On Lévy's Brownian motion indexed by the elements of compact groups
aa r X i v : . [ m a t h . P R ] O c t ON L ´EVY’S BROWNIAN MOTION INDEXED BY THEELEMENTS OF COMPACT GROUPS
PAOLO BALDI AND MAURIZIA ROSSI
Abstract.
We investigate positive definiteness of the Brownian kernel K ( x, y ) = ( d ( x, x ) + d ( y, x ) − d ( x, y )) on a compact group G and in particular for G = SO ( n ). Introduction
In 1959 P.L´evy [6] asked the question of the existence of a process X indexed bythe points of a metric space ( X , d ) and generalizing the Brownian motion, i.e. of areal Gaussian process which would be centered, vanishing at some point x ∈ X andsuch that E ( | X x − X y | ) = d ( x, y ). By polarization, the covariance function of such aprocess would be(1.1) K ( x, y ) = 12 ( d ( x, x ) + d ( y, x ) − d ( x, y ))so that this question is equivalent to the fact that the kernel K is positive definite.Positive definiteness of K for X = R m and d the Euclidean metric had been provedby Schoenberg [14] in 1938 and P.L´evy itself constructed the Brownian motion on X = S m − , the euclidean sphere of R m , d being the distance along the geodesics.Later Gangolli [12] gave an analytical proof of the positive definiteness of the kernel(1.1) for the same metric space ( S m − , d ), in a paper that dealt with this question fora large class of homogeneous spaces.Finally Takenaka in [13] proved the positive definiteness of the kernel (1.1) for theRiemannian metric spaces of constant sectional curvature equal to − , H m = { ( x , x , . . . , x m ) ∈ R m +1 : x + . . . x m − x = 1 } , the distance underconsideration is the unique, up to multiplicative constants, Riemannian distance thatis invariant with respect to the action of G = L m , the Lorentz group.In this short note we investigate this question for the cases X = SO ( n ). Theanswer is that the kernel (1.1) is not positive definite on SO ( n ) for n >
2. Thisis somehow surprising as, in particular, SO (3) is locally isometric to SU (2), wherepositive definiteness of the kernel K is immediate as shown below. Mathematics Subject Classification.
Primary 43A35; Secondary 60G60, 60B15.
Key words and phrases. positive definite functions, Brownian motion, compact groups.Research supported by ERC grant 277742
Pascal . We have been led to the question of the existence of the Brownian motion indexedby the elements of these groups - in particular of SO (3) - in connection with theanalysis and the modeling of the Cosmic Microwave Background which has becomerecently an active research field (see [5], [7], [8], [9] e.g.) and that has attractedthe attention to the study of random fields ([1], [2], [11] e.g.). More precisely, inthe modern cosmological models the CMB is seen as the realization of an invariantrandom field in a vector bundle over the sphere S and the analysis of its components(the polarization e.g.) requires the spin random fields theory. This leads naturally tothe investigation of invariant random fields on SO (3) enjoying particular propertiesand therefore to the question of the existence of a privileged random field i.e. L´evy’sBrownian random field on SO (3).In § § G = SU (2), recalling well known facts about theinvariant distance and Haar measure of this group. Positive definiteness of K for SU (2) is just a simple remark, but these facts are needed in § SO (3) and deduce from the case SO ( n ), n ≥ Some elementary facts
In this section we recall some well known facts about Lie groups (see mainly [3]and also [4], [15]).2.1.
Invariant distance of a compact Lie group.
From now on we denote by G acompact Lie group. It is well known that G admits at least a bi-invariant Riemannianmetric (see [4] p.66 e.g.), that we shall denote {h· , ·i g } g ∈ G where of course h· , ·i g isthe inner product defined on the tangent space T g G to the manifold G at g andthe family {h· , ·i g } g ∈ G smoothly depends on g . By the bi-invariance property, for g ∈ G the diffeomorphisms L g and R g (resp. the left multiplication and the rightmultiplication of the group) are isometries. Since the tangent space T g G at any point g can be translated to the tangent space T e G at the identity element e of the group,the metric {h· , ·i g } g ∈ G is completely characterized by h· , ·i e . Moreover, T e G being theLie algebra g of G , the bi-invariant metric corresponds to an inner product h· , ·i on g which is invariant under the adjoint representation Ad of G . Indeed there is a one-to-one correspondence between bi-invariant Riemannian metrics on G and Ad -invariantinner products on g . If in addition g is semisimple, then the negative Killing form of G is an Ad -invariant inner product on g itself.If there exists a unique (up to a multiplicative factor) bi-invariant metric on G (for asufficient condition see [4], Th. 2 .
43) and g is semisimple, then this metric is necessarilyproportional to the negative Killing form of g . It is well known that this is the case for SO ( n ) , ( n = 4) and SU ( n ); furthermore, the (natural) Riemannian metric on SO ( n )induced by the embedding SO ( n ) ֒ → R n corresponds to the negative Killing form of so ( n ).Endowed with this bi-invariant Riemannian metric, G becomes a metric space, witha distance d which is bi-invariant. Therefore the function g ∈ G → d ( g, e ) is a class N L´EVY’S BROWNIAN MOTION 3 function as(2.1) d ( g, e ) = d ( hg, h ) = d ( hgh − , hh − ) = d ( hgh − , e ) , g, h ∈ G .
It is well known that geodesics on G through the identity e are exactly the oneparameter subgroups of G (see [10] p.113 e.g.), thus a geodesic from e is the curve on G γ X ( t ) : t ∈ [0 , → exp( tX )for some X ∈ g . The length of this geodesic is L ( γ X ) = k X k = p h X, X i . Therefore d ( g, e ) = inf X ∈ g :exp X = g k X k . Brownian kernels on a metric space.
Let ( X , d ) be a metric space. Lemma 2.1.
The kernel K in (1.1) is positive definite on X if and only if d is arestricted negative definite kernel, i.e., for every choice of elements x , . . . , x n ∈ X and of complex numbers ξ , . . . , ξ n with P ni =1 ξ i = 0(2.2) n X i,j =1 d ( x i , x j ) ξ i ξ j ≤ . Proof.
For every x , . . . , x n ∈ X and complex numbers ξ , . . . , ξ n (2.3) X i,j K ( x i , x j ) ξ i ξ j = 12 (cid:16) a X i d ( x i , x ) ξ i + a X j d ( x j , x ) ξ j − X i,j d ( x i , x j ) ξ i ξ j (cid:17) where a := P i ξ i . If a = 0 then it is immediate that in (2.3) the l.h.s. is ≥ ≤
0. Otherwise set ξ := − a so that P ni =0 ξ i = 0. The followingequality(2.4) n X i,j =0 K ( x i , x j ) ξ i ξ j = n X i,j =1 K ( x i , x j ) ξ i ξ j is then easy to check, keeping in mind that K ( x i , x ) = K ( x , x j ) = 0, which finishesthe proof. (cid:3) For a more general proof see [12] p. 127 in the proof of Lemma 2.5.If X is the homogeneous space of some topological group G , and d is a G -invariantdistance, then (2.2) is satisfied if and only if for every choice of elements g , . . . , g n ∈ G and of complex numbers ξ , . . . , ξ n with P ni =1 ξ i = 0(2.5) n X i,j =1 d ( g i g − j x , x ) ξ i ξ j ≤ x ∈ X is a fixed point. We shall say that the function g ∈ G → d ( gx , x ) isrestricted negative definite on G if it satisfies (2.5). P. BALDI AND M. ROSSI
In our case of interest X = G a compact (Lie) group and d is a bi-invariant distanceas in § .
1. The Peter-Weyl development (see [3] e.g.) for the class function d ( · , e ) on G is(2.6) d ( g, e ) = X ℓ ∈ b G α ℓ χ ℓ ( g )where b G denotes the family of equivalence classes of irreducible representations of G and χ ℓ the character of the ℓ -th irreducible representation of G . Remark 2.2.
A function φ with a development as in (2.6) is restricted negativedefinite if and only if α ℓ ≤ φ is restricted neg-ative definite if and only if for every continuous function f : G → C with 0-mean (i.e.orthogonal to the constants)(2.7) Z G Z G φ ( gh − ) f ( g ) f ( h ) dg dh ≤ dg denoting the Haar measure of G . Choosing f = χ ℓ in the l.h.s. of (2.7) and denoting d ℓ the dimension of the corresponding representation, a straightforward computationgives(2.8) Z G Z G φ ( gh − ) χ ℓ ( g ) χ ℓ ( h ) dg dh = α ℓ d ℓ so that if φ restricted negative definite, α ℓ ≤ α ℓ ≤ φ is restricted negativedefinite, as the characters χ ℓ ’s are positive definite and orthogonal to the constants.3. SU (2)The special unitary group SU (2) consists of the complex unitary 2 × g such that det( g ) = 1. Every g ∈ SU (2) has the form(3.1) g = (cid:18) a b − b a (cid:19) , a, b ∈ C , | a | + | b | = 1 . If a = a + ia and b = b + ib , then the mapΦ( g ) = ( a , a , b , b )(3.2)is an homeomorphism (see [3], [15] e.g.) between SU (2) and the unit sphere S of R .Moreover the right translation R g : h → hg, h, g ∈ SU (2)of SU (2) is a rotation (an element of SO (4)) of S (identified with SU (2)). Thehomeomorphism (3.2) preserves the invariant measure, i.e., if dg is the normalized N L´EVY’S BROWNIAN MOTION 5
Haar measure on SU (2), then Φ( dg ) is the normalized Lebesgue measure on S . Asthe 3-dimensional polar coordinates on S are(3.3) a = cos θ,a = sin θ cos ϕ,b = sin θ sin ϕ cos ψ,b = sin θ sin ϕ sin ψ , ( θ, ϕ, ψ ) ∈ [0 , π ] × [0 , π ] × [0 , π ], the normalized Haar integral of SU (2) for an inte-grable function f is(3.4) Z SU (2) f ( g ) dg = 12 π Z π sin ϕ dϕ Z π sin θ dθ Z π f ( θ, ϕ, ψ ) dψ The bi-invariant Riemannian metric on SU (2) is necessarily proportional to the nega-tive Killing form of its Lie algebra su (2) (the real vector space of 2 × Ad -invariant inner product on su (2) h X, Y i = −
12 tr( XY ) , X, Y ∈ su (2) . Therefore as an orthonormal basis of su (2) we can consider the matrices X = (cid:18) − (cid:19) , X = (cid:18) ii (cid:19) , X = (cid:18) i − i (cid:19) The homeomorphism (3.2) is actually an isometry between SU (2) endowed with thisdistance and S . Hence the restricted negative definiteness of the kernel d on SU (2)is an immediate consequence of this property on S which is known to be true asmentioned in the introduction ([12], [6], [13]). In order to develop a comparison with SO (3), we shall give a different proof of this fact in § SO ( n )We first investigate the case n = 3. The group SO (3) can also be realized as aquotient of SU (2). Actually the adjoint representation Ad of SU (2) is a surjectivemorphism from SU (2) onto SO (3) with kernel {± e } (see [3] e.g.). Hence the wellknown result(4.1) SO (3) ∼ = SU (2) / {± e } . Let us explicitly recall this morphism: if a = a + ia , b = b + ib with | a | + | b | = 1and e g = (cid:18) a b − b a (cid:19) then the orthogonal matrix Ad ( e g ) is given by(4.2) g = a − a − ( b − b ) − a a − b b − a b − a b )2 a a − b b ( a − a ) + ( b − b ) − a b + a b )2( a b + a b ) − − a b + a b ) | a | − | b | P. BALDI AND M. ROSSI
The isomorphism in (4.1) might suggest that the positive definiteness of the Browniankernel on SU (2) implies a similar result for SO (3). This is not true and actually itturns out that the distance ( g, h ) → d ( g, h ) on SO (3) induced by its bi-invariantRiemannian metric is not a restricted negative definite kernel (see Lemma 2.1).As for SU (2), the bi-invariant Riemannian metric on SO (3) is proportional tothe negative Killing form of its Lie algebra so (3) (the real 3 × Ad -invariant inner product on so (3) defined as h A, B i = −
12 tr( AB ) , A, B ∈ so (3) . An orthonormal basis for so (3) is therefore given by the matrices A = −
10 1 0 , A = − , A = − Similarly to the case of SU (2), it is easy to compute the distance from g ∈ SO (3) tothe identity. Actually g is conjugated to the matrix of the form∆( t ) = cos t sin t − sin t cos t
00 0 1 = exp( tA )where t ∈ [0 , π ] is the rotation angle of g . Therefore if d still denotes the distanceinduced by the bi-invariant metric, d ( g, e ) = d (∆( t ) , e ) = t i.e. the distance from g to e is the rotation angle of g .Let us denote { χ ℓ } ℓ ≥ the set of characters for SO (3). It is easy to compute thePeter-Weyl development in (2.6) for d ( · , e ) as the characters χ ℓ are also simple func-tions of the rotation angle. More precisely, if t is the rotation angle of g (see [8]e.g.), χ ℓ ( g ) = sin (2 ℓ +1) t sin t = 1 + 2 ℓ X m =1 cos( mt ) . We shall prove that the coefficient α ℓ = Z SO (3) d ( g, e ) χ ℓ ( g ) dg is positive for some ℓ ≥
1. As both d ( · , e ) and χ ℓ are functions of the rotation angle t , we have α ℓ = Z π t (cid:16) ℓ X j =1 cos( jt ) (cid:17) p T ( t ) dt where p T is the density of t = t ( g ), considered as a r.v. on the probability space( SO (3) , dg ). The next statements are devoted to the computation of the density p T . N L´EVY’S BROWNIAN MOTION 7
This is certainly well known but we were unable to find a reference in the literature.We first compute the density of the trace of g . Proposition 4.1.
The distribution of the trace of a matrix in SO (3) with respect tothe normalized Haar measure is given by the density (4.3) f ( y ) = 12 π (3 − y ) / ( y + 1) − / [ − , ( y ) . Proof.
The trace of the matrix (4.2) is equal totr( g ) = 3 a − a − b − b . Under the normalized Haar measure of SU (2) the vector ( a , a , b , b ) is uniformlydistributed on the sphere S . Recall the normalized Haar integral (3.4) so that, takingthe corresponding marginal, θ has density(4.4) f ( θ ) = 2 π sin ( θ ) dθ . Now 3 a − a − b − b = 4 cos θ − . Let us first compute the density of Y = cos X , where X is distributed according tothe density (4.4). This is elementary as F Y ( t ) = P (cos X ≤ t ) = P (arccos( √ t ) ≤ X ≤ arccos( −√ t )) = 2 π arccos( −√ t ) Z arccos( √ t ) sin ( θ ) dθ . Taking the derivative it is easily found that the density of Y is, for 0 < t < F ′ Y ( t ) = 2 π (1 − t ) / t − / . By an elementary change of variable the distribution of the trace 4 Y − (cid:3) Corollary 4.2.
The distribution of the rotation angle of a matrix in SO (3) is p T ( t ) = 1 π (1 − cos t ) 1 [0 ,π ] ( t ) . Proof.
It suffices to remark that if t is the rotation angle of g , then its trace is equalto 2 cos t + 1. p T is therefore the distribution of W = arccos( Y − ), Y being distributedas (4.3). The elementary details are left to the reader. (cid:3) Now it is easy to compute the Fourier development of the function d ( · , e ). Proposition 4.3.
The kernel d on SO (3) is not restricted negative definite. P. BALDI AND M. ROSSI
Proof.
It is enough to show that in the Fourier development d ( g, e ) = X ℓ ≥ α ℓ χ ℓ ( g ) α ℓ > ℓ ≥ α ℓ = Z SO (3) d ( g, e ) χ ℓ ( g ) dg = 1 π Z π t (cid:16) ℓ X m =1 cos( mt ) (cid:17) (1 − cos t ) dt == 1 π Z π t (1 − cos t ) dt | {z } := I + 2 π ℓ X m =1 Z π t cos( mt ) dt | {z } := I − π ℓ X m =1 Z π t cos( mt ) cos t dt | {z } := I . Now integration by parts gives I = π , I = ( − m − m ,whereas, if m = 1, we have I = Z π t cos( mt ) cos t dt = m + 1( m − (( − m + 1)and for m = 1, I = Z π t cos t dt = π . Putting things together we find α ℓ = 2 π (cid:16) ℓ X m =1 ( − m − m + ℓ X m =2 m + 1( m − (( − m + 1) (cid:17) . If ℓ = 2, for instance, we find α = π >
0, but it is easy to see that α ℓ > ℓ even. (cid:3) Consider now the case n > SO ( n ) contains a closed subgroup H that is isomor-phic to SO (3) and the restriction to H of any bi-invariant distance d on SO ( n ) isa bi-invariant distance e d on SO (3). By Proposition 4.3, e d is not restricted negativedefinite, therefore there exist g , g , . . . , g m ∈ H , ξ , ξ , . . . , ξ m ∈ R with P mi =1 ξ i = 0such that(4.5) X i,j d ( g i , g j ) ξ i ξ j = X i,j e d ( g i , g j ) ξ i ξ j > . We have therefore
Corollary 4.4.
Any bi-invariant distance d on SO ( n ) , n ≥ is not a restrictednegative definite kernel. N L´EVY’S BROWNIAN MOTION 9
Remark that the bi-invariant Riemannian metric on SO (4) is not unique, meaningthat it is not necessarily proportional to the negative Killing form of so (4). In thiscase Corollary 4.4 states that every such bi-invariant distance cannot be restrictednegative definite. 5. Final remarks
We were intrigued by the different behavior of the invariant distance of SU (2) and SO (3) despite these groups are locally isometric and decided to compute also for SU (2) the development(5.1) d ( g, e ) = X ℓ α ℓ χ ℓ ( g ) . This is not difficult as, denoting by t the distance of g from e , the characters of SU (2)are χ ℓ ( g ) = sin(( ℓ + 1) t )sin t , t = kπ and χ ℓ ( e ) = ℓ + 1 if t = 0, χ ℓ ( − ) = ( − ℓ ( ℓ + 1) if t = π . Then it is elementary tocompute, for ℓ > α ℓ = 1 π Z π t sin(( ℓ + 1) t ) sin t dt = ( − π m +1 m ( m +2) ℓ odd0 ℓ eventhus confirming the restricted negative definiteness of d (see Remark 2.2). Remarkalso that the coefficients corresponding to the even numbered representations, thatare also representations of SO (3), here vanish. Acknowledgements.
The authors wish to thank A.Iannuzzi and S.Trapani for valu-able assistance.
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