Equivariant Diffusions on Principal Bundles

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  • Advanced Studies in Pure Mathematics 3?, 200?Stochastic Analysis and Related Topics in Kyotopp. 00

    Equivariant Diffusions on Principal Bundles

    K. D. Elworthy, Yves Le Jan, and Xue-Mei Li

    Let : P M be a smooth principal bundle with structure group G.This means that there is a C right multiplication P G P , u 7 u gsay, of the Lie group G such that identifies the space of orbits of G with themanifold M and is locally trivial in the sense that each point of M has anopen neighbourhood U with a diffeomorphism

    U : 1(U) - U GHHHHj

    U

    overU , which is equivariant with respect to the right action ofG, i.e. if u(b) =((b), k) then u(bg) = ((b), kg). Assume for simplicity thatM is compact.Set n = dimM . The fibres, 1(x), x M are diffeomorphic to G and theirtangent spaces V TuP (= kerTu), u P , are the vertical tangent spacesto P . A connection on P , (or on ) assigns a complementary horizontalsubspace HTuP to V TuP in TuP for each u, giving a smooth horizontal sub-bundle HTP of the tangent bundle TP to P . Given such a connection it is aclassical result that for any C1 curve: : [0, T ] M and u0 1((0))there is a unique horizontal : [0, T ] P which is a lift of , i.e. ((t)) =(t) and has (0) = u0.

    In his startling ICM article [8] Ito showed how this construction could beextended to give horizontal lifts of the sample paths of diffusion processes. Infact he was particularly concerned with the case when M is given a Riemann-ian metric , x, x M , the diffusion is Brownian motion on M , and P is theorthonormal frame bundle : OM M . Recall that each u OM withu 1(x) can be considered as an isometry u : Rn TxM , , x and a

    Research supported by EPSRC grants GR/NOO 845 and GR/S40404/01, NSFgrant DMS 0072387, and EU grant ERBF MRX CT 960075 A.

  • Equivariant Diffusions on Principal Bundles 1

    horizontal lift determines parallel translation of tangent vectors along

    //t //()t : T()M T(t)Mv 7 (t)((0))1v.

    The resulting parallel translation along Brownian paths extends also to paral-lel translation of forms and elements of pTM . This enabled Ito to use hisconstruction to obtain a semi-group acting on differential forms

    Pt = E(//1t )() = E(//t).

    As he pointed out this is not the semi-group generated by the Hodge-KodairaLaplacian, . To obtain that generated by the Hodge-Kodaira Laplacian, ,some modification had to be made since the latter contains zero order terms,the so called Weitzenbock curvature terms. The resulting probabilistic expres-sion for the heat semi-groups on forms has played a major role in subsequentdevelopment.

    In [5] we go in the opposite direction starting with a diffusion with smoothgenerator B on P , which isG-invariant and so projects to a diffusion generatorA on M . We assume the symbol A has constant rank so determining a sub-bundle E of TM , (so E = TM if A is elliptic). We show that this set-upinduces a semi-connection on P over E (a connection if E = TM ) withrespect to which B can be decomposed into a horizontal component AH and avertical part BV . Moreover any vertical diffusion operator such as BV inducesonly zero order operators on sections of associated vector bundles.

    There are two particularly interesting examples. The first when : GLM M is the full linear frame bundle and we are given a stochastic flow {t : 0 t < } on M , generator A, inducing the diffusion {ut : 0 t < } onGLM by

    ut = Tt(u0).

    Here we can determine the connection on GLM in terms of the LeJan-Watanabeconnection of the flow [12], [1], as defined in [6], [7], in particular giving con-ditions when it is a Levi-Civita connection. The zero order operators arisingfrom the vertical components, can be identified with generalized Weitzenbockcurvature terms.

    The second example slightly extends the above framework by letting :P M be the evaluation map on the diffeomorphism group DiffM of Mgiven by (h) := h(x0) for a fixed point x0 in M . The group G correspondsto the group of diffeomorphisms fixing x0. Again we take a flow {t(x) : x M, t 0} on M , but now the process on DiffM is just the right invariantprocess determined by {t : 0 t < }. In this case the horizontal lift tothe diffeomorphism group of the diffusion {t(x0) : 0 t < } on M is

  • 2 K. D. Elworthy, Yves Le Jan, and Xue-Mei Li

    obtained by removal of redundant noise, c.f. [7] while the vertical process isa flow of diffeomorphisms preserving x0, driven by the redundant noise.

    Here we report briefly on some of the main results to appear in [5] and givedetails of a more probabilistic version Theorem 2.5 below: a skew productdecomposition which, although it has a statement not explicitly mentioningconnections, relates to Itos pioneering work on the existence of horizontallifts. The derivative flow example and a simplified version of the stochasticflow example are described in 3.

    The decomposition and lifting apply in much more generality than withthe full structure of a principal bundle, for example to certain skew productsand invariant processes on foliated manifolds. This will be reported on later.Earlier work on such decompositions includes [4] [13].

    1. Construction

    A. If A is a second order differential operator on a manifold X , denoteby A : T X TX its symbol determined by

    df(A(dg)

    )=

    12A (fg) 1

    2A (f) g 1

    2fA (g) ,

    forC2 functions f, g. The operator is said to be semi-elliptic if df(A(df)

    ) 0

    for each f C2(X), and elliptic if the inequality holds strictly. Ellipticity isequivalent to A being onto. It is called a diffusion operator if it is semi-ellipticand annihilates constants, and is smooth if it sends smooth functions to smoothfunctions.

    Consider a smooth map p : N M between smooth manifolds M andN . By a lift of a diffusion operator A on M over p we mean a diffusionoperator B on N such that

    (1) B(f p) = (Af) p

    for all C2 functions f on M . Suppose A is a smooth diffusion operator on Mand B is a lift of A.

    Lemma 1.1. Let B and A be respectively the symbols for B andA. Thefollowing diagram is commutative for all u p1(x), x M :

    T uNBu - TuN

    6(Tp)

    T xM?TxM .-

    Ax

    Tp

  • Equivariant Diffusions on Principal Bundles 3

    B. Semi-connections on principal bundles. Let M be a smooth fi-nite dimensional manifold and P (M,G) a principal fibre bundle over M withstructure group G a Lie group. Denote by : P M the projection and Rathe right translation by a.

    Definition 1.2. LetE be a sub-bundle of TM and : P M a principalG-bundle. An E semi-connection on : P M is a smooth sub-bundleHETP of TP such that

    (i) Tu maps the fibres HETuP bijectively onto E(u) for all u P .(ii) HETP is G-invariant.

    Notes.(1) Such a semi-connection determines and is determined by, a smooth hori-zontal lift:

    hu : E(u) TuP, u P

    such that

    (i) Tu hu(v) = v, for all v Ex TxM ;(ii) hua = TuRa hu.

    The horizontal subspace HETuP at u is then the image at u of hu, and thecomposition hu TuP is a projection onto HETuP .(2) Let F = P V/ be an associated vector bundle to P with fibre V . Anelement of F is an equivalence class [(u, e)] such that (ug, g1e) (u, e).Set u(e) = [(u, e)]. An E semi-connection on P gives a covariant derivativeon F . Let Z be a section of F and w Ex TxM , the covariant derivativewZ Fx is defined, as usual for connections, by

    wZ = u(dZ(hu(w)), u 1(x) = Fx.

    Here Z : P V is Z(u) = u1Z ((u)) considering u as an isomorphismu : V F(u). This agrees with the semi-connections on E defined inElworthy-LeJan-Li [7] when P is taken to be the linear frame bundle of TMand F = TM . As described there, any semi-connection can be completed toa genuine connection, but not canonically.

    Consider on P a diffusion generator B, which is equivariant, i.e.

    Bf Ra = B(f Ra), f, g C2(P,R), a G.

    The operator B induces an operator A on the base manifold M by setting

    (2) Af(x) = B (f ) (u), u 1(x), f C2(M),

    which is well defined since

    B (f ) (u a) = B ((f )) (u).

  • 4 K. D. Elworthy, Yves Le Jan, and Xue-Mei Li

    LetEx := Image(Ax ) TxM , the image of Ax . Assume the dimensionof Ex = p, independent of x. Set E = xEx. Then : E M is a sub-bundle of TM .

    Theorem 1.3. Assume A has constant rank. Then B gives rise to asemi-connection on the principal bundle P whose horizontal map is given by

    (3) hu(v) = B ((Tu))

    where T (u)M satisfies Ax () = v.

    Proof. To prove hu is well defined we only need to show (B(Tu())) =0 for every 1-form on P and for every in ker Ax . Now

    A = 0 impliesby Lemma 1.1 that

    0 = A() = (T)()B((T)()).

    Thus T()B(T()) = 0. On the other hand we may consider B

    as a bilinear form on T P and then for all T uP ,

    B( + t(T)(), + t(T)())

    = B(, ) + 2tB(, (T)()) + t2B((T), (T))

    = B(, ) + 2tB(, (T)()).

    Suppose B(, (T)()) 6= 0. We can then choose t such that

    B( + t(T)(), + t(T)()) < 0,

    which contradicts the semi-ellipticity of B.We must verify (i) Tu hu(v) = v, v Ex TxM and (ii) hua =

    TuRa hu. The first is immediate by Lemma 1.1 and for the second use thefact that T TRa = T for all a G and the equivariance of B.

    2. Horizontal lifts of diffusion operators and decompositions of equi-variant operators

    A. Denote by Cp the space of smooth differential p-forms on a mani-fold M . To each diffusion operator A we shall associate a unique operator A.The horizontal lift of A can be defined to be the unique operator such that theassociated operator vanishes on vertical 1-forms and such that and A areintertwined by the lift map acting on 1-forms.

    Proposition 2.1. For each smooth diffusion operator A there is a uniquesmooth differential operator A : C(1) C0 such that