prev

next

out of 17

View

217Download

1

Embed Size (px)

Transcript

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