aa r X i v : . [ m a t h . P R ] D ec A decomposition of Markov processes via group actions
Ming Liao Summary
We study a decomposition of a general Markov process in a manifoldinvariant under a Lie group action into a radial part (transversal to orbits) and anangular part (along an orbit). We show that given a radial path, the conditionedangular part is a nonhomogeneous L´evy process in a homogeneous space, weobtain a representation of such processes, and as a consequence, we extend thewell known skew-product of Euclidean Brownian motion to a general setting.
Primary 60J25, Secondary 58J65.
Key words and phrases
Markov processes, L´evy processes, Lie groups, ho-mogeneous spaces.
It is well known that a Brownian motion x t in R n ( n ≥
2) may be expressed as a skewproduct of a Bessel process and a spherical Brownian motion. The Bessel process r t = | x t | isthe radial part of x t and the angular part θ t = x t /r t is a timed changed spherical Brownianmotion. This decomposition is naturally related to the action of the rotation group SO ( n ),the group of n × n orthogonal matrices of determinant 1, as the Brownian motion x t hasan SO ( n )-invariant distribution with its radial part transversal to the orbits of SO ( n ) andangular part contained in an orbit, namely the unit sphere. More generally, it is shown inGalmarino [3] that a continuous Markov process in R n with an SO ( n )-invariant distributionis a skew product of its radial motion and an independent spherical Brownian motion witha time change. Such a skew-product structure in connection with a group action has beennoticed in literature. For example, Pauwels and Rogers [11] considered a skew-product ofBrownian motion in a manifold.In this paper, we will consider a general Markov process x t in a smooth manifold X that has a distribution invariant under the smooth action of a Lie group K . Let Y be asubmanifold of X transversal to the orbits of K . The radial part and the angular part of x t are respectively its projections to Y and to a typical K -orbit. It is easy to show that theradial part is a Markov process in Y . Our main purpose is to study the conditioned angularprocess given a radial path.In the next section, we provide the easy proof that the radial part y t of x t is a Markovprocess in Y . We also briefly discuss to what extend the process x t is determined by itsradial part. Several examples are mentioned here. Department of Mathematics, Auburn University, Auburn, AL 36849, USA. Email: [email protected]
1n order to study the angular part, we need first to introduce, in section 3, the notionof nonhomogeneous L´evy processes (processes of independent but not necessarily stationaryincrements) in a homogeneous space, and establish their representation in terms of a drift, acovariance operator and a L´evy measure function. This is an extension of Feinsilver’s result[2] for nonhomogeneous L´evy processes in Lie groups. The arguments in [2] may be suitablymodified to work on homogeneous spaces, but requires a careful formulation of a productstructure on homogeneous spaces.The angular part of x t is introduced in section 4 as a process in a typical K -orbit whichmay be identified with a homogeneous space K/M . We prove that given a radial path, theconditioned angular process z t is a nonhomogeneous L´evy process in K/M . As a consequence,we show that if x t is continuous and if K/M is irreducible, then x t is a skew product of itsradial part and an independent Brownian motion in K/M with a time change. This is anextension of Galmarino’s result to a more general setting by a conceptually more transparentproof.In section 5, we study a class of K -invariant Markov processes in X obtained by inter-lacing a diffusion process with jumps. In this case, we may obtain explicit expressions forthe covariance operator and the L´evy measure function of the conditioned angular process.The skew-product decomposition of Brownian motion in Euclidean spaces or manifoldsmay be well known, but this is perhaps the first time when it is studied in the general settingof a Markov process in a manifold under the action of a Lie group, assuming only a simpleMarkov property with a possibly finite life time and some technical condition on the groupaction. We like to mention that it may be possible to extend our results to the action of alocally compact group K , using an extension of the representation of nonhomogeneous L´evyprocesses in such groups, obtained in Heyer and Pap [6], to homogeneous spaces. Throughout this paper, let X be a (smooth) manifold and let K be a Lie group acting(smoothly) on X . Let C b ( X ), C c ( X ) and C ( X ) be the spaces of continuous functions on X that are respectively bounded, compactly supported and convergent to 0 at infinity in theone-point compactification of X . When a superscript ∞ is added, such as C ∞ c ( X ), it willdenote the subspace of smooth functions.Let x t be a Markov process in X with transition semigroup P t . By this we mean aprocess x t with rcll paths (right continuous paths with left limits) that has the followingsimple Markov property: E [ f ( x t + s ) | F t ] = P s f ( x t ) (1)almost surely for s < t and f ∈ C b ( X ), where F t is the natural filtration of process x t , and2or t ≥ P t is a sub-probability kernel from X to X (that is, P t ( x, · ) is a sub-probabilitymeasure on X for x ∈ X and P t ( x, B ) is measurable in x for measurable B ⊂ X ), with P ( x, · ) = δ x (unit point mass at x ), such that P t + s ( x, · ) = R P t ( x, dy ) P s ( y, · ). Note that theMarkov process x t is allowed to have a finite life time as P t ( x, X ) may be less than 1.We will assume the Markov process x t or equivalently its transition semigroup P t is K -invariant in the sense that ∀ f ∈ C b ( X ) and k ∈ K, P t ( f ◦ k ) = ( P t f ) ◦ k. (2)This means that for k ∈ K , kx t is the same Markov process started at kx (in the sense ofdistribution).The Markov process x t in X is called a Feller process if P t f ∈ C ( X ) for f ∈ C ( X )and P t f → f uniformly as t →
0. In this case, P t is completely determined by its generator L given by Lf = lim t → (1 /t ) P t f with domain D ( L ) consisting of f ∈ C ( X ) for which thelimit exists under the sup norm. A continuous Feller process in X will be called a diffusionprocess in X if its generator L restricted to C ∞ c ( X ) is a differential operator with smoothcoefficients that annihilates constants.Let Y be a submanifold of X , possibly with a boundary, that is transversal to the actionof K in the sense that it intersects each orbit of K at exactly one point, that is, ∀ y ∈ Y, ( Ky ) ∩ Y = { y } and X = ∪ y ∈ Y Ky. (3)Let J : X → Y be the projection map J ( x ) = y for x ∈ Ky , which is continuous if K iscompact. Note that J ◦ k = J for k ∈ K . Theorem 1 y t = J ( x t ) is a Markov process in Y with transition semigroup Q t given by Q t f ( y ) = P t ( f ◦ J )( y ) , y ∈ Y and f ∈ C b ( Y ) . (4) Moreover, if x t is a Feller process in X with generator L and if K is compact, then so is y t in Y with generator L Y given by ( L Y f ) ◦ J = L ( f ◦ J ) for f ∈ D ( L Y ) = { h ◦ J ; h ∈ D ( L ) } . Proof
For f ∈ C b ( Y ) and y ∈ Y , E [ f ( y t + s ) | F t ] = E [ f ◦ J ( x t + s ) | F t ] = P s ( f ◦ J )( x t )= P s ( f ◦ J ◦ k − )( kx t ) (where k ∈ K is chosen such that kx t = y t )= P s ( f ◦ J )( y t ) . This proves that y t is a Markov process in Y with transition semigroup Q t . If K is compact, f ◦ J ∈ C ( X ) for f ∈ C ( Y ), the Feller property of y t follows from that of x t . ✷ y t = J ( x t ) in Theorem 1 will be called the radial part of process x t (relativeto K and Y ). Note that for a diffusion process x t with generator L , the generator L Y of y t is the radial part of the differential operator L as defined in [5].For a measure µ and a function f on X , the integral R f ( x ) µ ( dx ) maybe written as µ ( f ).For a measurable map g : X → X , let gµ be the measure on X defined by gµ ( f ) = µ ( f ◦ g ).If K is compact and the distribution µ of x is K -invariant, that is, if kµ = µ for k ∈ K ,then the marginal distributions of x t are completely determined by those of the radial part y t . Indeed, For f ∈ C b ( X ) and k ∈ K , E [ f ( x t )] = µ ( P t f ) = µ [( P t f ) ◦ k ] = µ [ P t ( f ◦ k )] = µ ( P t ˜ f ) = J µ ( Q t ˜ f ) = E [ ˜ f ( y t )] , where ˜ f = R dk ( f ◦ k ), with dk being the normalized Haar measure on K , may be regardedeither as a K -invariant function on X or a function on Y .However, the distribution of the process x t is not in general determined by the distributionof its radial part y t . Consider a process in R n , when starting at a point different from theorigin, it is a Bessel process in the ray containing the starting point, but when starting at theorigin, it immediately chooses a ray with a uniform distribution and then performs Besselmotion along the ray. It is easy to see that this is a continuous Feller process in R n that hasthe same radial part as a Brownian motion in R n , but not a Brownian motion in distribution.The radial part determines the process x t if the transition semigroup P t is determined by Q t , which will require an additional assumption. For example, let G be a Lie group actingtransitively on X and containing K as a subgroup, and assume there is a point o in Y fixedby K . Then the transition semigroup P t of a G -invariant Markov process in X is determinedby Q t ( o, · ), where Q t is the transition semigroup of the radial part. To prove this, let x t startat o . Then P t ( o, · ) is determined by Q t ( o, · ). Now the G -invariance of P t and the transitivityof the G -action on X imply that P t is determined by P t ( o, · ) and hence by Q t .A Feller process x t in a Lie group G with an infinite life time, invariant under lefttranslations, will be called a L´evy process in G . Such a process possesses independent andstationary increments in the sense that for any s < t , x − s x t is independent of the processup to time s and its distribution depends only on t − s (see [10]). This notion extends theusual definition of L´evy processes in R n regarded as an additive group.More generally, a Feller process x t in a homogeneous space G/K , with an infinite lifetime and invariant under the natural (left) action of G on G/K , where K is a compactsubgroup of G , will also be called a L´evy process in G/K . It is clearly K -invariant, and bythe discussion in the last paragraph, its distribution is determined by its radial part relativeto any Y transversal to K that has a point fixed by K .An explicit formula for the generator of a L´evy process in G or G/H is obtained by Hunt[7], see also [10, chapter 2]. A L´evy process in G may be characterized by a stochastic integral4quation driven by a Brownian motion and a Poisson random measure, see Applebaum andKunita [1].For x ∈ X , K x = { k ∈ K ; kx = x } is a closed subgroup of K , called the isotropysubgroup of K at x . Let Y ◦ be Y minus its boundary. Let X ◦ be the union of the K -orbitsthat intersect Y ◦ . This is an open dense subset of X . In order to introduce an angular partof the Markov process x t later in section 4, we will assume that K y is the same compactsubgroup M of K as y varies over Y ◦ . This assumption is often satisfied when the transversalsubmanifold Y is properly chosen. Then X ◦ = Y ◦ × ( K/M ) as a product manifold.
Example 1:
We have mentioned earlier that the radial part of a Brownian motion x t in X = R n ( n ≥ K = SO ( n ), is a Bessel process in a fixed ray Y fromthe origin. We may take Y to be the positive half of x -axis, which is transversal to K withboundary containing only the origin. Then M = diag { , SO ( n − } . Example 2:
Let X be the space of n × n real symmetric matrices ( n ≥
2) with K = SO ( n )acting on X by conjugation. The set Y of all n × n diagonal matrices with non-ascendingdiagonal elements is a submanifold of X transversal to SO ( n ), its boundary consists ofdiagonal matrices with two identical diagonal elements, and M is the finite subgroup of SO ( n ) consisting of diagonal matrices with ± J : X → Y mapsa symmetric matrix to the diagonal matrix of its eigenvalues in non-ascending order. Notethat X = GL ( n, R ) /SO ( n ), where GL ( n, R ) is the group of n × n real invertible matrices. Example 3:
Let Y be a manifold and K be a Lie group with a compact subgroup M , andlet X = Y × ( K/M ) as a product manifold. Then K acts on X as its natural action on K/M , Y is transversal to K and M is the isotropy subgroup of K at all y ∈ Y . For example, X = R n + m = Y × K with Y = R n , K = R m (additive group) and M = { } . Example 4:
Let X = S n be the n -dimensional sphere, regarded as the unit sphere in R n +1 , under the natural action of K = diag { , SO ( n ) } . The half circle Y connecting twopoles ( ± , , . . . , t, sin t, , . . . ,
0) for 0 ≤ t ≤ π , is transversal to K , and M = diag { , , SO ( n − } . Example 5:
Let X = G/K be a symmetric space of noncompact type, where G is asemisimple Lie group of noncompact type and of a finite center, and K is a maximal compactsubgroup. Using the standard notation and results in [4], let g and k be respectively the Liealgebras of G and K , let p be an Ad( K )-invariant subspace of g complementary to k , let a be a maximal abelian subspace of p , and let a + be a fixed (open) Weyl chamber ( ⊂ a ). Then Y = exp( a + ) (overline denotes the closure) is a submanifold of X transversal to the actionof K on G/K , its boundary is exp( ∂ a + ), where ∂ a + is the boundary of a + , and the isotropy5ubgroup M of K at any y ∈ Y ◦ = exp( a + ) is the centralizer M of A in K . Let G be a Lie group. The convolution of two finite measures µ and ν on G is the measure µ ∗ ν defined by µ ∗ ν ( f ) = R f ( xy ) µ ( dx ) ν ( dy ) for f ∈ C b ( G ). A family of probabilitymeasures µ t , t ∈ R + , on G is called a convolution semigroup if µ t + s = µ t ∗ µ s and µ = δ e ,the unit point mass at the identity element e of G . It is called continuous if µ t → µ weaklyas t →
0. If x t is a L´evy process in G with transition semigroup P t , then µ t = P t ( e, · ) isa continuous convolution semigroup. Conversely, any continuous convolution semigroup isassociated to a L´evy process in G in this way (see [10, chapter 1]).Let H be a compact subgroup of G . The convolution product may be extended tothe homogeneous space G/H as follows. Let π : G → G/H be the natural projection. Ameasurable map S : G/H → G is called a section map if π ◦ S = id G/H , the identity map on
G/H . The convolution of two H -invariant finite measures µ and ν on G/H is the measure µ ∗ ν defined by µ ∗ ν ( f ) = R f ( S ( x ) y ) µ ( dx ) ν ( dy ), which is independent of the choice of S and is H -invariant. Moreover, the convolution product is associative and hence the n -foldproduct µ ∗ µ ∗· · ·∗ µ n is well defined. The convolution semigroup of H -invariant probabilitymeasures on G/H is defined as on G but replacing e by o = eH (the origin of G/H ). Thecontinuous convolution semigroups on
G/H are associated to L´evy processes as on G .A process x t in a Lie group G with rcll paths and an infinite life time is called a nonho-mogeneous L´evy process if for s < t , its increment x − s x t is independent of process up to time s . The distributions µ s,t of the increments x − s x t , s ≤ t , form a two-parameter convolutionsemigroup in the sense that for s < t < u , µ s,t ∗ µ t,u = µ s,u and µ t,t = δ e , which is continuousin the sense that µ s,t → µ s,s weakly as t ↓ s . In fact, a nonhomogeneous L´evy process in G may be defined as a rcll process x t such that for 0 < t < t < · · · < t n and f ∈ C b ( G n +1 ), E [ f ( x t , x t , x t . . . , x t n )] = Z f ( x , x x , x x x , . . . , x x · · · x n ) µ ( dx ) µ ,t ( dx ) µ t ,t ( dx ) · · · µ t n − ,t n ( dx n ) (5)for a probability measure µ (distribution of x ) and a continuous two-parameter convolutionsemigroup µ s,t on G .Feinsilver [2] obtained a martingale representation of such processes. Let g be the Liealgebra of G , whose elements are identified with left invariant vector fields on G as usual,and let ξ , . . . , ξ n be a basis of g . Choose local coordinates φ , . . . , φ n ∈ C ∞ c ( G ) to satisfy x = exp( P i φ i ( x ) ξ i ) for x near e . A covariance function A is a continuous n × n symmetricmatrix valued function such that A (0) = 0 and for s < t , A ( t ) − A ( s ) is nonnegative definite.A L´evy measure function Π( t, dx ) is a measure valued function on G such that Π(0 , · ) = 0,6( t, { e } ) = 0 and for f ∈ C ∞ b ( G ) with f ( e ) = ξ i f ( e ) = 0, Π( t, f ) is finite and continuousin t . Let x t be a nonhomogeneous L´evy process in G with x = e . Assume x t is stochasticcontinuous (that is, x t = x t − almost surely for each fixed t ). Then by [2], there are unique G -valued (non-random) continuous function b t with b = e , covariance function A and L´evymeasure function Π, such that x t = z t b t and for f ∈ C ∞ c ( G ), f ( z t ) − Z t Z G { f ( z s b s τ b − s ) − f ( z s ) − X i φ i ( τ )[Ad( b s ) ξ i ] f ( z s ) } Π( ds, dτ ) − Z t X i,j [Ad( b s ) ξ i ][Ad( b s ) ξ j ] f ( z s ) dA ij ( s ) (6)is a martingale. Moreover, given ( b, A, Π) as above, there is a rcll process x t = z t b t in G with x = e such that (6) is a martingale for f ∈ C ∞ c ( G ). Furthermore, such a process x t isunique in distribution and is a stochastic continuous nonhomogeneous L´evy process in G .Note that because the exponential coordinates φ i are used, ρ ijk = ξ i ξ j φ k ( e ) satisfies ρ ijk = − ρ jik , so they will not appear in (6) as in [2]. Note also that the integrand of Π-integral in(6) is the remainder of a first order Taylor expansion and hence is integrable.Now consider the homogeneous space G/H . A point b ∈ G/H or a subset B of G/H iscalled H -invariant if hb = b or hB = B for all h ∈ H . For x ∈ G/H , H -invariant b and B , xb = S ( x ) b ∈ G/H and xB = S ( x ) B ⊂ G/H are well defined because they are independentof choice for the section map S . Note that g ∈ G with go H -invariant is characterized by g − Hg ⊂ H , and hence by g − Hg = H . Therefore, the set of H -invariant points in G/H isthe natural projection of a closed subgroup of G containing H as a normal subgroup, andhence has a natural group structure with product b b = S ( b ) b and inverse b − = S ( b ) − o (independent of S ). In general, the product x x · · · x n − x n = S ( x ) S ( x ) · · · S ( x n − ) x n isnot well defined because it depends on the choice of S . However, if x , . . . , x n are indepen-dent random variables with H -invariant distributions, then the distribution of the product x x · · · x n and also that of the sequence y i = x x · · · x i for i = 1 , , . . . are independent of S , and hence such a product or sequence is meaningful in the sense of distribution.Note that for H -invariant finite measures µ and ν on G/H , an integral like Z f ( xy, xyz ) µ ( dy ) ν ( dz ) = Z f ( S ( x ) y, S ( x ) S ( y ) z ) µ ( dy ) ν ( dz )is well defined (independent of choice of section map S ). So is R f ( xbyb − ) µ ( dy ) if b is an H -invariant point.A process x t in G/H with rcll paths and an infinite life time will be called a non-homogeneous L´evy process if there is a continuous two-parameter convolution semigroups µ s,t of H -invariant probability measures on G/H such that (5) holds. Then for s < t , x − s x t = S ( x s ) − x t has distribution µ s,t (independent of choice for section map S ) and isindependent of the process up to time s . 7ecause H is compact, there is a subspace p of g that is complementary to the Lie algebra h of H and is Ad( H )-invariant in the sense that Ad( h ) p = p for h ∈ H . Choose a basis ξ , . . . , ξ m of p and local coordinates φ , . . . , φ m ∈ C ∞ c ( G/H ) around o on G/H such that x = exp( P mi =1 φ i ( x ) ξ i ) o for x near o . Then ∀ h ∈ H, m X i =1 φ i Ad( h ) ξ i = m X i =1 ( φ i ◦ h ) ξ i (7)near o . The functions φ i may be suitably extended so that (7) holds globally on G/H .Any ξ ∈ g is a left invariant vector field on G . If ξ is Ad( H )-invariant, it may also beregarded as a vector field on G/H given by ξf ( x ) = ddt f ( xe tξ o ) | t =0 for f ∈ C ∞ ( G/H ) and x ∈ G/H (note that e tξ o is H -invariant), which is G -invariant in the sense that ξ ( f ◦ g ) = ( ξf ) ◦ g for g ∈ G . In fact, any G -invariant vector field on G/H is given by an Ad( H )-invariant ξ ∈ g .Note that if ξ ∈ g is Ad( H )-invariant and b ∈ G/H is H -invariant, then Ad( b ) ξ = Ad( S ( b )) ξ is Ad( H )-invariant and is independent of section map S . By (7), for any H -invariant measure µ on G/H , R µ ( dx ) P i φ i ( x ) ξ i is Ad( H )-invariant, and so is R µ ( dx ) P i φ i ( x )Ad( b ) ξ i .Let ξ, η ∈ g . With a choice of section map S , ξη may be regarded as a second orderdifferential operator on G/H by setting ξηf ( x ) = ∂ ∂t ∂s f ( S ( x ) e tξ e sη o ) | t = s =0 . As in [2], itcan be shown that ξ i ξ j f ( x ) = ∂ ∂t ∂s f ( S ( x ) e tξ i + sξ j o ) | t = s =0 + P mk =1 ρ ijk ξ k f ( x ) with ρ ijk = − ρ jik .Thus, if a ij is a symmetric matrix, then m X i,j =1 a ij ξ i ξ j f ( x ) = m X i,j =1 a ij ∂ ∂t i ∂t j f ( S ( x ) e P mp =1 t p ξ p o ) | t = ··· = t m =0 . The matrix a ij is called Ad( H )-invariant if a ij = P p,q a pq [Ad( h )] pi [Ad( h )] qj for h ∈ H ,where [Ad( h )] ij is the matrix representing Ad( h ), that is, Ad( h ) ξ j = P i [Ad( h )] ij ξ j . Thenthe operator P i,j a ij ξ i ξ j is independent of section map S and is G -invariant. In fact,any second order G -invariant differential operator on G/H is such an operator plus a G -invariant vector field. Note that if b ∈ G/H is H -invariant, then P i,j a ij [Ad( b ) ξ i ][Ad( b ) ξ j ] = P i,j a ij [Ad( S ( b )) ξ i ][Ad( S ( b )) ξ j ] is a G -invariant operator on G/H (independent of S ).A covariance function A and a L´evy measure function Π on G/H are defined as on G with the additional requirements that A ( t ) is Ad( H )-invariant and Π( t, · ) is H -invariant. Bythe preceding discussion, the expression in (6) is meaningful on G/H and is independent ofthe choice of section map S in Ad( b s ) = Ad( S ( b s )). The following result is an extension ofFeinsilver’s martingale representation to nonhomogeneous L´evy processes in G/H . Theorem 2
Let x t be a stochastic continuous nonhomogeneous L´evy process in G/H with x = o . Then there is a unique triple ( b, A, Π) of a continuous H -invariant function b t with b = o , a covariance function A and a L´evy measure function Π on G/H such that x t = z t b t and (6) is a martingale for f ∈ C ∞ c ( G/H ) . Moreover, given ( b, A, Π) as above, there is a cll process x t = z t b t in G/H with x = o and represented by ( b, A, Π) . Furthermore, such x t is unique in distribution and is a stochastic continuous nonhomogeneous process in G/H . Proof
Fix
T >
0. Let 0 = t < t < · · · < t n ≤ T be a partition of [0 , T ] with t i +1 − t i = 1 /n and let µ ni = µ t i − ,t i for 1 ≤ i ≤ n . By the stochastic continuity of x t , R φ i dµ ni is uniformlysmall in i as n → ∞ , and b ni ∈ G/H given by x i ( b ni ) = R φ i dµ ni is well defined and is H -invariant because the H -invariance of µ ni .Define an H -invariant function b n ( t ) in G/H by b n ( t ) = b n b n · · · b n [ nt ] for 0 < t ≤ T and b n (0) = o , where [ nt ] is the integer part of nt , a measure function Π n by Π n ( t, · ) = P [ nt ] i =1 µ ni for 0 < t ≤ T and Π(0 , · ) = 0, and a matrix valued function A n ( t, U ) of time t and measurable U ⊂ G/H by A n (0 , U ) = 0 and for 0 < t ≤ T ,[ A n ( t, U )] ij = [ nt ] X k =1 Z U [ φ i ( x ) − φ i ( b nk )][ φ j ( x ) − φ j ( b nk )] µ nk ( dx ) . Let x ni , 1 ≤ i ≤ n , be independent random variables in G/H with distributions µ ni .Define x n ( t ) = x n x n · · · x n [ nt ] and z n ( t ) = x n ( t ) b n ( t ) − . Then as n → ∞ , the process x n ( t ) converges in distribution to x t , and it can be shown that b n ( t ) converges uniformlyto a continuous H -invariant function b t , Π n converges to a L´evy measure function Π inthe sense that for any f ∈ C b ( G/H ) vanishing near o , Π n ( t, f ) → Π( t, f ) uniformly for0 ≤ t ≤ T , for any H -invariant neighborhood U of o which is a continuity set of Π( T, · ),[ A n ( t, U )] ij → A ij ( t ) + R U φ i ( x ) φ j ( x )Π( t, dx ) uniformly for 0 ≤ t ≤ T , for some covariancefunction A ij ( t ), and z n ( t ) converges in distribution to a stochastic continuous process z t forwhich (6) is a martingale. Thus, ( b, A, Π) are the parameters in the representation of x t .These statements and the rest of theorem may be proved by essentially repeating the proofin [2] with suitable changes, such as properly interpreting the product on G/H as discussedearlier and using H -invariant sets on G/H for various neighborhoods used in [2]. ✷ Let x t be a stochastic continuous nonhomogeneous L´evy process in G/H . Fix
T >
0. Wenow show that for f ∈ C b ( G/H ) with support supp( f ) not containing o ,Π( T, f ) = E [ X T, B ) is the expected numbers of jumps x − t − x t = S ( x t − ) − x t in B ⊂ G/H for0 < t ≤ T (independent of S ). Thus, x t is continuous if and only if Π = 0. It can be shownthat the pairs ( t, x − t − x t ) with x − t − x t = o , 0 < t ≤ T , form a Poisson random measure N on[0 , T ] × ( G/H ) with intensity measure Π( dt, dx ).Note that if Π( t, | f | ) < ∞ for any f ∈ C b ( G/H ) with f ( o ) = 0 and t > 0, in particular, ifΠ( t, · ) is a finite measure, then by suitably changing the drift b t , the term P i x i [Ad( b s ) ξ i ] f ( z s )in the integrand of Π-integral in (6) may be dropped. To prove this, apply Itˆo’s formula to f ( z t u t ), where u t is the H -invariant function in G/H determined by the ordinary differentialequation du t = R t R G/H Π( ds, dτ ) P i φ i ( τ )[Ad( b s ) ξ i ]( u s ) and u = o .The L´evy measure function Π is clearly independent of the choice for basis ξ , . . . , ξ m of p .Although the covariance function A depend on the basis, its uniqueness under a given basisimplies that the differential operator V ( t ) = (1 / P mi,j =1 A ij ( t ) ξ i ξ j on G/H is independentof the basis. The operator V ( t ) will be called the covariance operator of x t , which togetherwith b t and Π( t, · ) determines the distribution of the process x t completely.In general, a nonhomogeneous L´evy process x t in G/H may have a fixed jump, that is, P ( x t − = x t ) > t > 0. Suppose x t have only finitely many fixed jumps at times t < t < · · · < t k . As jumps x − t − x t for t = t j are independent of the rest of process x t ,they may be easily removed to obtain a stochastic continuous nonhomogeneous L´evy process x ′ t . The triple ( b, A, Π) (or ( b, V, Π) with V in place of A ) of x ′ t will also be called the drift,covariance function (or covariance operator) and L´evy measure function of x t , which togetherwith the distributions of fixed jumps determine the distribution of process x t completely.A consequence of Theorem 2 is that a nonhomogeneous L´evy process x t in G/H is theprojection of such a process g t in G which is H -conjugate invariant in the sense that for any h ∈ H , the process c h ( g t ) has the same distribution as g t , where c h : G ∋ g hgh − ∈ G is the conjugation map. However, such g t is not unique in distribution. A similar result for(homogeneous) L´evy processes are obtained in [10, chapter 2].10 orollary 1 Let x t be a stochastic continuous nonhomogeneous L´evy process in G/H with x = o . Then there is a stochastic continuous H -conjugate invariant L´evy process g t in G with g = e such that the two processes x t and g t o are identical in distribution. Proof Let ( b, A, Π) be the representation of x t in Theorem 2. There is a continuous G -valued function b ′ t in G with b ′ = e and b t = b ′ t o . Let ˆ b t = R H hb ′ t h − dh , where dh is thenormalized Haar measure on H . Then ˆ b t is a continuous H -conjugate invariant function in G with ˆ b = e and b t = ˆ b t o . The basis ξ , . . . , ξ m of p may be extended to be a basis ξ , . . . , ξ n of g such that ξ m +1 , . . . , ξ n form a basis of h . The covariance function A ij ( t ) on G/H maybe regarded as covariance function on G with A ij ( t ) = 0 if either i > m or j > m . Let S bea section map satisfying S ( x ) = exp[ P mi =1 φ i ( x ) ξ i ] for x near o , and let ˆΠ( t, · ) be defined byˆΠ( t, f ) = R H f ( hS ( x ) h − ) dh Π( t, dx ) for f ∈ C b ( G ). Then ˆΠ( t, · ) is a H -conjugate invariantL´evy measure function on G (that is, c h ˆΠ( t, · ) = ˆΠ( t, · ) for h ∈ H ). The nonhomogeneousL´evy process g t in G with g = e and representation (ˆ b, A, ˆΠ) satisfies x t = g t o in distributionbecause (ˆ b, A, ˆΠ) on G project to ( b, A, Π) on G/H , and is H -conjugate invariant becauseboth ˆ b t and ˆΠ( t, · ) are H -conjugate invariant, and A ( t ) is Ad( H )-invariant. ✷ Recall that x t is a Markov process in a manifold X invariant under the action of a Lie group K , Y is a submanifold transversal to K with interior Y ◦ , M is the compact isotropy subgroupat every point of Y ◦ , and X ◦ is the union of K -orbits through Y ◦ with X ◦ = Y ◦ × ( K/M ).The exit time ζ of process x t from X ◦ is the stopping time when x t together with its leftlimit first leaves X ◦ or reaches its life time ξ . More precisely, it is defined by ζ = inf { t > x t X ◦ , x t − X ◦ or t ≥ ξ } , (9)with inf of an empty set defined to be ∞ . The exit time of y t from Y ◦ is also denoted by ζ .Fix T > 0. Because the process x t has rcll paths, it may be regarded as a random variablein the space D T ( X ) of rcll maps: [0 , T ] → X ◦ , equipped with Skorohod topology. Let P x be the distribution on D T ( X ) associated to the process x t starting at x ∈ X . Its total massmay be less than 1 because x t may have a finite life time.For y ∈ Y ◦ and z ∈ K/M , zy = S ( z ) y ∈ X ◦ is well defined and is independent of choiceof section map S : K/M → K . Let x t = z t y t be the decomposition of the process x t with x ∈ X ◦ and t < ζ . Then y t is the radial part as defined before, and z t is a process in K/M with rcll paths and will be called the angular part of x t .Recall J is the projection map X ∋ x y ∈ Y . Let J be the projection map X ◦ ∋ x z ∈ K/M associated to the decomposition x = zy . We will also use J and J to denote11he maps J : D T ( X ) ∋ x ( · ) y ( · ) ∈ D T ( Y ) and J : D T ( X ◦ ) ∋ x ( · ) z ( · ) ∈ D T ( K/M )respectively given by the decomposition x ( · ) = z ( · ) y ( · ).Let F Y ,T = σ { y t ; 0 ≤ t ≤ T } be the σ -algebra generated by the radial process y t for0 ≤ t ≤ T , which may be regarded as a σ -algebra on D T ( Y ) and induces the σ -algebra J − ( F Y ,T ) on D T ( X ). By the existence of regular conditional distributions (see for example[8, chapter 5]), there is a probability kernel R y ( · ) z from D T ( Y ◦ ) × ( K/M ) to D T ( K/M ) suchthat for any x ∈ X ◦ and measurable F ⊂ D T ( K/M ), R J [ x ( · )] J ( x ) ( F ) = P x [ J − ( F ) | J − ( F Y ,T )] for P x -almost all x ( · ) in [ ζ > T ] ⊂ D T ( X ◦ ) . (10)The probability measure R y ( · ) z is the conditional distribution of the angular process z t givena radial path y ( · ) in D T ( Y ◦ ) and z = z . Theorem 3 Fix T > . Almost surely on [ ζ > T ] , given a radial path y t for ≤ t ≤ T , theconditioned angular process z t is a nonhomogeneous L´evy process in K/M . More precisely,this means that for y ∈ Y ◦ , z ∈ K/M , and J P y -almost all y ( · ) in [ ζ > T ] ⊂ D T ( Y ◦ ) , theangular process z t is a nonhomogeneous L´evy process under R y ( · ) z . Proof For x ∈ X ◦ , let ˜ P t ( x, B ) = P x { [ x t ∈ B ] ∩ [ ζ > t ] } for measurable B ⊂ X ◦ . Bythe simple Markov property of x t , it is easy to show that ˜ P t is the transition semigroup ofthe Markov process x t for t < ζ and it is K -invariant. Similarly, let ˜ Q t be the transitionsemigroup of y t for t < ζ . Then ˜ P t ( x, · ) and ˜ Q t ( y, · ) are respectively sub-probability kernelsfrom X ◦ to X ◦ = Y ◦ × ( K/M ) and from Y ◦ to Y ◦ . By the existence of a regular conditionaldistribution, there is a probability kernel R t ( y, y , · ) from ( Y ◦ ) to K/M such that for y ∈ Y ◦ ,˜ P t ( y, dy × dz ) = ˜ Q t ( y, dy ) R t ( y, y , dz ). The K -invariance of ˜ P t implies that the measure R t ( y, y , · ) is M -invariant for Q t ( y, · )-almost all y . Modifying R t on an exceptional set ofzero Q t ( y, · )-measure, we may assume R t ( y, y , · ) is M -invariant for all y, y ∈ Y ◦ . Therefore,for z ∈ K/M , it is meaningful to write R t ( y, y , z − dz ) = R t ( y, y , S ( z ) − dz ) because it isindependent of choice of section map S . We then have ∀ y ∈ Y ◦ and z ∈ K/M, ˜ P t ( zy, dy × dz ) = ˜ Q t ( y, dy ) R t ( y, y , z − dz ) . (11)For 0 < s < s < · · · < s k < ∞ , y ∈ Y ◦ , z ∈ K/M , h ∈ C b ( Y k ) and f ∈ C b (( K/M ) k ), E zy [ h ( y s , . . . , y s k ) f ( z s , . . . , z s k ); ζ > s k ]= Z Z ˜ P s ( zy, dy × dz ) ˜ P s − s ( z y , dy × dz ) · · · ˜ P s k − s k − ( z k − y k − , dy k × dz k ) h ( y , y , . . . , y k ) f ( z , z , . . . , z k )= Z ˜ Q s ( y, dy ) ˜ Q s − s ( y , dy ) · · · ˜ Q s k − s k − ( y k − , dy k ) h ( y , y , . . . , y k ) Z R s ( y, y , dz ) R s − s ( y , y , dz ) · · · R s k − s k − ( y k − , y k , dz k ) f ( zz , zz z , . . . , zz · · · z k )12 E y [ h ( y s , y s , . . . , y s k ) Z R s ( y, y s , dz ) R s − s ( y s , y s , dz ) · · · R s k − s k − ( y s k − , y s k , dz k ) f ( zz , zz z , . . . , zz · · · z k ); ζ > s k ] . This implies that on [ ζ > s k ], E zy [ f ( z s , . . . , z s k ) | y s , . . . , y s k ] = Z R s ( y, y s , dz ) R s − s ( y s , y s , dz ) · · · R s k − s k − ( y s k − , y s k , dz k ) f ( zz , zz z , . . . , zz · · · z k )] . (12)Let Γ be the set of dyadic numbers i/ m for integers i ≥ m > 0. For the moment,assume T ∈ Γ. For s, t ∈ Γ with s < t ≤ T , let s = s < s < · · · < s k be a partition of[0 , T ] spaced by 1 / m with s = s i and t = s j , and let µ ms,t = R s i +1 − s i ( y s i , y s i +1 , · ) ∗ R s i +2 − s i +1 ( y s i +1 , y s i +2 , · ) ∗ · · · ∗ R s j − s j − ( y s j − , y s j , · ) . By (12), M -invariance of P t ( y, y , · ) and the measurability of µ ms,t in y s i , . . . , y s j , µ ms,t ( f ) = E zy [ f ( z − s z t ) | y s , . . . , y s k ] = E zy [ f ( z − s z t ) | y s i , . . . , y s j ] on [ ζ > T ] (13)for f ∈ C b ( K/M ), which is independent of the choice for section map S to represent z − s z t = S ( z s ) − z t . By the right continuity of y t , as m → ∞ , σ ( y s , . . . , y s k } ↑ F Y ,T and σ { y s i , . . . , y s j } ↑ F Ys,t , it follows that as m → ∞ , almost surely, µ ms,t → µ s,t weakly for some M -invariant probability measure µ s,t on K/M such that ∀ f ∈ C b ( K/M ) , µ s,t ( f ) = E zy [ f ( z − s z t ) | F Y ,T ] = E zy [ f ( z − s z t ) | F Ys,t ] on [ ζ > T ] . (14)Note that µ s,t is an F Ys,t -measurable random measure independent of starting point zy . Be-cause Γ is countable, the exception set of probability zero in the above almost sure conver-gence may be chosen simultaneously for all s < t in Γ. Moreover, for t < t < · · · < t n of[0 , T ] in Γ, it can be shown from (12) and by choosing a partition s < s < · · · < s k fromΓ containing all t i that almost surely on [ ζ > T ], for f ∈ C b (( K/M ) n ), E zy [ f ( z t , . . . , z t n ) | F Y ,T ] = lim m →∞ E zy [ f ( z t , . . . , z t n ) | y s , . . . , y s k ]= lim m →∞ Z f ( zz , zz z , . . . , zz · · · z n ) µ m ,t ( dz ) µ mt ,t ( dz ) · · · µ mt n − ,t n ( dz n ) . This implies that almost surely on [ ζ > T ], for 0 ≤ t < · · · < t n ≤ T in Γ, E zy [ f ( z t , . . . , z t n ) | F Y ,T ]= Z f ( zz , zz z , . . . , zz · · · z n ) µ ,t ( dz ) µ t ,t ( dz ) · · · µ t n − ,t n ( dz n ) . (15)In particular, µ s,t for s < t in Γ form a two-parameter convolution semigroup on K/M . Toprove that the conditioned process z t is a nonhomogeneous L´evy processes in K/M , it remainsto extend µ s,t in (14) to all real s < t ≤ T and prove (15) for real 0 ≤ t < · · · < t n ≤ T .13et f ∈ C b ( K/M ). By the right continuity of z t and (14), for s ∈ Γ and real t with s < t < T , µ s,t may be defined as the weak limit of µ s,t n = P [ S ( z s ) − z t n ∈ · | F Y ,T ] asΓ ∋ t n ↓ t , which is independent of section map S . For a real s , choose Γ ∋ s n ↓ s . Wecan show µ s n ,t ( f ) = E [ f ( S ( z s n ) − z t ) | F Y ,T ] → E [ f ( S ( z s ) − z t ) | F Y ,T ] by using a partition ofunity { ψ j } and section map S j continuous on supp( ψ j ) as in the proof of (8) in section 3. Wecan then define µ s,t ( f ) = lim n µ s n ,t ( f ). Note that no additional exceptional set is producedin taking these limits. It is easy to see that µ s,t for real 0 < s < t form a two-parameterconvolution semigroup on K/M for which (14) and (15) hold. To show the conclusions arevalid for any real T > 0, we may choose T m ∈ Γ with T m ↓ T . ✷ Because the natural action of M on K/M fixes o , it induces an action on the tangentspace T o ( K/M ) at o . The homogeneous space K/M will be called irreducible if the action of M on T o ( K/M ) is irreducible (that is, it has no nontrivial invariant subspace). Among theexamples mentioned in section 2, K/M is irreducible in Examples 1 and 4, and in Example 3if it is chosen to be so, and in Example 5 if the symmetric space G/K is of rank 1 (see [5]).If K/M is irreducible, then, up to a constant multiple, there is a unique M -invariant innerproduct on T o ( K/M ) (see for example Appendix 5 in [9]), and hence, there is a unique secondorder K -invariant differential operator on K/M . By choosing a K -invariant Riemannianmetric on K/M , which is unique up to a constant factor, any such operator is a multipleof the Laplace-Beltrami operator ∆ K/M on K/M . The following result is an extension ofGalmarino’s result mentioned in section 1. Theorem 4 Assume K/M is irreducible. If x t is a continuous K -invariant Markov processin X with radial part y t in Y , then there are a Brownian motion B ( t ) in K/M undera K -invariant Riemannian metric, independent of process x t , and a real continuous non-decreasing process a t , with a = 0 and a t − a s F Ys,t -measurable for s < t , such that the twoprocesses x t and B ( a t ) y t , t < ζ , are identical in distribution. Proof By Theorem 3, given a radial path y t for 0 ≤ t ≤ T with [ ζ > T ], the conditionedangular process z t is a continuous nonhomogeneous L´evy process in K/M . Let ( b, A, Π)be its representation in Theorem 2. By the irreducibility of K/M , there is no nonzero M -invariant tangent vector at o , and hence, there is no M -invariant point near o except o . Thisimplies that the drift b t = o . Because z t is continuous, Π = 0. Because P i,j A ij ( t ) ξ i ξ j is a K -invariant second order differential operator on K/M , it must be equal to a t ∆ K/M for acontinuous non-decreasing function a t with a = 0. By (14), µ s,t is F Ys,t -measurable. Fromthe construction of A ( t ) from µ s,t in the proof of Theorem 2, it is seen that a t − a s is F Ys,t -measurable. Let B ( t ) be a Brownina motion in K/M independent of process x t . It is enoughto show that the conditioned process z t is equal to the time-changed Brownian motion B ( a t )in distribution. For f ∈ C b ( K/M ), f ( B ( t )) − R t ∆ K/M f ( B ( s )) ds is a martingale, and hence,14 ( B ( a t )) − R t ∆ K/M f ( B ( a s )) da s is a martingale. On the other hand, for the conditionedprocess z t , f ( z t ) − R t ∆ K/M f ( z s ) da s is a martingale. The uniqueness of the process withthe given representation ( b, A, Π) implies z t = B ( a t ) in distribution. ✷ K -invariant Markov processes In this section, we will study a type K -invariant Markov process x t in X which is obtainedfrom a K -invariant diffusion process interlaced with jumps.We first consider a K -invariant diffusion process x t in X . Its generator takes the followingform in local coordinates x , . . . , x n on X : L = 12 n X i,j =1 c ij ( x ) ∂ ∂x i ∂x j + n X i =1 c i ( x ) ∂∂x i (16)for some c ij , c i ∈ C ∞ ( X ) with c ij forming a nonnegative definite symmetric matrix.At x = zy ∈ X ◦ , we may assume y = x , . . . , y q = x q form local coordinates in Y ◦ around y and z = x q +1 , . . . , z p = x n form local coordinates in Z = K/M around z , where n = p + q , q = dim( Y ) and p = dim( Z ). Recall that the radial part L ,Y of the operator L is given by ( L ,Y f ) ◦ J = L ( f ◦ J ) for f ∈ C ∞ c ( Y ). This impliesthat L ,Y = (1 / P qi,j =1 c ij ( y ) ∂ / ( ∂y i ∂y j ) + P qi =1 c i ( y ) ∂/∂y i , thus, c ij , c i ∈ C ∞ ( Y ◦ ) for i, j = 1 , , . . . , q . Note that the sum of mixed partials in L , P qi =1 P pj =1 c ij ( y, z ) ∂ / ( ∂y i ∂z j ),is independent of the choice of local coordinates y , . . . , y q on Y ◦ and z , . . . , z p on Z .At least locally, there are (smooth) vector fields ζ , ζ , . . . , ζ q on Y ◦ such that L ,Y =(1 / P qi =1 ζ i ζ i + ζ . We will assume L ,Y is nondegenerate on Y ◦ in the sense that thesymmetric matrix formed by its second order coefficients c ij ( y ), i, j = 1 , , . . . , q , is positivedefinite at all y ∈ Y ◦ . This is independent of choice for local coordinates. Then ζ , . . . , ζ q are linearly independent at each point of Y ◦ , and hence the sum of mixed partials in L , P qi =1 P pj =1 c ij ( y, z ) ∂ / ( ∂y i ∂z j ), may be written as P qi =1 η i ζ i for uniquely determined vectorfields η , . . . , η q on Z that may depend on y ∈ Y . The K -invariance of L and the linearindependence of ζ , . . . , ζ q imply that η i are K -invariant vector fields on Z . We have L =(1 / P qi =1 ( ζ i + η i ) + ζ + L ′ for some K -invariant second order differential operator L ′ on Z that may depend on y ∈ Y .We may choose vector fields θ , . . . , θ q , θ q +1 , . . . , θ q + p , θ on Y ◦ × Z such that L =(1 / P q + pi =1 θ i θ i + θ . Let ( θ i ) Y and ( θ i ) Z be respectively the components of θ i tangent to Y ◦ and Z . After an orthogonal transformation of θ , . . . , θ q + p , which may depends on ( y, z ),we may assume that ( θ i ) Y = 0 for i = q + 1 , . . . , q + p . Then (1 / P qi =1 ( θ i ) Y ( θ j ) Y + ( θ ) Y =(1 / P qi =1 ζ i ζ j + ζ for i, j = 1 , , . . . , q . It follows that after an orthogonal transformationof θ , . . . , θ q , which may depend on y , we may assume that ( θ i ) Y = ζ i for 1 ≤ i ≤ q . Then15y the discussion in the previous paragraph, ( θ i ) Z = η i for 1 ≤ i ≤ q . It follows that thesecond order terms in L ′ are the same as those in (1 / P q + pi = q +1 θ i θ i and hence the coefficientmatrix of the second order terms in L ′ is nonnegative definite.Let k and m be respectively the Lie algebras of K and M , and let p be an Ad( M )-invariantsubspace of k complementary to m . Choose a basis ξ , . . . , ξ p of p . As discussed in section 3,Ad( M )-invariant vectors in k may be identified with K -invariant vector fields on Z = K/M ,and as a K -invariant second order differential operator, L ′ = (1 / P pi,j =1 a ij ξ i ξ j + ξ for anAd( M )-invariant matrix a ij (symmetric and nonnegative definite) and an Ad( M )-invariant ξ ∈ p , both may depend on y ∈ Y . The generator L may now be written as L = 12 q X i =1 ( ζ i + η i ) + 12 p X i,j =1 a ij ξ i ξ j + ζ + ξ = L ,Y + L Y Z + L Z + L Z (17)where L ,Y = (1 / P qi =1 ζ i + ζ is the radial part of L , η , . . . , η q , ξ are K -invariant vectorfields on Z which may depend on y ∈ Y , L Y Z = P qi =1 η i ζ i is the part of L containingmixed partials, L Z = (1 / P qi =1 η i + ξ and L Z = (1 / P pi,j =1 a ij ( y ) ξ i ξ j . The components L Y Z , L Z and L Z of operator L are uniquely determined. To show this, note that a differentchoice for ζ , . . . , ζ q leads to an orthogonal transform of these vector fields and the transposedtransform of η , . . . , η q . This implies that L Y Z and L Z , and hence L Z , are not changed. Wemay write L Z ( y ) for L Z to indicate its dependence on y ∈ Y and similarly for L Y Z and L Z .Let σ ij be the square root matrix of a ij , that is, a ij = P pk =1 σ ki σ kj . Then L Z =(1 / P pi =1 ( P pj =1 σ ij ξ j ) . The K -invariant vector fields η , . . . , η q , ξ on Z = K/M are theprojections of left invariant vector fields ˆ η , . . . , ˆ η q , ˆ ξ on K which are Ad( M )-invariant at e .Replacing η i and ξ by ˆ η i and ˆ ξ , then L given by (17) becomesˆ L = (1 / q X i =1 ( ζ i + ˆ η i ) + (1 / p X i =1 ( p X j =1 σ ij ξ j ) + ζ + ˆ ξ . This is a differential operator on Y ◦ × K and is the generator of the diffusion process ( y t , k t )that solves the following sde (stochastic differential equation) of Stratonovich form on Y ◦ × K . dy t = P qi =1 ζ i ( y t ) ◦ dB it + ζ ( y t ) dtdk t = P qi =1 ˆ η i ( y t , k t ) ◦ dB it + ˆ ξ ( y t , k t ) dt + P pi =1 [ P pj =1 σ ij ( y t ) ξ j ( k t )] ◦ dB q + it (18)with y = y and k = e , where B t = ( B t , . . . , B nt ) is an n -dim standard Brownian motion.Because, for f ∈ C ∞ c ( Y ◦ × Z ), ( Lf ) ◦ (id Y × π ) = ˆ L [ f ◦ (id Y × π )], where π : K → Z = K/M is the natural projection, it follows that x t = k t y t with radial part y t and angular part z t = k t o . The radial part y t is assumed to be nondegenerate in the sense that its generator L ,Y is nondegenerate on Y ◦ .Let k t = u t a t . Suppose u t and a t , with a = u = e , solve the following sde on K . da t = P qi =1 ˆ η i ( y t , a t ) ◦ dB it + ˆ ξ ( y t , a t ) dtdu t = P pi =1 { P pj =1 σ ij ( y t )[Ad( a t ) ξ j ]( u t ) } ◦ dB q + it , (19)16hen, writing kξ or ξk for left or right translations of a left invariant vector field ξ on K by k ∈ K , and noting kξ ( k ′ ) = kk ′ ξ ( e ) = ξ ( kk ′ ) and [Ad( k ) ξ ]( k ′ ) = k ′ kξ ( e ) k − , we have dk t = ( ◦ du t ) a t + u t ◦ da t = p X i =1 { p X j =1 σ ij ( y t )[Ad( a t ) ξ j ]( u t ) } a t ◦ dB q + it + q X i =1 u t ˆ η i ( y t , a t ) ◦ dB it + u t ˆ ξ ( y t , a t ) dt = p X i =1 { p X j =1 σ ij ( y t ) ξ j ( u t a t ) } ◦ dB q + it + q X i =1 ˆ η i ( y t , u t a t ) ◦ dB it + ˆ ξ ( y t , u t a t ) dt. Therefore, the second equation in (18) is equivalent to (19).Because ˆ η i and ˆ ξ are left invariant vector fields on K that are also right M -invariant, a t in the first equation of (19) may be replaced by either a t m or ma t for any m ∈ M . Theuniqueness of solution implies that a t m = ma t for all m ∈ M . Therefore, b t = a t o is an M -invariant continuous process in Z with b = o . Let v t = u t o . Then z t = u t a t o = v t b t .By the first equations in (18) and (19), and the nondegeneracy of L ,Y , the σ -algebrasgenerated by the radial process y t and by the q -dim Brownian motion ( B t , . . . , B qt ) are thesame and contain the σ -algebra generated by a t . Thus, given radial process y t , a t andhence b t become non-random. By the second equation in (19), the Itˆo formula, and theindependence between ( B t , . . . , B qt ) and ( B q +1 t , . . . , B nt ), for f ∈ C ∞ c ( K ), f ( u t ) − Z t (1 / p X i,j =1 a ij ( y s )[Ad( a s ) ξ i ][Ad( a s ) ξ j ] f ( u s ) ds (20)is a martingale given process y t . Then, replacing u t and a t by v t and b t , the above is still amartingale for f ∈ C ∞ c ( Z ) given process y t . Recall ζ is the exit time of the radial process from Y ◦ . The above shows that almost surely on [ ζ > T ], given y t for 0 ≤ t ≤ T , the conditionedangular process z t , as a continuous nonhomogeneous L´evy process in Z = K/M , has drift b t and covariance function A ij ( t ) = R t a ij ( y s ) ds under basis ξ , . . . , ξ p on p . Its covarianceoperator is thus R t L Z ( y s ) ds = R t ds P i,j a ij ( y s ) ξ i ξ j .Let G be a topological group acting on X continuously and containing K as a topologicalsubgroup. A measure µ on G is said to be K -conjugate invariant if c k µ = µ for k ∈ K , where c k is the conjugation map: G ∋ g → kgk − ∈ G .Now let x t be a process in X obtained from a K -invariant diffusion process x t in X interlaced with jumps determined by a Poisson random measure N on R + × G of intensitymeasure dtη ( dσ ), where η is a finite K -conjugate invariant measure on G . The process x t may be constructed as follows. Let γ n be positive random variables of a common exponentialdistribution of mean 1 /η ( G ) and let σ n be random variables in G of common distribution η/η ( G ), all independent of each other and of process x t , and let T n = γ + · · · + γ n with T = 0. Then the process x t is obtained by setting x t = x t for t < T and inductively setting x t = x nt for T n ≤ t < T n +1 , where x nt is x t determined by the initial condition x nT n = σ n ( x T n − ).17t is easy to see that x t defined above is a Markov process in X and its transitionsemigroup P t is the solution of the following integral equation: for f ∈ C ( X ), ∀ x ∈ X, P t f ( x ) = e − λt P t f ( x ) + Z t e − λu du Z G Z X P u ( x, dx ) P t − u f ( σx ) η ( dσ ) (21)with P f = f , where λ = η ( G ) and P t is the transition semigroup of x t (a Feller semigroup).The existence of the solution P t f and the Feller property of P t can be established by a routineargument of successive approximation, and the uniqueness of solution can be shown by asimple estimate on the difference between two possible solutions. By the uniqueness of P t ,the K -invariance of P t and the K -conjugate invariance of η , it is easy to show that P t is K -invariant. Thus, x t is a K -invariant Feller process in X . Moreover, by differentiating (21)at t = 0, we obtain its generator L on f ∈ C ∞ c ( X ): Lf ( x ) = L f ( x ) + Z G [ f ( σx ) − f ( x )] η ( dσ ) , (22)where L is the generator of the K -invariant diffusion process x t .We need some notation in order to describe the L´evy measure function of the conditionedangular process of x t . Recall J and J are the projection maps X → Y and X ◦ → Z = K/M .For a finite measure µ on X and y ∈ Y , let µ ( · | y ) be the conditional distribution of µ giventhe orbit Ky , which is the regular conditional distribution of a random variable x withdistribution µ/µ ( X ) given J x = y and is a probability measure supported by Ky . Note thatif µ ( Ky ) > 0, then µ ( · | y ) = µ ( · ; Ky ) /µ ( Ky ), where µ ( · ; Ky ) is the restriction of µ to Ky ,but even if µ ( Ky ) = 0, µ ( · | y ) is still defined for J µ -almost all y . If y ∈ Y ◦ , then µ ( · | y )is supported by X ◦ and hence J may be applied to obtain a probability measure J µ ( · | y )on Z . For x ∈ X , let i x be the evaluation map: G ∋ g gx ∈ X and for any measure η on G , let η ( x ) be the measure i x η on X , that is, η ( x )( f ) = η ( f ◦ i x ) for f ∈ C b ( X ). Note thatif η is K -conjugate invariant, then kη ( x ) = η ( kx ) for k ∈ K , and hence, η ( y ) is M -invariantfor y ∈ Y ◦ . It then follows that η ( y )( · | y ) is M -invariant and hence J η ( y )( · | y ) is an M -invariant probability measure on Z .The points in the Poisson random measure N are the pairs ( τ, σ ), where τ > σ ∈ G is the associated jump size of process x t in the sense that x τ = σx τ − . Underthe decomposition x = zy for x ∈ X ◦ with y = J ( x ) and z = J ( x ), z τ y τ = σz τ − y τ − = S ( z τ − ) σ ′ y τ − = S ( z τ − )[ J ( σ ′ y τ − )][ J ( σ ′ y τ − )] , (23)where σ ′ = S ( z τ − ) − σS ( z τ − ) and S is a section map on Z = K/M . Because σ is independentof x τ − and its distribution η/η ( G ) is K -conjugate invariant, σ ′ has the same distribution as σ and is independent of x τ − , and hence ( σ ′ y τ − ) has distribution η ( y τ − )( · ) /η ( y τ − )( X ). By (23)and the uniqueness of the decomposition x = zy , J ( σ ′ y τ − ) = y τ and S ( z τ − )[ J ( σ ′ y τ − )] = z τ .18hus, the jump of process z t at time τ is z − τ − z τ = S ( z τ − ) − z τ = J ( σ ′ y τ − ). Here the producton Z = K/M should be understood in the sense described in section 3,Note that almost surely, x t and hence y t have finitely many jumps for 0 ≤ t ≤ T . Given y t in Y ◦ for 0 ≤ t ≤ T with jumps at fixed times t < t < · · · < t k means three things:First, the Poisson random measure N is conditioned to have points ( t i , σ i ) for 1 ≤ i ≤ k with σ i satisfying J ( S ( z t − ) − σ i S ( z t − ) y t − ) = y t for t = t i ; secondly, at all other points ( τ, σ )of N , J ( S ( z τ − ) − σS ( z τ − ) y τ ) = y τ ; and finally, y t is given on each time interval [ t i − , t i ).Note that when N is conditioned to have points ( t i , σ i ) for 1 ≤ i ≤ k , its remaining pointsform a Poisson random measure N ′ of same distribution as N . On the time interval [ t i − , t i ),the conditioned angular process z t may be obtained from the diffusion process z t by addingjumps from N ′ . It follows that given y t in Y ◦ for 0 ≤ t ≤ T , the conditioned process z t , as a nonhomogeneous L´evy process in Z = K/M , has possible fixed jumps z − τ − z τ , at τ = t , t , . . . , t k , of distribution J η ( y τ − )( · | y τ ) (this measure may be equal to δ o andthen there is no fixed jump at τ ), its remaining jumps form a Poisson random measure N Z on [0 , T ] × Z with intensity measure µ ( dt, dz ) = dt [ J η ( y t )( · ; Ky t )]( dz ). The L´evy measurefunction of z t is Π( t, · ) = R t µ ( ds, · ). Its covariance operator is the same as that of z t obtainedearlier (but with y t replaced by y t ). To summarize, we have the following result. Theorem 5