏 lim 𝜀 → 𝜀 ∫ 𝜀 −2 ̄ ,𝜏𝑠 ( 𝛽 ( ⋅ ) ( ̄ ,𝜏 𝛽 ( ⋅ ) ) T + ̄ ,𝜏 𝛽 ( ⋅ ) ( 𝛽 ( ⋅ ) ) T ) ( 𝑥 ) d 𝑠 π (d 𝑥 )= − π ( 𝛽 ( ̄ ,𝜏 𝛽 ) T + ( ̄ ,𝜏 𝛽 ) 𝛽 T ) . For 𝑘 = 𝑙 = 2 it follows from [4, Proposition 2.5] that ̄ π ( ̄𝑎 ) = π ( 𝑎 ) . For mixed terms wehave ⟨ 𝑀 ( 𝑥, ⋅ ) , 𝑀 ( 𝑥, ⋅ ) ⟩ 𝑡 = ⟨ 𝑀 ( 𝑥, ⋅ ) , 𝑀 ( 𝑥, ⋅ ) ⟩ T 𝑡 , 𝑡 ≥ . Therefore ̄𝑎 ( 𝑥 ) ∶= ̄𝑎 ( 𝑥 ) = ( ̄𝑎 ( 𝑥 ) ) T ,which completes the proof. (cid:3) Let us now give several remarks.
ERIODIC HOMOGENIZATION OF LINEAR DEGENERATE PDEs 12
Remark 3.2. (i) Notice that 𝖺 = lim 𝑡 → ∞ 𝑡 ∫ (cid:84) n 𝜏 (cid:69) [( ̄𝑋 ( 𝑧 𝑥 , 𝑡 ) − 𝑧 𝑥 − π ( 𝑏 ) 𝑡 )( ̄𝑋 ( 𝑧 𝑥 , 𝑡 ) − 𝑧 𝑥 − π ( 𝑏 ) 𝑡 ) T ] π (d 𝑥 ) . (ii) In the proof of Theorem 3.1 we did not use the full strength of the assumptions in(A1)-(A3). It is straightforward to check that the assertion of the theorem holds iffor 𝜏 -periodic σ ∈ 𝑏 ( (cid:82) n , (cid:82) n×m ) and 𝑏 ∈ 𝑏 ( (cid:82) n , (cid:82) n ) the SDE in eq. (2.3) admits aunique strong solution which is a time-homogeneous non-explosive strong Markov pro-cess whose projection on (cid:84) n 𝜏 (under Π 𝜏 ( 𝑥 ) ) satisfies the conclusion of Proposition 2.4(for 𝜀 = 0 ).(iii) In the second part of the proof of Theorem 3.1 we show that(3.7) { 𝜀 ∫ 𝜀 −2 𝑡 ̄ 𝛽 ( ̄𝑋 𝑠 ) d 𝑠 } 𝑡 ≥ ⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇒ 𝜀 → { 𝑊 , (0 , 𝑡 )} 𝑡 ≥ , where { 𝑊 , (0 , 𝑡 )} 𝑡 ≥ is a 𝑛 -dimensional zero-drift Brownian motion (starting from theorigin) determined by covariance matrix = − π ( 𝛽 ( ̄ ,𝜏 𝛽 ) T + ( ̄ ,𝜏 𝛽 ) 𝛽 T ) . The proof isbased on (a version of) Motoo’s theorem given in [8, Proposition 3.56], combined withProposition 2.4 and Birkhoff ergodic theorem. It also follows from [4, Remark 2.1.1](together with Proposition 2.4 and [4, Proposition 2.5]), which is based on a CLT forstationary ergodic sequences given in [6, Chapter 4.19]. Furthermore, by setting ℎ ( 𝑥 ) ∶= ̄𝑎 ( 𝑥 ) + 𝛽 ( 𝑥 ) ( ̄ 𝛽 ( 𝑥 ) ) T + ̄ 𝛽 ( 𝑥 ) ( 𝛽 ( 𝑥 ) ) T , 𝑥 ∈ (cid:82) n , in [26, Corollary to Theorem 3.1] it has been shown that { 𝛽 ( ̄𝑋 ( 𝑥, 𝑡 ) )( 𝛽 ( ̄𝑋 ( 𝑥, 𝑡 ) )) T − 𝛽 ( 𝑥 ) ( 𝛽 ( 𝑥 ) ) T − ∫ 𝑡 ℎ ( ̄𝑋 ( 𝑥, 𝑠 ) ) d 𝑠 } 𝑡 ≥ is a { 𝑡 } 𝑡 ≥ -local martingale. Let { τ 𝑥𝑘 } 𝑘 ≥ be the corresponding localizing sequence.Then, (cid:69) [ 𝛽 ( ̄𝑋 ( 𝑥, 𝑡 ∧ τ 𝑥𝑘 ) )( 𝛽 ( ̄𝑋 ( 𝑥, 𝑡 ∧ τ 𝑥𝑘 ) )) T ] − 𝛽 ( 𝑥 ) ( 𝛽 ( 𝑥 ) ) T = (cid:69) [ ∫ 𝑡 ∧ τ 𝑥𝑘 ℎ ( ̄𝑋 ( 𝑥, 𝑠 ) ) d 𝑠 ] for all 𝑘 ≥ and 𝑡 ≥ . Since ℎ ( 𝑥 ) is 𝜏 -periodic and satisfies π (‖ ℎ ‖ HS ) < ∞ and ∫ 𝑡 (cid:69) [‖ ℎ ( ̄𝑋 ( 𝑥, 𝑠 ) )‖ HS ] d 𝑠 < ∞ , 𝑡 ≥ , by taking 𝑘 → ∞ in the previous relation, and employing dominated convergence the-orem, it follows that π ( ℎ ) = 0 , which again proves that π ( ̄𝑎 ) = − π ( 𝛽 ( ̄ ,𝜏 𝛽 ) T +( ̄ ,𝜏 𝛽 ) 𝛽 T ) . See also [20, VIII.3.65] for an analogous result.Let us also remark that the CLT of this type is a very well studied problem in theliterature, and it is known that it holds for general ergodic Markov processes (see [4],[20, Chapter VIII.3] and [24, Chapter 2]). To the best of our knowledge [2] and [19]are the only two works discussing this problem in the context of (ergodic) diffusionprocesses with possibly degenerate diffusion coefficient. However, in both works certain“incremental type” assumptions on the coefficients have been imposed, which excludediffusion processes with periodic coefficients. Therefore, it seems that Theorem 3.1 isthe first result in the literature showing the relation in eq. (3.7) in the case of a periodicdiffusion process with degenerate diffusion coefficient.
ERIODIC HOMOGENIZATION OF LINEAR DEGENERATE PDEs 13
The case 𝑐 ( 𝑥 ) ≢ . In this case, it is not clear that we can perform an analogous analysisas in Theorem 3.1. The difficulty is that the equality in distribution in eq. (3.3) does not holdanymore, which implies that the function ̄𝑎 ( 𝑥 ) (appearing in Theorem 3.1) might also depend onthe parameter 𝜀 . In [18, Lemma 3.2] the authors suggest a solution to this problem but, unfor-tunately, there seems to be a doubt about its proof. Namely, in the proof it is assumed that thefunction ̂𝑏 ( 𝑥 ) is twice continuously differentiable, but it is shown that it is continuously differ-entiable only. In what follows we resolve this issue, or at least suggest an alternative approachto the problem. We impose an additional assumption on the coefficients σ ( 𝑥 ) and 𝑏 ( 𝑥 ) , which istaken from [18] (see [18, Assumption H.3]). Let σ 𝑗 ( 𝑥 ) ∶= ( σ 𝑗 ( 𝑥 ) , … , σ n 𝑗 ( 𝑥 )) T , 𝑗 = 1 , … , m ,and let 𝒰 ⊆ [0 , 𝜏 ] be the set where the parabolic Hörmander condition holds, that is, the set of 𝑥 ∈ [0 , 𝜏 ] for which the Lie algebra generated by ( 𝑏 ( 𝑥 ) ,
1) ∪ {( σ ( 𝑥 ) , , … , ( σ m ( 𝑥 ) , spans (cid:82) n+1 . Observe that 𝒪 ⊆ 𝒰 . Assume the following (A4): σ ∈ ∞ ( (cid:82) n , (cid:82) 𝑚 ) , 𝑏, 𝑐 ∈ ∞ ( (cid:82) n , (cid:82) n ) , and inf 𝑡> sup 𝑥 ∈ (cid:82) n (cid:69) [‖ 𝐽 𝑥 ( 𝑡 ) ‖ HS (cid:49) [ 𝑡, ∞] ( ̄ τ ,𝑥 𝒰 + 𝜏 ) ] < , where { 𝐽 𝑥 ( 𝑡 )} 𝑡 ≥ is the Jacobian of the stochastic flow associated to { ̄𝑋 ( 𝑥, 𝑡 )} 𝑡 ≥ , that is, asolution to d 𝐽 𝑥 ( 𝑡 ) = D 𝑏 ( ̄𝑋 ( 𝑥, 𝑡 ) ) 𝐽 𝑥 ( 𝑡 ) d 𝑡 + m ∑ 𝑗 =1 D σ 𝑗 ( ̄𝑋 ( 𝑥, 𝑡 ) ) 𝐽 𝑥 ( 𝑡 ) d 𝐵 𝑗 ( 𝑡 ) 𝐽 𝑥 (0) = (cid:73) n . As it has been commented in [18, Remark 2.1], a simple condition ensuring the above relationto hold is the existence of 𝑡 > such that (cid:80) ( ̄ τ ,𝑥 𝒰 + 𝜏 < 𝑡 ) = 1 for all 𝑥 ∈ (cid:82) n . According to[17, Lemma II.9.2 and Theorem II.9.5], smoothness of σ ( 𝑥 ) , 𝑏 ( 𝑥 ) and 𝑐 ( 𝑥 ) implies that ̄ 𝑡 𝑓 ∈ 𝑘 ( (cid:82) n , (cid:82) ) for any 𝑡 ≥ and 𝑓 ∈ 𝑘𝑏 ( (cid:82) n , (cid:82) ) , 𝑘 = 0 , , . Also, under (A1)-(A4), in [18, Lemma2.6] it has been shown that there are ̄𝛾 > and ̄𝛤 > , such that(3.8) ‖ ∇ ̄ 𝑡 𝑓 ( ⋅ ) ‖ ∞ ≤ ̄𝛤 (‖ 𝑓 ‖ ∞ + ‖ ∇ 𝑓 ( ⋅ ) ‖ ∞ ) e − ̄𝛾𝑡 for all 𝑡 ≥ and 𝜏 -periodic 𝑓 ∈ ( (cid:82) n , (cid:82) ) with π ( 𝑓 ) = 0 . In particular, 𝛽 ∈ ( (cid:82) n , (cid:82) n ) . Inwhat follows we derive an Itô-type formula for the process { 𝛽 ( ̄𝑋 𝜀 ( 𝑥, 𝑡 ))} 𝑡 ≥ . Let ( ̄ 𝜀,𝜏 , ̄ 𝜀,𝜏 ) bethe 𝑏 -infinitesimal generator of { ̄𝑋 𝜀,𝜏 ( 𝑥, 𝑡 )} 𝑡 ≥ . We start with the following auxiliary lemma. Lemma 3.3.
Assume ( A1 ) - ( A3 ) , and let 𝑓 ∈ ( (cid:82) n , (cid:82) ) be 𝜏 -periodic. Then, 𝑓 ∈ ̄ 𝜀,𝜏 , and ̄ 𝜀,𝜏 𝑓 ( 𝑥 ) = ̄ 𝜀 𝑓 ( 𝑧 𝑥 ) for all 𝑥 ∈ (cid:84) n 𝜏 and 𝑧 𝑥 ∈ Π −1 𝜏 ({ 𝑥 }) .Proof. As we have already commented, 𝑓 ∈ ̄ 𝜀 and ̄ 𝜀 𝑓 ( 𝑥 ) = 12 Tr ( 𝑎 ( 𝑥 )∇∇ T 𝑓 ( 𝑥 ) ) + ( 𝑏 ( 𝑥 ) + 𝜀𝑐 ( 𝑥 ) ) T ∇ 𝑓 ( 𝑥 ) . Thus, lim 𝑡 → ‖‖‖( ̄ 𝜀,𝜏𝑡 𝑓 ( ⋅ ) − 𝑓 ( ⋅ ) ) ∕ 𝑡 − ̄ 𝜀 𝑓 ( 𝑧 ⋅ ) ‖‖‖ ∞ = lim 𝑡 → ‖‖‖( ̄ 𝜀𝑡 𝑓 − 𝑓 ) ∕ 𝑡 − ̄ 𝜀 𝑓 ‖‖‖ ∞ = 0 . (cid:3) Let 𝑓 ∈ ( (cid:82) n , (cid:82) ) be 𝜏 -periodic. Define 𝜙 ( 𝑥 ) ∶= − ∫ ∞0 ̄ 𝑡 ( 𝑓 − π ( 𝑓 ) ) ( 𝑥 ) d 𝑡 , 𝑥 ∈ (cid:82) n . According to eq. (2.9) this function is well defined, 𝜏 -periodic, continuous, and satisfies 𝜙 ∈ ̄ and ̄ 𝜙 ( 𝑥 ) = 𝑓 ( 𝑥 ) − π ( 𝑓 ) . ERIODIC HOMOGENIZATION OF LINEAR DEGENERATE PDEs 14
Lemma 3.4.
Assume 𝑓 ∈ ( (cid:82) n , (cid:82) ) . Under ( A1 ) - ( A4 ) it holds that 𝜙 ( ̄𝑋 𝜀 ( 𝑥, 𝑡 ) ) = 𝜙 ( 𝑥 ) + ∫ 𝑡 ( 𝑓 − π ( 𝑓 )) ( ̄𝑋 𝜀 ( 𝑥, 𝑠 ) ) d 𝑠 + 𝜀 ∫ 𝑡 (( ∇ 𝜙 ) T 𝑐 )( ̄𝑋 𝜀 ( 𝑥, 𝑠 ) ) d 𝑠 + ∫ 𝑡 (( ∇ 𝜙 ) T σ )( ̄𝑋 𝜀 ( 𝑥, 𝑠 ) ) d 𝐵 𝜀 ( 𝑠 ) ∀ 𝑡 ≥ . Proof.
As we have already commented, 𝑥 ↦ ̄ 𝑠 ( 𝑓 − π ( 𝑓 ))( 𝑥 ) is twice continuously differen-tiable for any 𝑠 ≥ . Itô’s formula then gives(3.9) ̄ 𝑠 ( 𝑓 − π ( 𝑓 )) ( ̄𝑋 𝜀 ( 𝑥, 𝑡 ) ) = ̄ 𝑠 ( 𝑓 − π ( 𝑓 ))( 𝑥 ) + ∫ 𝑡 ̄ 𝜀 ̄ 𝑠 ( 𝑓 − π ( 𝑓 )) ( ̄𝑋 𝜀 ( 𝑥, 𝑢 ) ) d 𝑢 + ∫ 𝑡 (( ∇ ̄ 𝑠 ( 𝑓 − π ( 𝑓 )) ) T σ )( ̄𝑋 𝜀 ( 𝑥, 𝑢 ) ) d 𝐵 𝜀 ( 𝑢 )= ̄ 𝑠 ( 𝑓 − π ( 𝑓 ))( 𝑥 ) + ∫ 𝑡 ̄ ̄ 𝑠 ( 𝑓 − π ( 𝑓 )) ( ̄𝑋 𝜀 ( 𝑥, 𝑢 ) ) d 𝑢 + 𝜀 ∫ 𝑡 (( ∇ ̄ 𝑠 ( 𝑓 − π ( 𝑓 )) ) T 𝑐 )( ̄𝑋 𝜀 ( 𝑥, 𝑢 ) ) d 𝑢 + ∫ 𝑡 (( ∇ ̄ 𝑠 ( 𝑓 − π ( 𝑓 )) ) T σ )( ̄𝑋 𝜀 ( 𝑥, 𝑢 ) ) d 𝐵 𝜀 ( 𝑢 ) . By integrating the previous relation with respect to the time variable 𝑠 ∈ [0 , ∞) (and recallingthe definition of the function 𝜙 ( 𝑥 ) ), we arrive at(3.10) 𝜙 ( ̄𝑋 𝜀 ( 𝑥, 𝑡 ) ) = 𝜙 ( 𝑥 ) − ∫ ∞0 ∫ 𝑡 ̄ ̄ 𝑠 ( 𝑓 − π ( 𝑓 )) ( ̄𝑋 𝜀 ( 𝑥, 𝑢 ) ) d 𝑢 d 𝑠 + 𝜀 ∫ 𝑡 (( ∇ 𝜙 ) T 𝑐 )( ̄𝑋 𝜀 ( 𝑥, 𝑢 ) ) d 𝑢 + ∫ 𝑡 (( ∇ 𝜙 ) T σ )( ̄𝑋 𝜀 ( 𝑥, 𝑢 ) ) d 𝐵 𝜀 ( 𝑢 ) ∀ 𝑡 ≥ . The last two integrals on the right-hand side in eq. (3.10) are well defined, and follow from thelast two terms in eq. (3.9), because of eq. (3.8). By observing that ̄ ̄ 𝑠 𝑓 ( 𝑥 ) = ̄ 𝑠 ̄ 𝑓 ( 𝑥 ) = ̄ ,𝜏𝑠 ̄ ,𝜏 𝑓 (Π 𝜏 ( 𝑥 )) (the last equality follows from Lemma 3.3), and π ( ̄ ,𝜏 𝑓 ) = 0 , eq. (2.9)implies that the second term on the right-hand side in eq. (3.10) is well defined. It remains toprove that − ∫ ∞0 ̄ ̄ 𝑠 ( 𝑓 − π ( 𝑓 ))( 𝑥 ) d 𝑠 = ( 𝑓 − π ( 𝑓 ))( 𝑥 ) ∀ 𝑥 ∈ (cid:82) n . We have ∫ ∞0 ̄ ̄ 𝑡 ( 𝑓 − π ( 𝑓 ))( 𝑥 ) d 𝑡 = ∫ ∞0 lim 𝑠 → ̄ 𝑠 + 𝑡 ( 𝑓 − π ( 𝑓 ))( 𝑥 ) − ̄ 𝑡 ( 𝑓 − π ( 𝑓 ))( 𝑥 ) 𝑠 d 𝑡 . By employing Itô’s formula, Lemma 3.3 and eq. (2.9), we have ‖ ̄ 𝑠 + 𝑡 ( 𝑓 − π ( 𝑓 )) − ̃ 𝑡 ( 𝑓 − π ( 𝑓 )) ‖ ∞ 𝑠 ≤ ∫ 𝑠 + 𝑡𝑡 ‖ ̄ 𝑢 ̄ ( 𝑓 − π ( 𝑓 )) ‖ ∞ d 𝑢𝑠 ≤ 𝛤 ‖ ̄ ( 𝑓 − π ( 𝑓 )) ‖ ∞ e − 𝛾𝑡 for all 𝑠, 𝑡 ∈ (0 , ∞) . The result now follows from the dominated convergence theorem. (cid:3) We are now ready to prove the main result of this subsection.
ERIODIC HOMOGENIZATION OF LINEAR DEGENERATE PDEs 15
Theorem 3.5.
Under ( A1 ) - ( A4 ) , the relation in eq. (3.1) holds with 𝖺 and 𝖻 given in eq. (3.2) .Proof. By combining Lemma 3.4 (applied to 𝑏 ( 𝑥 ) ) with eq. (2.2) we have 𝜀 ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 ) − 𝜀 −1 π ( 𝑏 ) 𝑡 − 𝑥 − 𝜀𝛽 ( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 ) ) + 𝜀𝛽 ( 𝑥 ∕ 𝜀 )= 𝜀 ∫ 𝜀 −2 𝑡 ( ( (cid:73) n − D 𝛽 ) 𝑐 )( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑠 ) ) d 𝑠 + 𝜀 ∫ 𝜀 −2 𝑡 ( ( (cid:73) n − D 𝛽 ) σ )( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑠 ) ) d 𝐵 𝜀 ( 𝑠 ) ∀ 𝑡 ≥ . Recall that 𝑋 𝜀 ( 𝑥, 𝑡 ) = 𝜀 ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 ) , 𝑡 ≥ . Hence, due to boundedness of 𝛽 ( 𝑥 ) , { 𝑋 𝜀 ( 𝑥, 𝑡 ) − 𝜀 −1 π ( 𝑏 ) 𝑡 } 𝑡 ≥ converges in law if, and only if, { 𝜀 ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 ) − 𝜀 −1 π ( 𝑏 ) 𝑡 − 𝜀𝛽 ( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 )) + 𝜀𝛽 ( 𝑥 ∕ 𝜀 )} 𝑡 ≥ converges, and if this is the case the limit is the same. Clearly, { 𝜀 ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 )− 𝜀 −1 π ( 𝑏 ) 𝑡 − 𝜀𝛽 ( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 ))+ 𝜀𝛽 ( 𝑥 ∕ 𝜀 )} 𝑡 ≥ is a semimartingale with boundedvariation and predictable quadratic covariation parts { 𝜀 ∫ 𝜀 −2 𝑡 ( ( (cid:73) n − D 𝛽 ) 𝑐 )( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑠 ) ) d 𝑠 } 𝑡 ≥ , and { 𝜀 ∫ 𝜀 −2 𝑡 ( ( (cid:73) n − D 𝛽 ) 𝑎 ( (cid:73) n − D 𝛽 ) T )( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑠 ) ) d 𝑠 } 𝑡 ≥ , respectively. From [20, Theorem VI.3.21] we see that both these processes are tight. Conse-quently, [20, Theorem VI.4.18] implies tightness of { 𝜀 ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 ) − 𝜀 −1 π ( 𝑏 ) 𝑡 − 𝜀𝛽 ( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 ) ) + 𝜀𝛽 ( 𝑥 ∕ 𝜀 )} 𝑡 ≥ . To this end, it remains to prove finite-dimensional conver-gence in law of { 𝜀 ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 )− 𝜀 −1 π ( 𝑏 ) 𝑡 − 𝜀𝛽 ( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 ))+ 𝜀𝛽 ( 𝑥 ∕ 𝜀 )} 𝑡 ≥ to { 𝑊 𝖺 , ( 𝑥, 𝑡 )} 𝑡 ≥ .According to [20, Theorem VIII.2.4] this will hold if 𝜀 ∫ 𝜀 −2 𝑡 ( ( (cid:73) n − D 𝛽 ) 𝑐 )( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑠 ) ) d 𝑠 (cid:80) ←←←←←←←←←←←←←←←→ 𝜀 → 𝖻 𝑡 , and 𝜀 ∫ 𝜀 −2 𝑡 ( ( (cid:73) n − D 𝛽 ) 𝑎 ( (cid:73) n − D 𝛽 ) T )( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑠 ) ) d 𝑠 (cid:80) ←←←←←←←←←←←←←←←→ 𝜀 → 𝖺 𝑡 for all 𝑡 ≥ . Due to 𝜏 -periodicity, we have that 𝜀 ∫ 𝜀 −2 𝑡 ( ( (cid:73) n − D 𝛽 ) 𝑐 − 𝖻 )( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑠 ) ) d 𝑠 = 𝜀 ∫ 𝜀 −2 𝑡 ( ( (cid:73) n − D 𝛽 ) 𝑐 − 𝖻 )( ̄𝑋 𝜀,𝜏 (Π 𝜏 ( 𝑥 ∕ 𝜀 ) , 𝑠 ) ) d 𝑠 ∀ 𝑡 ≥ , and an analogous relation holds for the predictable quadratic covariation part. We now have 𝜀 (cid:69) [ ( ∫ 𝜀 −2 𝑡 ( ( (cid:73) n − D 𝛽 ) 𝑐 − π 𝜀 ( ( (cid:73) n − D 𝛽 ) 𝑐 ))( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑠 ) ) d 𝑠 ) T ( ∫ 𝜀 −2 𝑡 ( ( (cid:73) n − D 𝛽 ) 𝑐 − π 𝜀 ( ( (cid:73) n − D 𝛽 ) 𝑐 ))( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑠 ) ) d 𝑠 ) ] = 𝜀 (cid:69) [ ( ∫ 𝜀 −2 𝑡 ( ( (cid:73) n − D 𝛽 ) 𝑐 − π 𝜀 ( ( (cid:73) n − D 𝛽 ) 𝑐 ))( ̄𝑋 𝜀,𝜏 (Π 𝜏 ( 𝑥 ∕ 𝜀 ) , 𝑠 ) ) d 𝑠 ) T ERIODIC HOMOGENIZATION OF LINEAR DEGENERATE PDEs 16 ( ∫ 𝜀 −2 𝑡 ( ( (cid:73) n − D 𝛽 ) 𝑐 − π 𝜀 ( ( (cid:73) n − D 𝛽 ) 𝑐 ))( ̄𝑋 𝜀,𝜏 (Π 𝜏 ( 𝑥 ∕ 𝜀 ) , 𝑠 ) ) d 𝑠 ) ] = 2 𝜀 ∫ 𝜀 −2 𝑡 ∫ 𝑠 (cid:69) [(( ( (cid:73) n − ∇ 𝛽 ) 𝑐 − π 𝜀 ( ( (cid:73) n − D 𝛽 ) 𝑐 ))( ̄𝑋 𝜀,𝜏 (Π 𝜏 ( 𝑥 ∕ 𝜀 ) , 𝑠 ) )) T (( ( (cid:73) n − D 𝛽 ) 𝑐 − π 𝜀 ( ( (cid:73) n − D 𝛽 ) 𝑐 ))( ̄𝑋 𝜀,𝜏 (Π 𝜏 ( 𝑥 ∕ 𝜀 ) , 𝑢 ) ))] d 𝑢 d 𝑠 = 2 𝜀 ∫ 𝜀 −2 𝑡 ∫ 𝑠 (cid:69) [( ̄ 𝜀,𝜏𝑠 − 𝑢 ( ( (cid:73) n − D 𝛽 ) 𝑐 − π 𝜀 ( ( (cid:73) n − D 𝛽 ) 𝑐 ))( ̄𝑋 𝜀,𝜏 (Π 𝜏 ( 𝑥 ∕ 𝜀 ) , 𝑢 ) )) T (( ( (cid:73) n − D 𝛽 ) 𝑐 − π 𝜀 ( ( (cid:73) n − D 𝛽 ) 𝑐 ))( ̄𝑋 𝜀,𝜏 (Π 𝜏 ( 𝑥 ∕ 𝜀 ) , 𝑢 ) ))] d 𝑢 d 𝑠 ≤ 𝜀 𝛤 ‖ ( (cid:73) n − D 𝛽 ) 𝑐 ‖ ∫ 𝜀 −2 𝑡 ∫ 𝑠 e − 𝛾 ( 𝑠 − 𝑢 ) d 𝑢 d 𝑠 = 8 𝜀 𝛤 ‖ ( (cid:73) n − D 𝛽 ) 𝑐 ‖ 𝛾 ( 𝜀 −2 𝑡 + e − 𝛾𝜀 −2 𝑡 − 1 ) , where in the fourth step we employed eq. (2.9). Thus, 𝜀 ( (cid:69) [ ( ∫ 𝜀 −2 𝑡 ( ( (cid:73) n − D 𝛽 ) 𝑐 − 𝖻 )( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑠 ) ) d 𝑠 ) T ( ∫ 𝜀 −2 𝑡 ( ( (cid:73) n − D 𝛽 ) 𝑐 − 𝖻 )( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑠 ) ) d 𝑠 ) ]) ≤ 𝜀 ( (cid:69) [ ( ∫ 𝜀 −2 𝑡 ( ( (cid:73) n − D 𝛽 ) 𝑐 − π 𝜀 ( ( (cid:73) n − D 𝛽 ) 𝑐 ))( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑠 ) ) d 𝑠 ) T ( ∫ 𝜀 −2 𝑡 ( ( (cid:73) n − D 𝛽 ) 𝑐 − π 𝜀 ( ( (cid:73) n − D 𝛽 ) 𝑐 ))( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑠 ) ) d 𝑠 ) ]) + | π 𝜀 ( ( (cid:73) n − D 𝛽 ) 𝑐 ) − 𝖻 | 𝑡 . Analogous estimate holds for for the predictable quadratic covariation part. Finally, by letting 𝜀 → the result follows from Proposition 2.5. (cid:3) Let us now give several remarks.
Remark 3.6. (i) Observe that when 𝑏 ( 𝑥 ) ≡ 𝑏 ∈ (cid:82) n then 𝛽 ( 𝑥 ) ≡ and, in this case, theconclusion of Theorem 3.5 holds under (A1)-(A3) (that is, assumption (A4) is not nec-essary).(ii) By additionally assuming (A4) in Theorem 3.1 we can derive a more explicit form of thecovariance matrix 𝖺 . According to Lemma 3.4 it holds that 𝛽 ( ̄𝑋 ( 𝑥, 𝑡 )) = 𝛽 ( 𝑥 ) + ∫ 𝑡 ( 𝑏 − π ( 𝑏 )) ( ̄𝑋 ( 𝑥, 𝑠 ) ) d 𝑠 + ∫ 𝑡 ( D 𝛽 σ )( ̄𝑋 ( 𝑥, 𝑠 ) ) d 𝐵 ( 𝑠 ) ∀ 𝑡 ≥ . Now, from eq. (3.4), Proposition 2.4, Birkhoff ergodic theorem and [4, Proposition 2.5](analogously as in eq. (3.6)) it follows that π ( ̄𝑎 ) = − π (D 𝛽 𝑎 ) . Thus, 𝖺 = π ( 𝑎 + ( D 𝛽 ) 𝑎 + 𝑎 ( D 𝛽 ) T − 𝛽 ( ̄ 𝛽 ) T − ( ̄ 𝛽 ) 𝛽 T ) . ERIODIC HOMOGENIZATION OF LINEAR DEGENERATE PDEs 17
Let us remark that an analogous representation of the covariance matrix 𝖺 in the uni-formly elliptic case has been derived in [5]. More precisely, under the same assumptionsas in [3, Theorem 3.4.3] ( 𝑎 ( 𝑥 ) is uniformly elliptic and twice continuously differentiable,and 𝑏 ∈ ( (cid:82) n , (cid:82) ) ) in [5, Theorem 3] it has been shown that π (d 𝑥 ) admits a 𝜏 -periodiccontinuously differentiable density function 𝜋 ( 𝑥 ) (with respect to the Lebesgue measureon (cid:84) n 𝜏 ), and 𝖺 has the following representation(3.11) π ( 𝑎 − 𝛽 ( ̄ 𝛽 ) T − ( ̄ 𝛽 ) 𝛽 T ) + ( ∫ (cid:84) n 𝜏 ( 𝛽 𝑖 ( 𝑥 ) n ∑ 𝑘 =1 𝜕 𝑘 ( 𝜋 ( 𝑥 ) 𝑎 𝑘𝑗 ( 𝑥 ) ) + 𝛽 𝑗 ( 𝑥 ) n ∑ 𝑘 =1 𝜕 𝑘 ( 𝜋 ( 𝑥 ) 𝑎 𝑘𝑖 ( 𝑥 ) )) d 𝑥 ) ≤ 𝑖,𝑗 ≤ n . Recall that in this situation 𝛽 ∈ ( (cid:82) n , (cid:82) n ) . Also, a direct computation shows thateq. (3.11) transforms to eq. (3.2), and vice versa .4. H OMOGENIZATION OF LINEAR
PDE S Denote by { ̂ 𝜀𝑡 } 𝑡 ≥ and ( ̂ 𝜀 , ̂ 𝜀 ) , and { ̂ 𝑡 } 𝑡 ≥ and ( ̂ , ̂ ) the operator semigroup and 𝑏 -infinitesimal generator of { 𝑋 𝜀 ( 𝑥, 𝑡 ) − 𝜀 −1 π ( 𝑏 ) 𝑡 } 𝑡 ≥ and { 𝑊 𝖺 , 𝖻 ( 𝑥, 𝑡 )} 𝑡 ≥ , respectively. Observethat for any 𝑓 ∈ 𝑢,𝑏 ( (cid:82) n , (cid:82) ) , ̂ 𝜀 𝑓 ( 𝑥 ) = 𝜀 𝑓 ( 𝑥 ) − 𝜀 −1 π ( 𝑏 ) T ∇ 𝑓 ( 𝑥 ) , and ̂ 𝑓 ( 𝑥 ) = 2 −1 Tr ( 𝖺 ∇∇ T 𝑓 ( 𝑥 ) ) + 𝖻 T ∇ 𝑓 ( 𝑥 ) . As a consequence of [25, Theorem 1.1], [22, Theorem 17.25] and Theorems 3.1 and 3.5 we havethe following.
Proposition 4.1.
Assume ( A1 ) - ( A4 ) (or ( A1 ) - ( A3 ) if 𝑐 ( 𝑥 ) ≡ or 𝑏 ( 𝑥 ) ≡ 𝑏 ∈ (cid:82) n ). Then, for any 𝑡 ≥ and 𝑓 ∈ ∞ ( (cid:82) n , (cid:82) ) ∪ { 𝑓 ∈ ( (cid:82) n , (cid:82) ) ∶ 𝑓 ( 𝑥 ) is 𝜏 -periodic } it holds that lim 𝜀 → sup ≤ 𝑡 ≤ 𝑡 ‖ ̂ 𝜀𝑡 𝑓 − ̂ 𝑡 𝑓 ‖ ∞ = 0 , and for any 𝑓 ∈ 𝑐 ( (cid:82) n , (cid:82) ) ∪ { 𝑓 ∈ ( (cid:82) n , (cid:82) ) ∶ 𝑓 ( 𝑥 ) is 𝜏 -periodic } , lim 𝜀 → ‖ ̂ 𝜀 𝑓 − ̂ 𝑓 ‖ ∞ = 0 . Let us now turn to the problem of homogenization of the problems in eqs. (1.2) and (1.3). Inthe sequel, we assume that π ( 𝑏 ) = 0 . This, in particular, implies that { ̂ 𝜀𝑡 } 𝑡 ≥ = { 𝜀𝑡 } 𝑡 ≥ and ̂ 𝜀 = 𝜀 for 𝜀 > . In the case when 𝑏 ( 𝑥 ) ≡ 𝑏 ≠ (thus π ( 𝑏 ) ≠ ) one can easily constructexamples satisfying (A3). Recall that in this case (A4) is not required. On the other hand, when 𝑏 ( 𝑥 ) vanishes (A3) in general does not have to hold. A typical example satisfying π ( 𝑏 ) = 0 and(A1)-(A4) can be constructed as follows. For 𝑖 = 1 , … , n put 𝑏 𝑖 ( 𝑥 ) ∶= 2 −1 n ∑ 𝑗 =1 𝜕 𝑗 𝑎 𝑖𝑗 ( 𝑥 ) + ̄𝑏 𝑖 ( 𝑥 ) , 𝑥 ∈ (cid:82) n , where ̄𝑏 𝑖 ( 𝑥 ) is 𝜏 -periodic, of class ∞ , does not depend on 𝑥 𝑖 , and satisfies ∫ [0 ,𝜏 ] ̄𝑏 𝑖 ( 𝑥 ) d 𝑥 = 0 . Itis then easy to see that π (d 𝑥 ) is the Lebegues measure on (cid:84) n 𝜏 and π ( 𝑏 ) = 0 , and it is not hardto construct examples satisfying (A1)-(A4).For instance, let n = 2 and 𝜏 = (10 , T , and take 𝜏 -periodic σ ∈ 𝐶 ∞ ( (cid:82) , (cid:82) ) such that 𝑎 ( 𝑥, 𝑦 ) = σ ( 𝑥, 𝑦 ) σ ( 𝑥, 𝑦 ) T is positive definite on ℬ (5 ,
5) + 𝜏 and 𝑎 ( 𝑥, 𝑦 ) ≡ on ([0 , ⧵ ERIODIC HOMOGENIZATION OF LINEAR DEGENERATE PDEs 18 ℬ (5 , 𝜏 . Here, ℬ 𝑟 ( 𝑥 ) stands for the open ball of radius 𝑟 > around 𝑥 ∈ (cid:82) n . For examplewe can take σ ( 𝑥, 𝑦 ) = ( (cid:49) ℬ (5 , ( 𝑥, 𝑦 ) e −19−( 𝑥 −5)2−( 𝑦 −5)2 ) (cid:73) . It remains to choose ̄𝑏 , ̄𝑏 ∈ 𝐶 ∞ ( (cid:82) , (cid:82) ) . Observe that this is enough to satisfy condition (A1),condition (A2) with 𝒪 = ℬ (5 , , and that in condition (A4) we have 𝒰 = 𝒪 = ℬ (5 , .Notice also that for conditions (A3) and (A4) to be satisfied it is enough to take such 𝑏 ( 𝑥, 𝑦 ) and 𝑐 ( 𝑥, 𝑦 ) that there exists 𝑡 > such that (cid:80) ( ̄ τ 𝜀, ( 𝑥,𝑦 ) ℬ (5 , 𝜏 < 𝑡 ) = 1 for all ( 𝜀, ( 𝑥, 𝑦 )) ∈ [0 , 𝜀 ] ×([0 , ⧵ ℬ (5 , for some 𝜀 > . Take for instance ̄𝑏 ( 𝑥, 𝑦 ) = ̃𝑏 ( 𝑦 ) and ̄𝑏 ( 𝑥, 𝑦 ) = ̃𝑏 ( 𝑥 ) such that ̃𝑏 ( 𝑥 ) is 𝜏 -periodic and positive for 𝑥 ∈ [0 ,
4] ∪ [6 , and on [4 , define it so that ∫ ̃𝑏 ( 𝑥 ) d 𝑥 = 0 . For example we can take ̃𝑏 ( 𝑥 ) = { , 𝑥 ∈ [0 ,
4] ∪ [6 , , 𝛽 e −11−( 𝑥 −5)2 , 𝑥 ∈ (4 , , where 𝛽 > is such that ∫ ̃𝑏 ( 𝑥 ) d 𝑥 = 0 . Notice that with such definition of 𝑏 ( 𝑥, 𝑦 ) we have thatthere exists 𝑡 > such that for all ( 𝑥, 𝑦 ) ∈ ([0 , ⧵ ℬ (5 , 𝜏 , (cid:80) ( ̄ τ , ( 𝑥,𝑦 ) ℬ (5 , 𝜏 < 𝑡 ) = 1 . Indeedsuppose that we take ( 𝑥, 𝑦 ) from the central white area in Figure 1, while we remain in white areaF IGURE
1. Visualization of different areas of domain for drift term 𝑏 ( 𝑥, 𝑦 ) .we move at the constant speed diagonally up and to the right (see Figure 2). We either hit theupper right circle, right pink strip or upper blue strip. If we hit the circle, we are done. If we hitthe right pink strip (if we hit the upper blue strip we reason in an analogous way), we continuemoving to the right but start to go down. Therefore we either hit the lower right circle of exit thepink strip to the right between two circles. But the later is not possible because ∫ ̃𝑏 ( 𝑥 ) d 𝑥 = 0 and 𝜏 -periodicity of ̃𝑏 ( 𝑥 ) imply that if the process moves horizontally for it must verticallyreturn to the same height (see Figure 2). Observe that in the previous example 𝑐 ( 𝑥, 𝑦 ) ≡ . However, one can easily see that the same assertion holds with (appropriately chosen) 𝑐 ( 𝑥, 𝑦 ) ≢ by choosing 𝜀 small enough.4.1. The elliptic problem in eq. (1.2).
Assume that 𝒟 is an open bounded subset of (cid:82) n satis-fying the following:(i) 𝒟 = { 𝑥 ∶ 𝒹 ( 𝑥 ) < for some 𝒹 ∈ 𝑏 ( (cid:82) n , (cid:82) ) ,(ii) | ∇ 𝒹 ( 𝑥 ) | ≥ 𝛿 > for all 𝑥 ∈ 𝜕 𝒟 . ERIODIC HOMOGENIZATION OF LINEAR DEGENERATE PDEs 19 F IGURE
2. Visualization of the vector field 𝑏 ( 𝑥, 𝑦 ) outside of supp σ .Further, suppose that 𝑒 ∶ 𝒟 → (−∞ , − 𝛼 ] , 𝛼 > , 𝑓 ∶ 𝒟 → (cid:82) and 𝑔 ∶ 𝜕 𝒟 → (cid:82) are continuous,and assume that { 𝑥 ∈ 𝜕 𝒟 ∶ (cid:80) ( ̂ τ 𝜀,𝑥 >
0) = 0} is a (topologically) closed set for all 𝜀 ∈ [0 , 𝜀 ] , where ̂ τ 𝜀,𝑥 ∶= inf { 𝑡 ≥ 𝑋 𝜀 ( 𝑥, 𝑡 ) ∉ 𝒟 } (recallthat π ( 𝑏 ) = 0 ) and ̂ τ ,𝑥 ∶= inf { 𝑡 ≥ 𝑊 𝖺 , 𝖻 ( 𝑥, 𝑡 ) ∉ 𝒟 } . Then, according to [32, Theorem3.49],(4.1) 𝑢 𝜀 ( 𝑥 ) ∶= (cid:69) [ 𝑔 ( 𝑋 𝜀 ( 𝑥, ̂ τ 𝜀,𝑥 ) ) e ∫ ̂ τ 𝜀,𝑥 𝑒 ( 𝑋 𝜀 ( 𝑥,𝑠 )∕ 𝜀 ) d 𝑠 + ∫ ̂ τ 𝜀,𝑥 𝑓 ( 𝑋 𝜀 ( 𝑥, 𝑠 ) ) e ∫ 𝑠 𝑒 ( 𝑋 𝜀 ( 𝑥,𝑠 )∕ 𝜀 ) d 𝑢 d 𝑠 ] is a unique continuous viscosity solution (see [32, Section 6.5] for the definition of viscositysolutions) to eq. (1.2). Theorem 4.2.
In addition to the above assumptions, assume ( A1 ) - ( A4 ) (or ( A1 ) - ( A3 ) if 𝑐 ( 𝑥 ) ≡ or 𝑏 ( 𝑥 ) ≡ ), that (cid:80) (( ∇ 𝒹 ( 𝑊 𝖺 , 𝖻 ( 𝑥, ̂ τ ,𝑥 ) )) T 𝖺 ∇ 𝒹 ( 𝑊 𝖺 , 𝖻 ( 𝑥, ̂ τ ,𝑥 ) ) > ) = 1 ∀ 𝑥 ∈ 𝒟 and that 𝑒 ( 𝑥 ) is 𝜏 -periodic (here we consider 𝑒 ( 𝑥 ) as a 𝜏 -periodic, continuous and boundedfunction on (cid:82) n such that 𝑒 ( 𝑥 ) ≤ − 𝛼 for all 𝑥 ∈ (cid:82) n ). Then, lim 𝜀 → 𝑢 𝜀 ( 𝑥 ) = 𝑢 ( 𝑥 ) ∀ 𝑥 ∈ 𝒟 , where 𝑢 ( 𝑥 ) ∶= (cid:69) [ 𝑔 ( 𝑊 𝖺 , 𝖻 ( 𝑥, ̂ τ ,𝑥 ) ) e π ( 𝑒 ) ̂ τ ,𝑥 + ∫ ̂ τ ,𝑥 𝑓 ( 𝑊 𝖺 , 𝖻 ( 𝑥, 𝑠 ) ) e π ( 𝑒 ) 𝑠 d 𝑠 ] is a solution to ̂ 𝑢 ( 𝑥 ) + π ( 𝑒 ) 𝑢 ( 𝑥 ) + 𝑓 ( 𝑥 ) = 0 , 𝑥 ∈ 𝒟 ,𝑢 ( 𝑥 ) = 𝑔 ( 𝑥 ) , 𝑥 ∈ 𝜕 𝒟 . Proof.
We follow the approach from [3, Theorem 3.4.5]. Define 𝜁 𝜀 ( 𝑥, 𝑡 ) ∶= ∫ 𝑡 𝑒 ( 𝑋 𝜀 ( 𝑥, 𝑠 )∕ 𝜀 ) d 𝑠 = 𝜀 ∫ 𝜀 −2 𝑡 𝑒 ( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑠 ) ) d 𝑠 . ERIODIC HOMOGENIZATION OF LINEAR DEGENERATE PDEs 20
Analogously as in the proof of Theorem 3.5 we see that 𝜁 𝜀 ( 𝑥, 𝑡 ) = 𝜀 ∫ 𝜀 −2 𝑡 𝑒 ( ̄𝑋 𝜀,𝜏 (Π 𝜏 ( 𝑥 ∕ 𝜀 ) , 𝑠 ) ) d 𝑠 L ( (cid:80) ) ←←←←←←←←←←←←←←←←←←→ 𝜀 → π ( 𝑒 ) 𝑡 , where L 𝑝 ( (cid:80) ) ←←←←←←←←←←←←←←←←←←→ stands for the convergence in L 𝑝 ( (cid:80) ) , 𝑝 ≥ . Set 𝜁 ( 𝑥, 𝑡 ) ∶= π ( 𝑒 ) 𝑡 . This, togetherwith the fact that the process { 𝜁 𝜀 ( 𝑥, 𝑡 )} 𝑡 ≥ is tight (due to [20, Theorem VI.3.21]), implies that { 𝜁 𝜀 ( 𝑥, 𝑡 )} 𝑡 ≥ ⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇒ 𝜀 → { 𝜁 ( 𝑥, 𝑡 )} 𝑡 ≥ . Since { 𝜁 ( 𝑥, 𝑡 )} 𝑡 ≥ is a constant in the space ([0 , ∞) , (cid:82) ) , usingTheorem 3.5 and [6, Theorem 3.9] we conclude(4.2) {( 𝑋 𝜀 , 𝜁 𝜀 )( 𝑥, 𝑡 )} 𝑡 ≥ ⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇒ 𝜀 → {( 𝑊 𝖺 , 𝖻 , 𝜁 )( 𝑥, 𝑡 )} 𝑡 ≥ . We next endow the space ([0 , ∞) , (cid:82) n ) × ([0 , ∞) , (cid:82) ) with the Borel 𝜎 -algebra generated bythe sets { ( 𝑊 𝖺 , 𝖻 , 𝜉 )( 𝑥, ⋅ ) ∈ ([0 , ∞) , (cid:82) n ) × ([0 , ∞) , (cid:82) ) ∶ ( 𝑊 𝖺 , 𝖻 , 𝜃 )( 𝑥, 𝑠 ) ∈ 𝐵 } where 𝑠 ∈ [0 , ∞) and 𝐵 ∈ 𝔅 ( (cid:82) n+1 ) . The processes {( 𝑋 𝜀 , 𝜉 𝜀 )( 𝑥, 𝑡 )} 𝑡 ≥ and {( 𝑊 𝖺 , 𝖻 , 𝜉 )( 𝑥, 𝑡 )} 𝑡 ≥ introduce on ([0 , ∞) , (cid:82) n ) × ([0 , ∞) , (cid:82) ) probability measures µ 𝜀𝑥 and µ 𝑥 , respectively. Observe that µ 𝜀𝑥 (w) ⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇐⇒ 𝜀 → µ 𝑥 . We now define F ∶ ([0 , ∞) , (cid:82) n ) × ([0 , ∞) , (cid:82) ) → (cid:82) ∪ {∞} with F( 𝑌 , 𝜂 ) = ⎧⎪⎨⎪⎩ 𝑔 ( 𝑌 ( τ ( 𝑌 ) )) e 𝜂 ( τ ( 𝑌 )) + ∫ τ ( 𝑌 )0 𝑓 ( 𝑌 ( 𝑡 ) ) e 𝜂 ( 𝑡 ) d 𝑡 , τ ( 𝑌 ) < ∞ and 𝜂 ( 𝑡 ) ≤ − 𝛼𝑡 ∀ 𝑡 ≥ , ∫ ∞0 𝑓 ( 𝑌 ( 𝑡 ) ) e 𝜂 ( 𝑡 ) d 𝑡 , τ ( 𝑌 ) = ∞ and 𝜂 ( 𝑡 ) ≤ − 𝛼𝑡 ∀ 𝑡 ≥ , ∞ , otherwise , where τ ( 𝑌 ) ∶= inf { 𝑡 ≥ 𝑌 ( 𝑡 ) ∉ 𝒟 } . Clearly, 𝑢 𝜀 ( 𝑥 ) = (cid:69) [ F ( {( 𝑋 𝜀 , 𝜁 𝜀 )( 𝑥, 𝑡 )} 𝑡 ≥ )] and 𝑢 ( 𝑥 ) = (cid:69) [ F ( {( 𝑊 𝖺 , 𝖻 , 𝜁 )( 𝑥, 𝑡 )} 𝑡 ≥ )] . The function F( 𝑌 , 𝜂 ) has the following properties:(i) it is measurable and bounded a.s. with respect to µ 𝜀𝑥 and µ 𝑥 ;(ii) it is continuous a.s. with respect to µ 𝑥 .This, together with eq. (4.2), gives lim 𝜀 → 𝑢 𝜀 ( 𝑥 ) = 𝑢 ( 𝑥 ) ∀ 𝑥 ∈ 𝒟 , which concludes the proof.To this end, let us verify (i) and (ii). To see that (i) holds, note that if 𝜂 ( 𝑡 ) ≤ − 𝛼𝑡 for all 𝑡 ≥ ,then | F( 𝑌 , 𝜂 ) | ≤ ‖ 𝑔 ‖ ∞ + ‖ 𝑓 ‖ ∞ ∫ ∞0 e − 𝛼𝑡 d 𝑡 = ‖ 𝑔 ‖ ∞ + ‖ 𝑓 ‖ ∞ 𝛼 < ∞ . Due to the definition of processes { 𝜁 𝜀 ( 𝑥, 𝑡 )} 𝑡 ≥ and { 𝜁 ( 𝑥, 𝑡 )} 𝑡 ≥ , and the fact that for all 𝑥 ∈ (cid:82) n we have 𝑒 ( 𝑥 ) ≤ − 𝛼 , property (i) follows.To see property (ii), we need to check that if { 𝑌 𝑛 } 𝑛 ∈ (cid:78) converges to 𝑌 uniformly on compactintervals, then(4.3) lim 𝑛 → ∞ F( 𝑌 𝑛 , 𝜂 ) = F( 𝑌 , 𝜂 ) for 𝜂 ( 𝑡 ) = 𝜁 ( 𝑥, 𝑡 ) . Recall that 𝜁 ( 𝑥, 𝑡 ) ≤ − 𝛼𝑡 for all 𝑥 ∈ (cid:82) n and 𝑡 ≥ . The relation in eq. (4.3) willfollow from the proof of [3, Lemma 3.4.3] where it has been shown that if { 𝑌 𝑛 } 𝑛 ∈ (cid:78) convergesto 𝑌 uniformly on compact intervals and (cid:80) ((∇ 𝒹 ( 𝑊 𝖺 , 𝖻 ( 𝑥, ̂ τ ,𝑥 ))) T 𝖺 ∇ 𝒹 ( 𝑊 𝖺 , 𝖻 ( 𝑥, ̂ τ ,𝑥 )) >
0) = 1
ERIODIC HOMOGENIZATION OF LINEAR DEGENERATE PDEs 21 for all 𝑥 ∈ 𝒟 (see the definition of the function 𝛽 ( 𝑡 ) in [3, pp. 412]), then lim 𝑛 → ∞ τ ( 𝑌 𝑛 ) = τ ( 𝑌 ) .Namely, from this fact, and the dominated convergence theorem, we immediately see that lim 𝑛 → ∞ ∫ τ ( 𝑌 𝑛 )0 𝑓 ( 𝑌 𝑛 ( 𝑡 ) ) e 𝜂 ( 𝑡 ) d 𝑡 = ∫ τ ( 𝑌 )0 𝑓 ( 𝑌 ( 𝑡 ) ) e 𝜂 ( 𝑡 ) d 𝑡 . To this end, we need to prove that lim 𝑛 → ∞ 𝑔 ( 𝑌 𝑛 ( τ ( 𝑌 𝑛 ) )) e 𝜂 ( τ ( 𝑌 𝑛 )) (cid:49) { τ ( 𝑌 𝑛 ) < ∞} = { 𝑔 ( 𝑌 ( τ ( 𝑌 ) )) e 𝜂 ( τ ( 𝑌 )) , τ ( 𝑌 ) < ∞ , , τ ( 𝑌 ) = ∞ . If τ ( 𝑌 ) = ∞ , then lim 𝑛 → ∞ τ ( 𝑌 𝑛 ) = ∞ and since 𝑔 ( 𝑥 ) is bounded and 𝜂 ( 𝑡 ) ≤ − 𝛼𝑡 for all 𝑡 ≥ ,the assertion follows. If τ ( 𝑌 ) < ∞ , then there exist 𝑇 > ad 𝑛 𝑌 ∈ (cid:78) , such that τ ( 𝑌 𝑛 ) ∈ [0 , 𝑇 ] for all 𝑛 ≥ 𝑛 𝑌 . Therefore, ||| 𝑔 ( 𝑌 𝑛 ( τ ( 𝑌 𝑛 ) )) e 𝜂 ( τ ( 𝑌 𝑛 )) − 𝑔 ( 𝑌 ( τ ( 𝑌 ) )) e 𝜂 ( τ ( 𝑌 )) ||| ≤ ‖ 𝑔 ‖ ∞ ||| e 𝜂 ( τ ( 𝑌 𝑛 )) − e 𝜂 ( τ ( 𝑌 )) ||| + e 𝜂 ( τ ( 𝑌 )) ||| 𝑔 ( 𝑌 𝑛 ( τ ( 𝑌 𝑛 ) )) − 𝑔 ( 𝑌 ( τ ( 𝑌 𝑛 ) ))||| + e 𝜂 ( τ ( 𝑌 )) ||| 𝑔 ( 𝑌 ( τ ( 𝑌 𝑛 ) )) − 𝑔 ( 𝑌 ( τ ( 𝑌 ) ))||| ≤ ‖ 𝑔 ‖ ∞ ||| e 𝜂 ( τ ( 𝑌 𝑛 )) − e 𝜂 ( τ ( 𝑌 )) ||| + e 𝜂 ( τ ( 𝑌 )) sup ≤ 𝑡 ≤ 𝑇 ||| 𝑔 ( 𝑌 𝑛 ( 𝑡 ) ) − 𝑔 ( 𝑌 ( 𝑡 ) )||| + e 𝜂 ( τ ( 𝑌 )) ||| 𝑔 ( 𝑌 ( τ ( 𝑌 𝑛 ) )) − 𝑔 ( 𝑌 ( τ ( 𝑌 ) ))||| . Clearly, the first and last terms in the above inequality tend to zero as 𝑛 tends to infinity. Sup-pose that lim sup 𝑛 → ∞ sup ≤ 𝑡 ≤ 𝑇 | 𝑔 ( 𝑌 𝑛 ( 𝑡 )) − 𝑔 ( 𝑌 ( 𝑡 )) | > . Then there exist 𝜖 > and sequences { 𝑛 𝑘 } 𝑘 ∈ (cid:78) ⊆ (cid:78) and { 𝑡 𝑘 } 𝑘 ∈ (cid:78) ⊆ [0 , 𝑇 ] , such that lim 𝑘 → ∞ 𝑡 𝑘 = 𝑡 ∈ [0 , 𝑇 ] and | 𝑔 ( 𝑌 𝑛 𝑘 ( 𝑡 𝑘 ))− 𝑔 ( 𝑌 ( 𝑡 𝑘 )) | >𝜖 for all 𝑘 ∈ (cid:78) . However, this is not possible since lim 𝑘 → ∞ 𝑔 ( 𝑌 ( 𝑡 𝑘 )) = 𝑔 ( 𝑌 ( 𝑡 )) , and lim 𝑘 → ∞ ||| 𝑌 𝑛 𝑘 ( 𝑡 𝑘 ) − 𝑌 ( 𝑡 ) ||| ≤ lim 𝑘 → ∞ ||| 𝑌 𝑛 𝑘 ( 𝑡 𝑘 ) − 𝑌 ( 𝑡 𝑘 ) ||| + lim 𝑘 → ∞ || 𝑌 ( 𝑡 𝑘 ) − 𝑌 ( 𝑡 ) || ≤ lim 𝑘 → ∞ sup ≤ 𝑠 ≤ 𝑇 ||| 𝑌 𝑛 𝑘 ( 𝑠 ) − 𝑌 ( 𝑠 ) ||| + lim 𝑘 → ∞ || 𝑌 ( 𝑡 𝑘 ) − 𝑌 ( 𝑡 ) || = 0 . From this we conclude that lim 𝑛 → ∞ ||| 𝑔 ( 𝑌 𝑛 ( τ ( 𝑌 𝑛 ) )) e 𝜂 ( τ ( 𝑌 𝑛 )) − 𝑔 ( 𝑌 ( τ ( 𝑌 ) )) e 𝜂 ( τ ( 𝑌 )) ||| = 0 , which proves the assertion. (cid:3) The parabolic problem in eq. (1.3).
Let 𝑑, 𝑒 ∈ 𝑏 ( (cid:82) n , (cid:82) ) , and let 𝑓 , 𝑔 ∈ ( (cid:82) n , (cid:82) ) besuch that(4.4) | 𝑓 ( 𝑥 ) | + | 𝑔 ( 𝑥 ) | ≤ 𝐾 (1 + | 𝑥 | 𝜅 ) for some 𝜅, 𝐾 > and all 𝑥 ∈ (cid:82) n . Then, according to [32, Theorem 3.43] (see also [30, Remark2.5]), for any 𝜀 > ,(4.5) 𝑢 𝜀 ( 𝑥, 𝑡 ) ∶= (cid:69) [ 𝑔 ( 𝑋 𝜀 ( 𝑥, 𝑡 ) ) e ∫ 𝑡 ( 𝜀 −1 𝑑 ( 𝑋 𝜀 ( 𝑥,𝑠 )∕ 𝜀 )+ 𝑒 ( 𝑋 𝜀 ( 𝑥,𝑠 )∕ 𝜀 ) ) d 𝑠 + ∫ 𝑡 𝑓 ( 𝑋 𝜀 ( 𝑥, 𝑠 ) ) e ∫ 𝑠 ( 𝜀 −1 𝑑 ( 𝑋 𝜀 ( 𝑥,𝑢 )∕ 𝜀 )+ 𝑒 ( 𝑋 𝜀 ( 𝑥,𝑢 )∕ 𝜀 ) ) d 𝑢 d 𝑠 ] is a viscosity solution to eq. (1.2). Assume further that 𝑑 ( 𝑥 ) is 𝜏 -periodic, continuously differen-tiable and such that π ( 𝑑 ) = 0 (otherwise we can just replace 𝑑 ( 𝑥 ) by 𝑑 ( 𝑥 ) − π ( 𝑑 ) in eqs. (1.2) ERIODIC HOMOGENIZATION OF LINEAR DEGENERATE PDEs 22 and (4.5)). Then, (A1)-(A4) imply that 𝛿 ( 𝑥 ) ∶= − ∫ ∞0 ̄ 𝑡 𝑑 ( 𝑥 ) d 𝑡 , 𝑥 ∈ (cid:82) n , is well defined, 𝜏 -periodic, continuously differentiable, and satisfies 𝛿 ∈ ̄ and ̄ 𝛿 ( 𝑥 ) = 𝑑 ( 𝑥 ) . Theorem 4.3.
In addition to the above assumptions, assume ( A1 ) - ( A4 ) (or ( A1 ) - ( A3 ) if 𝑐 ( 𝑥 ) ≡ or 𝑏 ( 𝑥 ) ≡ , and 𝑑 ( 𝑥 ) ≡ ), 𝑑 ∈ ( (cid:82) n , (cid:82) ) and that 𝑒 ( 𝑥 ) is 𝜏 -periodic. Then, lim 𝜀 → 𝑢 𝜀 ( 𝑡, 𝑥 ) = 𝑢 ( 𝑡, 𝑥 ) ∀ ( 𝑡, 𝑥 ) ∈ [0 , ∞) × (cid:82) n , where 𝑢 ( 𝑡, 𝑥 ) ∶= (cid:69) [ 𝑔 ( ̄𝑊 ( 𝑥, 𝑡 )) e π ( −1 (∇ 𝛿 ) T 𝑎 ∇ 𝛿 + 𝑒 −(∇ 𝛿 ) T 𝑐 ) 𝑡 + ∫ 𝑡 𝑓 ( ̄𝑊 ( 𝑥, 𝑠 )) e π ( −1 (∇ 𝛿 ) T 𝑎 ∇ 𝛿 + 𝑒 −(∇ 𝛿 ) T 𝑐 ) 𝑠 d 𝑠 ] is a solution to 𝜕 𝑡 𝑢 ( 𝑥, 𝑡 ) = 𝑢 ( 𝑥, 𝑡 ) + π ( −1 (∇ 𝛿 ) T 𝑎 ∇ 𝛿 + 𝑒 − (∇ 𝛿 ) T 𝑐 ) 𝑢 ( 𝑥, 𝑡 ) + 𝑓 ( 𝑥 ) 𝑢 ( 𝑥,
0) = 𝑔 ( 𝑥 ) , 𝑥 ∈ (cid:82) n , and { ̄𝑊 ( 𝑥, 𝑡 )} 𝑡 ≥ is a n -dimensional Brownian motion with 𝑏 -infinitesimal generator , de-termined by drift vector ̄ 𝖻 ∶= 𝖻 − π (( (cid:73) n − D 𝛽 ) 𝑎 ∇ 𝛿 ) and covariance matrix 𝖺 .Proof. We first show that lim 𝜀 → (cid:69) [ 𝑔 ( 𝑋 𝜀 ( 𝑥, 𝑡 ) ) e ∫ 𝑡 ( 𝜀 −1 𝑑 ( 𝑋 𝜀 ( 𝑥,𝑠 )∕ 𝜀 )+ 𝑒 ( 𝑋 𝜀 ( 𝑥,𝑠 )∕ 𝜀 ) ) d 𝑠 ] = (cid:69) [ 𝑔 ( ̄𝑊 ( 𝑥, 𝑡 )) ] e π ( −1 (∇ 𝛿 ) T 𝑎 ∇ 𝛿 + 𝑒 −(∇ 𝛿 ) T 𝑐 ) 𝑡 . From Lemma 3.4 we have that 𝛿 ( ̄𝑋 𝜀 ( 𝑥, 𝑡 ) ) = 𝛿 ( 𝑥 ) + ∫ 𝑡 𝑑 ( ̄𝑋 𝜀 ( 𝑥, 𝑠 ) ) d 𝑠 + 𝜀 ∫ 𝑡 ( (∇ 𝛿 ) T 𝑐 )( ̄𝑋 𝜀 ( 𝑥, 𝑠 ) ) d 𝑠 + ∫ 𝑡 ( (∇ 𝛿 ) T σ )( ̄𝑋 𝜀 ( 𝑥, 𝑠 ) ) d 𝐵 𝜀 ( 𝑠 ) ∀ 𝑡 ≥ . From this we have that 𝜀 −1 ∫ 𝑡 𝑑 ( 𝜀 −1 𝑋 𝜀 ( 𝑥, 𝑠 ) ) d 𝑠 = 𝜀 ∫ 𝜀 −2 𝑡 𝑑 ( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑠 ) ) d 𝑠 = 𝜀𝛿 ( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 ) ) − 𝜀𝛿 ( 𝑥 ∕ 𝜀 ) − 𝜀 ∫ 𝜀 −2 𝑡 ( (∇ 𝛿 ) T 𝑐 )( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑠 ) ) d 𝑠 − 𝜀 ∫ 𝜀 −2 𝑡 ( (∇ 𝛿 ) T σ )( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑠 ) ) d 𝐵 𝜀 ( 𝑠 ) . By assumption, (cid:69) [||| 𝑔 ( 𝜀 ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀 , 𝑡 ∕ 𝜀 ) )||| ] ≤ 𝐾 ( 𝜀 𝜅 (cid:69) [||| ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 ) )||| 𝜅 ]) . (4.6) ERIODIC HOMOGENIZATION OF LINEAR DEGENERATE PDEs 23
Without loss of generality we may assume that 𝜅 ∈ (cid:78) . By combining eq. (2.2) and Lemma 3.4we have(4.7) 𝜀 ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 ) = 𝑥 + 𝜀𝛽 ( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 ) ) − 𝜀𝛽 ( 𝑥 ∕ 𝜀 )+ 𝜀 ∫ 𝜀 −2 𝑡 ( 𝑐 − D 𝛽 𝑐 ) ( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑠 ) ) d 𝑠 + 𝜀 ∫ 𝜀 −2 𝑡 ( σ − D 𝛽 σ ) ( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑠 ) ) d 𝐵 𝜀 ( 𝑠 ) , 𝑡 ≥ . Thus, | 𝜀 ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 ) | 𝜅 ≤ ̄𝐾 (| 𝑥 | 𝜅 + 𝜀 𝜅 + 𝑡 𝜅 + 𝜀 𝜅 ||||| ∫ 𝜀 −2 𝑡 ( σ − D 𝛽 σ ) ( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑠 ) ) d 𝐵 𝜀 ( 𝑠 ) ||||| 𝜅 ) , for some ̄𝐾 > which does not depend on 𝜀 . By employing Itô’s formula and Doob’s inequality,we conclude(4.8) 𝜀 𝜅 (cid:69) [| ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 ) | 𝜅 ] ≤ ̃𝐾 (| 𝑥 | 𝜅 + 𝜀 𝜅 + 𝑡 𝜅 ) , for some ̃𝐾 > which does not depend on 𝜀 . Consequently, ||||| 𝑢 𝜀 ( 𝑥, 𝑡 ) − (cid:69) [ 𝑔 ( 𝜀 ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 ) ) e − 𝜀 ∫ 𝜀 −2 𝑡 ( (∇ 𝛿 ) T 𝑐 − 𝑒 )( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀,𝑠 ) ) d 𝑠 − 𝜀 ∫ 𝜀 −2 𝑡 ( (∇ 𝛿 ) T σ )( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀,𝑠 ) ) d 𝐵 𝜀 ( 𝑠 ) ]||||| ≤ (cid:69) [||| 𝑔 ( 𝜀 ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 ) )||| ] (cid:69) [||| e 𝜀𝛿 ( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀,𝑡 ∕ 𝜀 ) ) − 𝜀𝛿 ( 𝑥 ∕ 𝜀 ) − 1 ||| ] (cid:69) [ e −8 𝜀 ∫ 𝜀 −2 𝑡 ( (∇ 𝛿 ) T 𝑎 ∇ 𝛿 )( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀,𝑠 ) ) d 𝑠 −4 𝜀 ∫ 𝜀 −2 𝑡 ( (∇ 𝛿 ) T σ )( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀,𝑠 ) ) d 𝐵 𝜀 ( 𝑠 ) ] e 𝑡 ‖ 𝛿 ) T 𝑎 ∇ 𝛿 −(∇ 𝛿 ) T 𝑐 + 𝑒 ‖ ∞ ≤ (cid:69) [||| 𝑔 ( 𝜀 ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝜀 −2 𝑡 ) )||| ] (cid:69) [||| e 𝜀𝛿 ( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀,𝜀 −2 𝑡 ) ) − 𝜀𝛿 ( 𝑥 ∕ 𝜀 ) − 1 ||| ] e 𝑡 ‖ 𝛿 ) T 𝑎 ∇ 𝛿 −(∇ 𝛿 ) T 𝑐 + 𝑒 ‖ ∞ . Thus, 𝑢 𝜀 ( 𝑥, 𝑡 ) converges as 𝜀 → if, and only if, (cid:69) [ 𝑔 ( 𝜀 ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 ) ) e − 𝜀 ∫ 𝜀 −2 𝑡 ( (∇ 𝛿 ) T 𝑐 − 𝑒 )( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀,𝑠 ) ) d 𝑠 − 𝜀 ∫ 𝜀 −2 𝑡 ( (∇ 𝛿 ) T σ )( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀,𝑠 ) ) d 𝐵 𝜀 ( 𝑠 ) ] ERIODIC HOMOGENIZATION OF LINEAR DEGENERATE PDEs 24 converges, and if this is the case the limit is the same. Next,(4.9) |||| (cid:69) [ 𝑔 ( 𝜀 ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 ) ) e − 𝜀 ∫ 𝜀 −2 𝑡 ( (∇ 𝛿 ) T 𝑐 − 𝑒 )( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀,𝑠 ) ) d 𝑠 − 𝜀 ∫ 𝜀 −2 𝑡 ( (∇ 𝛿 ) T σ )( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀,𝑠 ) ) d 𝐵 𝜀 ( 𝑠 ) ] − e − π ( (∇ 𝛿 ) T 𝑐 − 𝑒 −2 −1 (∇ 𝛿 ) T 𝑎 ∇ 𝛿 ) 𝑡 (cid:69) [ 𝑔 ( 𝜀 ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 ) ) e − 𝜀 ∫ 𝜀 −2 𝑡 ( (∇ 𝛿 ) T 𝑎 ∇ 𝛿 )( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀,𝑠 ) ) d 𝑠 − 𝜀 ∫ 𝜀 −2 𝑡 ( (∇ 𝛿 ) T σ )( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀,𝑠 ) ) d 𝐵 𝜀 ( 𝑠 ) ]|||| ≤ (cid:69) [ ||| 𝑔 ( 𝜀 ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 ) )||||||| e − 𝜀 ∫ 𝜀 −2 𝑡 ( (∇ 𝛿 ) T 𝑐 − 𝑒 −2 −1 (∇ 𝛿 ) T 𝑎 ∇ 𝛿 )( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀,𝑠 ) ) d 𝑠 − e − π ( (∇ 𝛿 ) T 𝑐 − 𝑒 −2 −1 (∇ 𝛿 ) T 𝑎 ∇ 𝛿 ) 𝑡 |||| e − 𝜀 ∫ 𝜀 −2 𝑡 ( (∇ 𝛿 ) T 𝑎 ∇ 𝛿 )( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀,𝑠 ) ) d 𝑠 − 𝜀 ∫ 𝜀 −2 𝑡 ( (∇ 𝛿 ) T σ )( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀,𝑠 ) ) d 𝐵 𝜀 ( 𝑠 ) ] ≤ (cid:69) [||| 𝑔 ( 𝜀 ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 ) )||| ] (cid:69) [ |||| e − 𝜀 ∫ 𝜀 −2 𝑡 ( (∇ 𝛿 ) T 𝑐 − 𝑒 −2 −1 (∇ 𝛿 ) T 𝑎 ∇ 𝛿 )( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀 ,𝑠 ) ) d 𝑠 − e − π ( (∇ 𝛿 ) T 𝑐 − 𝑒 −2 −1 (∇ 𝛿 ) T 𝑎 ∇ 𝛿 ) 𝑡 |||| e − 𝜀 ∫ 𝜀 −2 𝑡 ( (∇ 𝛿 ) T 𝑎 ∇ 𝛿 )( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀,𝑠 ) ) d 𝑠 −2 𝜀 ∫ 𝜀 −2 𝑡 ( (∇ 𝛿 ) T σ )( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀,𝑠 ) ) d 𝐵 𝜀 ( 𝑠 ) ] . From eqs. (4.6) and (4.8) we see that the first term on the right-hand side in eq. (4.9) is uniformlybounded for 𝜀 on finite intervals. For the second term we have that(4.10) (cid:69) [ |||| e − 𝜀 ∫ 𝜀 −2 𝑡 ( (∇ 𝛿 ) T 𝑐 − 𝑒 −2 −1 (∇ 𝛿 ) T 𝑎 ∇ 𝛿 )( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀,𝑠 ) ) d 𝑠 − e − π ( (∇ 𝛿 ) T 𝑐 − 𝑒 −2 −1 (∇ 𝛿 ) T 𝑎 ∇ 𝛿 ) 𝑡 |||| e − 𝜀 ∫ 𝜀 −2 𝑡 ( (∇ 𝛿 ) T 𝑎 ∇ 𝛿 )( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀,𝑠 ) ) d 𝑠 −2 𝜀 ∫ 𝜀 −2 𝑡 ( (∇ 𝛿 ) T σ )( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀,𝑠 ) ) d 𝐵 𝜀 ( 𝑠 ) ] ≤ (cid:69) [ |||| e − 𝜀 ∫ 𝜀 −2 𝑡 ( (∇ 𝛿 ) T 𝑐 − 𝑒 −2 −1 (∇ 𝛿 ) T 𝑎 ∇ 𝛿 )( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀,𝑠 ) ) d 𝑠 − e − π ( (∇ 𝛿 ) T 𝑐 − 𝑒 −2 −1 (∇ 𝛿 ) T 𝑎 ∇ 𝛿 ) 𝑡 |||| ] (cid:69) [ e −2 𝜀 ∫ 𝜀 −2 𝑡 ( (∇ 𝛿 ) T 𝑎 ∇ 𝛿 ) T )( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀,𝑠 ) ) d 𝑠 −4 𝜀 ∫ 𝜀 −2 𝑡 ( (∇ 𝛿 ) T σ )( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀,𝑠 ) ) d 𝐵 𝜀 ( 𝑠 ) ] . Clearly, (cid:69) [ e −2 𝜀 ∫ 𝜀 −2 𝑡 ( (∇ 𝛿 ) T 𝑎 ∇ 𝛿 )( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀,𝑠 ) ) d 𝑠 −4 𝜀 ∫ 𝜀 −2 𝑡 ( (∇ 𝛿 ) T σ )( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀,𝑠 ) ) d 𝐵 𝜀 ( 𝑠 ) ] ≤ e 𝑡 ‖ (∇ 𝛿 ) T 𝑎 ∇ 𝛿 ‖ ∞ . Analogously as in the proof of Theorem 3.5 we see that 𝜀 ∫ 𝜀 −2 𝑡 ( (∇ 𝛿 ) T 𝑐 − 𝑒 − 2 −1 (∇ 𝛿 ) T 𝑎 ∇ 𝛿 ) ( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑠 ) ) d 𝑠 = 𝜀 ∫ 𝜀 −2 𝑡 ( (∇ 𝛿 ) T 𝑐 − 𝑒 − 2 −1 (∇ 𝛿 ) T 𝑎 ∇ 𝛿 ) ( ̄𝑋 𝜀,𝜏 (Π 𝜏 ( 𝑥 ∕ 𝜀 ) , 𝑠 ) ) d 𝑠 L ( (cid:80) ) ←←←←←←←←←←←←←←←←←←→ 𝜀 → π ( (∇ 𝛿 ) T 𝑐 − 𝑒 − 2 −1 (∇ 𝛿 ) T 𝑎 ∇ 𝛿 ) 𝑡 . ERIODIC HOMOGENIZATION OF LINEAR DEGENERATE PDEs 25
Consequently, Skorohod representation theorem and dominated convergence theorem imply thatthe first term on the right-hand side in eq. (4.10) converges to zero as 𝜀 → . Thus, 𝑢 𝜀 ( 𝑥, 𝑡 ) converges as 𝜀 → if, and only if, e − π ( L 𝑐 − 𝑒 −2 −1 (∇ 𝛿 ) T 𝑎 ∇ 𝛿 ) 𝑡 (cid:69) [ 𝑔 ( 𝜀 ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 ) ) e − 𝜀 ∫ 𝜀 −2 𝑡 ( (∇ 𝛿 ) T 𝑎 ∇ 𝛿 )( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀,𝑠 ) ) d 𝑠 − 𝜀 ∫ 𝜀 −2 𝑡 ( (∇ 𝛿 ) T σ )( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀 ,𝑠 ) ) d 𝐵 𝜀 ( 𝑠 ) ] converges, and if this is the case the limit is the same. Martingale convergence theorem nowimplies that e − 𝜀 ∫ 𝜀 −2 𝑡 ( (∇ 𝛿 ) T 𝑎 ∇ 𝛿 )( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀,𝑠 ) ) d 𝑠 − 𝜀 ∫ 𝜀 −2 𝑡 ( (∇ 𝛿 ) T σ )( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀,𝑠 ) ) d 𝐵 𝜀 ( 𝑠 ) (cid:80) - a . s . and L ( (cid:80) ) ←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←→ 𝑡 → ∞ 𝑌 ( 𝑥, 𝜀 ) , where 𝑌 ( 𝑥, 𝜀 ) ∈ L ( (cid:80) ) satisfies (cid:69) [ 𝑌 ( 𝑥, 𝜀 ) | 𝜀 −2 𝑡 ] = e − 𝜀 ∫ 𝜀 −2 𝑡 ( (∇ 𝛿 ) T 𝑎 ∇ 𝛿 )( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀,𝑠 ) ) d 𝑠 − 𝜀 ∫ 𝜀 −2 𝑡 ( (∇ 𝛿 ) T σ )( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀,𝑠 ) ) d 𝐵 𝜀 ( 𝑠 ) , 𝑡 ≥ . Define (cid:80) 𝜀 (d 𝜔 ) ∶= 𝑌 ( 𝑥, 𝜀 )( 𝜔 ) (cid:80) (d 𝜔 ) . Clearly, (cid:69) 𝜀 [ 𝑔 ( 𝜀 ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 ) )] = (cid:69) [ 𝑔 ( 𝜀 ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 ) ) e − 𝜀 ∫ 𝜀 −2 𝑡 ( (∇ 𝛿 ) T 𝑎 ∇ 𝛿 )( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀,𝑠 ) ) d 𝑠 − 𝜀 ∫ 𝜀 −2 𝑡 ( (∇ 𝛿 ) T σ )( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀,𝑠 ) ) d 𝐵 𝜀 ( 𝑠 ) ] , and Girsanov theorem implies that ̄𝐵 𝜀 ( 𝑡 ) ∶= 𝐵 𝜀 ( 𝑡 ) + 𝜀 ∫ 𝑡 ( σ T ∇ 𝛿 ) ( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑠 ) ) d 𝑠 , 𝑡 ≥ , is a (cid:80) 𝜀 -Brownian motion. From eq. (4.7) we have 𝜀 ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 ) = 𝑥 + 𝜀𝛽 ( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 ) ) − 𝜀𝛽 ( 𝑥 ∕ 𝜀 )+ 𝜀 ∫ 𝜀 −2 𝑡 ( ( 𝑐 − D 𝛽 𝑐 ) − ( (cid:73) n − D 𝛽 ) 𝑎 ∇ 𝛿 ) ( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑠 ) ) d 𝑠 + 𝜀 ∫ 𝜀 −2 𝑡 ( σ − D 𝛽 σ ) ( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑠 ) ) d ̄𝐵 𝜀 ( 𝑠 ) . It is clear that { 𝜀 ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 )} 𝑡 ≥ converges in law as 𝜀 → if, and only if, { 𝜀 ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 ) − 𝜀𝛽 ( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 )) + 𝜀𝛽 ( 𝑥 ∕ 𝜀 )} 𝑡 ≥ , and if this is the case the limit is the same. The boundedvariation and predictable quadratic covariation parts of { 𝜀 ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 ) − 𝜀𝛽 ( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 )) + 𝜀𝛽 ( 𝑥 ∕ 𝜀 )} 𝑡 ≥ are given by { 𝜀 ∫ 𝜀 −2 𝑡 ( ( 𝑐 − D 𝛽 𝑐 ) − ( (cid:73) n − D 𝛽 ) 𝑎 ∇ 𝛿 )( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑠 ) ) d 𝑠 } 𝑡 ≥ , and { 𝜀 ∫ 𝜀 −2 𝑡 ( ( (cid:73) n − D 𝛽 ) 𝑎 ( (cid:73) n − D 𝛽 ) T )( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑠 ) ) d 𝑠 } 𝑡 ≥ , respectively. We will now show that finite-dimensional distributions of { 𝜀 ̃𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 )− 𝜀𝛽 ( ̃𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 )) + 𝜀𝛽 ( 𝑥 ∕ 𝜀 )} 𝑡 ≥ converge in law to finite-dimensional distributions of { ̄𝑊 ( 𝑥, 𝑡 )} 𝑡 ≥ . According to [20, Theorem VIII.2.4] this will hold if 𝜀 ∫ 𝜀 −2 𝑡 ( ( 𝑐 − D 𝛽 𝑐 ) − ( (cid:73) n − D 𝛽 ) 𝑎 ∇ 𝛿 ) ( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑠 ) ) d 𝑠 (cid:80) 𝜀 ←←←←←←←←←←←←←←←→ 𝜀 → ̄ 𝖻 𝑡 , ERIODIC HOMOGENIZATION OF LINEAR DEGENERATE PDEs 26 and 𝜀 ∫ 𝜀 −2 𝑡 ( ( (cid:73) n − D 𝛽 ) 𝑎 ( (cid:73) n − D 𝛽 ) T )( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑠 ) ) d 𝑠 (cid:80) 𝜀 ←←←←←←←←←←←←←←←→ 𝜀 → 𝖺 𝑡 for all 𝑡 ≥ . We now have (cid:69) 𝜀 [ 𝜀 ||||| ∫ 𝜀 −2 𝑡 (( ( 𝑐 − D 𝛽 𝑐 ) − ( (cid:73) n − D 𝛽 ) 𝑎 ∇ 𝛿 ) − ̄ 𝖻 ) ( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑠 ) ) d 𝑠 |||||] = (cid:69) [ 𝜀 ||||| ∫ 𝜀 −2 𝑡 (( ( 𝑐 − D 𝛽 𝑐 ) − ( (cid:73) n − D 𝛽 ) 𝑎 ∇ 𝛿 ) − ̄ 𝖻 ) ( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑠 ) ) d 𝑠 ||||| e − 𝜀 ∫ 𝜀 −2 𝑡 ( (∇ 𝛿 ) T 𝑎 ∇ 𝛿 )( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀,𝑠 ) ) d 𝑠 − 𝜀 ∫ 𝜀 −2 𝑡 ( (∇ 𝛿 ) T σ )( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀,𝑠 ) ) d 𝐵 𝜀 ( 𝑠 ) ] ≤ 𝜀 (cid:69) [ ( ∫ 𝜀 −2 𝑡 (( ( 𝑐 − D 𝛽 𝑐 ) − ( (cid:73) n − D 𝛽 ) 𝑎 ∇ 𝛿 ) − ̄ 𝖻 ) ( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑠 ) ) d 𝑠 ) T ( ∫ 𝜀 −2 𝑡 (( ( 𝑐 − D 𝛽 𝑐 ) − ( (cid:73) n − D 𝛽 ) 𝑎 ∇ 𝛿 ) − ̄ 𝖻 ) ( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑠 ) ) d 𝑠 ) ] e 𝑡 ‖ (∇ 𝛿 ) T 𝑎 ∇ 𝛿 ‖ ∞ . Now, as in the proof of Theorem 3.5 follows that 𝜀 ∫ 𝜀 −2 𝑡 ( ( 𝑐 − D 𝛽 𝑐 ) − ( (cid:73) n − D 𝛽 ) 𝑎 ∇ 𝛿 ) ( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑠 ) ) d 𝑠 L ( (cid:80) ) ←←←←←←←←←←←←←←←←←←→ 𝜀 → ̄ 𝖻 𝑡 , which implies 𝜀 ∫ 𝜀 −2 𝑡 ( ( 𝑐 − D 𝛽 𝑐 ) − ( (cid:73) n − D 𝛽 ) 𝑎 ∇ 𝛿 ) ( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑠 ) ) d 𝑠 L ( (cid:80) 𝜀 ) ←←←←←←←←←←←←←←←←←←←←←→ 𝜀 → ̄ 𝖻 𝑡 . Analogous result holds for the predictable quadratic covariation part. Thus, finite-dimensionaldistributions of { 𝜀 ̃𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 )} 𝑡 ≥ converge in law to finite-dimensional distributions of { ̄𝑊 ( 𝑥, 𝑡 )} 𝑡 ≥ . It remains to prove that lim 𝜀 → (cid:69) 𝜀 [ 𝑔 ( 𝜀 ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 ) )] = (cid:69) [ 𝑔 ( ̄𝑊 ( 𝑥, 𝑡 ) )] ∀ 𝑡 ≥ . Without loss of generality we may assume that 𝑓 ( 𝑥 ) is non-negative. From Skorohod represen-tation theorem and Fatou’s lemma we conclude lim inf 𝜀 → (cid:69) 𝜀 [ 𝑔 ( 𝜀 ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 ) )] ≥ (cid:69) [ 𝑔 ( ̄𝑊 ( 𝑥, 𝑡 ) )] ∀ 𝑡 ≥ . To prove the reverse inequality we proceed as follows. For any 𝑡 ≥ we have lim sup 𝜀 → (cid:69) 𝜀 [ 𝑔 ( 𝜀 ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 ) )] ≤ lim sup 𝑚 → ∞ lim sup 𝜀 → (cid:69) 𝜀 [ ( 𝑔 ∧ 𝑚 ) ( 𝜀 ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 ) )] + lim sup 𝑚 → ∞ lim sup 𝜀 → (cid:69) 𝜀 [ 𝑔 ( 𝜀 ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 ) ) (cid:49) { 𝑔 ( 𝜀 ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀,𝑡 ∕ 𝜀 ) ) ≥ 𝑚 } ] ≤ lim sup 𝑚 → ∞ (cid:69) [ ( 𝑔 ∧ 𝑚 ) ( ̄𝑊 ( 𝑥, 𝑡 ) )] + lim sup 𝑚 → ∞ lim sup 𝜀 → (cid:69) 𝜀 [ 𝑔 ( 𝜀 ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 ) ) (cid:49) { 𝑔 ( 𝜀 ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀,𝑡 ∕ 𝜀 ) ) ≥ 𝑚 } ] = (cid:69) [ 𝑔 ( ̄𝑊 ( 𝑥, 𝑡 ) )] + lim sup 𝑚 → ∞ lim sup 𝜀 → (cid:69) 𝜀 [ 𝑔 ( 𝜀 ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 ) ) (cid:49) { 𝑔 ( 𝜀 ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀,𝑡 ∕ 𝜀 ) ) ≥ 𝑚 } ] . ERIODIC HOMOGENIZATION OF LINEAR DEGENERATE PDEs 27
Finally, we show that lim sup 𝑚 → ∞ lim sup 𝜀 → (cid:69) 𝜀 [ 𝑔 ( 𝜀 ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 ) ) (cid:49) { 𝑔 ( 𝜀 ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀,𝑡 ∕ 𝜀 ) ) ≥ 𝑚 } ] = 0 ∀ 𝑡 ≥ . We have (cid:69) 𝜀 [ 𝑔 ( 𝜀 ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 ) ) (cid:49) { 𝑔 ( 𝜀 ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀,𝑡 ∕ 𝜀 ) ) ≥ 𝑚 } ] ≤ (cid:69) 𝜀 [||| 𝑔 ( 𝜀 ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 ) )||| ] ( (cid:80) 𝜀 ( 𝑔 ( 𝜀 ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 ) ) ≥ 𝑚 )) ≤ 𝑚 (cid:69) 𝜀 [||| 𝑔 ( 𝜀 ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 ) )||| ] . Now, as in eq. (4.6) we get (cid:69) 𝜀 [||| 𝑔 ( 𝜀 ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 ) )||| ] ≤ ̂𝐾 (1 + | 𝑥 | 𝜅 + 𝜀 𝜅 + 𝑡 𝜅 ) for some ̂𝐾 > which does not depend on 𝜀 . The assertion now follows.To this end it remains to show lim 𝜀 → (cid:69) [ ∫ 𝑡 𝑓 ( 𝑋 𝜀 ( 𝑥, 𝑠 ) ) e ∫ 𝑠 ( 𝜀 −1 𝑑 ( 𝑋 𝜀 ( 𝑥,𝑢 )∕ 𝜀 )+ 𝑒 ( 𝑋 𝜀 ( 𝑥,𝑢 )∕ 𝜀 ) ) d 𝑢 d 𝑠 ] = (cid:69) [ ∫ 𝑡 𝑓 ( ̄𝑊 ( 𝑥, 𝑠 )) e π ( −1 (∇ 𝛿 ) T 𝑎 ∇ 𝛿 + 𝑒 −(∇ 𝛿 ) T 𝑐 ) 𝑠 d 𝑠 ] . From the first part of the proof we see that lim 𝜀 → (cid:69) [ 𝑓 ( 𝑋 𝜀 ( 𝑥, 𝑠 ) ) e ∫ 𝑠 ( 𝜀 −1 𝑑 ( 𝑋 𝜀 ( 𝑥,𝑢 )∕ 𝜀 )+ 𝑒 ( 𝑋 𝜀 ( 𝑥,𝑢 )∕ 𝜀 ) ) d 𝑢 ] = (cid:69) [ 𝑓 ( ̄𝑊 ( 𝑥, 𝑠 )) e π ( −1 (∇ 𝛿 ) T 𝑎 ∇ 𝛿 + 𝑒 −(∇ 𝛿 ) T 𝑐 ) 𝑠 ] ∀ 𝑠 ≥ , and (cid:69) [ 𝑓 ( 𝑋 𝜀 ( 𝑥, 𝑠 ) ) e ∫ 𝑠 ( 𝜀 −1 𝑑 ( 𝑋 𝜀 ( 𝑥,𝑢 )∕ 𝜀 )+ 𝑒 ( 𝑋 𝜀 ( 𝑥,𝑢 )∕ 𝜀 ) ) d 𝑢 ] ≤ (cid:69) [||| 𝑓 ( 𝑋 𝜀 ( 𝑥, 𝑠 ) )||| ] (cid:69) [ e ∫ 𝑠 ( 𝜀 −1 𝑑 ( 𝑋 𝜀 ( 𝑥,𝑢 )∕ 𝜀 )+ 𝑒 ( 𝑋 𝜀 ( 𝑥,𝑢 )∕ 𝜀 ) ) d 𝑢 ] ≤ ̌𝐾 (1 + | 𝑥 | 𝜅 + 𝜀 𝜅 + 𝑠 𝜅 ∕2 ) (cid:69) [ e 𝜀𝛿 ( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀,𝑠 ∕ 𝜀 ) ) −2 𝜀𝛿 ( 𝑥 ∕ 𝜀 )−2 𝜀 ∫ 𝜀 −2 𝑠 ( (∇ 𝛿 ) T 𝑐 − 𝑒 ) ) ( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀,𝑢 ) ) d 𝑢 −2 𝜀 ∫ 𝜀 −2 𝑠 ( (∇ 𝛿 ) T σ )( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀,𝑢 ) ) d 𝐵 𝜀 ( 𝑢 ) ] ≤ ̌𝐾 (1 + | 𝑥 | 𝜅 + 𝜀 𝜅 + 𝑠 𝜅 ∕2 ) e 𝜀 ‖ 𝛿 ‖ ∞ + ‖ (∇ 𝛿 ) T 𝑐 − 𝑒 −(∇ 𝛿 ) T 𝑎 ∇ 𝛿 ‖ ∞ 𝑠 , for some ̌𝐾 > which does not depend on 𝜀 . The result now follows from the dominatedconvergence theorem. (cid:3) Remark 4.4.
Grant the assumptions in Theorem 4.2. Under (cid:80) 𝜀 (where (cid:80) ∶= (cid:80) ) the process { ̄𝑋 𝜀 ( 𝑥, 𝑡 )} 𝑡 ≥ solves d ̄𝑋 𝜀 ( 𝑥, 𝑡 ) = 𝑏 ( ̄𝑋 𝜀 ( 𝑥, 𝑡 ) ) d 𝑡 + 𝜀 ( 𝑐 − 𝑎 ∇ 𝛿 ) ( ̄𝑋 𝜀 ( 𝑥, 𝑡 ) ) d 𝑡 + σ ( ̄𝑋 𝜀 ( 𝑥, 𝑡 ) ) d ̄𝐵 𝜀 ( 𝑡 ) ̄𝑋 𝜀 ( 𝑥,
0) = 𝑥 ,
ERIODIC HOMOGENIZATION OF LINEAR DEGENERATE PDEs 28 and satisfies ̄𝑋 𝜀 ( 𝑥, 𝑡 ) = 𝑥 + 𝛽 ( ̄𝑋 𝜀 ( 𝑥, 𝑡 ) ) − 𝛽 ( 𝑥 ) + 𝜀 ∫ 𝑡 ( ( 𝑐 − D 𝛽 𝑐 ) − ( (cid:73) n − D 𝛽 ) 𝑎 ∇ 𝛿 )( ̄𝑋 𝜀 ( 𝑥, 𝑠 ) ) d 𝑠 + ∫ 𝑡 ( σ − D 𝛽 σ ) ( ̄𝑋 𝜀 ( 𝑥, 𝑠 ) ) d ̄𝐵 𝜀 ( 𝑠 ) ∀ 𝑡 ≥ . We now easily see that eq. (2.4) and Proposition 2.1 hold under (cid:80) 𝜀 . Furthermore, if there is 𝜀 > such that(4.11) (cid:80) 𝜀 ( ̄ τ 𝜀,𝑥 𝒪 + 𝜏 < ∞ ) > 𝜀, 𝑥 ) ∈ [0 , 𝜀 ] × (cid:82) n , then Propositions 2.3 to 2.5 also hold under (cid:80) 𝜀 . Assume 𝑓 ( 𝑥 ) = 𝑓 ( 𝑥 ) + 𝑓 ( 𝑥 ∕ 𝜀 ) and 𝑔 ( 𝑥 ) = 𝑔 ( 𝑥 ) + 𝑔 ( 𝑥 ∕ 𝜀 ) , where 𝑓 , 𝑔 ∈ ( (cid:82) n , (cid:82) ) satisfy eq. (4.4) and 𝑓 , 𝑔 ∈ ( (cid:82) n , (cid:82) ) are 𝜏 -periodic.Then, lim 𝜀 → (cid:69) 𝜀 [ 𝑓 ( 𝜀 ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 ) )] = (cid:69) [ 𝑓 ( ̄𝑊 ( 𝑥, 𝑡 ) )] + lim 𝜀 → (cid:69) 𝜀 [ 𝑓 ( Π 𝜏 ( ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 ) ))] = (cid:69) [ 𝑓 ( ̄𝑊 ( 𝑥, 𝑡 ) )] + π ( 𝑓 ) ∀ 𝑡 ≥ , and lim 𝜀 → (cid:69) 𝜀 [ 𝑓 ( 𝜀 ̄𝑋 𝜀 ( 𝑥 ∕ 𝜀, 𝑡 ∕ 𝜀 ) )] = (cid:69) [ 𝑓 ( ̄𝑊 ( 𝑥, 𝑡 ) )] + π ( 𝑓 ) ∀ 𝑡 ≥ . Thus, under (A1)-(A4) (or (A1)-(A3) if 𝑐 ( 𝑥 ) ≡ or 𝑏 ( 𝑥 ) ≡ , and 𝑑 ( 𝑥 ) ≡ ) and eq. (4.11), lim 𝜀 → 𝑢 𝜀 ( 𝑡, 𝑥 ) = 𝑢 ( 𝑡, 𝑥 ) ∀ ( 𝑡, 𝑥 ) ∈ [0 , ∞) × (cid:82) n , where 𝑢 ( 𝑡, 𝑥 ) ∶= (cid:69) [ ( 𝑔 ( ̄𝑊 ( 𝑥, 𝑡 )) + π ( 𝑔 ) ) e π ( −1 (∇ 𝛿 ) T 𝑎 ∇ 𝛿 + 𝑒 −(∇ 𝛿 ) T 𝑐 ) 𝑡 + ∫ 𝑡 ( 𝑓 ( ̄𝑊 ( 𝑥, 𝑠 )) + π ( 𝑓 ) ) e π ( −1 (∇ 𝛿 ) T 𝑎 ∇ 𝛿 + 𝑒 −(∇ 𝛿 ) T 𝑐 ) 𝑠 d 𝑠 ] is a solution to 𝜕 𝑡 𝑢 ( 𝑥, 𝑡 ) = 𝑢 ( 𝑥, 𝑡 ) + π ( −1 (∇ 𝛿 ) T 𝑎 ∇ 𝛿 + 𝑒 − (∇ 𝛿 ) T 𝑐 ) 𝑢 ( 𝑥, 𝑡 ) + 𝑓 ( 𝑥 ) + π ( 𝑓 ) 𝑢 ( 𝑥,
0) = 𝑔 ( 𝑥 ) + π ( 𝑔 ) , 𝑥 ∈ (cid:82) n . A CKNOWLEDGEMENTS
Financial support through the
Alexander-von-Humboldt Foundation and
Croatian ScienceFoundation under project 8958 (for N. Sandrić), and Croatian Science Foundation under project8958 (for I. Valentić) are gratefully acknowledged.R
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