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STOCHASTIC HOMOGENIZATION ON HOMOGENEOUS SPACES XUE-MEI LI A BSTRACT. Let G be a Lie group with a closed subgroup H and a left invariant Riemannian metric. Let g = h ⊕ m be a reductive decomposition. We propose a family of ‘interpolation equations’ arising from inhomogeneous scaling of the √ Riemannian metric, where the h directions are scaled by . Solutions of interpolation equations interpolate between translates of a one parameter subgroups of G generated by a horizontal vector Y0 and a hypoelliptic diffusion on H. Let m = m0 ⊕ m1 ⊕ · · · ⊕ mr be an AdH invariant decomposition. If is small and Y0 ∈ ml , l 6= 0, the ‘slow components’ of the solutions can be approximated on [0, 1 ] by a non-trivial diffusion process, describing the ‘effective dynamic’ across the orbits. We classify the scaling limits and examine whether the effective dynamic on G/H is Markovian. 1. I NTRODUCTION Let G be a smooth connected real Lie group of dimension n and H a closed proper subgroup of dimension at least one. Denote by g and h their respective Lie algebras. If m is a left invariant Riemannian metric on G, we consider a family of non-homogeneously scaled Riemannian metrics m . To define these let {X1 , . . . , Xn } be an orthonormal basis of g extending an orthonormal basis {X1 , . . . , Xp } of h and let 1 ∗ ∗ 1 ∗ ∗ E = √ X1 , . . . , √ Xp , Xp+1 , . . . Xn . The superscript ∗ above a letter denotes the corresponding left invariant vector field. By declaring E an orthonormal frame, we have a family of left invariant Riemannian metrics m on G. We study the stochastic dynamics associated with these inhomogeneous scalings of Riemannian metrics on G and propose an interpolation equation that, in the limit of → 0, describes the ‘effective motion’ across the ‘orbits’. In this article we are not concerned with the problem of keeping the sectional curvatures bounded and G needs not be compact. As an example let G = S 3 and H = S 1 . As tends to 0, the sequence of Riemannian manifolds (S 3 , m ) converges to S 2 ( 12 ), in the Gromov Hausdorff topology, while the sectional curvatures remain bounded. See M. Berger [2]. Let (Ω, F, Ft , P ) be a filtered probability space satisfying the usual assumptions and (bkt , k = 1, . . . , N, ) a family of independent real valued Brownian motions. AMS Mathematics Subject Classification : 60Gxx, 60Hxx, 58J65, 58J70 1 2 XUE-MEI LI Let γ, δ be positive real numbers, Xk ∈ h as above, and Y0 ∈ g. The interpolation equation is: dgt = p X γXk∗ (gt ) ◦ dbkt + δY0∗ (gt )dt. (1.1) k=1 Here ◦ denotes Stratonovich integration. For γ = 1 and δ = |Y10 | , these equations are driven by unit length vector fields on the Riemannian manifold (G, m ). If δ = 0 the solution from the identity is a scaled Brownian motion on the subgroup H. If γ = 0, the solutions are translates of the one parameter family subgroup generated by Y0 . If m = {Xp+1 , . . . , Xn } is AdH -invariant and Y0 ∈ m, these one parameter families are horizontal curves for the horizontal distribution determined by m. The solutions of equations (1.1) interpolate between translates of the one parameter group generated by δY0 on G and Brownian motion on H. Take γ → ∞ we study the limits of the solutions, suitably scaled, and investigate the problem whether the effective motion is a Brownian motion on the homogeneous manifold G/H. More generally let {Ak } be a Lie algebra generating subset of h. Take Y0 from a complement, m, of h in g. We study the stochastic differential equations (SDEs) N dgt 1 1 X ∗ Ak (gt ) ◦ dbkt + A∗0 (g0 )dt + Y0∗ (gt )dt, =√ g0 = g0 , (1.2) k=1 as → 0. Denote π the projection from G to M = G/H. If (yt ) are solutions to the P 1 ∗ k ∗ equations dyt = √1 N k=1 Ak (yt ) ◦ dbt + A0 (yt ) then π(yt ) = π(y0 ) for all t. The constant of motion, π, takes its value in the orbit manifold M . For small, we would expect that π(gt ) measures its deviation from the ‘orbit’ containing g0 . The inherited non-linearity from π causes some technical problems, for example the homogeneous manifold is in general not prarallelizable, so it is not easy to work 1 PN 1 ∗ ∗ ∗ ∗ directly with xt = π(gt ). The operator 2 k=1 LAk LAk + LA0 + LY0 does not satisfy Hörmander’s conditions, so we do not wish to work directly with (gt ) either. To overcome these difficulties we assume that g has a reductive decomposition, by which we mean an AdH -invariant subspace m such that g = h ⊕ m, where AdH denotes the adjoint representation restricted to H. If H is compact, such a decomposition always exists. With this we construct a stochastic process (ut ) on G which has the same projection as (gt ). In fact, since m is AdH invariant, G is a principal bundle over M with structure group H. We lift (xt ) to G to obtain a ‘horizontal’ stochastic process (x̃t ) covering (xt ). The horizontality is with respect to the Ehreshmann connection determined by m. As ‘perturbations’ to the random motions on the fibres, the horizontal lifts (x̃t ) of (xt ) describe transverse motions across the orbits of the action fields. This consideration has a bonus: our study does not simply lead to asymptotic analysis of the x processes, we also have information on the asymptotic properties of their horizontal lifts. This is more striking if we get out of the picture of STOCHASTIC HOMOGENIZATION ON HOMOGENEOUS SPACES 3 homogenization for a moment, and consider instead the three dimensional Heisenberg group as a fibre bundle over R2 . The horizontal lift of an R2 -valued Brownian motion to the Heisenberg group is their stochastic Levy area. Horizontal lifts of stochastic processes are standard tools in the study of Malliavin calculus in association with the study of the space of continuous paths over a Riemannian manifold. We will deduce an explicit equation for the horizontal lift in terms of the vertical component of (gt ), which makes further analysis possible. If Y0 ∈ m, it is fairly easy to see that xt moves fairly slowly and furthermore the speed at which xt crosses M depends on the specific vector Y0 . More precisely, m decomposes into two AdH -invariant components m0 ⊕ m̃ with the property that, m0 is the space of invariant vectors and for Y0 ∈ m̃, x̃t are approximated on [0, 1 ] by a stochastic process ūt . The main objectives of the article is to use an invariant irreducible decomposition of m and the position of Y0 in m to describe the scale at which the limit is taken and to classify the effective limit. If the Lie bracket between Ak and Y0 is not trivial, randomness is generated in the [Ak , Y0 ] direction and transmitted from the vertical to the horizontal directions. We ask the question weather the effective limits of xt are scaled Brownian motions? This is the case for S 2 = S 3 /S 1 , S n = SO(n + 1)/SO(n), and the non-compact homogeneous space H n . In these cases the speed of the effective Brownian motion is given by an eigenvalue of the Laplace-Beltrami operator on H. However it only takes a moment to figure out this cannot be always true. The noise cannot be transmitted to directions in an irreducible invariant subspace of m not containing Y0 . However, within an irreducible invariant subspace, the transmission of the noise should be ‘homogeneous’ and there is one candidate effective second order differential operator. In case the noise is ‘homogeneous’ and symmetric we prove that this candidate operator is indeed the ‘effective Markov operator’ up to a constant multiple λ and determine λ. If {Ak } is an o.n.b. of h, the effective limits depend only on the norm of Y0 and the component containing Y0 . If Y0 ∈ m0 then x̃t (ω) = g0 (ω) exp(tY0 ), which converges trivially if g0 does, and the study terminates here. For simplicity of the statements, we take the initial values g0 to be g0 ∈ G, independent of . The projection of the effective Markov process is a Markov process if the ‘effective Markov operator’ has suitable invariant property. This is shown to hold if the noise is ‘homogeneous elliptic’ and ‘symmetric’, in other words, if A0 = 0 and {Ak } is an orthonormal basis of h. This ‘invariant’ property also holds if the noise is symmetric, satisfies strong Hörmander’s conditions and the dimension of ml is not 2, 4, or 7. This work is reminiscent of that in M. Freidlin and A. D. Wentzell [19] where an explicit Markov process on a graph was obtained from the level sets of the Hamiltonian function. This model is somewhat related to that in X.-M. Li [37], which was motivated by the problem of approximating’ Brownian motions by solutions of Hamiltonian systems and was studied in the one dimensional case by E. Nelson [45]. Some ideas in this paper were developed from X.-M. Li [36]. We note that in M. Freidlin and M. Weber [18], real valued constants of motion are used in place of a Hamiltonian. Our reduction in complexity bears resemblance to aspects of 4 XUE-MEI LI geometric mechanics and Hamiltonian reduction, see e.g. D. Holm, J. Marsden, T. Ratiu, A. Weinstein [26], and J. Marsden, G. Misiolek, J.-P. Ortega, M. Perlmutter and T. Ratiu [41]. Our reduced equations are ODEs on manifolds with random coefficients. There are many studies of random ODEs Rn , see e.g. R. L. Stratonovich [53], R. Z. Khasminski [23] or for Hamiltonian system, see e.g. A. N. Borodin and M. Freidlin [9], M. Freidlin and A. D. Wentzell [20]. See also W. Kohler and G. C. Papanicolaou[47] and G. C. Papanicolaou and S. R. S. Varadhan [48]. A set of more complete references can be found in X.-M. Li [38]. We finally note that convergence of Riemannian metrics have been studied by N. Ikeda and Y. Ogura [29] and Y. Ogura and S. Taniguchi [46]. We would also like to refer to J.-M. Bismut [4, 5] whose philosophy is close to ours and whose studies were motivated by analysis on loop spaces. See also G. Lebeau and J.-M. Bismut [7] and a recent survey by J.-M. Bismut [6]. For stochastic averaging with geometric structures, we refer to [35], P. Gargate and P. Ruffino [22], M. Hogele and P. Ruffino [25] for studies of stochastic flows on foliated manifolds M. Liao and L. Wang [39], and R. Sowers [52]. 2. M AIN R ESULTS We endow g with an AdH invariant metric and let m = m0 ⊕ m1 ⊕ · · · ⊕ ml be an invariant decomposition for AdH where m0 consists of invariant vectors of AdH and each ml is an irreducible invariant space if l 6= 0. Let m̃ = m1 ⊕· · ·⊕ml . If H is compact we may and will assume that the decomposition is orthogonal. We state the program below, where in part (2-3) H is compact and Y0 ∈ m̃. (1) Deduce an equation for (x̃t ). We observe that the Ehreshman connection is independent of . The techniques here applies to SDEs on general principal fibre bundles, see §5 and §14. (2) If Y0 ∈ m̃ and {A1 , . . . , AN } generates h, we prove that the stochastic processes (x̃t , t ∈ [0, T ]) and (xt , t ∈ [0, T ]) converge weakly, as → 0, to ūt and x̄t respectively. A rate of convergence is given. (3) Describe these limits, the so called effective processes. In Proposition 5.3 and Corollary 5.6 we write down explicit formulas for the slow motions and at the same time we identify the ‘fast motion’ (ht ) to be an 1 Haar measure is its unique invariant L0 diffusion on H for which the normalized P ∗ ∗ ∗ probability measure. Here L0 = 12 N L L k=1 Ak Ak + LA0 . The Markov generator L̄ has an explicit Hörmander form, see Corollary 8.5. The first questions we ask are: (a) Is L̄ a semi-elliptic differential operator and if so what is its rank? (b) Does L̄ have an invariant form? Or, is (x̄t ) a Markov process? For question (a) we first observe that the action of AdH generates directions in ml , it however does not generate any direction in a component complementary to ml and so the rank of L̄ is at most dim(ml ). In the statements below, Y0 ∈ ml where l 6= 0, and A0 = 0. Theorem 9.3. Suppose that {Ak , k = 1, . . . , N } is an orthonormal basis of h. Then there exists a number λl > 0, independent of Y0 ∈ ml , such that for any STOCHASTIC HOMOGENIZATION ON HOMOGENEOUS SPACES 5 2 (G; R), f ∈ CK 1 L̄f = λl Z ∇L df ((Ad(h)(Y0 ))∗ , (Ad(h)(Y0 ))∗ ) dh. H Furthermore (x̄t ) is a Markov process, λP l is an eigenvalue of −L0 . LYi LYi where {Yi } is an orthonormal Denote by ∆ml the ‘round’ operator basis of ml . The round operator is independent of the choice of the basis, indeed ∆ml = traceml ∇L d where ∇L is the left invariant connection. In case G is of compact type and m is irreducible this is the ‘horizontal Laplacian’. Theorem 10.3. Suppose that {Ak } is an orthonormal basis of h and dim(ml ) 6= |Y0 |2 4, 7. Then L̄ = λl dim(m ∆ml and (x̄t ) is a Markov process. l) Corollary 10.4. Suppose that H is a maximal torus group of a semi-simple group G and {Ak } is an orthonormal basis of h. Then there exists λl such that L̄ = 2λ1 l |Y0 |2 ∆ml . If {Ak } is only a set of generator of the Lie algebra h, an invariant formula, such as in Theorem 9.3 may still exist. This is the content of Theorem 10.1 below. However the scale may depend on Y0 , see Example 9.4 and Example 9.5. Theorem 11.1. Suppose that {A1 , . . . , AN } is a set of generators of h and dim(ml ) 6= 2, 4, 7. Then for each vector Y0 ∈ ml there exists a number λ(Y0 ) such 2 0| that L̄ = λ(Y0|Y ) dim(ml ) ∆ml , and (x̄t ) is a Markov process. The above descriptions are algebraic, in §12 we discuss their differential geometric counterpart. Let G be endowed with a left invariant Riemannian metric that is AdH -invariant and let M be the Riemannian homogeneous space. The translates of the one parameter subgroups of G are not necessarily geodesics for the Levi-Civita connection. Their projections are not necessarily geodesics either, for the Riemannian homogeneous manifold. In general L̄ needs not be a multiple of the Laplace-Beltrami operator even if it is elliptic: it may have a nontrivial drift. Suppose the conclusion of Theorem 11.1 holds. If G is compact, or more generally 2 0| traceml ∇d where if traceml (X) vanishes for all X ∈ g, then L̄ = λl (Y0|Y) dim(m l) ∇ is the Levi-Civita connection on G. See Remark 12.1. If furthermore M is naturally reductive, see Definition 12.2, then (x̄t ) is a Markov process with gener2 0| ator λl (Y0|Y) dim(m trace ∇d where ∇ is the Levi-Civita connection on M and the l) trace is taken in a sub-bundle of the tangent bundle T M , see Theorem 12.4. The above conclusion holds under the more general condition ‘traceml U = 0’ where U : m × m → m is defined by 2hU (X, Y ), Zi = hX, [Z, Y ]m i + h[Z, X]m , Y i. In Examples 13.2 and 13.4, we consider the following Riemannian homogeneous spaces: the n- sphere S n = SO(n + 1)/SO(n) and the non-compact hyperbolic space H n = Oo (1, n)/SO(n). If we take the reductive symmetric decompositions the effective operator L̄ on SO(n) (or the identity component of O(1, n)) is a multiple of the horizontal Laplacian and satisfies the one step Hörmander con8 . The effective stochastic processes on these symmetric dition. The scale is n(n−1) manifolds are scaled Brownian motions. The Hopf fibration example is given in 6 XUE-MEI LI Example 13.1. Further examples include the Stiefel manifolds, the Grassmannian manifolds and SU (n). 3. E XAMPLES AND N OTATION If g ∈ G denote by Lg and Rg the left and right multiplications on G. Denote by dLg and dRg their differential. The differentials are also denoted by T Lg and T Rg where these are traditionally used. Denote by Ad the adjoint action on G and ad the adjoint action on g. If m is a subspace of g, gm denotes the set of left translates of elements of m to Tg G. From time to time, we identify the set of left invariant vector fields with elements of the Lie algebra. Let h be a Lie sub-algebra of g. A complement m of h in g is said to be a reductive structure if AdH (m) ⊂ m where AdH : H → L(g; g) denotes the restriction of the adjoint action Ad to H. This implies that [h, m] ⊂ m and we say that g = h ⊕ m is a reductive decomposition. If H is connected the reductive property AdH (m) ⊂ m is equivalent to [h, m] ⊂ m. We list a number of examples. (1) If G is a Lie group, let H = {1} and m = g. (2) Let G be a connected compact Lie group, H = {(g, g) : g ∈ G} and h = {(X, X) : X ∈ g}. Then G = (G × G)/H is a reductive homogeneous space in three ways: m+ = {(0, X), X ∈ g}, (3) (4) (5) (6) m0 = {(X, −X), X ∈ g}, m− = {(X, 0), X ∈ g}. The middle one, h ⊕ m0 , is the symmetric decomposition. If g has an AdH invariant inner product take m = h⊥ with respect to this inner product. Then if Y ∈ m and X ∈ h, Ad(h)(Y ) ∈ m. In fact Ad(h−1 )(X) ∈ h, so it follows that hY, Ad(h−1 )(X)i = 0 and hAd(h)(Y ), Xi = 0 for all X ∈ h. Thus the decomposition is reductive. If g admits an AdH invariant non-degenerate bilinear form such that its restriction to h is non-degenerate, let m = {Y ∈ g : B(X, Y ) = 0, ∀X ∈ h}. In [34, Chap. 10], Kobayashi and Nomizu considered the case where B is Ad(G) invariant. If H is compact there exists an AdH -invariant inner product on g and hence an AdH -invariant subspace m such that g = h ⊕ m is a direct sum. If G/H is a Riemannian homogeneous space and G is connected then by a theorem of van Danzig and van der Waerden (1928), the isotropy isometry groups at every point is compact. See Kobayashi [32] for discussions. Let E(n) be the space of rigid motions on Rn , R v n E(n) = : R ∈ SO(n), v ∈ R 0 1 and H its subgroup of rotations. Elements of H fix the point o = (0, 1)T and E(n)/H = {(x, 1)T , x ∈ Rn }. The projection of a matrix in E(n) is the last column. Let A 0 0 v h= , A ∈ so(n) , m = , v ∈ Rn . 0 0 0 0 STOCHASTIC HOMOGENIZATION ON HOMOGENEOUS SPACES 7 Then h ⊕ m is a reductive structure for the Lie algebra of E(n). Notation. If ζ is a random variable Law(ζ) denotes its probability distribution. Let T be a positive number. If M is a manifold, denote by C([0, T ]; M ) the space of continuous functions from [0, T ] to M and Cx ([0, T ]; M ) its subspace with initial value x. If µ is a measure on the path space, µt = (evt )∗ µ denotes the pushed forward measure by the evaluation map at t. Denote by C r (M ; R) the space of real r (M ; R) its subspace of functions valued C r smooth functions on M , and by CK Pr−1 (r) with compact support. For f ∈ C r (M ; R), define |f |r,∞ := k=1 |∇ df |∞ . The connection ∇ will be specified later. For y in a Riemannian manifold, ρy denotes the Riemannian distance from y. Also ∆N denotes the Laplace-Beltrami operator on a Riemannian manifold N with sign convention ∆N = div grad. 4. T HE M OTIVATING E XAMPLE As a motivating example, we take G = SU (2), H = U (1), and the bi-invariant Riemannian metric. For A, B ∈ g, hA, Bi = 21 trace AB ∗ . Let i 0 0 1 0 i X1 = , X2 = , X3 = (4.1) 0 −i −1 0 i 0 be the Pauli matrices. They form an orthonormal basis of the Lie algebra g with respect to the bi-invariant metric. The Lie algebra h is generated by X1 . We take m to be the vector space generated by the remaining two Pauli matrices and obtain the family of Berger’s metric, which we denote as before by {m , > 0}. The manifolds (G, m ) are Berger’s spheres. With the original metric, G is the canonical three sphere in R4 . As tends to zero, the Berger’s spheres converge in the Gromov-Hausdorff topology to the lower dimensional two sphere S 2 ( 12 ) of radius 12 . We refer to J. Cheeger, M. Gromov [10], K. Fukaya [21], and A. Kasue, H. Kumura [31] for further studies and other types of convergences of Riemannian manifolds. Let us first take the Brownian motions on Bergers’ spheres. A Brownian motion on (S 3 , m ) is determined by the Hörmander type operator ∆ = 1 LX1∗ LX1∗ + P3 ∗ ∗ i=2 LXi LXi . Although no longer associated with the round metric, there are many symmetries in the following stochastic differential equation (SDE), 1 dgt = √ X1∗ (gt ) ◦ db1t + X2∗ (gt ) ◦ db2t + X3∗ (gt ) ◦ db3t . In particular the probability distributions of its slow components at time t are independent of , see Example 5.5 in Section 5. A more interesting equation is perhaps the following 1 dgt = √ X1∗ (gt ) ◦ db1t + X2∗ (gt ) ◦ db2t , in which the incoming noise is 2-dimensional. We will go one step further and use a one dimensional noise. Let (bt ) be a one dimensional Brownian motion on a given probability space. If Y0 = c2 X2 + c3 X3 , where c2 , c3 are numbers not 8 XUE-MEI LI simultaneously zero, the infinitesimal generator of the equation 1 dgt = √ X1∗ (gt ) ◦ dbt + (Y0 )∗ (gt )dt, (4.2) satisfies weak Hörmander’s conditions. Indeed by the structural equations, [Y0 , X1 ] = 2c2 X3 − 2c3 X2 , and the matrix 1 √ 0 0 0 c2 −2c3 . 0 c3 2c2 is not degenerate. In general, equation (1.2) need not satisfy Hörmander’s condition. See Corollary 10.4 below for the scaling limits. 5. R EDUCTION Let M = G/H be a homogeneous space. The sub-group H acts on G on the right by group multiplication and we have a principal bundle structure P (G, H, π) with base space M and structure group H. Each fibres π −1 (x) is diffeomorphic to H and the kernel of dπg is gh = {T Lg (X) : X ∈ h}. For a ∈ G, denote by L̄a the left action on M : L̄a (gH) = agH. Then π ◦ La = L̄a ◦ π and (dπ)a dLa |To M = (dL̄a )(dπ)1 . We identify m with To M as below: d dL̄ o. dt t=0 exp(tX) An Ehreshmann connection is a choice of a set of complements of gh that is right invariant by the action of H. If g = h ⊕ m is reductive, (gh)m = T Rh (gm) and Tg G = gh ⊕ gm is an Ehreshmann connection. See Kobayashi, Nomizu [33]. An Ehreshmann connection determines and is uniquely determined by horizontal lifting maps hu : TgH M → Tu G where u ∈ π −1 (gH). Furthermore, to every piecewise C 1 curve c on M and every initial value c̃(0) ∈ π −1 (c(0)), there is a d c̃(t) ∈ c̃(t)m. See Besse [3]. unique curve c̃ covering c with the property that dt We can also horizontally lift a sample continuous semi-martingale. The case of the linear frame bundle is specially well known, see J. Eells and D. Elworthy [13], P. Malliavin [40]. See also M. Arnaudon [1], M. Emery [17] and D. Elworthy [14], and N. Ikeda, S. Watanabe [30]. This study has been taken further in D. Elworthy, Y. LeJan, X.-M. Li [16], in connection with horizontal lifts of intertwined diffusions. A continuous time Markov process, whose infinitesimal generator is in the form of the sum of squares of vector fields, is said to be horizontal if the vector fields are horizontal vector fields. Let {blt , wtk , k = 1, . . . , N1 , l = 1, . . . , N2 } be real valued, not necessarily independent, Brownian motions. Let g = h ⊕ m be a reductive structure . Let {Ai , 1 ≤ i ≤ p} be a basis of h, {Xj , p + 1 ≤ j ≤ n} a basis of m, and {cik , cjl } is a family of real valued smooth functions on G. Let $ be the canonical connection 1-form on the principal bundle P (G, H, π), determined by $(A∗k ) = Ak whenever Ak ∈ h and $(Xj∗ ) = 0 whenever Xj ∈ m. X 7→ (dπ)1 (X) = STOCHASTIC HOMOGENIZATION ON HOMOGENEOUS SPACES 9 Definition 5.1. A semi-martingale (x̃t ) in G is horizontal if $(◦dx̃t ) = 0. If (xt ) is a semi-martingale on M , denote by (x̃t ) its horizontal lift. P P Let Ykh (g) = ni=p+1 cik (g)Xi∗ (g) and Ylv (g) = pj=1 cjl (g)A∗j (g). Denote by (gt , t < ζ) the maximal solution to the following system of equations, dgt = Y0h (gt )dt + N1 X Ykh (gt ) ◦ dwtk + Y0v (gt )dt + k=1 N2 X Ylv (gt ) ◦ dblt , (5.1) l=1 0 bt = with initial value g0 and xt = π(gt ). For simplicity let t and wt0 = t. In the lemma below, we split (gt ) into its ‘horizontal’ and ‘vertical part’ and describe the horizontal lift of the projection of (gt ) by an explicit stochastic differential equation where the role played by the vertical part is transparent. Lemma 5.2. Let x0 = π(g0 ). Take u0 ∈ π −1 (x0 ) and define a0 = u−1 0 g0 . Let (ut , at , t < η) be the maximal solution to the following system of equations dut = N1 X n X cik (ut at ) (Ad(at )Xi )∗ (ut ) ◦ dwtk (5.2) k=0 i=p+1 dat = p N2 X X cjl (ut at )A∗j (at ) ◦ dblt . (5.3) l=0 j=1 Then the following statements hold. (1) (ut at , t < η) solves (5.1). Furthermore ζ ≥ η and ζ is the life time of (gt ). (2) (ut , t < η) is a horizontal lift of (xt , t < ζ). Consequently ζ = η a.s. (3) If (x̃t , t < ζ) is an horizontal lift of (xt , t < ζ), it is a solution of (5.2) with u0 = x̃0 . Proof. (1) Define g̃t := ut at . On {t < η}, we have ∗ dg̃t = dRat ◦ dut + (a−1 t ◦ dat ) (g̃t ). Here dRg denotes the differential of the right translation Rg ;, a−1 t in the last term denotes the action of the differential, dL(at )−1 , of the left multiplication. See page 66 of S. Kobayashi, K. Nomizu [33]. The stochastic differential d on both the left and right hand side denotes Stratonovich integration. Then (g̃t , t < η) is a solution of (5.1), which follows from the computations below. dg̃t = N1 X n X cik (ut at )dRat (Ad(at )Xi )∗ (ut ) ◦ dwtk k=0 i=p+1 + Since p N2 X X cjl (ut at )A∗j (g̃t ) ◦ dblt . l=0 j=1 dRat (Ad(at )(Xj ))∗ (ut ) dg̃t = N1 X n X k=0 j=p+1 = Xj∗ (g̃t ), cjk (g̃t ) (Xk )∗ (ut ) ◦ dwtk + N2 X l=0 Ylv (g̃t ) ◦ dblt , 10 XUE-MEI LI which is equation (5.1). Since the coefficients of (5.1) are smooth, pathwise uniqueness holds. In particular gt = ũt at and the life time ζ of (5.1) must be greater or equal to η. (2) It is clear that a0 ∈ H and $(◦dut ) = 0. Let yt = π(ut ). Then dyt = N1 X n X cik (ut at )dL̄ut at (dπ(Xi )) ◦ dwtk , k=0 i=p+1 following from the identity dπ ((Ad(a)Xi )∗ (u)) = dL̄u dL̄a (dπ(Xi )). By the same reasoning (xt ) satisfies the equation dxt = N1 X dπ(Ykh (gt )) ◦ dwtk . k=0 P = ni=p+1 cik (gt )dπ(Xi∗ (gt )) and dπ(Xi∗ (gt )) = By the definition, T L̄gt dπ(Xi ). Using part (1), gt = ut at , we conclude that the two equations above are the same and π(ut ) = xt . This concludes that (ut ) is a horizontal lift of (xt ) up to time η. It is well known that through each u0 there is a unique horizontal lift (x̃t ) and the life time of (x̃t ) is the same as the life time of (xt ). See I. Shigekawa [51] and R. Darling [11]. The life time of (xt ) is ζ. Let ãt be the process such that gt = ut ãt for t < η. On {t < η}, ãt = at and ut = x̃t . If η < ζ, as t → η, limt→η ut leaves every compact set. This is impossible as it agrees with x̃t . Similarly (at ) cannot explode before ζ. (3) Let (gt , t < ζ) be a solution of (5.1) and set xt = π(gt ). For each t, gt and x̃t belong to the same fibre. Define kt = x̃−1 t gt , which takes values in H and is defined for all t < ζ. Then, ∗ dx̃t = dR(kt )−1 ◦ dgt + kt ◦ d(kt )−1 (x̃t ). dπ(Ykh (gt )) From this and equation (5.1) we obtain the following, dx̃t = dR(kt )−1 N1 X n X cik (gt )Xi∗ (gt ) ◦ dwtk k=0 i=p+1 + dR(kt )−1 p N2 X X (5.4) cjl (gt )A∗j (gt ) ◦ dblt + ((kt ) ◦ d(kt )−1 )∗ (x̃t ). l=0 j=1 We apply the connection 1-form $ to equation (5.4), observing ω(◦dx̃t ) = 0 and Ad(kt )(Xi ) ∈ m, 0= p N2 X X cjl (gt )$x̃t dR(kt )−1 A∗j (gt ) ◦ dblt + (kt ) ◦ d(kt )−1 l=0 j=1 = p N2 X X l=0 j=1 cjl (gt )Ad(kt )Aj ◦ dblt + (kt ) ◦ d(kt )−1 . STOCHASTIC HOMOGENIZATION ON HOMOGENEOUS SPACES 11 We have used the fact that Xj∗ are horizontal, dR(kt )−1 ((Xj )∗ (gt )) = (Ad(at )(Xj ))∗ (x̃t ), and $ga−1 (R(a−1 )∗ w) = Ad(a)$g (w) for any w ∈ Tg G. It follows that d(kt )−1 = P 2 Pp j l − N j=1 cl (gt )R(kt )−1 Aj ◦ dbt . By the product rule, l=0 dkt = p N2 X X cjl (gt )A∗j (kt ) ◦ dblt . l=0 j=1 Thus (kt ) solves equation (5.3) and we take this back to (5.4). Since the vertical vector field associated to kt ◦ dkt−1 evaluated at x̃t is given by the formula (kt ◦ d(kt ) −1 ∗ ) (x̃t ) = − p N2 X X cjl (gt )(Ad(kt )Aj )∗ (x̃t ) ◦ dblt , l=0 j=1 the second term and the third term on the right hand side of (5.4) cancel. Using the same computation given earlier, we see that N1 X n X dx̃t = dR(k )−1 cik (gt )Xi∗ (gt ) ◦ dwtk t k=0 i=p+1 = N1 X n X cik (x̃t kt ) (Ad(kt )Xi )∗ (x̃t ) ◦ dwtk , k=0 i=p+1 proving that (x̃t , t < ζ) is a solution of (5.2) and concludes the proof. In particular ζ ≥ τ. If the stochastic process on H does not explode, the following proposition follows immediately. Proposition 5.3. Suppose that ut is a solution of the following equation dut = N1 X n X cik (ut at ) (Ad(at )Xi )∗ (ut ) ◦ dwtk k=0 i=p+1 P 2 Pp j ∗ where (at , t ≥ 0) is a global solution of the equation dat = N j=1 cl (ut at )Aj (at )◦ l=0 dblt . If (gt ) is a solution of (5.1) with g0 = u0 then π(ut ) = π(gt ) and (ut ) is the horizontal lift of (xt ). An immediate consequence of Lemma 5.2 is that a solution to equation (5.5) below factorizes and is described by equation (5.6). Given appropriate scaling, the stochastic process (at ) is a fast rotation in the fibre. Corollary 5.4. Let l > 0 and Y0 ∈ m. Let (gt ) be a solution to the equation dgt = Y0h (gt )dt + N1 X k=1 g0 N Ykh (gt ) ◦ dwtk + 2 X 1 v 1 v Y0 (gt )dt + Y (g ) ◦ dblt , 0 l k t l=1 = g0 . (5.5) 12 XUE-MEI LI Then the horizontal lift of its projection satisfies the following system of equations dx̃t = N1 X n X cik (x̃t at ) (Ad(at )Xi )∗ (x̃t ) ◦ dwtk k=1 i=p+1 n X ci0 (x̃t at ) (Ad(at )Xi )∗ (x̃t ) dt, + (5.6) i=p+1 dat p p N2 X X 1 j ∗ 1X j ∗ l c (x̃ a )A (a ) dt, cl (x̃t at )Aj (at ) ◦ dbt + = l 0 0 t t j t l=1 j=1 j=1 up to an explosion time. Here x̃0 = g0 and a0 is the identity. Proof. We only need to observe that the Ehreshmann connection induced by the reductive decomposition is independent of the scaling m given by E . Example 5.5. Let us take the Hopf fibration π : G → S 2 ( 21 ) where G = SU (2) is given the bi-invariant metric hA, Bi = 12 trace AB ∗ . Let H = U (1) and M = S 2 ( 12 ). If we represent SU (2) by the unit sphere in C2 and S 2 ( 21 ) as a subset in R ⊕ C, the Hopf map is given by the formula π(z, w) = ( 21 (|w|2 − |z|2 ), z w̄). It is a Riemannian submersion. Let {X1 , X2 , X3 } be Pauli matrices defined by (4.1) and let m = hX2 , X3 i. Then [m, h] ⊂ m. This is easily seen from the structure of the Lie bracket: [X1 , X2 ] = −2X3 , [X2 , X3 ] = −2X1 , [X3 , X1 ] = −2X2 . Let (gt ) be a solution to the equation dgt = 3 X k=2 1 Xk∗ (gt ) ◦ dbkt + √ X1∗ (gt ) ◦ db1t . Let xt = π(gt ), denote by (ut ) the horizontal lift of (xt ). By Corollary 5.4, (ut ) satisfies dut = (Ad(at )X2 )∗ (ut ) ◦ db2t + (Ad(at )X3 )∗ (ut ) ◦ db3t and (at ) satisfies dat = √1 X1∗ (at ) ◦ db1t . Since the metric is invariant by AdH , (ut , t ≥ 0) is a horizontal Brownian motion, its Markov generator is the horizontal Laplacian ∆h = trace ∇m d. Furthermore {π∗ ((Ad(at )X2 )∗ ), π∗ ((Ad(at )X3 )∗ )} is an orthonormal frame in S 2 ( 21 ) and then for each > 0, (xt , t ≥ 0) is a Brownian motion on S 2 ( 12 ). Corollary 5.6. Let Y0 ∈ m, A0 ∈ h and {Ak , 1 ≤ k ≤ N } ⊂ h. Suppose that there is an AdH invariant inner product on g, then for all > 0 the following SDEs STOCHASTIC HOMOGENIZATION ON HOMOGENEOUS SPACES 13 are conservative: u̇t = dht = (5.7) A∗k (ht ) ◦ dbkt + A∗0 (ht )dt (5.8) N X ∗ (ut ), Ad(h t )Y0 k=1 N dgt = Y0∗ (gt )dt 1 1 X ∗ Ak (gt ) ◦ dbkt + A∗0 (gt )dt. +√ (5.9) k=1 Furthermore π(gt ) = π(ut ) and gt = ut h t a.s.. Proof. This is straight forward from Proposition 5.3. We only need to observe that the equations (5.7) and (5.8) are conservative for every . It is clear that (5.8) does not explode, see Lemma 7.1 below. With respect to a left invariant Riemannian metric on G generated by an AdH -invariant inner product on g, the random vector fields (Ad(ht )Y0 )∗ are bounded, so (5.7) is conservative for almost surely all ω and for all . 6. D ISCUSSION ON C ONDITIONS Let G be a Lie group with a left invariant Riemannian metric and H a connected proper subgroup of G. Let Y0 ∈ g and {Ak , 0 ≤ k ≤ p} ⊂ h. Let L0 = 1 Pp ∗ ∗ ∗ k=1 LAk LAk + LA0 . We make a formal multi-scale analysis of the family of 2 equations p dgt = Y0∗ (gt )dt 1 1 X ∗ Ak (gt ) ◦ dbkt + A∗0 (gt )dt. +√ k=1 The infinitesimal generator of (g ) t is L = 1 L 2 0 + 1 LY0∗ . For simplicity take A0 = 0, assume H compact and denote by dh the normalised Haar measure on H. Let F : R+ × G → R be a solution of the parabolic equation Ḟ = L F . Expand F in , F (t) = F0 (t) + F1 (t) + 2 F2 (t) + o(2 ), and expand both sides of the parabolic equation in . Equating coefficients of r we have L0 F1 +LY0∗ F0 = 0 and Ḟ0 = LY0∗ F1 +L0 F2 . The first equation can be solved for∗ mally for F1 with solution denoted by −L−1 0 (LY0 F0 ). We will interpret this as an ∗ equation on H. The second equation reduces to Ḟ0 = −LY0∗ L−1 0 LY0 F0 + L0 F2 . Integrate the equation with respect to dh, since L0 is symmetric by Lemma 7.1 and L0 F2 averages to zero. We obtain the equation for the effective motion: Z Z d ∗ F0 dh = − LY0∗ L−1 0 (LY0 F0 )dh. dt H H From the computation above we extract an essential condition for the above formal computation to hold. Let dh be the right invariant Haar measure on H. R Assumption 6.1. Y0 6= 0, Ȳ0 ≡ H Ad(h)(Y0 )dh = 0. 14 XUE-MEI LI If g = h ⊕ m is a reductive decomposition, AdH is the direct sum of sub representations AdH = AdH |h ⊕AdH |m . When H is compact, there is an AdH invariant inner product on both h and m and we obtain an AdH -invariant inner product on g in which case we may take m = h⊥ . Let m0 := {X ∈ m : AdH (X) = X} be the subspace on which AdH acts trivially. Lemma 6.2. Suppose that H is unimodular. Let m̃ be an irreducible AdH -invariant sub-space of m. If Ȳ = 0 for a non-trivial vector Y ∈ m̃, then m̃ ∩ m0 = {0}. Conversely if m̃ ∩ m0 = {0}, Ȳ = 0 for all Y ∈ m̃. Proof. If m̃ ⊂ m0 and if Y ∈ m̃ is non-trivial then Ȳ = Y 6= 0, proving the ‘if part’. R On the other hand, for any Y ∈ g, the integral Ȳ := H Ad(h)(Y )dh is an invariant vector of AdH . Suppose that there exists Y ∈ m̃ such that Ȳ 6= 0, then Ȳ ∈ m̃ and dim(m̃) = 1. If Y is not an invariant vector,R there exists h0 s.t. 0 )(Y ) = aY for a number a 6= 1. This implies that Ȳ = Ad(h H Ad(h)(Y )dh = R R 0 H Ad(h)(Ad(h )(Y ))dh = aȲ , hence H Ad(h)(Y )dh = 0. We used the assumption that H is unimodular, i.e. the Haar measure dh is bi-invariant. This contradicts with the assumption that Ȳ 6= 0, completing the proof. In particular, if AdH |m has no non-trivial invariant vectors, the assumption holds for all Y ∈ m. Lemma 6.3. Suppose that m̃ ∩ m0 = {0}, dim(m̃) 6= 4, 7 and H is connected. If there is an AdH -invariant scalar product on m̃ then AdH acts transitively on the unit sphere of m̃. Proof. Denote M the Riemannian homogeneous space such that π : G → M is a Riemannian submersion. The action of AdH on m̃ is equivalent to the action of H on π∗ (m̃), which are orientation preserving rotations. Since m̃ is irreducible, AdH does not leave any one dimensional subspace of m̃ invariant. By a theorem of D. Montgomery and H. Samelson [43], H is the special orthogonal group SO(dim(m̃)) and the conclusion holds. Example 6.4. Let M be the Stiefel manifold S(k, n) of oriented k frames in Rn . The orthogonal group takes a k-frame to a k frame and acts transitively. The isotropy group of the k-frame o = (e1 , . . . , ek ), the first k vectors from the standard basis of Rn , contains rotation matrices that keep the first k frames fixed rotates the rest. Hence, S(k, n) = SO(n)/SO(n − k). Then h = and 0 0 where A ∈ so(n − k). Denote by Mn−k,k the set of (n − k) × k 0 A S −C T matrices and let m = where S ∈ so(k) and C ∈ Mn−k,k . Since C 0 1 0 S −C T S −C T RT Ad = for R ∈ SO(n − k), we 0 R C 0 RC 0 S 0 0 −C T see m0 = . Define m̃ = . Let us identify m̃ with 0 0 C 0 STOCHASTIC HOMOGENIZATION ON HOMOGENEOUS SPACES 15 (n − k) × k matrices. For C ∈ Mn−k,k R denote by RY0 the corresponding skew symmetric matrix in m̃. It is clear that H RCdR and H Ad(R)(Y0 )dR vanish. 7. E LEMENTARY L EMMAS We first state two elementary lemmas, which will be used later. Lemma 7.1. Let G be a Lie group with left invariant Riemannian metric. Then a stochastic differential equation driven by left invariant vector fields is conservative. If µ is a right invariant measure on G and {Xi , i = 1, . . . , m} ∈ g then the P ∗ LX ∗ is symmetric on L2 (G; dµ). operator B = 21 m L X i=1 i i Proof. The SDE is conservative follows from the fact that a Lie group with left invariant metric is geodesically complete and has positive injectivity radius. The left invariant vector fields and their covariant derivatives are bounded, so by localisation or the uniform cover criterion in D. Elworthy [14, Chapt vii], solutions of equation (5.8) from any initial point exist for all time. If f ∈ BC 1 (G; R), Z Z Z d d ∗ f (g exp(tX)) µ(dg) = f (g)µ(dg) = 0. (LX f )(g)µ(dg) = dt dt t=0 t=0 G G G Consequently the divergence of X ∗ with respect to µ vanishes. Let f1 , f2 ∈ BC 1 (G.; R) then Z Z f2 LX ∗ f1 dµ = − f1 LX ∗ f2 dµ and B is symmetric on G L2 (G, µ). G Definition 7.2. A family of vectors {A1 , . . . , AN } in h ⊂ g is Lie algebra generating if {A1 , . . . , AN } and their iterated brackets generate h. Let {Xk , k = 1, . . . , m} be smooth vector P fields on a smooth manifold N . Denote by L the Hörmander type operator 21 m k=1 LXk LXk + LX0 . If X1 , . . . , Xm and their Lie brackets generate Tx N at each x ∈ N , L is said to satisfy strong Hörmander’s condition. A Hörmander type operator on a compact manifold satisfying strong Hörmander’s condition has a unique invariant probability R measure π; furthermore for f ∈ C ∞ (N ; R), LF = f is solvable if and only if f dπ = 0. We denote by L−1 f a solution to the Poisson equation LF = f whenever it exists. If a Markov operator RL has a unique invariant probability measure π and f ∈ L1 (N ; π) we write f¯ = N f dπ. P ∗ ∗ ∗ Define L0 = 21 N k=1 LAk LAk + LA0 . We record the above statements for this special case in the lemma below, the proof is included for the convenience of the reader. Lemma 7.3. If {A1 , . . . , AN } is Lie algebra generating, the normalised Haar measure is the unique invariant probability measure for L0 . Let m̃ be an invariant subspace of m with m̃ ∩ m0 = {0} and Y0 ∈ m̃. Then for any Y ∈ g there is a unique function F ∈ C ∞ (H; R) satisfying the equation L0 F = hAd(·)(Y0 ), Y i. 16 XUE-MEI LI Proof. We restrict A∗k to the compact manifold H and treat L0 as an operator on H, which satisfies strong Hörmander’s condition. Denote by L∗0 the dual of L0 on L2 (H; and so P R). By Lemma 7.1 the second order part of L0 is symmetric ∗ ∗ ∗ ∗ ∗ ∗ L0 = k LAk LAk − LA0 . The maximal principle holds for L0 and L0 u = 0 has only constant solutions, see J.-M. Bony [8] and A. Sanchez-Calle [50]. Since R L f dh vanishes for all f ∈ C ∞ , dh is the unique invariant probability measure. H 0 Also L0 satisfies a sub-elliptic estimate: kuks ≤ kL0 ukL2 + ckukL2 , see L. Hörmander [27]. This implies that L0 has compact resolvent, and L0 is a Fredholm operator. In particular L0 has closed range, see L. Hörmander [28] and L0 u = hAd(h)(Y0 ), Y i is solvable if and only if hAd(h)(Y0 ), Y i annihilates the kernel of L∗0 , i.e. h̄Ad(h)(Y0 ), Y i = 0, which holds by Lemma 6.2. 8. C ONVERGENCE AND E FFECTIVE L IMITS In this section we assume that H is a connected compact subgroup of a connected group G, A0 , A1 , . . . , AN ∈ h and Y0 ∈ g. We assume h, i is an AdH invariant scalar product on g. Let (u· , h· ) be solutions to the following system of equations on G × H: ∗ u̇t = Ad(h t )Y0 (ut ), (8.1) dht = N X A∗k (ht ) ◦ dbkt + A∗0 (ht )dt, (8.2) k=1 u0 with initial values = u0 and h0 = 1. By Lemma 7.1, for each there is a unique global solution to the equations (8.1-8.2). Below let {Yj , j = 1, . . . N } be an orthonormal basis of m and define αj (Y0 )(h) = hAd(h)(Y0 ), Yj i . (8.3) Let x· = π(u· ) and x0 = π(u0 ). Let m̃ be an AdH invariant subspace of m. Theorem 8.1. Suppose that H is compact and {A1 , . . . , AN } a Lie algebra generating subset of h. Suppose that m̃ ∩ m0 = {0} and Y0 ∈ m̃. Then as → 0, the stochastic processes (us , s ≤ T ) converge weakly to a Markov process (ūs , s ≤ T ), whose Markov generator is L̄ = − m X ∗ ∗ αi (Y0 ) (L−1 0 αj (Y0 )) LYi LYj . i,j=1 Also, (x t , t ≤ T ) converges weakly to a stochastic process (x̄t , t ≤ T ). Proof. Since G has a left invariant Riemannian metric, Equation (8.1) is equivalent P ∗ to ut = N j=1 αj (h t )Yj (ut ). So (Ad(h)(Y0 ))∗ (g) = N X j=1 N X (Ad(h)(Y0 ))∗ , Yj∗ Yj∗ (g) = αj (Y0 )(h)Yj∗ (g), j=1 STOCHASTIC HOMOGENIZATION ON HOMOGENEOUS SPACES 17 P and (8.1) is equivalent to the following equation: u̇t = N j=1 Yj (ut )αj (h t ). We observe that G has positive injectivity radius and bounded geometry. By Lemma 7.3, L−1 0 αj exists and is smooth for each j. Furthermore by Lemma 7.1, the L̄ diffusions exist for all time. Since the Riemannian metric on G is left invariant, the Riemannian distance function is also left invariant, ρ(gg1 , gg2 ) = ρ(g1 , g2 ) for any g, g1 , g2 ∈ G. Furthermore, for sufficiently small δ > 0, supp:ρ(p,y)<δ ∇2 ρ(y, p) is finite and is independent of y, where ∇ is the Levi-Civita connection. By [38, Theorem 5.4] we conclude that (us , s ≤ t) converges weakly as approaches 0. As any continuous real valued function f on M lifts to a continuous function on G, the weak convergence passes, trivially, to the processes (xt ). Below we discuss the rate of convergence. For p ≥ 1, denote by Wp or Wp (X) the Wasserstein distance on probability measures over a metric space X. For T a positive number, let Cx ([0, T ]; N ) denote the space of continuous curves defined on the interval [0, T ] starting from x. Given two probability measures µ1 , µ2 and Cx ([0, T ]; N ), !1 Z p Wp (µ1 , µ2 ) := inf sup ρp (σ1 (s), σ2 (s))dν(σ1 , σ2 ) . 0≤s≤T Here σ1 and σ2 take values in Cx ([0, T ]; N ) and the infimum is taken over all probability measures ν whose marginals are µ1 , µ2 . The definition of BC r functions on a manifold depend on the linear connection on T M , in general. We use the flat connection ∇L for BC r functions on the Lie group G. The canonical connection ∇c is a convenient connection for defining BC r functions on M = G/H. The parallel transports for ∇c are differentials of the left actions of G on M . See §12 below for more discussions on the canonical connection. In the theorem below, M is given the induced G-invariant Riemannian metric. Set r X r r BC (M ; R) = {f ∈ C (M ; R) : |f |∞ + |∇f |∞ + |(∇c )(k) df |∞ < ∞} k=1 BC r (G; R) = {f ∈ C r (G; R) : |f |∞ + |∇f |∞ + r X |(∇L )(k) df |∞ < ∞}. k=1 belongs to BC r (G) if and only if for an orthonormal ba- Remark 8.2. A function f sis {Yi } of g, the following functions are bounded for any set of indices: f , LYi∗ f , 1 P P . . . , LYi∗ . . . LYi∗r f . In fact ∇f = i df (Yi∗ )Yi∗ and ∇L ∇f = i Yi∗ L· (LYi∗ f ), · 1 and so on. P If F is a real valued function on G or on M , denote |f |r,∞ = rk=0 |∇(k) df |∞ where ∇ is one of the the connections described above. Denote by µ̄ the probability measure of (ū· ) and P̄t the associated probability semigroup. Assumption 8.3. Assume one of the following conditions. (1) G is compact. 18 XUE-MEI LI (2) There exist V ∈ C 2 (V ; R+ ), c > 0, K > 0 and q ≥ 1 such that |∇V | ≤ C + KV, |∇dV | ≤ c + KV, |∇dρ2o | ≤ c + KV q . (3) For some point o ∈ G, ρ2o is smooth and |∇L dρ2o | ≤ C + Kρ2o . Theorem 8.4. In addition to the assumptions of Theorem 8.1, suppose also Assumption 8.3. Then the following statements hold. (1) Both (ut , t ≤ T ) and (xt , t ≤ T ) converge in Wp for an p > 1. (2) There are numbers c, C, K and a function γ such that γ ≤ c + KV q for some q and for all f ∈ BC 4 (G; R), p sup Ef ut − P̄t f (u0 ) ≤ C | log | γ(u0 )(1 + |f |4,∞ ), u0 ∈ G. 0≤t≤T An analogous statement hold for xt . (3) For any r ∈ (0, 41 ), sup W1 (Law(ut ), µ̄t ) ≤ Cr . 0≤t≤T An analogous statement holds for xt . ∗ Proof. If ρ2 is bounded or smooth, LYi∗ LYj∗ ρ = ∇L dρ(Yi∗ , Yj∗ ) + ∇L Y ∗ Yj is i bounded. Denote T L the torsion of ∇L and ∇ the Levi-Civita connection. The following equation for the derivative flow of (8.1) holds: X ∇vt = αj (Y0 )(ht ) T L (vt , Yj (ut )). j Since T L (u, v) = −[u, v], the torsion tensor and their covariant derivatives are bounded. Thus all covariant derivatives of Yj∗ with respect to the Levi-Civita connection are bounded. By [38, Theorem 5.4, Corollary5.5, Proposition 8.2], the conclusions for the u process hold. For the process on the homogeneous manifold, take f ∈ BC 4 (M ; R) and let f˜ = f ◦ π. Then LYi∗ LYj∗ f˜ = ∇c df (π∗ Yi∗ , π∗ Yj∗ ) + df ∇cYi∗ dπ(Yj∗ ) . The last term vanishes, as (dπ)u (Yi∗ ) = T L̄u ((dπ)1 Yi )), c.f. Lemma 12.3 below. Since π∗ is a Riemannian isometry, LYi∗ LYj∗ f˜ is bounded if ∇c df is. The same assertion holds for higher order derivatives. Thus f ◦ π ∈ BC 4 (G; R) and p sup Ef xt − π∗ P̄t f (x0 ) ≤ C | log | γ(x0 ) (1 + |f |4,∞ ), 0≤t≤T where γ is a function in Bρ,o . For the convergence in the Wasserstein distance, denote by ρ̃ and ρ respectively the Riemannian distance function on G and on M . For i = 1, 2, let xi ∈ M and ui ∈ π −1 (xi ). If u1 and u2 are the end points of a horizontal lift of the unit speed geodesic connecting x1 and x2 , then ρ̃(u1 , u2 ) = ρ(x1 , x2 ). Otherwise STOCHASTIC HOMOGENIZATION ON HOMOGENEOUS SPACES 19 ρ̃(u1 , u2 ) ≥ ρ(x1 , x2 ). If c1 and c2 are C 1 curves on M with c1 curves c̃1 and c̃2 on G covering c1 and c2 respectively, then ρ(c1 , c2 ) = sup ρ(c1 (t), c2 (t)) ≤ ρ̃(c̃1 , c̃2 ). t∈[0,T ] Since Cu0 ([0, T ]; G) is a Polish space, there is an optimal coupling of the probability law of u· and µ̄ which we denote by µ. Then π∗ µ̄ is a coupling of Law(x· ) and π∗ µ̄ and !1 Z p p Wp (Law(u · ), µ̄) = ρ̃ (γ1 , γ2 )dµ(γ1 , γ2 ) Cu0 ([0,T ];G) !1 p Z ρp (π(γ1 ), π(γ2 ))dµ(γ1 , γ2 ) Cx0 ([0,T ];M ) !1 p Z sup ρp (σ1 (t), σ2 (t))dπ∗ µ̄(σ1 , σ2 ) = ≥ Cx0 ([0,T ];M ) 0≤t≤T Wp (Law(x· ), π∗ µ̄). Consequently x · converges in W2 . Similarly the rate of convergence passes from the u process to the x process. For r < 1 4 there is a number C such that, W1 (Law(xt ), π∗ µ̄t ) ≤ Wp (Law(ut ), µ̄) ≤ Cγ(x0 )r . We remark that the Lie group G is compact if and only if the homogeneous space M is compact (given that H is compact). The statements of the theorem hold for compact homogeneous spaces. We return to equation (1.2): N dgt 1 X ∗ 1 =√ Ak (gt ) ◦ dbkt + A∗0 (g0 ) + Y0∗ (gt )dt, g0 = g0 . k=1 Corollary 8.5. Suppose that H is compact and g = h ⊕ m is a reductive decomposition. Suppose that {A1 , . . . , AN } is Lie algebra generating and Y0 ∈ m̃ where m̃ ⊂ m is an irreducible AdH -invariant subspace transversal to m0 . Let xt = π(gt ) and (ut ) the horizontal lift of (xt ) through g0 . Then (ut ) converges weakly to the diffusion process (ut ) determined by L̄ = − K X ∗ ∗ αi (L−1 0 αj ) LYi LYj , i,j=1 where {Yj , j = 1, . . . , K} is an orthonormal basis of m̃. Also (xt ) converges weakly to a stochastic process (x̄t ). 20 XUE-MEI LI Proof. By Corollary 5.6 the horizontal lift process of (xt ) solves equations (8.18.2). By Lemma 6.2, Ȳ0 = 0. Since m̃ is an invariant space of AdH and Y0 ∈ m̃, Ad(h)(Y0 ) ∈ m̃. In Theorem 8.1 it is sufficient to take {Yj } to be an orthonormal basis of the invariant subspace m̃. 9. E FFECTIVE L IMITS AND T HE M ARKOV P ROPERTY In the rest of the paper we assume that H is a compact connected proper subgroup, and h, i an AdH -invariant inner product on g such that h and m are orthogonal. There is a complete decomposition of AdH by orthogonal invariant subspaces: m = m0 ⊕ m1 ⊕ · · · ⊕ mr such that, for j 6= 0, mj is irreducible. This invariant decomposition of m is also an invariant decomposition for adh . P ∗ ∗ ∗ If {Ak , k = 1, . . . , N } ⊂ g and A = 12 N k=1 LXk LXk + LX0 we define the 2 1 PN linear map c(A) = 2 k=1 ad (Xk ) + ad(X0 ). Let Y0 , Y ∈ g. If c(A)Y0 = λ0 Y0 for some number λ0 , it is trivial that hY, Ad(g)(Y0 )i is an eigenfunction of A corresponding to λ0 . Lemma 9.1. Let g = h ⊕ m be a reductive decomposition, Ak ∈ h and L0 = 1 Pp ∗ ∗ ∗ L k= Ak LAk + LA0 . If Y0 ∈ m is an eigenvector of c(L0 ) corresponding to an 2 eigenvalue −λ(Y0 ) then for any Y ∈ m, hY, Ad(h)(Y0 )i is an eigenfunction of L0 corresponding to −λ(Y0 ). The converse holds. Proof. Just note that LA∗k Ad(h)(Y0 ) = d |t=0 Ad(h exp(tAk ))(Y0 ) = Ad(h)([Ak , Y0 ]), dt which by iteration gives LA∗k LA∗k Ad(h)(Y0 ) = Ad(h)([Ak , [Ak , Y0 ]]). Following the notation in Theorem 8.1, we take an orthonormal basis {Yj } of ml and define αj (Y0 )(h) = hAd(h)(Y0 ), Yj i. (9.1) Recall that L−1 0R (αj ) is the solution to the Poisson equation L0 u = αj when it exists, also f = H f dh. Let L̄ = − ml X ∗ ∗ αi (Y0 ) (L−1 0 αj (Y0 )) LYi LYj . i,j=1 Lemma 9.2. In Lemma 9.1 suppose that {A1 , . . . , AN } is a Lie algebra generating subset of h. Let Y0 ∈ m̃ where m̃ is an irreducible AdH -invariant subspace s.t. m̃ ∩ m0 = {0}. If Y0 is an eigenvector of c(L0 ) corresponding to the eigenvalue −λ(Y0 ), then 2 (G; R), for any f in CK Z 1 L̄f = ∇L df ((Ad(h)(Y0 ))∗ , (Ad(h)(Y0 ))∗ ) dh. (9.2) λ(Y0 ) H STOCHASTIC HOMOGENIZATION ON HOMOGENEOUS SPACES Equivalently, L̄f = basis of m̃ and Pm i,j=1 ai,j (Y0 )∇ 1 ai,j (Y0 ) = λ(Y0 ) L df (Y ∗ , Y ∗ ) i j 21 where {Yi } is an orthonormal Z hYi , Ad(h)Y0 ihYj , Ad(h)Y0 i dh. H Proof. Since αj (Y0 ) = 0, by Lemma 6.2 and the fact that {Ak } is a Lie algebra generating subset of h, there is a unique smooth solution to the Poisson equation L0 βj = αj . By Lemma 9.1 and the assumption on Y0 , βj := − λ(Y1 0 ) hYj , Ad(h)Y0 i. Consequently, Z 1 hYi , Ad(h)Y0 )ihYj , Ad(h)(Y0 )i dh. (9.3) αi βj = − −λ(Y0 ) H ∗ Since ∇L Xi Yj = 0 for any i, j, the limit operator is L̄f = − m X αi βj LYi∗ LYj∗ f = − i,j=1 = 1 λ(Y0 ) m X αi βj ∇L df (Yi , Yj ) i,j=1 Z ∇L df ((Ad(h)(Y0 )), (Ad(h)(Y0 ))) dh. H Theorem 9.3. Let {A1 , . . . , Ap } be an orthonormal basis of h, A0 = 0, and Y0 ∈ ml where l 6= 0. The following statements hold. 2 (G; R), (1) There exists a number λl such that for any f ∈ CK Z 1 ∇L df ((Ad(h)(Y0 ))∗ , (Ad(h)(Y0 ))∗ ) dh. (9.4) L̄f = λl H P Furthermore, 21 pk=1 ad2 (Ak )|ml = −λl Idml , where Idml denotes the identity map on ml . (2) The limiting process (x̄t ) is Markovian. Proof. The statements follow from Lemma 9.2 and two additional lemmas. In Lemma 9.6 below we observe that hYj , Ad(h)Y0 i is an eigenfunction of L0 corresponding to the eigenvalue −λl . By Lemma 9.2, formula (9.4) holds. The second statement follows from Lemma 9.7. In the theorem above, L0 = 21 ∆H , where ∆H is the Laplacian for the biinvariant metric on the compact Lie group H, and −2λl is an eigenvalue of ∆H . If {Ak } is not an orthonormal basis of h, the constant λl in Theorem 9.3 may depend on the Y0 . Example 9.4. Take G = SO(4) and let Ei,j be the elementary 4 × 4 matrices and Ai,j = √12 (Eij − Eji ). For k = 1, 2, 3 let Yk = Ak4 and consider the equations, 1 1 dgt = √ A∗1,2 (gt ) ◦ db1t + √ A∗1,3 (gt ) ◦ db2t + Yk∗ (gt )dt. Let H = SO(3). Then h = {A1,2 , A1,3 , A2,3 }, and its orthogonal complement m is irreducible w.r.t. AdH , in fact [A1,2 , A1,3 ] = − √12 A2,3 . Furthermore 22 XUE-MEI LI {A1,2 , A1,3 } is a set of generators and L0 = 12 LA∗1,2 LA∗1,2 + 21 LA∗1,3 LA∗1,3 satisfies strong Hörmander’s conditions. It is easy to check that, 1 2 1 1 ad (A1,2 )(Yk ) + ad2 (A1,3 )(Yk ) = − (2δ1,k + δ2,k + δ3,k )Yk . 2 2 4 For any Y ∈ m, hY, Ad(h)Yk i is an eigenfunction for L0 corresponding to the eigenvalue −λ(Yk ), where 1 1 1 λ(Y1 ) = , λ(Y2 ) = , λ(Y3 ) = . 2 4 4 4 Define π : g ∈ SO(4) → S so π(g) is the last column of g. Set xt = π(gt ) and denote by x̃t its horizontal lift. Then Z 1 hYi , Ad(h)Yk )iL−1 ai,j (Yk ) := 0 (hYj , Ad(h)(Yk )i)dh. λ(Yk ) H 1 For for i 6= j, ai,j (Yk ) = 0 and ai,i (Yk ) := 3λ(Y . In Lemma 9.2, take Y0 = Yk k) we see that Z 3 1 1 X L L̄ = ∇L df ((Ad(h)(Yk ))∗ , (Ad(h)(Yk ))∗ )dh = ∇ df (Yi , Yi ). λ(Yk ) SO(3) 3λ(Yk ) i=1 On the other hand, the effective diffusion for the equations 1 1 1 dgt = √ A∗1,2 (gt ) ◦ db1t + √ A∗1,3 (gt ) ◦ db2t + √ A∗2,3 (gt ) ◦ db3t + Yk∗ (gt )dt 2 P3 is the same for all Yk . It is easy to see that L̄ = 3 i=1 ∇L df (Yi , Yi ). Further symmetries can, of course, be explored. P Example 9.5. Let Y0 = 3k=1 ck Yk be a mixed vector. We compute the coeffiP cients of the operator 3i,j=1 αi βj LYi∗ LYj∗ : αi βj = 3 X ck cl αi (Yk )βj (Yl ) = k,l=1 If c1 = 0 and c2 = c3 , αi βj = 4(c2 ) αi (Yk )αj (Yl ) vanishes for i 6= j. 3 X k,l=1 P3 2 ck cl 1 αi (Yk )αj (Yl ). λl k,l=2 αi (Yk )αj (Yl ). By symmetry, We proceed with the proof of the remaining lemmas. Let Badh ,ml (X, Y ) be the symmetric associative bilinear form of the adjoint sub-representation of h on ml . The symmetric bilinear form for a representation ρ of a Lie algebra g on a finite dimensional vector space is given by the formula Bρ (X, Y ) = trace ρ(X)ρ(Y ) for X, Y ∈ h. A bilinear form B on a Lie algebra is associative if B([X, Y ], Z]) = B(X, [Y, Z]). P Lemma 9.6. Let L0 = 21 pk=1 (LA∗k )2 where {Ak , k = 1, . . . , p} is an orthonormal basis of h (for the AdH invariant metric). The following statements hold. P (1) There exists λl such that 12 pk=1 ad2 (Ak )|ml = −λl Idml . Furthermore λl = 0 if and only if l = 0. STOCHASTIC HOMOGENIZATION ON HOMOGENEOUS SPACES 23 (2) For any Y0 , Y ∈ ml , h ∈ H define β(h) := hY, Ad(h)Y0 i. Then β is an eigenfunction of L0 corresponding to the eigenvalue −λl . (3) Suppose that adh : h → L(m; m) is faithful. Then Badh ,ml is non-degenerate, negative definite, and AdH -invariant. If l 6= 0, λl = −Badh ,ml (X, X) dim(h) , 2 dim(ml ) where X is any unit vector in h. If the inner product on h agrees with −Badh ,ml then λl = 2 dim(h) dim(ml ) . Proof. Part (1). For Z ∈ h and X, Y ∈ g, we differentiate the identity hAd(exp(tZ))(X), Ad(exp(tZ))(Y )i = hX, Y i at t = 0 to see that had(Z)(X), Y i+hX, ad(Z)(Y )i = 0. If Y0 ∈ m, ad(Ak )(Y0 ) ∈ m and by the skew symmetry from above, X X h (ad(Ak ))2 (Y0 ), Y0 i = − had(Ak )(Y0 ), ad(Ak )(Y0 )i. k k Pp Thus k=1 (ad(Ak ))2 (Y0 ) = 0, where p = dim(h), implies that ad(Ak )(Y0 ) = 0 for in turn implies that Y0 ∈ m0 . Conversely if Y0 ∈ m0 it is clear that Ppall k which 2 ad (A ) k = 0. k=1 Let l 6= 0. It is clear P that c(L0 ) is a multiplication operator on ml . Indeed if X ∈ h let [X, Ak ] = l0 akl0 Al0 . Then X akl = ak,l0 hAl0 , Al i = h[X, Ak ], Al i = −h([X, Al ], Ak i = −alk , l0 [ad(X), p X k=1 = X k ad2 (Ak )] = X [ad(X), ad(Ak )]ad(Ak ) + k ad([X, Ak ])ad(Ak ) + X ad(Ak )[ad(X), ad(Ak )] k X ad(Ak )ad([X, Ak ]) = 0. k Since each ad(Ak ) is skew symmetric, c(L0 ) is self-adjoint. Since ρ is an irreducible representation of the Lie algebra h on the finite dimensional vector space ml and the endomorphism c(L0 ) commutes with ρ, it must be a scalar. Let λl be a real number such that 21 c(L0 ) = −λl Idm . It is clear that λl > 0, completing the proof of part (1). Part (2). By restricting the discussion to an irreducible invariant subspace of Ad(H) we may assume that m0 = {0} and m is irreducible. Let Xk ∈ h. Apply Lemma 9.1 to L0 and use part (1) we conclude that hY, Ad(h)Y0 i is an eigenfunction for L0 with eigenvalue −λl . (3) Firstly, Badh ,ml is non-degenerate. If {Yj } is a basis of ml with respect to the adH invariant inner product and X ∈ h, then X Badh ,ml (X, X) = − |[X, Yj ]|2 , j 24 XUE-MEI LI which vanishes only if [X, Yj ] = 0 for all j. Suppose that Badh ,ml (X, X) = 0. By the skew symmetry of adh , for any Y ∈ m, 0 = h[X, Yj ], Y i = −hYj , [X, Y ]i for all j. Since ad(X)(ml ) ⊂ ml we conclude that [X, Y ] = 0. Since adh is faithful, X must vanish. It is clear that Badh ,ml is AdH -invariant. Let X ∈ h and h ∈ H, then X Badh ,ml (Ad(h)X, Ad(h)(X)) = had(Ad(h)(X))ad(Ad(h)(X))Yj , Yj i j =− X h[Ad(h)(X), Yj ], [Ad(h)(X), Yj ]i j =− X Ad(h)[X, Ad(h−1 )Yj ], Ad(h)[X, Ad(h−1 )Yj ] j X =− [X, Ad(h−1 )Yj ], [X, Ad(h−1 )Yj ] = Badh ,ml (X, X). j Since there is a unique, up to a scalar multiple, AdH -invariant inner product on a compact manifold, Badh ,ml is essentially the inner product on H. There is a positive number al such that Badh ,ml = −al h, i. It is clear that al > 0. For the orthonormal basis {Ak }, Badh ,ml (Ai , Aj ) = −al δi,j . We remark that { a1l Ak } is a dual basis of P {Ak } with respect to Badh ,ml and a1l k ad2 (Ak ) is the Casimir element. Pdim(h) By part (1), there is a number λl such that 21 k=1 ad2 (Ak ) = −λl Idml . The ration between the symmetric form Badh ,ml and the inner products on ml can be determined by any unit length vector in h. It follows that dim(h) traceml X ad2 (Ak ) = traceml (−2λl Idml ) = −2λl dim(ml ). k=1 On the other hand, dim(h) traceml ( X k=1 dim(h) 2 ad (Ak )) = X k=1 dim(h) 2 traceml (ad (Ak )) = X Badh ,ml (Ak , Ak ). k=1 Consequently λl = −Badh ,ml (A1 , A1 ) 2 dim(h) dim(ml ) . We completed part (3). If h = h0 ⊕ h1 splits s.t. h0 acts trivially on ml and h1 a sub Lie-algebra, we take h1 in place of h, the statement in part (3) of the lemma still holds. In this case {Ak } is taken as an orthonormal basis of h1 . Let ∇c be the canonical connection on the Riemannian homogeneous manifold M , see Section 12 for detail. Lemma 9.7. If L̄ has the invariant form (9.2) then x̄t is a Markov process with Markov generator Z 1 L̄(f ◦ π)(u) = (∇c df )π(u) dL̄uh Y0 , dL̄uh Y0 dh. λ(Y0 ) H STOCHASTIC HOMOGENIZATION ON HOMOGENEOUS SPACES 25 Proof. Denote (ut ) the L̄ diffusion. For any f ∈ C 2 (M ; R), Z tZ 1 f ◦π(ut )−f ◦π(u0 )− ∇L d(f ◦π) ((Ad(h)(Y0 ))∗ (us ), (Ad(h)(Y0 ))∗ (us )) dh ds λ(Y0 ) 0 H is a local martingale. ∇L d(f ◦ π) ((Ad(h)(Y0 ))∗ (g), (Ad(h)(Y0 ))∗ (g)) = LAd(h)(Y0 )∗ df dπ (Ad(h)Y0 )∗ (g) = ∇c df (dπ(Ad(h)(Y0 )∗ ), dπ(Ad(h)(Y0 )∗ )) + df ∇cπ∗ Ad(h)(Y0 )∗ dπ (Ad(h)Y0 )∗ . By part (4) of Lemma 12.3 below the last term on the right hand side vanishes. It is clear that, Z (∇c df )π(g) (dπ(Ad(h)(Y0 )∗ ), dπ(Ad(h)(Y0 ))) dh H Z = (∇c df )π(g) dL̄gh dπ(Y0 ), dL̄gh dπ(Y0 ) dh. H Since dh is right invariant, the above formulation is independent of the choice of g in π −1 (x). That the stochastic process π(ut ) is a Markov process follows from Dynkin’s criterion which states that if (ut ) is a Markov process with semigroup Pt and if π : G → M is a map such that Pt (f ◦ π)(y) depends only on π(y) then π(ut ) is a Markov process. See e.g. E. Dynkin[12] and M. Rosenblatt [49]. Remark 9.8. The operator L̄ in the above Lemma provides an example of a cohesive operator, as defined in D. Elworthy, Y. LeJan, X.-M. Li [15]. 10. T HE ROUND O PERATORS Let H be a connected subgroup. We fix an AdH invariant Pminner product on g and let {Yj } be an orthonormal basis of ml . Denote ∆ml = j=1 LYj∗ LYj∗ , this is the ‘round’ operator on ml . For the left invariant connection ∇L , ∆ml = traceml ∇L d is a ‘generalised’ horizontal Laplacian and is independent of the choice of the basis. If G is isomorphic to the Cartesian product of a compact group and an additive vector group, there is a bi-invariant metric. If ∇ is the Levi-Civita connection for a bi-invariant metric, ∇X X ∗ = 0 and ∆ml = traceml ∇d. In the irreducible case, this operator ∆m is the horizontal Laplacian and we denote the operator by ∆hor . Its corresponding diffusion is a horizontal Brownian motion. In the reducible case, we abuse the notation and define a similar concept. Definition 10.1. A sample continuous Markov process is a (generalized) horizontal Brownian motion if its Markov generator is 21 ∆ml ; it is a (genralized) scaled horizontal Brownian motion with scale c if its Markov generator is 21 c∆ml for some constant c 6= 0. 26 XUE-MEI LI Since the limit operator in Theorem 8.1 is given by averaging the action of AdH , we expect that the size of the isotropy group H is correlated with the ‘homogeneneity’ of the diffusion operator, which we explore in the remaining of the section. Lemma 10.2. Let m̃ be an irreducible adH -invariant subspace. Assume that m̃ ∩ m0 = {0} and dim(m̃) ≥ 3. Let Y0 ∈ m̃. Let {Yj } be an orthonormal basis of m̃. Suppose that for any pair of i 6= j there is hi,j ∈ H such that Ad(hi,j )(Yi ) = Yi and Ad(hi,j )(Yj ) = −Yj . Suppose that there is a number λ(Y0 ) such that Z 1 L̄f = ∇L df ((Ad(h)(Y0 ))∗ , (Ad(h)(Y0 ))∗ ) dh. (10.1) λ(Y0 ) H Then |Y0 |2 L̄ = ∆m̃ . λ(Y0 ) dim(m̃) Proof. If we expand (Ad(h)(Y0 ))∗ by the orthonormal basis {Yj , j = 1, . . . , K} P L ∗ ∗ where K = dim(m̃), we see that L̄f = K i,j=1 ai,j (Y0 )∇ df (Yi , Yj ) where Z 1 ai,j (Y0 ) = hYi , Ad(h)Y0 )ihYj , Ad(h)(Y0 )i dh. λ(Y0 ) H Take i 6= j. By the assumption there exists h̃ ∈ H with the property Ad(h̃)(Yi ) = Yi and Ad(h̃)(Yj ) = −Yj . Using the bi-invariance of the Haar measure and the AdH -invariance of the inner product, we obtain Z Z hYi , Ad(h)Y0 ihYj , Ad(h)Y0 idh = hYi , Ad(h̃)Ad(h)(Y0 )ihYj , Ad(h̃)Ad(h)Y0 idh H H Z = − hYi , Ad(h)(Y0 )ihYj , Ad(h)Y0 idh. H If i 6= j, ai,j (Y0 ) = 0. By the same reasoning, a11 (Y0 ) = · · · = aKK (Y0 ). Since m̃ is an invariant space of AdH and since the inner product is AdH -invariant, we sum over the basis vectors Yj and obtain Z X K Z K X hYj , Ad(h)Y0 i2 dh = hAd(h−1 )Yj , Y0 i2 dh = |Y0 |2 . j=1 H H j=1 R It follows that for any unit vector in m̃, H hY, Ad(h)Y0 i2 dh = clude that |Y0 |2 L̄ = ∆m̃ . λ(Y0 ) dim(m̃) |Y0 |2 dim(m̃) . We con- Let L0 = 1 PN 2 k=1 (LAk ) ∗ 2. Theorem 10.3. Suppose that l 6= 0 and dim(ml ) is not 4 or 7. Let L̄ be given |Y0 |2 by (10.1). Then L̄ = λl dim(m ∆ml . If M is furthermore isotropy irreducible, l) L̄ = |Y0 |2 hor . λl dim(M ) ∆ STOCHASTIC HOMOGENIZATION ON HOMOGENEOUS SPACES 27 Proof. It is more intuitive to study the action of AdH through the isotropy representation τ of H on To M which is defined by the formula τh (π∗ X) := dL̄h (π∗ X). Since left translation on To M corresponds to adjoint action on m, π∗ Ad(h)(X) = (L̄h )∗ (π∗ X), the linear representation is equivalent to the isotropy representation. We identify ml with Rdim(ml ) . Since adh consists of skew symmetric matrices, with respect to an invariant Riemannian metric on the homogeneous manifold G/H, H acts as a group of isometries. So do the left action by elements of H on π∗ (ml ). Since π takes the identity to the coset H, we may rule out translations and every element of AdH (ml ) is a rotation. Since H is connected, we see that H can be identified with a sub-group of SO(n) where n = dim(ml ). By a theorem of D. Montgomery and H. Samelson [43], if H is a connected closed proper subgroup of SO(n), and if n 6= 4, 7, then it leaves a one dimensional subspace of Rn -invariant. Since ml is irreducible and in particular does not contain a one dimensional invariant subspace, H can be identified with SO(n). If n ≥ 3, for any pair of natural numbers i 6= j from {1, . . . , dim(ml )}, there is hi,j ∈ H such that Ad(hi,j )(Yi ) = Yi and Ad(hi,j )(Yj ) = −Yj . For dim(ml ) 6= 2, 4, 7, the conclusion follows from Theorem 9.3 and Lemma 10.2. Suppose that dim(ml ) = 2 we identify ml with R2 . By the same argument as before, H is SO(2). Let {Y1 , Y2 } be an orthonormal basis of ml . Then Z Z hY1 , Ad(h)Y0 ihY2 , Ad(h)Y0 idh = hY1 , gY0 ihY2 , gY0 idg H SO(2) where dg is the pushed forward measure Ad∗ (dh). Since dg is bi-invariant it is the standard measure on SO(2), normalised to have volume 1. Thus the above integral vanishes. This can be computed explicitly. The same proof as in Lemma R |Y0 |2 . We have proved the claim that L̄ = 10.2 shows H hY, Ad(h)Y0 i2 dh = dim(m l) 1 2 2λl |Y0 | ∆ml . Dimension 4 is special: the space S 3 of unit quaternions is a Lie group and acts on itself transitively. Note that we may also restrict to a subgroup H 0 of H which acts transitively on ml and in which case Ak would be from the lie algebra of H 0 . Corollary 10.4. Let H be a maximal torus group of a semi-simple group G. Let 2 {Ak } be an orthonormal basis of h. Let Y0 ∈ ml . Then L̄ = |Y2λ0 |l ∆ml where λl is P determined by 12 pk=1 ad2 (Ak ) = −λl Idml . Proof. If h is the Cartan sub-algebra of the semi-simple Lie algebra g the dimension of ml is 2 and Lemma 10.2 applies. It is clear that dim(ml ) ≥ 2, for otherwise it consists of invariant vectors. It is well known that g is semi-simple if and only if gC = g ⊗ C is semi-simple. Denote the complexification of h by t, it is a Cartan sub-algebra of g ⊗ C. Let α ∈ t∗ be a weight for gC . Take Y = Y1 + iY2 from the root space gα corresponding to α. Let X ∈ h, and write α = α1 + iα2 . Then [X, Y ] = [X, Y1 ] + i[X, Y2 ] = (α1 (X) + iα2 (X))(Y1 + iY2 ), 28 XUE-MEI LI and Y1 and Y2 generate an invariant subspace for adH which follows from the following computation: [X, Y1 ] = α1 (X)Y1 − α2 (X)Y2 ; [X, Y2 ] = α1 (X)Y1 + α2 (X)Y2 . P Restricted on each vector space gα only one specific Ak in the sum pk=1 ad2 (Ak ) makes non-trivial contribution to the corresponding linear operation. Thus in Theorem 10.3 we can restrict ourselves to the one dimensional torus sub-group of H, which acts faithfully on gα . We may apply the theorem and conclude that |Y0 |2 L̄ = λl dim(m ∆ml = 2λ1 l |Y0 |2 ∆ml . This completes the proof. l) Example 10.5. Let G0 (k, n) = SO(n)/SO(k) × SO(n − k) be the oriented Grassmannian manifold of k oriented planes in n dimensions. It is a connected manifold of dimension k(n − k). The Lie group SO(n) act on it transitively. Let o be k-planes spanned by the first k vectors of the standard basis in Rn . Then SO(k) × O(n − k) keeps Rk fixed and as well as keeps the orientation of the first k-frame. Let π : SO(n) → G(k, n), then πO = {O1 , . . . , Ok }, the first k columns of the matrix O. Let σ(A) = SAS −1 , where S is the diagonal block matrix with −Ik and In−k as entries, be the symmetry map on G. The Lie algebra has the symmetric decomposition so(n) = h ⊕ m: 0 M so(k) 0 h= , m = YM := , M ∈ Mk,n−k , 0 so(n − k) −M T 0 The adjoint action on m is given by Ad(h)(YM ) = 0 −(RM QT )T RM QT 0 , R 0 for h = , R ∈ SO(k) and Q ∈ SO(n − k). We identify m with 0 Q R 0 To G0 (k, n). The isotropy action on To G0 (k, n) is now identified with ( , M) → 0 Q RM QT . Let M1 = (e1 , 0 . . . , 0). Then RM1 QT = R1 QT1 where R1 and Q1 are the first columns of R and Q respectively. The orbit of M1 by the Adjoint action generates a basis of Mk,n−k and M is isotropy irreducible. Let M2 = (0, e2 , 0, . . . , 0), dR and dQ the Haar measure on SO(k) and SO(n − k) respectively. Then if R = (rij ) and Q = (qij ). Then Z Z hAd(h)M1 , M1 ihAd(h)(M1 ), M2 idRdQ = (r11 )2 q11 q12 dQdR = 0. H H ∆hor . Hence L̄ is proportional to It is easy to see that L̄ satisfies the one step Hörmander condition and is hypoelliptic on G. Given M, N ∈ Mk×(n−k) whose corresponding elements in m denoted by M̃ , Ñ . Then [M̃ , Ñ ] = N T M − M N T . There is a basis of h in this form. If {ei } is the standard basis of Rk , Eij = ei eTj , {Eij − Eji , i < j} is a basis of so(k). STOCHASTIC HOMOGENIZATION ON HOMOGENEOUS SPACES 29 11. T HE D EGENERATE C ASE In this section we assume that A) = 0 and {A1 , . . . , AN } is a set of generators, not necessarily an orthonormal basis, of h. Denote h, i an AdH -invariant inner product on g. Suppose that H is connected and AdH acts faithfully on ml . Theorem 11.1. Suppose that l 6= 0, dim(ml ) 6= 2, 4, 7. Then for each vector 2 0| Y0 ∈ ml there is a number λ(Y0 ) such that L̄ = λ(Y0|Y ) dim(ml ) ∆ml . The number λ(Y0 ) can be computed from eigenvalues of L0 . In particular (x̄t ) is a Markov process. Proof. With respect to the AdH -invariant inner product on g, ad2 (Ak ) is a selfadjoint linear map on ml . Let {Y1 , . . . , Ydim(ml ) } be an orthonormal basis of ml P 2 consisting of eigenvectors of − 21 N k=1 ad (Ak ). Let λ(Yj ) be the corresponding P 2 eigenvalues: − 21 N k=1 ad (Ak )Yj = λ(Yj )Yj . For each j, l = 1, . . . , dim(ml ), let αj (Yl ) = hAd(h)(Yl ), Yj i. Then (L0 αj (Yl )) (h) = −λ(Yl )hAd(h)(Yl ), Yj i. Pdim(ml ) Take Y0 ∈ ml so Y0 = cl Yl for some numbers cl . Then αj (Y0 ) = l=1 Pdim(ml ) cl hAd(h)(Yl ), Yj i. For each i, j = 1, . . . , dim(ml ), l=1 dim(ml ) αi (Y0 )L−1 0 αj (Y0 ) =− X k,l=1 ck cl hAd(h)(Yk ), Yi ihAd(h)(Yl ), Yj i. λ(Yl ) We examine the cross terms in the summation. Firstly, observe that AdH preserves the inner product of ml and is an orthogonal transformation. Since H is connected and contains the identity it is a subgroup of the special orthogonal group. So H = SO(n). Since dim(ml ) ≥ 3, for any pair of numbers i, j from 1, dim(ml ), not equal, there is hi,j ∈ H such that Ad(hi,j )(Yi ) = Yi and Ad(hi,j )(Yj ) = −Yj . For such i 6= j and for any k, l = 1, . . . , dim(ml ), Z hAd(h)(Yk ), Yi ihAd(h)(Yl ), Yj idh H Z = hYk , Ad(h−1 )Ad(hi,j )(Yi )ihYl , Ad(h−1 )Ad(hi,j )(Yj )dh H Z = − hAd(h)(Yk ), Yi ihAd(h)(Yl ), Yj idh. H Thus αi (Yl )αj (Yk ) = 0 is i 6= j or if k 6= l. In particular all the cross terms αi (Y0 )L−1 0 αj (Y0 ) vanish and for i = j, dim(ml ) X (ck )2 Z αi (Y0 )αj (Y0 ) = − hAd(h)(Yk ), Yi i2 dh. λ(Yk ) H k=1 For k0 ∈ {1, . . . , dim(ml )}, Z Z hAd(h)(Yk ), Yi i2 dh = hAd(h)(Yk0 ), Yi i2 dh = H H 1 . dim(ml ) 30 XUE-MEI LI Thus αi (Y0 )L−1 0 αi (Y0 ) dim(ml ) X (ck )2 1 = . dim(ml ) λ(Yk ) k=1 Define dim(ml ) X (ck )2 1 = , λ(Y0 ) λ(Yk ) (11.1) k=1 then L̄ = 1 dim(ml )λ(Y0 ) ∆ml , concluding the proof. Finally we discuss the case of dim(ml ) = 2. The question whether (xt ) is a Markov process is investigated later in Example 12.5. Remark 11.2. Suppose that dim(ml ) = 2 then H is essentially SO(2). Let Y1 , Y2 1P be a pair of orthogonal unit length eigenvectors of the linear map 2 k ad2 (Ak ), P restricted to ml . The cross term in L̄ = 2i,j=1 ai,j (Y0 )LYi∗ LYj∗ vanishes if and P only if k ad2 (Ak ) is constant multiple of the identity. If so, L̄ = |Y0 |2 2λ1 l ∆ml . We prove this. Let Y0 = c1 Y1 + c2 Y2 . For j 0 , j, k, l = 1, 2, let Z ck cl k,l hAd(ei2πθ )(Yk ), Yj 0 ihAd(ei2πθ )(Yl ), Yj idθ. bj 0 ,j (Y0 ) = λ(Yl ) S 1 0 By direct computation, it is easy to see that bk,l j 0 ,j = 0 unless j = j, k = l or 0 j = k, j = l. In particular, a1,2 (Y0 ) = 2 X 2,1 1,2 bk,l 1,2 = b1,2 + b1,2 k,l=1 Z c1 c2 = hAd(ei2πθ )(Y1 ), Y1 ihAd(ei2πθ )(Y2 ), Y2 idθ λ(Y2 ) S 1 Z c1 c2 + hAd(ei2πθ )(Y2 ), Y1 ihAd(ei2πθ )(Y1 ), Y2 idθ. λ(Y1 ) S 1 Direct computation shows that Z Z c1 c2 c1 c2 2 a1,2 (Y0 ) = ± cos (i2πθ)dθ − ± sin2 (i2πθ)dθ λ(Y2 ) λ(Y1 ) c1 c2 1 c1 c2 =± ( − ), 2 λ(Y2 ) λ(Y1 ) where the sign depends on the relevant positions of the pair of vectors Y1 and Y2 in the plane. Thus, a1,2 (Y0 ) vanishes if and only if λ(Y2 ) = λ(Y1 ), concluding the first statement. STOCHASTIC HOMOGENIZATION ON HOMOGENEOUS SPACES 31 Also, assuming λ(Y2 ) = λ(Y1 ), Z Z (c1 )2 c22 2 a1,1 (Y0 ) = cos (i2πθ)dθ + sin2 (i2πθ)dθ λ(Y1 ) λ(Y2 ) 1 c2 c22 = ( 1 + ) 2 λ(Y1 ) λ(Y2 ) c22 1 c2 ) a2,2 (Y0 ) = ( 1 + 2 λ(Y1 ) λ(Y2 ) We see that a11 (Y0 ) = a22 (Y0 ) = λ(Y1 0 ) |Y0 |2 . Set λl = λ(Y1 ) then L0 = completing the proof of the second statement. |Y0 |2 2λl ∆, 12. L IMITS O N R IEMANNIAN H OMOGENEOUS M ANIFOLDS Let M be a smooth manifold with a transitive action by a Lie group G. Denote by L̄a the action of a ∈ G. A Riemannian metric on M is G-invariant if L̄a are isometries, in which case M is a Riemannian homogeneous space and G is a subgroup of Iso(M ). We identify G with the group of actions and M with G/H where H = Go , the subgroup fixing a point o. By declaring dπ at the identity an isometry, an Ad(H)-invariant inner product on m induces a G invariant inner product on To M and vice versa. This extends to a G invariant Riemannian metric by defining: h(dL̄g )o π∗ Y1 , (dL̄g )o π∗ Y2 igo = hπ∗ Y1 , π∗ Y2 io . Furthermore, G-invariant metrics on M are in one to one correspondence with AdH -invariant metrics on m. We should mention that, by a theorem of S. B. Myers and N. E. Steenrod [44], the set of all isometries of a Riemannian manifold M is a Lie group under composition of maps, and furthermore the isotropy subgroup Isoo (M ) is compact. See also S. Kobayashi and K. Nomizu [33]. If Iso(M ) or a subgroup of it acts on M transitively, M is a Riemannian homogeneous space, as defined traditonally. A connected Lie group admits an Ad-invariant metric if and only if G is of compact type, J. Milnor [42, Lemma 7.5 ], in which case we choose to use the bi-invariant metric for simplicity. The existence of an AdH -invariant metric is less restrictive. If H is compact by averaging we can construct an AdH L invariant inner product in each irreducible invariant subspace of g. If m = m0 ⊕ ri=1 mi is an irreducible invariant decomposition of m, AdH -invariant inner products on m are P precisely of the form g0 + ri=1 ai gi where g0 is any inner product on m0 , ai are positive numbers and gi are AdH invariant inner products on mi . In particular an irreducible homogeneous space with compact H admits a G-invariant Riemannian metric that is unique up to homotheties. See J. A. Wolf [54] and A. L. Besse [3, Thm. 7.44]. In the remaining of the section we assume that G is given a left invariant Riemannian metric which induces a G-invariant Riemannian metric on M as constructed earlier. We ask the question whether the projection of the limiting stochastic process in Theorem 9.3 is a Brownian motion like process. Firstly, we note that the projections of ‘exponentials’, g exp(tX), are not necessarily Riemannian geodesics on M . They are not necessarily Riemannian exponential maps on G. Also, given an 32 XUE-MEI LI P orthonormal basis {Yi , i = 1, . . . , n} of g, ni=1 LYi∗ LYi∗ is not necessarily the Laplace-beltrami operator, which we explore below. Let X, Y, Z ∈ g, by Koszul’s formula for the Levi-Civita connection, h2∇X Y, Zi = h[X, Y ], Zi + h[Z, Y ], Xi + h[Z, X], Y i. By polarisation, ∇X X = 0 for all X if and only if h[Z, Y ], Xi + h[Z, X], Y i = 0, ∀X, Y, Z, (12.1) 1 2 [X, Y in which case ∇X Y = ]. In another word, (12.1) holds if and only if ∇L and ∇ have the same set of geodesics, they are translates of the one parameter subgroups. If the Riemannian metric on G is bi-invariant then translates of the one parameter subgroups are indeed geodesics for the Levi-Civita connection. Recall that a connection is torsion skew symmetric if its torsion T satisfies: hT (u, v), wi+ hT (u, w), vi = 0 for all u, v, w ∈ Tx M and x ∈ M . Denote by T L (X, Y ) the torsion for the flat connection ∇L . Since T L (X, Y ) = −[X, Y ], (12.1) is equivalent to ∇L being torsion skew symmetric. Remark 12.1. Let G be a connected Lie group with a left invariant metric. Then Pn ∗ ∗ i=1 LYi LYi = ∆ if and only if G is unimodular. If ml is a subspace of g, then traceml ad(X) = 0 for all X ∈ g if and only if traceml ∇d = traceml ∇L d. Proof. Recall that a Lie group G is unimodular if and only if its left invariant Haar measure is also right invariant. A Lie group is unimodular if and only if the absolute value of the determinant of Ad(g) is 1 for every g ∈ G. See S. Helgson [24]. Equivalently, ad(X) has J. Milnor [42, Lemma Pzero trace for every X ∈ g, seeP n ∗ ∗ only if 6.3]. On the other hand ni=1 LYi∗ LYi∗ = ∆ if and i=1 ∇Yi Yi = 0. The P latter condition is equivalent to trace ad(X) = i h[X, Yi ], Yi i = 0 for all X ∈ g. Similarly if {Yi } is an orthonormal basis of ml , then dim(ml ) dim(ml ) X X LYi∗ LYi∗ = i=1 dim(ml ) X ∇d + i=1 ∇Yi∗ Yi∗ . i=1 The second statement follows from the identity: dim(ml ) h X i=1 dim(ml ) ∇Yi∗ Yi∗ , Xi = X h[X, Yi ], Yi i. i=1 We discuss the relation between the Levi-Civita connection and the canonical connection on the Rieamnnian homogeneous manifold. A connection is G invariant if {L̄a , a ∈ G} are affine maps for the connection. For X ∈ g, define the derivation AX = LX − ∇X . A G-invariant connection is determined by a linear map (AX )o ∈ L(To M ; To M ). Each AX is in correspondence with an endomorphism Λm (X) on Rn , satisfying the condition that for all X ∈ m and h ∈ H, Λm (ad(h)X) = Ad(λ(h))Λm (X) where λ(h) is the isotropy representation of h, see S. Kobayashi and K. Nomizu [34]. Then ˜ X Y := LX Y − u0 Λm (X)u−1 Y ∇ 0 STOCHASTIC HOMOGENIZATION ON HOMOGENEOUS SPACES 33 defines a connection. We identified To M with Rn by a frame u0 . If Λm = 0 this defines the canonical connection ∇c whose parallel translation along a curve is left translation. c Denote by Ddt the corresponding covariant differentiation. Parallel translations along α(t) are given by dL̄γ(t) . This is due to the fact that left translations and π commute. If X ∈ m, let γt = γ0 exp(tX) and α(t) = γ0 exp(tX)o. Then γ is the horizontal lift of α and α(t) is a ∇c geodesic, Dc Dc Dc α̇ = π∗ T Lγ(t) (X) = T L̄γ(t) π∗ (X) = 0. dt dt dt For x ∈ g, denote by Xm and Xh the component of X in m and in h respectively. Definition 12.2. A reductive Riemannian homogeneous space is naturally reductive, if h[X, Y ]m , Zi = −hY, [X, Z]m i, for all X, Y, Z ∈ m. (12.2) It is clear that (12.1) implies (12.2). In other words, M is naturally reductive if ∇ and ∇L have the same set of geodesics. In particular if G admits an adG invariant inner product and m = h⊥ , then G/H is naturally reductive with respect to the induced Riemannian metric. This covers groups of compact type. A special reductive homogeneous space is a symmetric space with symmetry σ and the canonical decomposition g = h ⊕ m. For it, [h, m] ⊂ m and [m, m] ⊂ h. If the symmetric decomposition is orthogonal, it is naturally reductive. We collect in the lemma below useful information for the computations of the Markov generators whose proof is included for the convenience of the reader. Lemma 12.3. (1) ∇c is torsion skew symmetric precisely if M is naturally reductive. (2) The projections of the translates of one parameter family of subgroups of G are geodesics for the Levi-Civita connection if and only if M is naturally reductive. (3) Let U : m × m → m be defined by 2hU (X, Y ), Zi = hX, [Z, Y ]m i + h[Z, X]m , Y i. (12.3) Then traceml ∇c df = traceml ∇df if and only if traceml U = 0. In other words, for any Z ∈ ml , traceml ad(Z) = 0. In particular traceml ∇c df = traceml ∇df , if M is naturally reductive. (4) ∇c dπ = 0. Proof. (1) Let Y ∗ denote the action field generated by Y ∈ m, identified with d d |t=0 L̄exp(tY ) uo = dt π(exp(tY )u). If p = π∗ (Y ) ∈ To M , and Y ∗ (uo) = dt t=0 c ∗ ∗ ∗ uo, ∇X Y (o) = [X , Y ](o) = −[X, Y ]m . The torsion tensor for ∇c is left invariant, its value at o is Toc (X, Y ) = (∇cX Y ∗ )o − (∇cY X ∗ )o − [X ∗ , Y ∗ ]o = −[X, Y ]m , We see that T c is skew symmetric if and only if (12.2) holds. 34 XUE-MEI LI (2) A G-invariant connection on M has the same set of geodesics as ∇c if and only if Λm (X)(X) = 0, c.f. Kobayashi and Nomizu [34]. It is well known that the function 1 Λm (X)(Y ) = [X, Y ]m + U (X, Y ). 2 defines the Levi-Civita connection. In fact it is clear that 12 [X, Y ] − U (X, Y ) has vanishing torsion and hΛ(X, Y ), Zi + hY, Λ(X, Z)i = 0, which together with the fact ∇c is metric implies that it is a Riemannian connection. The Levi-Civita connection ∇ and ∇c have the same set of geodesics if and only if U (X, X) = 0. By polarisation, this is equivalent to M being naturally reductive. (3) For any X, Y ∈ m, 1 (∇X Y ∗ )o = ∇cX Y + [X, Y ]m + U (X, Y ). 2 If f : M → R is a smooth function, h∇X ∇f, Y i = LY (df (Y ∗ )) − h∇f, ∇X Y ∗ i = h∇cX ∇f, Y i + h∇f, ∇cX Y ∗ − ∇X Y ∗ i. Summing over the basis of ml we see that traceml ∇df − traceml ∇c df = h∇f, X U (Yi , Yi )i, i P which vanishes for all smooth f if and only if i U (Yi , Yi ) vanishes. If G is of compact type, it has an AdG -invariant metric for which (12.1) holds. This complets the proof. (4) We differentiate the map dπ : G 7→ L(T G; T M ) with respect to the connection ∇c . Let γ(t) be a curve in G with γ(0) = u and γ̇(0) = w. Then for X ∈ g, (∇cw dπ) (X) = //tc (π ◦ γ) d c (//t (π ◦ γ))−1 dπ //tL (γ)X , dt where //tc (π ◦ γ) and //tL (γ) denote respectively parallel translations along π ◦ γ and γ with respect to to ∇c and ∇L . Since //tc (π ◦ γ) = dL̄π◦γ(t) and //tL (γ) = dLγ(t) the covariant derivative vanishes. Indeed π and left translation commutes, (dπ)u dLu = (dL̄u )o (dπ)1 , ∇cw dπ = //tc (π ◦ γ) d (dπ)1 dLγ(−t) //tL (γ) = 0. t=0 dt In the propositions below we keep the notation in Corollary 8.5, Theorem 9.3 and Lemma 10.2. We identify ml with its projection to To M . Let x̄t := lim→0 xt . |2 0 Proposition 12.4. Suppose that L̄ = λ(Y0|Y ) dim(ml ) ∆ml and U is defined by (12.3). If traceml U = 0, in particular if M is naturally reductive, then (x̄t ) is a Markov STOCHASTIC HOMOGENIZATION ON HOMOGENEOUS SPACES 35 process with generator |Y0 |2 traceml ∇d. λ(Y0 ) dim(ml ) If M is furthermore isotropy irreducible, (x̄t ) is a scaled Brownian motion. Proof. Let {Ỹi , 1 ≤ i ≤ ml } be an orthonormal basis of ml and let 21 a2 = |Y0 |2 1 Pdim(ml ) LYk∗ LYk∗ , k=1 dim(ml )λl . Let Yi = aỸi . Let (ut ) be a Markov process with generator 2 Pml ∗ k and represented by a solution of the left invariant SDE: dut = k=1 Yk (ut ) ◦ dBt where {Btk } are independent one dimensional Brownian motions. Let yt = π(ut ). 2 (M ; R), Denote by φkt the integral flow of Yk , φit (g) = g exp(tYi ). If f ∈ CK ml Z t X d f ◦ π (ur exp(tYi )) t=0 dBri f ◦ π(ut ) =f ◦ π(u0 ) + dt i=1 0 ml Z t 2 1X d dr. + f ◦ π (u exp(tY ))) r i t=0 2 2 0 dt i=1 We compute the last term beginning with the first order derivative, d f ◦ (ur exp(tXi )) |t=0 = df (π∗ (Lur exp(tYi ) Yi ))|t=0 = df ((L̄ur )∗ Xi ). dt D Let dt denote covariant differentiation along the curve (xt ) with respect to the Levi-Civita connection. ml X d2 f ◦ π (ur exp(tYi )) |t=0 dt2 i=1 ml X d = df (L̄ur exp(tYi ) )∗ (π∗ (Yi )) |t=0 dt i=1 = ml X ml X ∇df (L̄ur )∗ (π∗ (Yi )), (L̄ur )∗ (π∗ (Yi ) + df i=1 i=1 D L̄ )∗ (π∗ (Yi ) |t=0 . dt ur exp(tYi ) Since left translations are isometries, for each u ∈ G, {(L̄u )∗ (π∗ (Yi ))} is an orthonormal basis of π∗ (uml ) ⊂ Tπ(u) M . Thus c ml ml X X d2 D 2 c ∗ (L̄ )∗ Y |t=0 . f ◦π (ur exp(tYi )) |t=0 = a traceml ∇ df + df dt2 dt ur exp(tYi ) i i=1 i=1 If traceml U = 0, traceml ∇c df = traceml ∇df by Lemma 12.3. We have seen c that the term involving Ddt vanishes. This gives: ml X d2 f ◦ π (ur exp(tYi )) |t=0 = a2 traceml ∇df. dt2 i=1 Put everything together we see that 1 Ef ◦ π(ut ) = f ◦ π(u0 ) + a2 2 Z t E(trace)ml ∇df (π(ur ))dr. 0 36 XUE-MEI LI For x = π(y), define Qt f (x) = Pt (f ◦ π)(y). Then Pt (f ◦ π) = (Qt f ) ◦ π. We apply Dynkin’s criterion for functions of a Markov process to see that (x̄t ) is Markovian. The infinitesimal generator associated to Qt is 21 a2 (trace)ml ∇df . If M is an irreducible Riemannian symmetric space, m is irreducible and dπ(m) = To M and the Markov process (x̄t ) is a scaled Brownian motion, concluding the Proposition. If M is naturally reductive, the proof is even simpler. In this case, σi (t) = d π (u exp(tYi )) is a geodesic with initial velocity Yi and σ̇(t) = dt π∗ (u exp(tYi )). Consequently, the following also vanishes: D (L̄ )∗ (Yi∗ )|t=0 = 0. dt ur exp(tYi ) P Example 12.5. Suppose that traceml U = 0, dim(ml ) = 2. Suppose that k ad2 (Ak ) restricted to ml is not a constant multiple of the identity. P Let Y0 = c1 Y1 + c2 Y2 where Y1 , Y2 is a pair of orthonormal eigenvectors of k ad2 (Ak ). By Remark c2 c2 c1 c2 11.2, a1,1 (Y0 ) = a2,2 (Y0 ) = 21 ( λ(Y11 ) + λ(Y22 ) ), a1,2 (Y0 ) = a2,1 (Y0 ) = ± 12 ( λ(Y − 2) c1 c2 λ(Y1 ) ). As in Theorem 12.4, we represent the L̄ diffusion as a solution of the left P invariant SDE: dut = 2j,k=1 σjk Yj∗ (ut )◦dBtk where (σjk ) is a square root matrix P of (ai,j ). Let Xk = 2j=1 σjk Yj . By the same computation as in Proposition 12.4, we have f ◦ π(ut ) =f ◦ π(u0 ) + 2 Z X k=1 + 1 2 2 Z t X k=1 0 0 t d f ◦ π (ur exp(Xk )) t=0 dBrk dt d2 dr. f ◦ π (u exp(tX ))) r k t=0 dt2 It follows that Z t 2 1X ai,i (Y0 ) E∇df (L̄ur )∗ Yi∗ , (L̄ur )∗ Yi∗ dr Ef ◦ π(ut ) =f ◦ π(u0 ) + 2 0 i=1 Z t + a1,2 (Y0 ) E∇df (L̄ur )∗ Y1∗ , (L̄ur )∗ Y2∗ dr 0 Z t 1 c21 c22 + ) Etraceml ∇df (π(ur ))dr =f ◦ π(u0 ) + ( 2 λ(Y1 ) λ(Y2 ) 0 Z t 1 c1 c2 c1 c2 ± ( − ) E∇df (L̄ur )∗ Y1∗ , (L̄ur )∗ Y2∗ dr. 2 λ(Y2 ) λ(Y1 ) 0 If λ(Y1 ) = λ(Y2 ), the second term vanishes and π(ut ) is a Markov process. If f is not rotationally invariant, the integrand in the second term depending only on π(ur ). STOCHASTIC HOMOGENIZATION ON HOMOGENEOUS SPACES 37 13. E XAMPLES Corollary 10.4 applies to the example in §4, where G = SU (2), H = U (1) and the AdH -invariant space m = hX2 , X3 i is irreducible. For any Y ∈ m, ad2 (X1 )(Y ) = −4Y . Note that 4 is the second non-zero eigenvalue of ∆S 1 , also the killing form of SU (2) is K(X, Y ) = 4 trace XY and Badh ,m (X1 , X1 ) = K(X1 , X1 ) = −8. Example 13.1. Let (bt ) be a one dimensional Brownian motion, g0 ∈ SU (2), Y0 ∈ hX2 , X3 i non-zero. Let (gt , ht ) be the solution to the following SDE on SU (2) × U (1), 1 dgt = gt Y0 dt + √ gt X1 dbt , (13.1) with g0 = g0 . Let π(z, w) = ( 21 (|w|2 − |z|2 ), z w̄) and xt = π(gt ). Let (x̃t ) be the horizontal lift of (xt ). Then (x̃t ) converges weakly to the hypoelliptic diffusion 2 with generator L̄ = |Y40 | ∆hor . Furthermore, motion on S 2 ( 21 ) scaled by 12 |Y0 |2 . xt converges in law to the Brownian The first part of the theorem follows from Theorem 8.1 and Corollary 10.4. The scaling 12 indicates the extra time needed for producing the extra direction [X1 , Y0 ]. For the second part we note G is compact, so h[X, Y ], [Z]i = −hY, [X, Z]i for any X, Y, Z ∈ m, and Proposition 12.4 applies. We observe that a similar equation can be set up on G = SU (n) with H the torus group. R 0 Example 13.2. Let n ≥ 2, G = SO(n + 1), H = , R ∈ SO(n) , 0 1 and S n = SO(n + 1)/SO(n). Then H fixes the point o = (0, . . . , 0, 1)T . The homogeneous space S n has the reductive decomposition: 0 C S 0 n h= , S ∈ so(n) , m = YC = ,C ∈ R . 0 0 −C T 0 I 0 Let σ(A) = S0 AS0−1 and S0 = . Then g = h ⊕ m is the symmetric 0 −1 space decomposition for σ. We identify YC with the vector C, and compute: 0 RC R 0 Ad YC = . 0 1 −(RC)T 0 This action is transitive on the unit tangent sphere and is irreducible. There is a matrix R ∈ SO(n) that sends C to −C. Assumption 6.1 is satisfied by every vector in m and the conditions of Lemma 10.2 are satisfied. Let Ai,j = √12 (Eij − Eji ), where i < j, and Eij is the elementary matrix with the nonzero entry at the (i, j)-th position. Let Y0 ∈ m be a non-trivial vector. Consider the equation, X 1 ∗ dgt = √ A∗i,j (gt ) ◦ dbi,j t + Y0 (gt )dt. 1≤i<j≤n 38 XUE-MEI LI We define λ = (n−1) 4 . By a symmetry argument it is easy to see that, X 1 ad2 (Aij ) = − (n − 1)I. 2 1≤i<j≤n For 1 ≤ i, j, k ≤ n, [Ai,j , Ak,n+1 ] = − √12 δk,i Aj,n+1 + follows from Eij Ekl = δjk Eil . Hence for i 6= j, √1 δj,k Ai,n+1 , 2 2ad2 (Ai,j )(Ak,n+1 ) = −(δi,k + δj,k )Ak,n+1 . x g o (x̃ which (13.2) (x Let t = t and t ) the horizontal lift of t ) through g0 . By Theorem 8.1, converges to a Markov process whose limiting Markov generator is, by symmetry, |Y0 |2 4|Y0 |2 ∆hor = ∆hor . λ dim(m) n(n − 1) The upper bound for the rate of convergence from Theorem 8.4 holds. By Theorem 8.1 and Theorem 12.4 the stochastic processes (xt ) converge to the Brownian L̄ = motion on S n with scale 8|Y0 |2 n(n−1) . By (13.2), for any 1 ≤ i, j, k ≤ n, Ak,n+1 are eigenvectors for ad2 (Ai,j ). Furthermore, for 1 ≤ i < j ≤ n, 1 [Ai,n+1 , Aj,n+1 ] = − √ Ai,j , 2 and ∆hor satisfies the one step Hörmander condition. Example 13.3. We keep the notation in the example above. Let Y0 be a unit vector from m and let (gt ) be the solutions of the following equation: X 1 n(n − 1) ∗ dgt = √ Y0 (gt )dt. (13.3) A∗i,j (gt ) ◦ dbi,j t + 8 1≤i<j≤n Then as → 0, π(g ) t converges to a Brownian motion on S n . We finally provide an example in which the group G is not compact. P Example 13.4. Let n ≥ 3. Let F (x, y) = −x0 y0 + nk=1 xk yk be a bilinear form on Rn+1 . Let O(1, n) be the set of (n + 1) × (n + 1) matrices preserving the indefinite form −1 0 T O(1, n) = A ∈ GL(n + 1) : A SA = S, S = . 0 In Let G denote the identity component of O(1, n), consisting of A ∈ O(1, n) with det(A) = 1 and a00 ≥ 1. This is a n(n + 1)/2 dimensional manifold, X ∈ g if and only X T S + SX = 0. So 0 ξT n g= , ξ ∈ R , S ∈ so(n) . ξ S The map σ(A) = SAS −1 defines an evolution on G and M = {x = (x0 , x1 , . . . , xn ) : F (x, x) = −1, x0 ≥ 1} STOCHASTIC HOMOGENIZATION ON HOMOGENEOUS SPACES 39 1 0 0 B where B ∈ SO(n). The −1 eigenspace of the Lie algebra involution is: 0 ξT m= . ξ 0 is a symmetric space. Its isometry group at e0 = (1, 0, . . . , 0)T is H = Let us take Y0 ∈ m and consider the equation, X 1 ∗ A∗i,j (gt ) ◦ dbi,j dgt = √ t + Y0 (gt )dt. 1≤i<j≤n It is clear that Theorem 8.1 and Theorem 9.3 apply. The isotropy representation of H is irreducible. The action of AdH on m is essentially the action of SO(n) on Rn : 1 0 0 ξT 0 (Bξ)T Ad( ) = . 0 B ξ 0 Bξ 0 The conclusion of Theorem 10.3 holds. The symmetric space M is naturally reductive and so Proposition 12.4 applies. Note that 0 0 0 ξT 0 ηT , = . ξ 0 η 0 0 ξη T − ηξ T Since ei eTj = Eij , [m, m] generates a basis of h. The effective process of the stochastic differential equation (1.2) is a scaled horizontal Brownian motion, which satisfies the one step Hörmander condition. The limit of π(g t ) is a scaled hyper bolic Brownian motion. The rate of convergence stated in Theorem 8.4 is valid 8 here, the scale is n(n−1) , as for the spheres. 14. F URTHER D ISCUSSIONS AND O PEN Q UESTIONS Singularly perturbed equations can be studied in the setting of a general principal fibre bundle. Let π : P → M be a principal bundle with group action H and assume that h is given a suitable inner product. Let n = dim(P ) and p = dim(h). For A ∈ h denote by A∗ the corresponding fundamental vertical vecd |t=0 u exp(tA). Let us fix an orthonormal basis {A1 , . . . , Ap } tor field: A∗ (u) = dt of h. Let {σj , j = p + 1, . . . , n} be smooth horizontal sections of T P , f0 a vertical 2 vector field, {wt1 , . . . , wtN1 , b1t , . . . , bN t } independent one dimensional Brownian k l motions. Let {aj , ci } be a family of smooth functions on P . A computation that is completely analogous to that in Lemma 5.2, should show that (x̃t , at ) satisfies a system of SDEs of Markovian type, which should lead to the following conclusion: Proposition 14.1. Let ut ≡ φt (u0 ) be a solution to dut =(σ0 + f0 )(ut )dt + N1 X n X (cjk σj )(ut ) k=1 j=p+1 ◦ dwtk + p N2 X X ajl A∗j (ut ) ◦ dblt . l=1 j=1 (14.1) 40 XUE-MEI LI Let xt = π(ut ) and (x̃t ) its horizontal lift. Then ut = x̃t at where dx̃t =(Ra−1 )∗ σ0 (x̃t )dt + N2 X n X cjk (x̃t at )(Ra−1 )∗ (σj )(x̃t ) ◦ dwtk t t k=1 j=p+1 dat =T Rat ($x̃t (f0 )) dt + p N1 X X ajl (x̃t at )A∗j (at ) ◦ dblt . l=1 j=1 Based on this proposition, the study of scaling limits should generalise to principal bundles and to more general foliated manifolds. We also expect this work generalise to general foliated manifolds. It would be also interesting to study the following questions. 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