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n For e0 R , the horizontal vector field H(e0) does not project to a vector field on M∈. It, however, induces a vector field X on T M which is a geodesic e0 spray. If (ut ) is the solution to the first-order differential equation
u˙(t)= Hu(t)(e0), u(0) = u0,
e0 then π(ut ) is the geodesic on M with initial velocity u0(e0) and initial value π(u0). n(n 1) Let N = 2− and let so(n) be the space of skew symmetric matrices in dimension n. It is the Lie algebra of the orthogonal group O(n). For A so(n), we denote by A∗ the fundamental vertical vector field on OM determined∈ by right actions of the exponentials of tA; see (2.1) below. If X is a vector field, we denote by LX Lie differentiation in the direction of X. Let us fix a time T > 0. Let ρ be the Riemannian distance function on M, the Levi–Civita connection and ∆ the Laplace–Beltrami operator. Let ∇ε a positive number. Our main theorems concern the convergence, as ε approaches zero, of the “horizontal part” of the solutions to a family of stochastic differential equations with parameter ε. The definitions for the horizontal and vertical vector fields and for the horizontal lift of a curve are n given in Section 2. Let e0 be a unit vector in R .
Theorem 1.1. Let M be a complete Riemannian manifold of dimen- sion n> 1 and of positive injectivity radius. Suppose that there are positive numbers C and a such that supρ(x,y) a dρ (x,y) C. Let x0 M and 1 ≤ |∇ | ≤ ∈ u0 π− (x0). Let A¯ so(n) and A1,...,AN be an orthonormal basis of so(∈n). Let (uε, 0 t ∈T ) be the solution{ to the} SDE t ≤ ≤ N ε 1 ε k ε du = H ε (e ) dt + A∗(u ) dw + A¯∗(u ) dt, (1.1) t ut 0 √ε k t ◦ t t k=1 ε X u0 = u0. Let xε = π(uε) and let (˜xε, 0 t T ) be the horizontal lift of (xε, 0 t T ) t t t ≤ ≤ t ≤ ≤ to OM through u0. Then the following statements hold: (1) The SDE does not explode. ε ε (2) The processes (xt/ε, 0 t T ) and (˜xt/ε, 0 t T ) converge in law, as ε 0. ≤ ≤ ≤ ≤ → ε (3) The limiting law of (xt/ε, 0 t T ) is independent of e0. It is a scaled ≤ 4≤ ε Brownian motion with generator n(n 1) ∆. The limiting law of (˜xt/ε, 0 t − 4 ≤ ≤ T ) is that associated to the generator n(n 1) ∆H . − If “ε = ” and A¯ =0, the SDE (1.1) reduces to the first-order differential ∞ equationu ˙(t)= Hu(t)(e0) whose solutions are geodesics. If “ε =0”, the SDE PERTURBATION TO GEODESIC EQUATION 3
“reduces” to the “vertical SDE”, duε = 1 N A (uε) dwk. This vertical t √ε k=1 k∗ t ◦ t equation does not have a meaning for ε = 0, nevertheless the “vertical SDE” P ε ε has a first integral π : OM M, that is, π(ut )= π(u0). By a preliminary → ε multi-scale analysis, we see that π(ut ) varies slowly with ε and there is a 1 visible effective motion in the time interval [0, ε ]. The first integral π is not a real-valued function. It is a function from a manifold to a manifold and ε the slow variables (xt ),ε> 0 are not Markov processes. Before further dis- cussions on conservation{ laws} related to the SDEs, we remark the following features: (1) the slow motion solves a first-order differential equation, (2) the “fast motion” on OM is not elliptic, (3) the limiting process is semi- elliptic. Another feature of Theorem 1.1 is that the pair of the intertwined ε ε family of stochastic processes (xt/ε, x˜t/ε) converge. We will explore (3) in a forthcoming article on homogeneous manifolds. For now, the following ob- servation indicates a potential application of (3): the stochastic area of two 1 2 linear Brownian motions wt , wt is the principal part of the horizontal lift { } 1 2 of the two-dimensional Brownian motion (wt , wt ) to the three-dimensional Heisenberg group. We remark also that the first-order horizontal geodesic equation on the orthonormal frame bundle corresponds a second-order differ- ential equation on the manifold, which explains the unusual scaling in (1.1). There have been many studies of limit theorems whose geometric settings or scalings or methodologies relate that in this article. For example, our phi- 1 losophy agrees with that in Bismut [3] where the equationx ¨ = T ( x˙ +w ˙ ) interpolates between classical Brownian motion (T 0) and the− geodesic flow (T ). In Ikeda [16] and Ikeda and Ochi [17→], the authors studied →∞ t limit theorems for line integrals of the form 0 φ(dxs), where φ is a differ- ential form and (xs) is a suitable process such as a Brownian motion. In Manabe and Ochi [23] the authors obtained centralR limit theorems for line integrals along geodesic flows. One of their tools is symbolic representations of geodesic flows. Another related work can be found in Pinsky [27], where a piecewise geodesic with a Poisson-type switching mechanism is shown to converge to the horizontal Brownian motion. We also note that geodesic flows perturbed by vertical Brownian motions were considered by Franchi and Le Jan [10], in the context of relativistic diffusions. The conclusion of (1.1) is consistent with the following central limit theo- rems for geodesic flows. Let M be a manifold of constant negative curvature and of finite volume. Let (γt(x, v)) denote the geodesic with initial value (x, v) in the unit tangent bundle STM and let θt(v) = (γt(x, v), γ˙t(x, v)), a stochastic process on STM . Let f be a bounded measurable function on STM with the property that it is centered with respect to the normalized Liouville measure m. Then there is a number σ with the property that t a 0 f(θs(ξ)) ds 1 y2/2 lim m ξ : a = e− dy. t σ√t ≤ √2π →∞ R Z−∞ 4 X.-M. LI
See Sinai [30], Ratner [28]; see Guivarch and Le Jan [12] and Enriquez, Franchi and Le Jan [8] for further developments. See also Helland [15] and Kipnis and Varadhan [19]. These results exploit the chaotic nature of the deterministic dynamical system on manifolds of negative curvature. In the homogenisation literature, the following works are particularly rel- evant: Khasminskii [14, 18], Nelson [24], Borodin and Freidlin [4], Freidlin and Wentzell [11] and Bensoussan, Lions and Papanicolaou [1]. We note in particular Theorem 2.1 in [4] which deals with the convergence of path in- tegrals of a suitable function along a family of ergodic Markov processes. In this article, such integrals are better understood as integrals of differential 1-forms along random paths. Finally, we mention the following work: Li [21] for averaging of integrable systems and Ruffino and Gonzales Gargate [29] for averaging on foliated manifolds. See also [22] for an earlier work on the orthonormal frame bundle. We also refer to Dowell [5] for a scaling limit of Ornstein–Uhlenbeck type. Open question. The local uniform bound on dρ is only used in Lemma 3.2 for the proof of tightness. This bound can be weakened,∇ for example, replaced dρ by a local uniform control over the rate of growth of the norms of ∇ρ and ρ ∇ρ . We remark that Brownian motion constructed in Theorem 1.1 is auto- matically complete. The conditions in Theorem 1.1 appear to be related to the uniform cover criterion on stochastic completeness and could be studied in connection with that in Li [20]. Also, much of the work in this article is valid for a connection with torsion, the horizontal tangent bundle and ∇ ∆H will then be induced by this connection with torsion. The effect of the torsion will generally lead to an additional drift to the Brownian motion downstairs. In this case the geodesic completeness of the manifold M may no longer be equivalent to the metric completeness of (M, ρ).
2. Preliminaries. Given a Riemannian metric on M, an orthonormal frame u = u1, . . ., un is an ordered basis of TxM that is orthonormal. We denote by {OM the set} of all orthonormal frames on M and π the map 1 that takes the frame u to the point x M. Let π− (x)= u OM : π(u)= ∈ j ∂{ ∈ x . If (O,x) is a coordinate system on M, ui = u x. This gives a } j i ∂xj | j 1 coordinate map on OM . The map (x, ui ) is a homeomorphismP from π− (O) to (x(O), O(n)). If we identify a frame u with the transformation u : Rn → TxM, then OM is a principal bundle with fibre O(n) and group G, acting on the right. We adopt the notation ue = u(e). For g O(n) let Rg denote right multiplication on O(n) and the right action of O(n∈) on OM . For A, B so(n) let A, B = tr ABT . ∈ Ah tangenti vector v in OM is vertical if Tπ(v)=0 where Tπ denotes the differential of π. If A belongs to the Lie algebra so(n), we denote by exp(tA) the exponential map. If u is a frame, the composition u exp(tA) is again PERTURBATION TO GEODESIC EQUATION 5 a frame in the same fibre. We define the fundamental vertical vector fields associated to A by A∗, d (2.1) A (u)= u exp(tA). ∗ dt t=0
By a linear connection on the principal bundle OM , we mean a splitting of the tangent bundle T OM with the following properties: (1) TuOM = HTuOM V TuOM (2) (Ra) HuT OM = HuaT OM for all u OM and ⊕ ∗ ∈ a G. The spaces HTuOM and V TuOM are, respectively, the horizontal tangent∈ spaces and the vertical tangent spaces. We will introduce a met- ric on OM such that π is an isometry between HuT OM and Tπ(u)M and such that HuT OM and V TuOM are orthogonal. The metric on so(n) is the bi-invariant metric introduced earlier. We will restrict our attention to the Levi–Civita connection. 1 Let hu(v) denote the horizontal lift of v TxM through u π− (x). n ∈ ∈ To each e R we denote Hu(e)= hu(ue) the basic vector field. Later, ∈ n we also use Hue for Hu(e). If e1,...,en is an orthonormal basis of R , then H (e ),...,H (e ) is an{ orthonormal} basis for the horizontal tan- { u 1 u n } gent space HTuOM . A piecewise C1 curve γ on OM is horizontal if the one-sided derivatives γ˙ ( ) are horizontal for all t. If c is a C1 curve on M, there is a horizontal curve± c ˜ on OM such thatc ˜ covers c, that is, π(˜c(t)) = c(t). In fact,c ˜(t) is the family of orthonormal frames along c that are obtained by parallel transporting the framec ˜(0). We say thatc ˜ is a horizontal lift of c. The map 1 c˜(t)(˜c(0))− : Tc(0)M Tc(t)M is the parallel translation along the curve c(t). In a coordinate chart→ (O,x), the principal part ofc ˜(t) is a n n matrix × whose column vectors c˜1(t),..., c˜n(t) form a frame. In components, write 1 n {T } c˜l(t)=(˜cl (t),..., c˜l (t)) . Then n ∂c˜k(t) ∂ci(t) l + Γk (c(t))˜cj(t) = 0. ∂t ∂t ij l i=1X,j=1 Take c(t) = (0,...,t,..., 0), where the nonzero entry is in the ith-place. We obtain the principal part of the horizontal lift of ∂ through u =c ˜(0) = (uj): ∂xi l
n T ∂ ∂c˜ 1 j n j hc˜(0) = (0) = Γijul ,..., Γijul . ∂xi l ∂t l − ! Xj Xj=1 b j Denote by Ai the matrix whose element at the (b, l) position is j Γijul . ∂ Then Ai is the principal part of Hu( ) and the horizontal space at u is ∂xi P ∂ spanned by the basis ( , Ai) . { ∂xi } 6 X.-M. LI
A basic object we use in our computation is the connection 1-form ̟ on OM . A connection 1-form assigns a skew symmetric matrix to every tangent vector on OM and it satisfies the following conditions: (1) ̟(A∗)= A for all A so(n); ∈ 1 (2) for all a O(n) and w OM , ̟(Ra w) = Ad(a− )̟(w). We re- ∈ 1 ∈ ∗ call that Ra (A∗) = (Ad(a− )A)∗ for all a O(n). It is convenient to con- sider horizontal∗ tangent vectors on OM as∈ elements of the kernel of ̟. If A1,...,AN is a basis of so(n), then the horizontal component of a vector { h } w is w = w j ̟(w), Aj Aj∗. The connection− h 1-form ̟iis basically the set of Christoffel symbols. Let P E = E1,...,En be a local frame; we define the Christoffel symbols relative to E{by E = } Γk dx E . Let θi be the set of dual differential 1-forms ∇ j ki ij i ⊗ k on M to E : θi(E )= δ . We define ωi =Γi θl. Then dθi = ωi θk. { i} P j ij k lk − k k ∧ Let Aj be a basis of g. To each moving frame E, we associate a 1-form, { i } P ω = ωi Aj, on M. If (O,x) is a chart of M and s : O OM is a local i,j j i → section of OM , let us denote by ωs the differential 1-form given above, then P ̟(s v)= ωs(v). Conditions (1) and (2) are equivalent to the following: if a : U∗ G is a smooth function, → 1 1 ̟((s a) v)= a− (x) da(v)+ a− (x)̟(s v)a(x). · ∗ ∗ This corresponds to the differentiation of s a and this type of consideration will be used in the next section. ·
3. Some lemmas.
Lemma 3.1. Let M be a geodesically complete Riemannian manifold. ε ε ε Let (ut ) be the solution to the SDE (1.1) on OM . Let xt = π(ut ), which has a unique horizontal lift, x˜ε, through u uε. Then t 0 ≡ 0 d ε ε x˜ = H ε (g e ), dt t x˜t t 0 m 1 dgε = gεA dwk + gεA¯ dt, t √ε t k ◦ t t Xk=1 ε where g0 is the unit matrix. Consequently the SDE (1.1) is conservative.
Proof. By the defining properties of the basic horizontal vector fields, ε ε x π H ε e u e h v ˙ t = ( ut ( 0)) = t 0. Let u( ) denote the horizontal lift of a tangent ∗ ε vector v through u OM . Since ut e0 has unit speed, the solution exists for ε ∈ all time if (ut ) does, and
d ε ε ε x˜ = h ε (x ˙ )= h ε (u e ). dt t x˜t t x˜t t 0 PERTURBATION TO GEODESIC EQUATION 7
ε ε At each time t, the horizontal lift (˜xt ) of the curve (xt ) through u0 and the ε ε original curve ut belong to the same fibre. Let gt be an element of G with ε ε ε ε the property that ut =x ˜t gt . Then g0 is the unit matrix and
d ε ε ε ε x˜ = h ε (˜x g e )= H ε (g e ). dt t x˜t t t 0 x˜t t 0 −1 1 1 d rat a˙ t If at is a C path with values in O(n), at− a˙ t = dr r=0e , its action on u gives rise to a fundamental vector field, |
d d 1 1 ua = ua a a = (a a˙ )∗(ua ). dt t dr t t− r+t t− t t t r=0
We denote by DL g and DRg , respectively, the differentials of the left mul- tiplication and of the right action. By Itˆo’s formula applied to the product ε ε x˜t gt , ε ε ε ε du = DRgε dx˜ + (DL ε −1 dg )∗(u ). t t ◦ t (gt ) ◦ t t Since right translation of horizontal vectors are horizontal, the connection ε ε 1-form vanishes on the first term and ̟( dut )= DL(gε)−1 dgt . We apply ε ◦ t ◦ ̟ to the SDE for ut , N ε ε 1 ε k ε dg ε ̟ du ε ̟ A u dw A¯ u dt t = DLgt ( t )= DLgt k∗( t ) t + ∗( t ) ◦ √ε ◦ ! Xk=1 m 1 = gεA dwk + gεA¯ dt. √ε t k ◦ t t Xk=1 d ε ε x H ε g e There is a global solution to the above equation. The ODE dt ˜t = x˜t ( t 0) ε has bounded right-hand side and has a global solution. It follows that ut = ε ε x˜t gt has a global solution.
ε Remark 3.1. Since the stochastic process (gt ) is sample continuous with initial value the unit matrix, it stays in the connected component SO(n) of O(n).
1 N If Ak is an orthonormal basis of so(n) let G = 2 k=1 LgAk LgAk . Then (gε){ is a} Markov process with infinitesimal generatorL t P ε 1 = + L ¯. L εLG gA Lemma 3.2. Let M be a complete Riemannian manifold with positive injectivity radius. Suppose that there are numbers C > 0 and a2 > 0 such that supρ(x,y) a2 dρ (x,y) C. Let T > 0. The probability distributions of ≤ |∇ | ≤ the family of stochastic processes x˜ε ,t T are tight. There is a metric d˜ { t/ε ≤ } on M such that (˜xε ) is equi-H¨older continuous with exponent α< 1 . { t/ε } 2 8 X.-M. LI
ε ε Proof. Let µ be the probability laws of (˜xt ) on the path space over ε OM with initial value u0, which we denote by C([0, T ]; OM ). Sincex ˜0 = u0, it suffices to estimate the modulus of continuity and show that for all positive numbers a, η, there exists δ> 0 such that for all ε sufficiently small (see Billingsley [2] and Ethier and Kurtz [9])
ε ε P ω : sup d(˜xt , x˜s) > a < δη. s t <δ | − | Here, d denotes a distance function on OM . We will choose a suitable dis- tance function. The Riemannian distance functionρ ˜(x,y) is not smooth in y if y is in the cut locus of x. To avoid any assumption on the cut locus of OM , we construct a new distance function that preserves the topology of OM . Let 2a be the minimum of 1, a2 and the injectivity radius of M. Let φ : R+ R+ be a smooth concave function such that φ(r)= r when r < a and φ(→r)=1 when r 2a. Let ρ andρ ˜ be, respectively, the Riemannian distance on M and on≥OM . Then φ ρ and d˜= φ ρ˜ are distance functions on M and on OM , respectively. Then◦ for r ε ε The tightness of the law of x˜t/ε follows. By Kolmogorov’s criterion, x˜t/ε { } 1 { } is H¨older continuous with exponent α for any α< 2 . The H¨older constants are independent of ε and, for any p′ We will need the following lemma in which we make a statement on the limit of a function of two variables, one of which is ergodic and the other one varies significantly slower. The result is straightforward, but we include the proof for completeness. If f : N R is a Lipschitz continuous function on a metric space (N, d) with distance→ function d, we denote by f its Lipschitz | |Lip semi-norm. If S is a subset of N, we let OscS(f) denote supx S f(x) | ∈ − infx S f(x) , the Oscillation of f over S. Let Osc(f)=OscN (f). Let∈ E(N|) be one of the following classes of real valued functions on a metric space (N, d): E(N)= f : N R : f < , Osc(f) < { → | |Lip ∞ ∞} or E (N)= E(N) Cr, where r = 0, 1,..., . Denote r ∩ ∞ f = f + Osc(f). | |E | |Lip Let d be the metric with respect to which the Lipschitz property is defined. We define d˜= d 1 to be a new metric on N. Then f C and Osc(f) C ∧ | |Lip ≤ ≤ is equivalent to f being Lipschitz with respect to d˜. Let p 1 and let Wp(N) denote the Wasserstein p-distance between two probability≥ measures on a metric space (N, d): p p (Wp(µ1,µ2)) = inf (d(x,y)) dν(x,y). ν : (π1)∗ν=µ1,(π2)∗ν=µ2 N N { } Z × Let µε,µ be a family of probability measures on the metric space (N, d). ε Then µ µ in Wp(N) if and only if they converge weakly and → p ˜ ˜ supx N (d(x,y)) dµε(y) is bounded for any x N. If d = d 1, then d and ∈d induce the same topology on N and the∈ concepts of weak∧ conver- R gence are equivalent. With respect to d˜, weak convergence is equivalent to Wasserstein p-convergence. Let (Ω, , ( t), P ) be a filtered probability space. Let (Y, ρ), (Z, d) be F F m ε metric spaces or C manifolds. Let (yt ,t T ),ε> 0 be a family of t- { ≤ } ε F adapted stochastic processes with state space Y . Let (zt ) be a family of sample continuous -Markov processes on Z. Ft 12 X.-M. LI Assumption 3.3. (1) The stochastic processes (yε ,t T ) are equi- t/ε ≤ uniformly continuous and converge weakly to a continuous process (¯yt,t T ). ≤ ε (2) For each ε, (ztε,t T ) has an invariant measure µε. There exists a function δ on R Z R≤ with the property that δ( ,z,ε) is nondecreasing + × × + · for each pair of (z, ε) and limε 0 supz Z δ(K,z,ε) = 0 for all K and for all f E (Z) and t> 0, → ∈ ∈ r ε t/ε ε E f(zε ) ds f(z) dµ (z) δ f , zε, . t sε − ε ≤ | |E 0 t Z0 ZZ (3) There exists a probability measure µ on W 1(C([0, T ]; Z)) s.t. limε 0 W1(µε,µ)=0. → ε (4) The processes (yt/ε) converges to (¯yt) in W1(Y ), and there exists an exponent α> 0 such that ρ(yε ,yε ) E t/ε s/ε sup sup α < . ε s=t t s ∞ 6 | − | We cannot assume that (¯yt) is adapted to the filtration with respect to ε ε which (zt/ε) is a Markov process. The process (zt/ε) is usually not conver- ε ε gent and we do not assume that (yt , zt ) and (¯yt) are realized in the same probability space. We denote by Pˆη the probability distribution of a random variable η and let T be a positive real number. If r is a positive number, let C([0, r]; Y ) denote the space of continuous paths, σ : [0, r] Y , on Y . If F : C([0, r]; Y ) R → ε → is a Borel measurable function, we use the shorter notation F (y /ε) for F ((yε , u r)). · u/ε ≤ Lemma 3.4. Let (Ω, , ( t), P ) be a filtered probability space. Let (Y, ρ), F mF ε (Z, d) be metric spaces or C manifolds in case m 1. Let (yt ,t T ),ε> 0 be a family of -adapted stochastic processes on≥Y . Let ({zε) be≤ a family } Ft t of sample continuous t-Markov processes on Z. Let G Em(Y Z). Let 0 r Assume (1)–(4) in Assumption 3.3. Then there is a constant c, s.t. for • ε< 1, ˆ ˆ W1(PA(ε), PA) ε c F max δ G E , z, + 2ε F min( G , Osc(G) ) ≤ | |∞ z Z | | t r | |∞ | |∞ | | ∈ − ˆ ε ˆ ε α + c(t r) F G Lip(W1(Py , Py¯· )+ W1(µ ,µ)) + cε F G Lip. − | |∞| | ·/ε | |∞| | Proof. Let us fix the functions F , G, r, t and define t t (r, t)= G(yε , zε ) ds G(yε , z) dµ (z) ds; E1 s/ε s/ε − s/ε ε Zr Zr ZZ t t = F (yε ) G(yε , z) dµ (z) ds G(yε , z) dµ(z) ds ; E2 /ε s/ε ε − s/ε · Zr ZZ Zr ZZ t ε ε I(ε)= F (y /ε) G(ys/ε, z) dµ(z) ds. · Zr ZZ ε The proof is split into three parts: (i) F (y /ε) 1(r, t) converges to zero in · E Lp(Ω) for any p> 1, (ii) 2 converges to zero in Lp(Ω) for any p> 1 and (iii) I(ε) converges to A weakly.E ε r We first prove that F (y ([0, ε ])) 1(r, t) converges to zero in Lp(Ω). Since F is bounded it is sufficient to takeE r =0 and F a constant, and to work with (0,t). Let us write E1 t t := G(yε , zε ) ds G(yε , z) dµ (z) ds. E1 s/ε s/ε − s/ε ε Z0 Z0 ZZ Let 0= t0 Mε 1 Mε 1 − ti+1 − ti+1 := G(yε , zε ) ds G(yε , zε ) ds 3 s/ε sε ti/ε sε E ti − ti i=0 Z i=0 Z X X Mε 1 − ti+1 G ρ(yε ,yε ) ds. Lip s/ε ti/ε ≤ | | ti Xi=0 Z 14 X.-M. LI By equi-uniform continuity of (yε ), for almost surely all ω, converges to s/ε E3 zero. Since is bounded the convergence is in L (Ω). If (yε ) is assumed E3 p s/ε to be equi-H¨older continuous as in condition (4), there is a convergence rate α p of ε G Lip for the L convergence. | | Mε 1 ti+1 ε ε We prove next that − G(y , z ) ds converges. We apply the i=0 ti ti/ε s/ε Markov property of (zε) and we use the fact that (yε) is adapted to the Pt R t filtration ( ), with respect to which (zε) is a Markov process: Ft t Mε 1 ti+1 − ε ε ε E G(y , z ) ds ∆ti G(y , z) dµε(z) ti/ε s/ε − ti/ε i=1 Zti ZZ X Mε 1 ti+1 − 1 ε ε ∆tiE E G(y , z ) ds ≤ ∆t ti/ε s/ε i=1 i Zti X G(yε , z) dµ (z) ti/ε ε ti/ε − Z F Z M 1 2 ε 2 ti+1/ε − E E ε ε = ∆ti G(y, zsε) ds ∆t 2 i=1 i Zti/ε X G(y, z) dµ (z) . ε ε − Z y=y Z ti/ε 2 Since ε = ε , we may now apply condition (2) and obtain ∆ti t 2 2 ti+1/ε E ε ε G(y, zsε) ds G(y, z) dµε(z) ∆t 2 − i Zti/ε ZZ ε ε ε ε ε δ G(y , ) , z , δ G E , z , . ≤ | ti/ε · |E ti/ε t ≤ | | ti/ε t We record that Mε 1 Mε 1 − ti+1 − := E G(yε , zε ) ds ∆t G(yε , z) dµ (z) 4 ti/ε s/ε i ti/ε ε E ti − Z i=0 Z i=0 Z (3.6) X X ε max δ G E, z, . ≤ z Z | | t ∈ Let us define Mε 1 − t := ∆t G(yε , z) dµ (z) G(yε , z) dµ (z) ds. 5 i ti/ε ε s/ε ε E Z − 0 Z Xi=0 Z Z Z PERTURBATION TO GEODESIC EQUATION 15 By the definition of Riemann integral Mε 1 − G ∆t Osc (yε ), E5 ≤ | |Lip i [si,si+1] s/ε Xi=0 where Osc[a,b](f) denotes the oscillation of a function f in the indicated ε interval. Since (ys/ε) is equi-uniform continuous on [0, T ], 5 0 in Lp. Given ε E → H¨older continuity of (ys/ε) from condition (4), we have the quantitative estimates: C G εα. To summarize, |E5|Lp(Ω) ≤ | |Lip (0,t) (t,t˜ ) + + + . |E1 |≤|E1 | E3 E4 E5 It follows that F (yε ) (r, t) converges to zero. r/ε E1 When condition (4) holds, there is a constant C such that F yε r, t ( /ε) 1( ) Lp(Ω) | · E | F (2ε min( G , Osc(G) )+ 3 + 4 + 5) (3.7) ≤ | |∞ | |∞ | | E E E α C F (ε + ε) G Lip + 2ε F min( G , Osc(G) ) ≤ | |∞ | | | |∞ | |∞ | | ε + C F max δ G E , z, . | |∞ z Z | | t r ∈ − For any two random variables on the same probability space and with the same state space, the Lp norm of their difference dominates their Wasserstein p-distance. The random variable t t W (N) F (yε ) G(yε , zε ) ds F (yε ) G(yε , z) dµ (z) ds p 0, r/ε s/ε s/ε − r/ε s/ε ε → Zr Zr ZZ with the same rate as indicated above. ε t ε We proceed to step (ii). It is clear that for almost all ω, F (y /ε) r G(ys/ε, z) ds is Lipschitz continuous in z. For any z , z Z, · 1 2 ∈ R t t F (yε ) G(yε , z ) ds F (yε ) G(yε , z ) ds /ε s/ε 1 − /ε s/ε 2 · Zr · Zr t ε F d(z1, z2) G(ys/ε, ) Lip ds (t r)d(z 1, z2) F G Lip. ≤ | |∞ | · | ≤ − | |∞| | Zr By the Kantorovich duality formula, for the distance between two probability measures µ1 and µ2, W (µ ,µ )=sup U dµ U dµ : U 1 , 1 1 2 1 − 2 | |Lip ≤ Z Z we have ε 2 (t r) F G Lip W1(µ ,µ). |E |≤ − · | |∞ · | | · 16 X.-M. LI For part (iii), let U be a continuous function on C([0, T ]; Y ). If σ C([0, T ]; Y ), let us denote by σ([0, r]) the restriction of the path to [0, r∈]. Since F is bounded continuous and G is Lipschitz continuous, t σ U F (σ([0, r])) G(σ , z) dµ(z) ds 7→ s Zr ZZ ε is a continuous function on C([0, T ]; Y ). By the weak convergence of (y /ε), E(U(I(ε))) converges to E(U(A(F, G))) and the random variables I(ε) con-· verge weakly to A(F, G). By now, we have proved that A(ε, F, G) converges to A(F, G) weakly; we thus conclude the first part of the lemma. ε Let us assume condition (4) from Assumption 3.3. In particular, (y /ε) · converges in W1(C([0, T ]; Y )). Let U be a Lipschitz continuous function on C([0, T ]; Y ). We define U˜ : C([0, T ]; Y ) R by → t U˜(σ)= U F (σ([0, r])) G(σs, z) dµ(z) ds . Zr ZZ Let σ1,σ2 are two paths on Y , U˜(σ ) U˜ (σ ) | 1 − 2 | t t 1 2 U Lip F G(σs , z) dµ(z) ds G(σs , z) dµ(z) ds ≤ | | · | |∞ − Zr ZZ Zr ZZ 1 2 (t r) U Lip F G Lip sup ρ(σs ,σs ). ≤ − | | · | |∞ · | | · 0 s T ≤ ≤ By the Kantorovitch duality and assumption (4), ˆ ˆ ˆ ε ˆ W1(PI(ε), PI ) (t r) F G Lip W1(Py , Py¯· ). ≤ − · | |∞ · | | · ·/ε We collect all the estimations together. Under assumptions (1)–(4), the following estimates hold: ˆ ˆ α ε W1(PA(ε), PA) C F G Lip(ε + ε)+ C F max δ G E, z, ≤ | |∞| | | |∞ z Z | | t r ∈ − ˆ ε ˆ + C(t r) F G Lip (W1(Py , Py¯· )+ W1(µε,µ)) − · | |∞ · | | · ·/ε + 2ε F min( G , Osc(G) ). | |∞ | |∞ | | We may now limit ourselves to ε 1 and conclude part 2 of the lemma. ≤ Remark 3.2. In the lemma above, we should really think that the zε process and process yε follow different clocks, the former is run at the fast 1 time scale ε and the latter at scale 1. PERTURBATION TO GEODESIC EQUATION 17 Example 3.5. Let (gs) be a Brownian motion on G = SO(n), solving N k dgt = LgtAk dwt . Xk=1 Here, A1,...,AN is an orthonormal basis of g. In Lemma 3.4 we take ε { } zt = gt/ε, then condition (2) holds. If f is a Lipschitz continuous function, t it is well known that the law of large numbers holds for 0 f(gs) ds, so does a central limit theorem. The remainder term in the central limit theorem is R of order √t and depends on f only through the Lipschitz constant f Lip. It is easy to see that the remainder term in the law of large num| | bers depends only on the Lipschitz constant of the function. Without loss of generality, we assume that f dg = 0. Let α solve the Poisson equation: ∆Gα = f. Then R t t 1 1 1 1 k f(gs) ds = α(gt) α(g0) (Dα)(gsAk) dws . t 0 t − t − t 0 Z Xk Z Since α is bounded, we are only concerned with the martingale term. By Burkholder–Davis–Gundy inequality, its L2 norm is bounded by 1/2 N t t 1/2 2 E 2 2 E 2 ((Dα)(gsAk)) ds Dα gs ds . t 0 ! ≤ t 0 | | Xk=1 Z Z By elliptic estimates, Dα is bounded by f L∞ . Since f is centered, it is bounded by Osc(f). In| summary,| | | t 2 1 1/2 2 E f(g ) ds f(g) dg C(Osc(f)t− ) . t s − ≤ Z0 ZN 1 In Theorem 1.1, we may wish to add an extra drift of the form ε A∗ where 1 G A g, so that G is 2 ∆ + LgA. Translations by orthogonal matrices are isometries,∈ so forL any A g the vector field gA is a killing field, and the Haar measure remains an invariant∈ measure for the diffusion with infinitesimal 1 G generator 2 ∆ + LgA. However, on a compact Lie group no left invariant 1 G vector field is the gradient of a function and 2 ∆ + LgA is no longer a symmetric operator. In this case, we do not know how to obtain the estimate in the example. 4. Proof. We are ready to prove the main theorem. In Lemma 3.2, we used a fundamental technique to split the integral t/ε ε ε ε (DF )x˜s (Hxs )(gse0) ds Zr/ε 18 X.-M. LI into the sum of a process of finite variation and a martingale. The computa- tion in the proof of Lemma 3.2 will be used to prove the weak convergence. A similar consideration was used in Li [21], which was inspired by a paper of Hairer and Pavliotis [13]. In the above-mentioned papers, the conver- gence is in probability; while here we can only expect weak convergence. To prove the convergence, we apply Stroock–Varadhan’s martingale method and Lemma 3.4; see also Borodin and Freidlin [4]; Papanicolaou, Stroock and Varadhan [25, 26] where the limit is given by a double integration in time. Our formulation for the limit is in terms of space averaging. Finally, we use explicit eigenfunctions of the Laplacian on SO(n) to compute the limiting generator. Proof of Theorem 1.1. We define a Markov generator ¯ on OM . If L F : OM R is bounded and Borel measurable and ei is an orthonormal basis of →Rn, we define { } n ¯F = ( DF ) (H (ge ),H (e ))h (g) dg L − ∇ u u 0 u i i i=1 ZG (4.1) X n (DF )u(Huei)LgA¯hi(g) dg, − G Xi=1 Z ε where hi is the solution to the Poisson equation (3.3). Since (˜xt/ε) is tight ε by Lemma 3.2, every sub-sequence of (˜xt/ε) has a sub-sequence that con- verges in distribution. We will prove that the probability distributions of ε ¯ ¯ (˜xt/ε) converge weakly to the probability measure, P , determined by . It ε L is sufficient to prove that if (¯yt) is a limit of (˜xt/ε), then t F (¯y ) F (u ) ¯F (¯y ) ds t − 0 − L s Z0 is a martingale. Since the convergence is weak, and the Markov process ε ε (˜xt , gt/ε) is not tight, we do not have a suitable filtration on Ω to work with. We formulate the above convergence on the space of continuous paths over OM on a given time interval [0, T ]. Let Xt be the coordinate process on the path space over OM , t = ε G σ (X ) : 0 s t and let Pˆ ε be the probability distribution of (˜x ) on the { s ≤ ≤ } x˜ t/ε path space over OM . By taking a subsequence if necessary, we may assume that Pˆx˜ε converges to P¯. Let{ F :}OM R be a smooth function with compact support. We will prove that with→ respect to P¯, t E F (Xt) F (Xr) ¯F (Xs) ds r = 0. − − r L G Z PERTURBATION TO GEODESIC EQUATION 19 ˆ ¯ Since Px˜ε P weakly, we only need to prove that for all bounded and continuous→ real value random variables ξ that are measurable with respect to , Gr t (4.2) lim ξ(F (Xt) F (Xr)) dPˆx˜ε = ξ ¯F (Xs) ds dP¯ . ε 0 − L → Z Z Zr By formula (3.4) in the proof of Lemma 3.2, for t r, ≥ F (˜xε ) F (˜xε ) t/ε − r/ε n t/ε ε ε ε ( DF ) ε (H ε (g e ),H ε (e ))h (g ) ds ∼− ∇ x˜s x˜s s 0 x˜s i i s i=1 Zr/ε (4.3) X n t/ε ε ε (DF ) ε (H ε e )L ε ¯h (g ) ds x˜s x˜s i gs A i s − r/ε Xi=1 Z n N t/ε ε k √ε DF ε H ε e Dh ε g A dw . ( )x˜s ( x˜s i)( i)(gs )( s k) s − r/ε Xi=1 Xk=1 Z Hence, up to a term of order ε, ξ(F (X ) F (X )) dPˆ ε t − r x˜ Z n t/ε ˆ ε = O(ε) ε ξ ( DF )Xs (HXs (Gse0),HXs (ei))hi(Gs) ds dPx˜ − r/ε ∇ Xi=1 Z Z n t/ε ˆ ε ε ξ (DF )Xs (HXs ei)LGsA¯hi(Gs) ds dPx˜ . − r/ε Xi=1 Z Z ε We prove this by working with the original processes. Let (˜xt ) denote a sub- sequence of the original sequence with limit (¯ys). For each i, l = 1,...,n, let us define β (u) = ( DF ) (H (e ),H (e )). li ∇ u u l u i By linearity of H and DF , u ∇ ( DF ) (H (ge ),H e )h (g) ∇ u u 0 u i i n n = ( DF ) (H (e ),H (e )) ge , e h (g)= β (u) ge , e h (g), ∇ u u l u i h 0 li i li h 0 li i Xl=1 Xl=1 for each i = 1,...,n; and t/ε ε ε ε ( DF ) ε (H ε (g e ),H ε (e ))h (g ) ds − ∇ x˜s x˜s s 0 x˜ (s) i i s Zr/ε 20 X.-M. LI n t/ε ε ε ε = ε βli(˜xs) gse0, el hi(gs) ds − r/ε h i Xl=1 Z n t ε ε ε = βli(˜xs/ε) gs/εe0, el hi(gs/ε) ds. − r h i Xl=1 Z We observe that (gε ) satisfies the equation dg = g A dwk with initial sε t k t k ◦ t value the identity element. The solution stays in the connected component SO(n). It is ergodic with the normalized Haar measureP dg on SO(n) as its invariant measure and it satisfies the Birkhoff ergodic theorem; see Exam- ε ple 3.5. By Lemma 3.2, (˜xs/ε) is tight, and equi-uniformly H¨older contin- ε ε ε ε uous on [0, T ]. In Assumption 3.3, we take zt = gt , dµε = dg, yt =x ˜t and check that conditions (1)–(4) are satisfied. In Lemma 3.4, we take G(u, g)= n β (u) ge , e h (g). Since the functions h : G R are smooth and G l=1 li h 0 li i i → is compact, also βli are smooth and bounded by construction, we may applyP Lemma 3.4. If φ is a bounded real valued continuous function on C([0, r]; OM ), let ξ = φ(˜xε , 0 u r). Then u/ε ≤ ≤ n t ε ε ε lim E ξ βli(˜x ) g e0, el hi(g ) ds ε 0 s/ε h s/ε i s/ε → r ! Xl=1 Z n t = E ξ βli(¯ys) ds ge0, el hi(g) dg r Gh i Xl=1 Z Z n t E = ξ DFy¯s (Hy¯s (el),Hy¯s (ei)) ge0, el hi(g) dg r ∇ Gh i Xl=1 Z Z n t E = ξ DFy¯s (Hy¯s (ge0),Hy¯s (ei))hi(g) dg . r G ∇ Xl=1 Z Z By the same reasoning, we also have t/ε E ε lim ε ξ (DF )x˜ε (Hx˜ε ei)LgεA¯hi(gs) ds ε 0 s s s → Zr/ε t E = ξ (DF )y¯s (Hy¯s ei) ds LgA¯hi(g) dg . Zr ZG We have proved (4.2). Since every sub-sequence of Pˆx˜ε has a sub-sequence that converges to the same limit, we have proved Pˆx˜ε P¯ weakly. Finally, we compute the limiting Markov generator→¯. We observe that L there is a family of eigenfunctions of the Laplacian on G with eigenvalue PERTURBATION TO GEODESIC EQUATION 21 n 1 n(n 1)/2 2 n 1 − . Indeed, since − (A ) = − I, − 2 k=1 k − 2 n(n 1)/2 n(n 1)/2 − P 4 4 − L L ge , e = g(A )2e , e gAk gAk −n 1h 0 ii −n 1 h k 0 ii Xk=1 − − Xk=1 = 2 ge , e . h 0 ii Thus, 4 h = ge , e i −n 1h 0 ii − is the solution to the Poisson equation (3.3): n(n 1)/2 1 − h = ge , e where = L L . LG i h 0 ii LG 2 gAk gAk Xk=1 4 ¯ We compute the second integral in (4.1). Since LgA¯hi = n 1 gAe0, ei , we have − − h i n (DF )u(Huei)LgA¯hi(g) dg G Xi=1 Z 4 = (DF ) (H gAe¯ ) dg −n 1 u u 0 − ZG 4 = (DF ) H gAe¯ dg = 0. −n 1 u u 0 − ZG Consequently, n ¯F = ( DF )u(Hu(ge0),Hu(ei))hi(g) dg L − G ∇ Xi=1 Z n = ( DF )u(Hu(ej),Hu(ei)) ge0, ej hi(g) dg. − G ∇ h i i,jX=1 Z In the last step, we use the fact that Hu( ) is linear and that ei is an o.n.b. of Rn. Let us define · { } a (e )= ge , e h (g) dg i,j 0 − h 0 ji i ZG 4 = ge , e ge , e dg. n 1 h 0 jih 0 ii − ZG Then n (4.4) ¯F = a ( DF ) (H (e ),H (e )). L − i,j ∇ u u j u i i,jX=1 22 X.-M. LI To further identify the limit, we first prove that ai,j(e0) is independent of e . Recall that G acts transitively on the unit sphere of Rn. Let e Rn 0 0′ ∈ we take O such that Oe0′ = e0. By the right invariant property of the Haar measure, ge′ , e ge′ , e dg = gOe , e gOe , e dg = ge , e ge , e dg. h 0 jih 0 ii h 0 jih 0 ii h 0 jih 0 ii ZG ZG ZG We first compute the case of i = j and n = 2: 6 2π a1,2(e1)= ge1, e1 ge1, e2 dg = cos(θ) sin(θ) dθ = 0. SO h ih i − Z (2) Z0 If n> 2, for any i = j, there is an orientation preserving rotation matrix 6 O such that Oei = ei and Oej = ej. For example, if i = 1, j = 2, we take O = ( e , e , e , e −,...,e ). So − 1 2 − 3 4 n ge , e ge , e dg = ge , Oe ge , Oe dg h 0 jih 0 ii − h 0 jih 0 ii ZG ZG = ge , e ge , e dg. − h 0 jih 0 ii ZG Thus, a =0 if i = j. Let i,j 6 C = ge , e 2 dg. i h 0 ii ZG For i = 1,...,n, C = ge , e 2 dg is independent of i and i Gh 0 ii n R 2 ge0, ei dg = 1 G h i Z Xi=1 1 and consequently Ci = n . The nonzero values of (ai,j) are 4 4 a = ge , e h (g) dg = ge , e 2 dg = . i,i − h 0 ii i n 1 h 0 ii (n 1)n ZG − ZG − By the definition, ∆ F (u)= n L L F . Since is the canonical H i=1 H(ei) H(ei) ∇ flat connection, H(ei)H(ei) = 0. See the paragraph before equation (3.4). By (4.4), we see∇ that P n ¯F (u)= a ( DF ) (H (e ),H (e )) L − i,j ∇ u u j u i i,jX=1 n 4 = ( DF ) (H (e ),H (e )) (n 1)n ∇ u u i u i − Xi=1 4 = ∆ F (u). (n 1)n H − PERTURBATION TO GEODESIC EQUATION 23 ε We conclude that (˜xt/ε) is a diffusion process with infinitesimal generator 4 ε ε (n 1)n ∆H . Since (xt/ε) is the projection of (˜xt/ε) it is also convergent. The − operators ∆H and ∆ are intertwined by π; for f : M R smooth, (∆H f) π = ∆(f π). 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