aa r X i v : . [ m a t h . P R ] D ec Inhomogeneous L´evy processes in Lie groups andhomogeneous spaces
Ming Liao (Auburn University)
Summary
We obtain a representation of an inhomogeneous L´evy process ina Lie group or a homogeneous space in terms of a drift, a matrix function anda measure function. Because the stochastic continuity is not assumed, our re-sult generalizes the well known L´evy-Itˆo representation for stochastic continuousprocesses with independent increments in R d and its extension to Lie groups. Primary 60J25, Secondary 58J65.
Key words and phrases
L´evy processes, Lie groups, homogeneous spaces.
Let x t , t ∈ R + = [0 , ∞ ), be a process in R d with rcll paths (right continuous paths withleft limits). It is said to have independent increments if for s < t , x t − x s is independent of F xs (the σ -algebra generated by x u , 0 ≤ u ≤ s ). The process is called a L´evy process if italso has stationary increments, that is, if x t − x s has the same distribution as x t − s − x . Itis well known that the class of L´evy processes in R d coincides with the class of rcll Markovprocesses with translation-invariant transition functions P t . The celebrated L´evy -Khinchinformula provides a useful representation of a L´evy process in R d by a triple of a drift vector,a covariance matrix and a L´evy measure.More generally, a rcll process in R d with independent but possibly non-stationary incre-ments may be called an inhomogeneous L´evy process. It is easy to show that such processes,also called additive processes in literature, coincide with rcll inhomogeneous Markov pro-cesses with translation invariant two-parameter transition functions P s,t (see [14]).A L´evy process x t is always stochastically continuous, that is, x t = x t − a.s. (almostsurely) for any fixed t >
0. An inhomogeneous L´evy process x t may not be stochasticallycontinuous, but if it is, then the well known L´evy-Itˆo representation holds ([8, chapter 15]): x t = x + b t + B t + Z [0 , t ] Z | x |≤ x ˜ N ( ds, dx ) + Z [0 , t ] Z | x | > xN ( ds, dx ) , (1)where b t is a continuous path in R d with b = 0, called a drift, B t = ( B t , . . . , B dt ) is a d -dim continuous Gaussian process of zero mean and independent increments, N ( dt, dx ) isan independent Poisson random measure on R + × R d with intensity η = E ( N ) being a L´evymeasure function (to be defined later), and ˜ N = N − η is the compensated form of N .The distribution of the Gaussian process B t is determined by its covariance matrix func-tion A ij ( t ) = E ( B it B jt ) and the distribution of the Poisson random measure N is determined1y its intensity measure η . Thus, the distribution of a stochastically continuous inhomoge-neous L´evy process x t in R d is completely determined by the time-dependent triple ( b, A, η ).In general, an inhomogeneous L´evy process x t may not be a semimartingale (see [7, II]).By Itˆo’s formula, it can be shown that the process z t = x t − b t is a semi-martingale and forany smooth function f ( x ) on R d of compact support, f ( z t ) − X i,j Z t f ij ( z s ) dA ij ( s ) − Z t Z R d [ f ( z s + x ) − f ( z s ) − X i x i f i ( z s )1 [ | x |≤ ] η ( ds, dx ) (2)is a martingale, where f i and f ij denote respectively the first and second order partialderivatives of f . This in fact provides a complete characterization for the distribution of astochastically continuous inhomogeneous L´evy process x t in R d .This martingale representation is extended to stochastically continuous inhomogeneousL´evy process in a general Lie group G in [3], generalizing an earlier result in [13] for continuousprocesses. A different form of martingale representation in terms of the abstract Fourieranalysis is obtained in [5], where the processes considered are also stochastically continuous.A L´evy process x t in a Lie group G is defined as a rcll process with independent andstationary (multiplicative) increments, that is, for s < t , x − s x t is independent of F xs andhas the same distribution as x − x t − s . Such a process may also be characterized as a rcllMarkov process in G whose transition function P t is invariant under left translations on G .The classical triple representation of L´evy processes was extended to Lie groups in [6] in theform of a generator formula. A functional form of L´evy-Itˆo representation for L´evy processesin Lie groups was obtained in [1].An inhomogeneous L´evy process x t in a Lie group G is defined to be a rcll process thathas independent but not necessarily stationary increments, which may also be characterizedas an inhomogeneous rcll Markov process with a left invariant transition function P s,t (seeProposition 1). As mentioned earlier, the stochastically continuous inhomogeneous L´evyprocesses in G may be represented by a martingale determined by a triple ( b, A, η ).The notion of L´evy processes as invariant Markov processes, including inhomogeneousones, may be extended to more general homogeneous spaces, such as a sphere. A homo-geneous space G/K may be regarded as a manifold X under the transitive action of a Liegroup G with K being the subgroup that fixes a point in X . As on a Lie group G , a Markovprocess in X with a G -invariant transition function P t will be called a L´evy process, andan inhomogeneous Markov process with a G -invariant transition function P s,t will be calledan inhomogeneous L´evy process. Although there is no natural product on X = G/K , theincrements of a process in X may be properly defined, and a L´evy process in X may becharacterized by independent and stationary increments, with inhomogeneous ones just byindependent increments, same as on G . See section § G or a homogeneous space G/K . Wewill show that such a process is represented by a triple ( b, A, η ) with possibly discontinuous b t and η ( t, · ) = η ([0 , t ] × · ). This is in contrast to the stochastically continuous case whenthe triple is continuous in t . A non-stochastically-continuous process may have a fixed jump,that is, a time t > P ( x t = x t − ) >
0. It turns out to be quite non-trivial tohandle fixed jumps which may form a countable dense subset of R + .Our result applied to R n leads to a martingale representation of inhomogeneous L´evyprocesses in R n , for which the martingale (2) contains an extra term: − X u ≤ t Z R n [ f ( z u − + x − h u ) − f ( z u − )] ν u ( dx ) , where ν u is the distribution of the fixed jump at time u with mean h u . This complementsthe Fourier transform representation on R n obtained in [7, II.5].We will obtain the representation of inhomogeneous L´evy processes not only on a Liegroup G but also on a homogeneous space G/K . For this purpose, we will formulate aproduct structure and develop certain invariance properties on
G/K so that the formulasobtained on G , and their proofs, may be carried over to G/K . We will also show that aninhomogeneous L´evy process in
G/K is the natural projection of an inhomogeneous L´evyprocess in G . On an irreducible G/K , such as a sphere, the representation takes a verysimple form: there is no drift and the covariance matrix function A ( t ) = a ( t ) I for somefunction a ( t ). This is even simpler than the representation on R n .Our interest in non-stochastically-continuous inhomogeneous L´evy processes lies in thefollowing application. Let x t be a Markov process in a manifold X invariant under theaction of a Lie group G . It is shown in [10] that x t may be decomposed into a radial part y t ,transversal to G -orbits, and an angular part z t , in a standard G -orbit Z . The y t can be anarbitrary Markov process in a transversal subspace, whereas given y t , the conditioned z t is aninhomogeneous L´evy process in the homogeneous space Z = G/K . For example, a Markovprocess in R n invariant under the group O ( n ) of orthogonal transformations is decomposedinto a radial Markov process in a half line and an angular process in the unit sphere.In [10], the representation of stochastically continuous inhomogeneous L´evy processes in G/K is used to obtain a skew product decomposition of a G -invariant continuous Markovprocess x t in which the angular part z t is a time changed Brownian motion in G/K , gener-alizing the well known skew product of Brownian motion in R n .When the G -invariant Markov process x t is discontinuous, its conditioned angular part z t is typically not stochastically continuous. For example, a discontinuous O ( n )-invariantL´evy process in R n is stochastically continuous, but its conditioned angular part is not. Thepresent result will provide a useful tool in this situation.We note that an important related problem, the weak convergence of convolution prod-3cts of probability measures to a two-parameter convolution semigroup, which is the distri-bution of an inhomogeneous L´evy process, is not pursued in this paper. The stochasticallycontinuous case is studied in [12] and [11].Our paper is organized as follows. The main results on Lie groups are stated in the nextsection with proofs given in the four sections to follow. In §
3, we establish the martingalerepresentation, associated to a triple ( b, A, η ), for a given inhomogeneous L´evy process, undertwo technical assumptions (A) and (B). These assumptions are verified in §
4. We then provethe uniqueness of the triple for a given process in §
5, and the uniqueness and the existence ofthe process for a given triple in §
6. The results on homogeneous spaces are presented in § R + = [0 , ∞ ), butit is clear that the results also hold on a finite time interval. For a manifold X , let B ( X )be the Borel σ -field on X and let B + ( X ) be the space of nonnegative Borel functions on X .Let C ( X ), C b ( X ) and C ∞ c ( X ) be respectively the spaces of continuous functions, boundedcontinuous functions, and smooth functions with compact supports on X . Let x t be an inhomogeneous L´evy process in a Lie group G . By definition, x t is a rcll processin G with independent increments, that is, x − s x t is independent of F xs for s < t . It becomesa L´evy process in G if it also has stationary increments, that is, if the distribution of x − s x t depends only on t − s . Let µ s,t be the distribution of x − s x t . Then for f ∈ B + ( G ), E [ f ( x t ) | F xs ] = E [ f ( x s x − s x t ) | F xs ] = Z G f ( x s y ) µ s,t ( dy ) . This shows that x t is an inhomogeneous Markov process with transition function P s,t givenby P s,t f ( x ) = R f ( xy ) µ s,t ( dy ).Note that x t is left invariant in the sense that its transition function P s,t is left invariant,that is, P s,t ( f ◦ l g ) = ( P s,t f ) ◦ l g for f ∈ B + ( G ) and g ∈ G , where l g is the left translation x gx on G . Conversely, if x t is a left invariant inhomogeneous Markov process in G , then E [ f ( x − s x t ) | F xs ] = P s,t ( f ◦ l x − s )( x s ) = P s,t f ( e ), where e is the identity element of G . Thisimplies that x t has independent increments. We have proved the following result. Proposition 1
A rcll process x t in G is an inhomogeneous L´evy process if and only if it isa left invariant inhomogeneous Markov process. G is a L´evy process if and only it is a Markov process with a left invariant transitionfunction P t (see also [9, Proposition 1.2]).A measure function on G is a family of σ -finite measures η ( t, · ) on G , t ∈ R + , such that η ( s, · ) ≤ η ( t, · ) for s < t , and η ( t, · ) ↓ η ( s, · ) as t ↓ s ≥
0. Here the limit is set-wise, that is, η ( t, B ) → η ( s, B ) for B ∈ B ( G ). The left limit η ( t − , · ) at t >
0, defined as the nondecreasinglimit of measures η ( s, · ) as s ↑ t , exists and is ≤ η ( t, · ).A measure function η ( t, · ) may be regarded as a σ -finite measure on R + × G and maybe written as η ( dt, dx ), given by η (( s, t ] × B ) = η ( t, B ) − η ( s, B ) for s < t and B ∈ B ( G ).Conversely, any measure η on R + × G such that η ( t, · ) = η ([0 , t ] × · ) is a σ -finite measureon G for any t > η ( t, · ).A measure function η ( t, · ) is called continuous at t > η ( t, · ) = η ( t − , · ), and continuousif it is continuous at all t >
0. In general, the set J = { t > η ( t, G ) > η ( t − , G ) } , ofdiscontinuity times, is at most countable, and η ( t, · ) = η c ( t, · ) + X s ≤ t, s ∈ J η ( { s } × · ) , (3)where η c ( t, · ) = R [0 , t ] ∩ J c η ( ds, · ) is a continuous measure function, called the continuous partof η ( t, · ), and η ( { s } × · ) = η ( s, · ) − η ( s − , · )Recall e is the identity element of G . The jump intensity measure of an inhomogeneousL´evy process x t is the measure function η ( t, · ) on G defined by η ( t, B ) = E { { s ∈ (0 , t ]; x − s − x s ∈ B and x − s − x s = e }} , B ∈ B ( G ) , (4)the expected number of jumps in B by time t . The required σ -finiteness of η ( t, · ) will be clearfrom Proposition 7 later, and then the required right continuity, η ( t, · ) ↓ η ( s, · ) as t ↓ s ≥ x t is continuous if and only if η = 0.Note that η (0 , · ) = 0 and η ( t, { e } ) = 0, and for t > η ( { t } × G ) ≤ ν t = η ( { t } × · ) + [1 − η ( { t } × G )] δ e (where δ e is the unit mass at e ) (5)is the distribution of x − t − x t , so η ( t, · ) is continuous if and only if x t is stochastically continuous.Let { ξ , . . . , ξ d } be a basis of the Lie algebra g of G . We will write e ξ for the exponentialmap exp( ξ ) on G . There are φ , . . . , φ d ∈ C ∞ c ( G ) such that x = e P di =1 φ i ( x ) ξ i for x near e ,called coordinate functions associated to the basis { ξ i } of g . Note that ξ i φ j ( e ) = δ ij . The φ · -truncated mean, or simply the mean, of a G -valued random variable x or its distribution µ is defined to be b = e P dj =1 µ ( φ j ) ξ j (where µ ( φ j ) = R φ j dµ ). (6)The distribution µ is called small if its mean b has coordinates µ ( φ ) , . . . , µ ( φ d ), that is, φ j ( b ) = µ ( φ j ) , ≤ j ≤ d. (7)5his is the case when µ is sufficiently concentrated near e .As defined in [3], a L´evy measure function on G is a continuous measure function η ( t, · )such that(a) η (0 , · ) = 0, η ( t, { e } ) = 0 and η ( t, U c ) < ∞ for any t ∈ R + and neighborhood U of e .(b) η ( t, k φ · k ) < ∞ for any t ≥
0, where k φ · k = P di =1 φ i ( x ) .The notion of L´evy measure functions is now extended. A measure function η ( t, · ) on G is called an extended L´evy measure function if (a) above and (b ′ ) below hold.(b ′ ) For t ≥ η c ( t, k φ · k ) < ∞ , η ( { t } × G ) ≤ ν t given by (5), X s ≤ t ν s ( k φ · − φ · ( h s ) k ) < ∞ , (8)where η c ( t, · ) is the continuous part of η ( t, · ) and h t = exp[ P j ν t ( φ j ) ξ j ] is the mean of ν t .Note that ν t = δ e and h t = e at a continuous point t of η ( t, · ), and hence the sum P s ≤ t in (8) has at most countably many nonzero terms. If η ( t, · ) is continuous, then (b ′ ) becomes(b), and hence a continuous extended L´evy measure function is a L´evy measure function.It can be shown directly that conditions (b) and (b ′ ) are independent of the choice forcoordinate functions φ i and basis { ξ i } . This is also a consequence of Theorem 2 below.A continuous path b t in G with b = e will be called a drift. A d × d symmetric matrix-valued function A ( t ) = { A jk ( t ) } will be called a covariance matrix function if A (0) = 0, A ( t ) − A ( s ) ≥ s < t , and t A jk ( t ) is continuous. A triple( b, A, η ) of a drift b t , a covariance matrix function A ( t ) and a L´evy measure function η ( t, · ) willbe called a L´evy triple on G . For g ∈ G , the adjoint map Ad( g ): g → g is the differentialof the conjugation x gxg − at x = e . It is shown in [3] that if x t is a stochasticallycontinuous inhomogeneous L´evy process in G with x = e , then there is a unique L´evy triple( b, A, η ) such that x t = z t b t and ∀ f ∈ C ∞ c ( G ) , f ( z t ) − Z t d X j,k =1 [Ad( b s ) ξ j ][Ad( b s ) ξ k ] f ( z s ) dA jk ( s ) − Z t Z G { f ( z s b s τ b − s ) − f ( z s ) − d X j =1 φ j ( τ )[Ad( b s ) ξ j ] f ( z s ) } η ( ds, dτ ) (9)is a martingale under F xt . Conversely, given a L´evy triple ( b, A, η ), there is a stochasticallycontinuous inhomogeneous L´evy process x t with x = e represented as above, unique indistribution.Therefore, a stochastically continuous inhomogeneous L´evy process in G is representedby a triple ( b, A, η ) just like its counterpart in R d . The complicated form of the martingale6n (9) with the presence of the drift b t , as compared with its counterpart (2) on R d , is causedby the non-commutativity of G . This representation is extended to all inhomogeneous L´evyprocesses in G , not necessarily stochastically continuous, in Theorem 2 below.An extended drift on G is a rcll path b t in G with b = e . A triple ( b, A, η ) of an extendeddrift b t , a covariance matrix function A ( t ) and an extended L´evy measure function η ( t, · )will be called an extended L´evy triple on G if b − t − b t = h t for any t > G = R d , our definition of an extended L´evy triple corresponds to theassumptions in [7, Theorem II 5.2]. In particular, (a) and (b ′ ) corresponds to (i) - (iii), and b − t − b t = h t to (v), but (iv) in [7] is redundant as it is implied by the other conditions.A rcll process x t in G is said to be represented by an extended L´evy triple ( b, A, η ) ifwith x t = z t b t , M t f = f ( z t ) − Z t d X j,k =1 [Ad( b s ) ξ j ][Ad( b s ) ξ k ] f ( z s ) dA jk ( s ) − Z t Z G { f ( z s b s xb − s ) − f ( z s ) − d X j =1 φ j ( x )[Ad( b s ) ξ j ] f ( z s ) } η c ( ds, dx ) − X u ≤ t Z G [ f ( z u − b u − xh − u b − u − ) − f ( z u − )] ν u ( dx ) (10)is a martingale under the natural filtration F xt of x t for any f ∈ C ∞ c ( G ).We note that in (10), the η c -integral is absolutely integrable and the sum P u ≤ t con-verges absolutely a.s., and hence M t f is a bounded random variable. This may be veri-fied by (b ′ ) and Taylor’s expansions of f ( z s b s xb − s ) = f ( z s b s e P j φ j ( x ) ξ j b − s ) at x = e and f ( z u − b u − xh − u b − u − ) at x = h u (see the computation in the proof of Lemma 15). Theorem 2
Let x t be an inhomogeneous L´evy process in G with x = e . Then there is aunique extended L´evy triple ( b, A, η ) on G such that x t is represented by ( b, A, η ) as definedabove. Moreover, η ( t, · ) is the jump intensity measure of process x t given by (4). Conse-quently, x t is stochastically continuous if and only if ( b, A, η ) is a L´evy triple.Conversely, given an extended L´evy triple ( b, A, η ) on G , there is an inhomogeneous L´evyprocess x t in G with x = e , unique in distribution, that is represented by ( b, A, η ) . Remark 1
As the jump intensity measure, η ( t, · ) is clearly independent of the choice forthe basis { ξ j } of g and coordinate functions φ j . By Lemma 11, A ( t ) is independent of { φ j } and the operator P dj,k =1 A jk ( t ) ξ j ξ k is independent of { ξ j } .In Theorem 2, the representation of process x t is given in the form of a martingaleproperty of the shifted process z t = x t b − t . By Theorem 3 below, when the drift b t has afinite variation, a form of martingale property holds directly for x t .7 rcll path b t in a manifold X is said to have a finite variation if for any f ∈ C ∞ c ( X ), f ( b t ) has a finite variation on any finite t -interval. Let ξ , . . . , ξ d be a family of smooth vectorfields on X such that ξ ( x ) , . . . , ξ d ( x ) form a basis of the tangent space T x X of X at any x .If b t is a continuous path in X with a finite variation, then there are uniquely defined realvalued continuous functions b j ( t ) of finite variation, 1 ≤ j ≤ d , with b j (0) = 0, such that ∀ f ∈ C ∞ c ( X ) , f ( b t ) = f ( b ) + Z t X j ξ j f ( b s ) db j ( s ) . (11)Indeed, ξ j ( x ) = P k α jk ( x )( ∂/∂ψ k ) under local coordinates ψ , . . . , ψ d on G , and df ( b t ) = P j ( ∂/∂ψ j ) f ( b t ) dψ j ( b t ) = P j,k β jk ( b t ) ξ k f ( b t ) dψ j ( b t ), where { β jk ( x ) } = { α jk ( x ) } − , then b j ( t )are determined by db j ( t ) = P k β kj ( b t ) dψ k ( b t ). Conversely, given b j ( t ) as above, a continuouspath b t of finite variation satisfying (11) may be obtained by solving the integral equation ψ j ( b t ) − ψ j ( b t ) = X k Z tt α kj ( b s ) db k ( s )for ψ j ( b t ) by the usual successive approximation method.The functions b j ( t ) above will be called components of the path b t under the vector fields ξ , . . . , ξ d . When X = G , these vector fields will be the basis of g chosen before.More generally, if b t is a rcll path in X of finite variation, then there is a unique continuouspath b ct in X of finite variation with b c = b such that, letting b j ( t ) be the components of b ct , ∀ f ∈ C ∞ c ( X ) , f ( b t ) = f ( b ) + Z t X j ξ j f ( b s ) db j ( s ) + X s ≤ t [ f ( b s ) − f ( b s − )] . (12)To prove this, cover the path by finitely many coordinate neighborhoods and then prove theclaim on a Euclidean space. The path b ct will be called the continuous part of b t . Theorem 3
Let x t be an inhomogeneous L´evy process in G with x = e , represented by anextended L´evy triple ( b, A, η ) . Assume b t is of finite variation. Then f ( x t ) − Z t X j ξ j f ( x s ) db j ( s ) − Z t X j,k ξ j ξ k f ( x s ) dA jk ( s ) − Z t Z G { f ( x s τ ) − f ( x s ) − X j φ j ( τ ) ξ j f ( x s ) } η c ( ds, dτ ) − X u ≤ t Z G [ f ( x u − τ ) − f ( x u − )] ν u ( dτ ) (13) is a martingale under F xt for any f ∈ C ∞ c ( G ) .Conversely, given an extended L´evy triple ( b, A, η ) with b t of finite variation, there is aninhomogeneous L´evy process x t in G with x = e , unique in distribution, such that (13) is amartingale under F xt for f ∈ C ∞ c ( G ) . b t has a finite variation and φ i ( b − u b u ) = ν u ( φ i ) for all but finitely many u ≤ t , P u ≤ t | ν u ( φ i ) | < ∞ and hence P u ≤ t | R G [ f ( x u − τ ) − f ( x u − )] ν u ( dτ ) | < ∞ . The absolute inte-grability of the η c -integral in (13) can be verified as in (10). Lemma 4
Let ( b, A, η ) be an extended L´evy triple on G with b t of finite variation. For aninhomogeneous L´evy process x t = z t b t in G with x = e , (10) being a martingale under F xt for all f ∈ C ∞ c ( G ) is equivalent to (13) being a martingale under F xt for all f ∈ C ∞ c ( G ) . Proof
Let us assume (10) is a martingale. Let ∆ n : 0 = t n < t n < · · · < t ni ↑ ∞ (as i ↑ ∞ )be a sequence of partitions of R + with mesh k ∆ n k = sup i ≥ ( t ni − t n i − ) → n → ∞ , andlet f ni ( z ) = f ( zb t ni ). Then f ni ( z t ) = f ni ( e ) + M nit + Z [0 , t ] L ( s, ds ) f ni , where M nit is a martingale with M ni = 0 and L ( t, dt ) f = 12 X j,k [Ad( b t ) ξ j ][Ad( b t ) ξ k ] f ( z t ) dA jk ( t ) + Z G { f ( z t b t τ b − t ) − f ( z t ) − X j φ j ( τ )[Ad( b t ) ξ j ] f ( z t ) } η c ( dt, dτ ) + X s ∈ dt Z G [ f ( z s − b s − τ h − s b − s − ) − f ( z s − )] ν s ( dτ ) . Let b nt = b t ni for t ∈ [ t ni , t n i +1 ). Let J be the set of fixed jump times of x t . We may assume J ⊂ ∆ n as n → ∞ in the sense that ∀ u ∈ J , u ∈ ∆ n for large n . Then b nt is a step functionand b nt → b t as n → ∞ uniformly for bounded t . It follows that for t ∈ [ t ni , t n i +1 ), f ( z t b nt ) = f ( z t b t ni ) = f ( e ) + i X j =1 [ f ( z t nj b t nj ) − f ( z t n j − b t n j − )] + [ f ( z t b t ni ) − f ( z t ni b t ni )]= f ( e ) + i X j =1 [ f ( z t nj b t n j − ) − f ( z t n j − b t n j − )] + [ f ( z t b t ni ) − f ( z t ni b t ni )]+ i X j =1 [ f ( z t nj b t nj ) − f ( z t nj b t n j − )]= f ( e ) + i X j =1 [ M n j − t nj − M n j − t n j − ] + i X j =1 Z ( t n j − , t nj ] L ( s, ds ) f n j − + [ M nit − M nit ni ]+ Z ( t ni , t ] L ( s, ds ) f ni + i X j =1 [ f ( z t nj b t nj ) − f ( z t nj b t n j − )]= f ( e ) + M ( n ) t + Z [0 , t ] L ( s, ds )( f ◦ r b ns − ) + i X j =1 [ f ( z t nj b t nj ) − f ( z t nj b t n j − )] , M ( n ) t = P ij =1 [ M n j − t nj − M n j − t n j − ] + [ M nit − M nit ni ], t ∈ [ t ni , t n i +1 ), is a martingale, and r b is the right translation x xb on G . As n → ∞ , by the uniform convergence b nt → b t , f ( z t b nt ) → f ( z t b t ) = f ( x t ), and by the continuity of A ( t ) and η c ( t, · ), Z [0 , t ] L ( s, ds )( f ◦ r b ns − ) → Z [0 , t ] L ( s, ds )( f ◦ r b s − ) = 12 Z t X p,q ξ p ξ q f ( x s ) dA pq ( s ) + Z t Z G [ f ( x s τ ) − f ( x s ) − X p φ p ( τ ) ξ p f ( x s )] η c ( ds, dτ ) + X s ≤ t Z G [ f ( x s − τ h − s ) − f ( x s − )] ν s ( dτ )and, by (12) and noting that b p ( t ) is continuous in t and J ⊂ ∆ n as n → ∞ , i X j =1 [ f ( z t nj b t nj ) − f ( z t nj b t n j − )] = i X j =1 { Z t nj t n j − X p ξ p f ( z t nj b s ) db p ( s ) + X t n j −
0, let J m = { u , u , . . . , u m } . 10or i = 1 , , , . . . , let x ni = x − t n i − x t ni . Then for each fixed n , x ni are independent randomvariables in G . Let µ ni be their distributions, and define the measure function η n ( t, · ) = X t ni ≤ t µ ni (setting η n (0 , · ) = 0) . (15) Proposition 5
For any
T > and any neighborhood U of e , η n ( T, U c ) is bounded in n and η n ( T, U c ) ↓ as U ↑ G uniformly in n . Proof
We first establish an equi-continuity type property for η n ( t, · ). Lemma 6
For any
T > , neighborhood U of e and ε > , there are integers n and m > ,and δ > , such that if n ≥ n and s, t ∈ ∆ n ∩ [0 , T ] with < t − s < δ and ( s, t ] ∩ J m = ∅ (the empty set), then η n ( t, U c ) − η n ( s, U c ) < ε . By Borel-Cantelli Lemma, the independent increments and rcll paths imply that for anyneighborhood U of e , X u ≤ T, u ∈ J P ( x − u − x u ∈ U c ) < ∞ . (16)Suppose the claim of the lemma is not true. Then for some ε >
0, and for any n , m and δ >
0, there are s n , t n ∈ ∆ n ∩ [0 , T ] with n ≥ n , t n − s n < δ and ( s n , t n ] ∩ J m = ∅ suchthat η n ( t n , U c ) − η n ( s n , U c ) ≥ ε . Letting n → ∞ and δ → n → ∞ such that s n and t n converge to a common limit t ≤ T as n → ∞ .Let s n = t ni and t n = t nj . In the following computation, we will write A n ≈ B n if thereis a constant c > /c ) A n ≤ B n ≤ cA n . Then η n ( t n , U c ) − η n ( s n , U c ) = j X p = i +1 P ( x − t n p − x t np ∈ U c ) ≈ − j X p = i +1 log[1 − P ( x − t n p − x t np ∈ U c )]= − log P [ ∩ jp = i +1 ( x − t n p − x t np ∈ U )] = − log P ( A n ) , where A n = ∩ jp = i +1 ( x − t n p − x t np ∈ U ). Because η n ( t n , U c ) − η n ( s n , U c ) ≥ ε , P ( A n ) ≤ − ε for some constant ε >
0. Then P ( A cn ) > ε . Note that on A cn , the process x t makes a U c -oscillation during the time interval [ s n , t n ].There are three possible cases and we will reach a contradiction in all these cases. Caseone, there are infinitely many s n ↓ t . This is impossible by the right continuity of paths attime t . Case two, there are infinitely many t n ↑ t . This is impossible by the existence ofpath left limit at time t . Case three, there are infinitely many s n < t ≤ t n . This implies P ( x − t − x t ∈ U c ) ≥ ε . Then t ∈ J . Because ( s n , t n ] ∩ J m = ∅ , t J m . By (16), m maybe chosen so that P u ≤ T, u ∈ J − J m P ( x − u − x u ∈ U c ) < ε , which is impossible. Lemma 6 is nowproved. 11o prove Proposition 5, fix ε > n , m, δ be as in Lemma 6. It suffices to provefor n ≥ n . For each n ≥ n , J m may be covered by no more than m sub-intervals of theform [ t n i − , t ni ], and the rest of the interval [0 , T ] may be covered by finitely many, say p ,sub-intervals [ s, t ] as in Lemma 6. Then η n ( T, U c ) ≤ pε + m , and hence η n ( t, U c ) is bounded.Now let V be a neighborhood of e such that V − V = { x − y ; x, y ∈ V } ⊂ U . Then η n ( T, U c ) = X t ni ≤ T P ( x − t n i − x t ni ∈ U c ) ≈ − X t ni ≤ T log[1 − P ( x − t n i − x t ni ∈ U c )]= − log P {∩ t ni ≤ T [ x − t n i − x t ni ∈ U ] } ≤ − log P {∩ t ni ≤ T [ x t n i − ∈ V and x t ni ∈ V ] }≤ − log P {∩ t ≤ T [ x t ∈ V ] } ↓ V ↑ G uniformly in n because P {∩ t ≤ T [ x t ∈ V ] } ↑ ✷ Let η ( t, · ) be the jump intensity measure of process x t defined by (4). Proposition 7
For any t > and f ∈ C b ( G ) vanishing in a neighborhood of e , η n ( t, f ) → η ( t, f ) . Moreover, η ( t, U c ) < ∞ for any neighborhood U of e . Proof
We may assume f ≥
0. Let F = P u ≤ t f ( x − u − x u ) and F n = P pi =1 f ( x ni ), where p isthe largest index i such that t ni ≤ t . Then η ( t, f ) = E ( F ) and η n ( t, f ) = E ( F n ). Because∆ n contains J as n → ∞ , F n → F a.s. as n → ∞ . By the independence of x n , x n , x n , . . . , E ( F n ) = E { [ p X i =1 f ( x ni )] } = p X i =1 µ ni ( f ) + X i = j µ ni ( f ) µ nj ( f )= p X i =1 µ ni ( f ) + [ p X i =1 µ ni ( f )] − p X i =1 µ ni ( f ) ≤ η n ( t, f ) + [ η n ( t, f )] , which is bounded by Proposition 5. Therefore, E ( F n ) are uniformly bounded in n , andhence F n are uniformly integrable. It follows that η n ( t, f ) = E ( F n ) → E ( F ) = η ( t, f ). Thefiniteness of η ( t, U c ) now follows from Proposition 5. ✷ Define η m ( t, B ) = E [ { s ≤ t ; x − s − x s ∈ B, x − s − x s = e and s J m } ] . (17)Then η ( t, · ) ≥ η m ( t, · ) ↓ η c ( t, · ) as m ↑ ∞ . Let ∆ mn be the subset of ∆ n consisting of t ni suchthat u ∈ ( t n i − , t ni ] for some u ∈ J m , and let η mn ( t, · ) = P t ni ≤ t, t ni ∆ mn µ ni . Lemma 8
Fix m > . Then as n → ∞ , η mn ( t, f ) → η m ( t, f ) for any t > and f ∈ C b ( G ) vanishing in a neighborhood of e . roof Let F ′ = P s ≤ t, s J m f ( x − s − x s ) and F ′ n = P t ni ≤ t, t ni ∆ mn f ( x ni ). Then η m ( t, f ) = E ( F ′ )and η mn ( t, f ) = E ( F ′ n ). Because ∆ n contains J as n → ∞ , F ′ n → F ′ a.s. as n → ∞ . Onemay assume f ≥
0. Because F ′ n ≤ F n in the notation of the proof of Proposition 7, so F ′ n are uniformly integrable and hence η mn ( t, f ) = E ( F ′ n ) → E ( F ′ ) = η m ( t, F ). ✷ The notion of measure functions is now extended to matrix-valued measures. Thusa family of d × d symmetric matrix valued functions A ( t, B ) = { A jk ( t, B ) } j,k =1 , ,...,d , for t ∈ R + and B ∈ B ( G ), is called a matrix-valued measure function on G if A jk ( t, · ) is a finitesigned measure on G , A (0 , B ) and A ( t, B ) − A ( s, B ) are nonnegative definte for s < t , and A jk ( t, B ) → A jk ( s, B ) as t ↓ s .The trace of a matrix-valued measure function A ( t, · ), q ( t, · ) = Trace[ A ( t, · )], is a finitemeasure function such that for s < t and j, k = 1 , , . . . , d , | A jk ( t, · ) − A jk ( s, · ) | ≤ q ( t, · ) − q ( s, · ) = q (( s, t ] × · ) . (18)Let A n ( t, · ) be the matrix-valued measure functions on G defined by A njk ( t, B ) = X t ni ≤ t Z B [ φ j ( x ) − φ j ( b ni )][ φ k ( x ) − φ k ( b ni )] µ ni ( dx ) , (19)where b ni = e P j µ ni ( φ j ) ξ j is the mean of x ni . Its trace is q n ( t, B ) = X t ni ≤ t Z B k φ · ( x ) − φ · ( b ni ) k µ ni ( dx ) , (20)where φ · ( x ) = ( φ ( x ) , . . . , φ d ( x )) and k · k is the Euclidean norm on R d .In the rest of this section, let η ( t, · ) be the jump intensity measure of an inhomogeneousL´evy process x t in G , defined by (4), with J being the set of its discontinuity times. Let ν t be defined by (5) with mean h t . Then ν t and h t are respectively the distribution and themean of x − t − x t , which are nontrivial only when t ∈ J . We will see that η ( t, · ) is an extendedL´evy measure function. This fact is now proved below under an extra assumption. Proposition 9
Assume q n ( t, G ) is bounded in n for each t > . Then the jump intensitymeasure η ( t, · ) is an extended L´evy measure function, and hence its continuous part η c ( t, · ) is a L´evy measure function. Proof
Let c q be a constant such that q n ( t, G ) ≤ c q for all n . Any u ∈ J is contained in( t n i − , t ni ] for some i = i n and ν u ( k φ · − φ · ( h u ) k ) = lim n →∞ µ ni ( k φ · − φ · ( b ni ) k ). It followsthat P u ≤ t,u ∈ J ν u ( k φ · − φ · ( h u ) k ) ≤ c q . It remains to prove the finiteness of η c ( t, k φ · k ). Let r mn = max t ni ≤ t, t ni ∆ mn k φ · ( b ni ) k , (21)13here ∆ mn is defined after (17). Then r mn = max t ni ≤ t, t ni ∆ mn k µ ni ( φ · ) k for large m , andlim n →∞ r mn = sup u ≤ t, u J m k ν u ( φ · ) k → m → ∞ . For two neighborhoods U ⊂ V of e , η c ( t, k φ · k V c ) ≤ η m ( t, k φ · k V c ) ≤ lim n →∞ Z U c k φ · ( x ) k η mn ( t, dx ) (by Lemma 8) ≤ n →∞ X t ni ≤ t, t ni J m Z U c k φ · ( x ) − φ · ( b ni ) k µ ni ( dx ) + 2lim n →∞ ( r mn ) η n ( t, U c ) ≤ c q + 2(lim n →∞ ( r mn ) )(lim n →∞ η n ( t, U c )) . Now letting m → ∞ and then V ↓ { e } shows η c ( t, k φ · k ) ≤ c q . ✷ Now consider the following equi-continuity type condition on q n ( t, · ).(A) For any T > ε >
0, there are integers n , m > δ > n ≥ n and s, t ∈ [0 , T ] with 0 < t − s < δ and ( s, t ] ∩ J m = ∅ , then q n ( t, G ) − q n ( s, G ) < ε . Lemma 10
Under (A), q n ( T, G ) is bounded in n , and q n ( T, U c ) ↓ uniformly in n as U ↑ G , where U is a neighborhood of e . Proof
The boundedness of q n ( T, G ) is derived from (A) in the same way as the boundednessof η n ( T, U c ) in Proposition 5 is derived from Lemma 6. Because the convergence η n ( T, U c ) ↓ U ↑ G , is uniform in n , so is q n ( T, U c ) ↓ ✷ Lemma 11
Assume (A). Then there is a matrix valued measure function A ( t, · ) on G suchthat along a subsequence of n → ∞ , A n ( t, f ) → A ( t, f ) for all t > and f ∈ C b ( G ) .Moreover, there is a covariance matrix function A ( t ) on G such that A jk ( t, f ) = f ( e ) A jk ( t ) + Z G f ( x ) φ j ( x ) φ k ( x ) η c ( t, dx )+ X u ≤ t, u ∈ J Z G f ( x )[ φ j ( x ) − φ j ( h u )][ φ k ( x ) − φ k ( h u )] ν u ( dx ) . (22) Furthermore, A ( t ) is independent of the choice for coordinate functions φ j and the operator P dj,k =1 A jk ( t ) ξ j ξ k is independent of the choice for the basis { ξ j } of g . Proof
Let Λ be a countable dense subset of [0 , T ] containing J ∩ [0 , T ] and let H be acountable subset of C b ( G ). By the boundedness of q n ( T, G ) and (18), along a subsequenceof n → ∞ , A n ( t, f ) converges for all t ∈ Λ and f ∈ H . Let K n be an increasing sequenceof compact subsets of G such that K n ↑ G . For each K n , there is a countable subset H n of C b ( K n ) that is dense in C ( K n ) under the supremum norm. We will extend functions in H n to be functions on G without increasing their suprenorms and let H = ∪ n H n .Because for any relatively compact neighborhood U of e , q n ( T, U c ) ↓ U ↑ G uniformlyin n , it follows that A n ( t, f ) converges for any f ∈ C b ( G ). By (A), A n ( t, f ) converges to14ome A ( t, f ) for any t ∈ [0 , T ] and f ∈ C b ( G ), and the convergence is uniform in t and f bounded by a fixed constant. Because A n ( t, f ) is right continuous with left limits in t , sois A ( t, f ) and hence A ( t, · ) is a matrix-valued measure function. Moreover, the jumps of A ( t, f ) are R G f ( x )[ φ j ( x ) − φ j ( h t )][ φ k ( x ) − φ k ( h t )] ν t ( dx ) at t ∈ J .Let the sum P t ni ≤ t in (19), which defines A n ( t, · ), be broken into two partial sums: P t ni ≤ t, t ni ∆ mn and P t ni ≤ t, t ni ∈ ∆ mn , and write A n ( t, · ) = A n,m ( t, · )+ B n,m ( t, · ), where A n,m ( t, · ) = P t ni ≤ t, t ni ∆ mn and B n,m ( t, · ) = P t ni ≤ t, t ni ∈ ∆ mn . Then for f ∈ C b ( G ), as n → ∞ , B n,mjk ( t, f ) → X s ≤ t, s ∈ J m Z G f ( x )[ φ j ( x ) − φ j ( h t )][ φ k ( x ) − φ k ( h t )] ν s ( dx ) , (23)and hence lim n →∞ A n,m ( t, f ) also exists. Note that the sum P s ≤ t, s ∈ J m in (23) contains thejumps of A ( s, f ) at s ∈ [0 , t ] ∩ J m and it converges to B ′ jk ( t, f ) = X s ≤ t, s ∈ J Z G f ( x )[ φ j ( x ) − φ j ( h t )][ φ k ( x ) − φ k ( h t )] ν s ( dx )as m → ∞ , and B ′ ( t, · ) is a matrix valued measure function. It follows that as m → ∞ ,lim n →∞ A n,m ( t, f ) converges to a matrix valued measure function A ′ ( t, f ) = A ( t, f ) − B ′ ( t, f ).Let ψ ∈ C ∞ c ( G ) with ψ = 1 near e and 0 ≤ ψ ≤ G . Because η n ( t, − ψ ) is boundedin n , and lim n →∞ r mn → m → ∞ for r mn defined by (21), by Lemma 8, A n,mjk ( t, (1 − ψ ) f ) = Z [1 − ψ ( x )] f ( x ) φ j ( x ) φ k ( x ) η c ( t, dx ) + r ′ nm (24)with lim n →∞ r ′ nm → m → ∞ . Then A ′ jk ( t, (1 − ψ ) f ) = Z [1 − ψ ( x )] f ( x ) φ j ( x ) φ k ( x ) η c ( t, dx ) . Let ψ = ψ p ↓ { e } with supp( ψ p ) ↓ { e } as p ↑ ∞ , and define A ( t ) = lim p →∞ A ′ ( t, ψ p ).Then lim p →∞ A ′ ( t, f ψ p ) = f ( e ) A ( t ). Because A ( t, f ) = A ′ ( t, f ) + B ′ ( t, f ) and A ′ ( t, f ) = A ′ ( t, ψ p f ) + A ′ ( t, (1 − ψ p ) f )), letting p → ∞ yields (22). Because the jumps of A ( t, f ) areaccounted for by the sum P u ≤ t, u ∈ J in (22), it follows that A ( t ) is continuous and hence is acovariance matrix function.Let { ˜ φ j } be another set of coordinate functions associated to the same basis { ξ j } of g .Then ˜ φ j = φ j in a neighborhood V of e . Let ˜ b ni , ˜ A n,m ( t, · ) and ˜ A ( t ) be b ni , A n,m ( t, · ) and A ( t ) for ˜ φ i . Because for t ni ≤ t with t ni ∆ mn and a large m , µ ni is small in the sense definedbefore (7), φ j ( b ni ) = µ ni ( φ j ) and ˜ φ j (˜ b ni ) = µ ni ( ˜ φ j ). Then for f ∈ C b ( G ) vanishing on V c ,˜ A n,mj,k ( t, f ) = X t ni ≤ t, t ni ∆ mn Z G Z G Z G f ( x )[ ˜ φ j ( x ) − ˜ φ j ( y )][ ˜ φ k ( x ) − ˜ φ k ( z )] µ ni ( dx ) µ ni ( dy ) µ ni ( dz )= X Z V Z V Z V f ( x )[ φ j ( x ) − φ j ( y )][ φ k ( x ) − φ k ( z )] µ ni ( dx ) µ ni ( dy ) µ ni ( dz ) + R, R = P [ R V R V c R V + R V c R V R V + R V c R V c R V ], and a similar expression holds for A n,mjk ( t, f ).Subtract the two expressions, it can be shown that | ˜ A n,mjk ( t, f ) − A n,mjk ( t, f ) | is controlledby max t ni ≤ t, t ni ∆ mn [ µ ni ( V c ) + R R µ ni ( dx ) µ ni ( dy ) k φ · ( x ) − φ · ( y ) k ] η n ( t, V c ), and it follows thatlim m →∞ lim n →∞ | ˜ A n,mjk ( t, f ) − A n,mjk ( t, f ) | = 0. This implies that ˜ A ( t ) = A ( t ).Now let { ˜ ξ j } be another basis of g such that ξ j = P dk =1 a jk ˜ ξ k . Then ˜ φ j = P dk =1 a kj φ k are the coordinate functions associated to { ˜ ξ j } , and from the above displayed expressionfor ˜ A n,m ( t, f ) in terms of ˜ φ j , ˜ A n,mjk ( t, f ) = P p,q a pj a qk A n,mpq ( t, f ). This implies that ˜ A jk ( t ) = P p,q a pj a qk A pq ( t ), and hence P j,k ˜ A jk ( t ) ˜ ξ j ˜ ξ k = P j,k A jk ( t ) ξ j ξ k . ✷ Let Y be a smooth manifold equipped with a compatible metric r , which will be takento be G × G in the proof of Lemma 15. Let y n and y be rcll functions: R + → Y . Assumefor any t > r ( y n ( t ni ) , y ( t ni )) → n → ∞ uniformly for t ni ≤ t . Let F ( y, b, x ) and F jk ( y, b, x ) be bounded continuous functions on Y × G × G . Lemma 12
Assume the above and (A), and let A ( t ) be the covariance matrix function inLemma 11. Then for any t > and neighborhood U of e with η ( T, ∂U ) = 0 , as n → ∞ , X t ni ≤ t Z U c F ( y n ( t n i − ) , b ni , x ) µ ni ( dx ) → Z t Z U c F ( y ( s ) , e, x ) η c ( ds, dx ) + X u ≤ t, u ∈ J Z U c F ( y ( u − ) , h u , x ) ν u ( dx ) , (25) and along the subsequence of n → ∞ in Lemma 11, X t ni ≤ t d X j,k =1 Z G F jk ( y n ( t n i − ) , b ni , x )[ φ j ( x ) − φ j ( b ni )][ φ k ( x ) − φ k ( b ni )] µ ni ( dx ) → d X j,k =1 { Z t F jk ( y ( s ) , e, e ) dA jk ( s ) + Z t Z G F jk ( y ( s ) , e, x ) φ j ( x ) φ k ( x ) η c ( ds, dx )+ X u ≤ t, u ∈ J Z G F jk ( y ( u − ) , h u , x )[ φ j ( x ) − φ j ( h u )][ φ k ( x ) − φ k ( h u )] ν u ( dx ) } . (26) Proof
Let V be a relatively compact neighborhood of e . By Proposition 5, η n ( t, V c ) ↓ V ↑ G uniformly in n . Because of the uniform convergence r ( y n ( t ni ) , y ( t ni )) → F ( y n ( t ni ) , b, x ) − F ( y ( t ni ) , b, x ) → n → ∞ uniformly for t ni ≤ t and for ( b, x ) in acompact set. Therefore, it suffices to prove (25) with y n and U c replaced by y and U c ∩ V for an arbitrary relatively compact neighborhood V of e with η ( T, ∂V ) = 0. Similarly, itsuffices to prove (26) with y n and G replaced by y and V .We now show that for s < t and f ∈ C b ( G ) vanishing in a neighborhood of e , as n → ∞ , X s 1, let x nt = x t ni = x n x n · · · x ni , t ni ≤ t < t n i +1 , ( x ni = x − t n i − x t ni ) (28) b nt = b n b n · · · b ni , t ni ≤ t < t n i +1 , (29) z nt = x nt ( b nt ) − = z n z n · · · z ni , where z ni = b nt n i − ( x ni b − ni )( b nt n i − ) − . (30)For f ∈ C ∞ c ( G ), let M nt f = f ( z nt ) = f ( e ) for 0 ≤ t < t n , and for t ≥ t n , let M nt f = f ( z nt ) − X t ni ≤ t Z G [ f ( z nt n i − b nt n i − xb − ni ( b nt n i − ) − ) − f ( z nt n i − )] µ ni ( dx ) . (31) Lemma 13 M nt f is a martingale under the natural filtration F xt of process x t . Proof The martingale property can be verified directly noting E [ Z G f ( z nt n i − b nt n i − xb − ni ( b nt n i − ) − ) µ ni ( dx ) | F xt n i − ] = E [ f ( z nt n i − z ni ) | F xt n i − ]= E [ f ( z nt ni ) | F xt n i − ] . ✷ r on G . A subset of G is relatively compact ifand only if it is bounded in r . Consider the following equi-continuity type condition on b nt .(B) For any T > ε > 0, there are integers n , m > δ > n ≥ n and s, t ∈ [0 , T ] with 0 < t − s < δ and ( s, t ] ∩ J m = ∅ , then r ( b ns , b nt ) < ε . Lemma 14 Under (B), b nt is bounded for ≤ t ≤ T and n ≥ . Moreover, there is a rcllpath b t in G with b = e such that along a subsequence of n → ∞ , b nt → b t uniformly for ≤ t ≤ T . Furthermore, b − t − b t = h t (the mean of x − t − x t ) for any t > . Proof The boundedness of b nt may be derived from (B) in the same way as the boundednessof η n ( t, U c ) in Proposition 5 is derived from Lemma 6.Because b nt is bounded, there is a subsequence of n → ∞ along which b nt converges tosome b t for t in a countable dense subset of [0 , T ] including J . The equi-continuity of b nt in(B) implies that the convergence holds for all t and is uniform for t ≤ T , and the limit b t isa rcll path with jump b − t − b t = h t at t ∈ J , noting h t = lim n →∞ b ni for t ∈ ( t n i − , t ni ]. ✷ Lemma 15 Assume (A) and (B), and let z t = x t b − t . Then for f ∈ C ∞ c ( G ) , M t f given in(10) is a martingale under the natural filtration F xt of x t . Proof We need only to show that for any t > M nt f is bounded in n and M nt f → M t f .We will assume, for the time being, that for any u > ν u is small in the meaning definedbefore (7). Then µ ni are small for t ni ≤ t when n is large.Let U be a neighborhood of e such that x = e P j φ j ( x ) ξ j for x ∈ U and η ( T, ∂U ) = 0.Let P t ni ≤ t be the sum in (31) that defines M nt f as f ( z nt ) − P t ni ≤ t . A typical term in P t ni ≤ t may be written as R G [ f ( zbxb ′− b − ) − f ( z )] µ ( dx ), where z = z nt n i − , b = b nt n i − , b ′ = b ni and µ = µ ni . By Taylor expansion of f ( zbxb ′− b − ) = f ( zb exp( P dj =1 φ j ( x ) ξ j ) b ′− b − ) at x = b ′ , Z G [ f ( zbxb ′− b − ) − f ( z )] µ ( dx ) = Z U c [ . . . ] µ ( dx ) + Z U [ . . . ] µ ( dx )= Z U c [ f ( zbxb ′− b − ) − f ( z )] µ ( dx ) + Z U { X j f j ( z, b, b ′ )[ φ j ( x ) − φ j ( b ′ )] } µ ( dx )+ Z U { X j,k f jk ( z, b, b ′ )[ φ j ( x ) − φ j ( b ′ )][ φ k ( x ) − φ k ( b ′ )] } µ ( dx ) + λ, where f j ( z, b, b ′ ) = ∂∂φ j f ( zbe P p φ p ( x ) ξ p b ′− b − ) | x = b ′ , (32) f jk ( z, b, b ′ ) = ∂ ∂φ j ∂φ k f ( zbe P p φ p ( x ) ξ p b ′− b − ) | x = b ′ , (33)18nd the remainder λ satisfies | λ | ≤ c U µ ( k φ · − φ · ( b ′ ) k U ) with constant c U → U ↓ { e } .Because b ′ is the mean of the small µ = µ ni , φ j ( b ′ ) = µ ( φ j ) and R U [ φ j ( x ) − φ j ( b ′ )] µ ( dx ) = R G − R U c = 0 − R U c [ φ j ( x ) − φ j ( b ′ )] µ ( dx ). This implies that the sum P t ni ≤ t in (31) is equal to X t ni ≤ t Z U c { f ( zbxb ′− b − ) − f ( z ) − X j f j ( z, b, b ′ )[ φ j ( x ) − φ j ( b ′ )] } µ ( dx )+ X t ni ≤ t Z U { X j,k f jk ( z, b, b ′ )[ φ j ( x ) − φ j ( b ′ )][ φ k ( x ) − φ k ( b ′ )] } µ ( dx ) + Λ , (34)where the remainder Λ satisfies | Λ | ≤ c U P t ni ≤ t µ ( k φ · − φ · ( b ′ ) k U ) = c U q n ( t, U ).Note that the two sums in (34) are bounded in absolute value by cη n ( t, U c ) and cq n ( t, U )respectively for some constant c > 0. Therefore, P t ni ≤ t and hence M nt f are bounded in n .By (28), x nt ni = x t ni . Because z nt = x nt ( b nt ) − , by the uniform convergence b nt → b t , r ( z nt ni , z t ni ) = r (( b nt ni ) − , ( b t ni ) − ) → n → ∞ uniformly for t ni ≤ t .The above fact allows us to apply (25) in Lemma 12 to the first sum in (34). Let Y = G × G , y = ( z, b ), and F ( y, b ′ , x ) = f ( zbxb ′− b − ) − f ( z ) − P j f j ( z, b, b ′ )[ φ j ( x ) − φ j ( b ′ )].Define y n ( t ) and y ( t ) by setting y n ( t ) = ( z nt ni , b nt ni ) for t ni ≤ t < t n i +1 , and y ( t ) = ( z t , b t ). Notethat f j ( z, b, e ) = [Ad( b ) ξ j ] f ( z ). It follows that as n → ∞ , the first sum in (34) converges to Z t Z U c { f ( z s b s xb − s ) − f ( z s ) − X j φ j ( x )[Ad( b s ) ξ j ] f ( z s ) } η c ( ds, dx )+ X u ≤ t,u ∈ J Z U c { f ( z u − b u − xh − u b − u − ) − f ( z u − ) − X j f j ( z u − , b u − , h u )[ φ j ( x ) − φ j ( h u )] } ν u ( dx ) . (35)Similarly, apply (26) in Lemma 12 to the second sum in (34) shows that it converges to12 X j,k Z t [Ad( b s ) ξ j ][Ad( b s ) ξ k ] f ( z s ) dA s + 12 X j,k Z t Z U [Ad( b s ) ξ j ][Ad( b s ) ξ k ] f ( z s ) φ j ( x ) φ k ( x ) η c ( ds, dx )+ 12 X j,k X u ≤ t,u ∈ J Z U f jk ( z u − , b u − , h u )[ φ j ( x ) − φ j ( h u )][ φ k ( x ) − φ k ( h u )] ν u ( dx ) . (36)By a computation similar to the one leading to (34), using the Taylor expansion of f ( z u − b u − xh − u b − u − ) at x = h u , one can show that P u ≤ t,u ∈ J R G [ f ( z u − b u − xh − u b − u − ) − f ( z u − )] ν u ( dx )is the sum of P u ≤ t,u ∈ J ( · · · ) in (35) and (1 / P j,k P u ≤ t,u ∈ J ( · · · ) in (36), plus an error termthat converges to 0 as U ↓ { e } . Letting U ↓ { e } in (35) and (36) shows that M nt f → M t f .For u ∈ J , let a u be the ν u -integral term in M t f given by (10) and let a nu be the µ ni -integral term in M nt f given by (31) with u ∈ ( t n i − , t ni ]. Then a nu → a u as n → ∞ . In thepreceding computation, we have assumed that all ν u are small. When ν u is not small, µ ni with u ∈ ( t n i − , t ni ] may not be small even for large n , so the computation leading to (34)19one for the µ ni -integral is not valid, but we may still take limit a nu → a u for this term andthe result is the same. Since there are only finitely many u ∈ J for which ν u are not small,the result holds even when not all ν u are small. ✷ Because q n ( t, · ) = q n ( t ni , · ) and b nt = b nt ni if t ni ≤ t < t n i +1 , and ∆ n contains J as n → ∞ , toverify (A) and (B), we may assume s, t in (A) and (B) are contained in [0 , T ] ∩ ∆ n .If either (A) or (B) does not hold, then for some ε > n , m, δ > 0, thereare s n , t n ∈ [0 , T ] ∩ ∆ n with n ≥ n , 0 < t n − s n < δ and ( s n , t n ] ∩ J m = ∅ such thateither q n ( t n , G ) − q n ( s n , G ) ≥ ε or r ( b ns n , b nt n ) ≥ ε . Because the jumps of q n ( t, G ) and b nt for t ∈ ( s n , t n ] become arbitrarily small when n and m are large, by decreasing t n if necessary,we may also assume q n ( t n , G ) − q n ( s n , G ) ≤ ε and r ( b ns n , b nt ) ≤ ε for s n ≤ t ≤ t n .Letting δ → m → ∞ yields a subsequence of n → ∞ such that s n and t n in[0 , T ] ∩ ∆ n converge to a common limit, ( s n , t n ] ∩ J m n = ∅ with m n ↑ ∞ , and either(i) ε ≤ q n ( t n , G ) − q n ( s n , G ) ≤ ε and r ( b ns n , b nt ) ≤ ε for t ∈ [ s n , t n ], or(ii) q n ( t n , G ) − q n ( s n , G ) ≤ ε , r ( b ns n , b nt n ) ≥ ε and r ( b ns n , b nt ) ≤ ε for t ∈ [ s n , t n ].We will derive a contradiction from (i) or (ii). Let ε n = sup s n 1. Then x γ n t is an inhomogeneousL´evy process in G starting at e , and on [0 , x t on [ s n , t n ] time changedby γ n . Because s n and t n converge to a common limit, and ( s n , t n ] ∩ J m n = ∅ with m n ↑ ∞ ,it follows that x γ n t → e as n → ∞ uniformly in t a.s.. Define x γ,nt , µ γni , η γn ( t, · ) , b γni , b γ,nt , A n,γ ( t, · ) , q γ,n ( t, · ) , z γ,nt , M γ,nt f, for the time changed process x γ n t and partition ∆ γn = { s ni } , where s ni = γ − n ( t ni ), in thesame way as x nt , µ ni , η n ( t, · ) , b ni , b nt , A n ( t, · ) , q n ( t, · ) , z nt , M nt f are defined for the process x t and partition ∆ n = { t ni } . Then for s, t ∈ [0 , | q γ,n ( t, G ) − q γ,n ( s, G ) | = | q n ( γ n ( t ) , G ) − q n ( γ n ( s ) , G ) | ≤ ε | t − s | + ε n . (37)Because ε n → 0, the above means that q γ,n ( t, G ) are equi-continuous in t for large n .Note that we now have a process x γ n t for each n and from which other objects, such as η γn and A γ,n , are defined, unlike before when we have a single process x t for all n , but as20 γ n t → e uniformly in t , the results established for η n and A n hold for η γn and A γ,n in simplerforms. For example, because η γn ( t, · ) = η n ( γ n ( t ) , · ) − η n ( s n , · ), by Proposition 7, η γn ( t, f ) → n → ∞ for f ∈ C b ( G ) vanishing in near e . Using (37), the proofs of Lemmas 11 and12 can be easily modified for A γ,n , and also simplified, to show that there is a covariancematrix function A γ ( t ) such that A γ,n ( t, f ) → f ( e ) A γ ( t ) for any f ∈ C b ( G ), and under theassumption of Lemma 12, X s ni ≤ t Z U c F ( y n ( s n i − ) , b γni , x ) µ γni ( dx ) → X s ni ≤ t d X j,k =1 F jk ( y n ( s n i − ) , b γni , x )[ φ j ( x ) − φ j ( b γni )][ φ k ( x ) − φ k ( b γni )] µ γni ( dx ) → d X j,k =1 Z t F jk ( y ( s ) , e, e ) dA γjk ( s ) . (39)Let D ( G ) be the space of rcll paths R + → G . Equipped with the Skorohod metric, D ( G ) is a complete separable metric space (see [2, chapter 3]). A rcll process x t in G may beregarded as a random variable in D ( G ). A sequence of processes x nt is said to converge weaklyin D ( G ) to a process x t if their distributions converge weakly on D ( G ) to the distributionof process x t .We will show that z γ,nt has a weakly convergent subsequence in D ( G ). Let U be aneighborhood of e and let σ be a stopping time. The amount of time it takes for a process x t to make a U c -displacement from time σ is denoted as τ σU , that is, τ σU = inf { t > x − σ x σ + t ∈ U c } (40)For a sequence of processes x nt in G , let τ σ,nU be the U c -displacement time for x nt from σ . Lemma 16 A sequence of processes x nt in G have a weakly convergent subsequence in D ( G ) if for any T > and any neighborhood U of e , lim n →∞ sup σ ≤ T P ( τ σ,nU < δ ) → δ → , (41) and lim n →∞ sup σ ≤ T P [( x nσ − ) − x nσ ∈ K c ] → as compact K ↑ G (in the topology on G ), (42) where sup σ ≤ T is taken over all stopping times σ ≤ T . roof This lemma is proved in [3]. We will provide a different proof as the argument willbe used to prove Lemma 24. Let r be a left invariant metric on G as before. As in [2,section 3.6], the measurement of δ -oscillation of a path x in D ( G ) on [0 , T ] is given by ω ′ ( x, δ, T ) = inf { t i } max ≤ i ≤ n sup s,t ∈ [ t i − , t i ) r ( x t , x s ) , (43)where the infimum inf { t i } is taken over all partitions 0 = t < t < · · · < t n − < T ≤ t n with min ≤ i ≤ n ( t i − t i − ) > δ . By Corollary 7.4 in [2, chapter 3], x nt have a weakly convergentsubsequence in D ( G ) if for any T > n →∞ P ( x nt ∈ K c for some t ≤ T ) → K ↑ G (in the r -topology) (44)and for any ε > n →∞ P [ ω ′ ( x n · , δ, T ) ≥ ε ] → δ → . (45)For a fixed ε > 0, the successive stopping times 0 = τ ε < τ ε < τ ε < · · · < τ εi < · · · whena rcll process x t makes an ε -displacement are defined inductively by τ εi = inf { t > τ εi − ; r ( x t , x τ εi − ) > ε } for i = 1 , , , . . . , setting inf ∅ = ∞ and τ εi = ∞ if τ εi − = ∞ . Let τ ε,ni be the ε -displacementtimes of the process x nt . It is easy to see that[min i ≥ { τ εi +1 − τ εi ; τ εi < T } > δ ] implies [ ω ′ ( x, δ, T ) ≤ ε ] , (46)and hence P [ ω ′ ( x, δ, T ) > ε ] ≤ P [min i ≥ { τ εi +1 − τ εi ; τ εi < T } ≤ δ ]. Thus, if ∀ T > ε > , lim n →∞ P [min i ≥ { τ ε,ni +1 − τ ε,ni ; τ ε,ni < T } < δ ] → δ → , (47)then (45) holds for any T > ε > F ( t ) = sup n sup i ≥ P ( τ ε,ni +1 − τ ε,ni < t, τ ε,ni < T ). By Lemma 8.2 in [2, chapter 3], F ( δ ) ≤ sup n P [min i ≥ { τ ε,ni +1 − τ ε,ni ; τ ε,ni < T } < δ ] ≤ LF ( δ ) + Z ∞ e − Lt F ( t/L ) dt (48)for any δ > L = 1 , , , . . . . It is clear that sup n in F ( t ) and (48) may be replaced bylim n →∞ . Consequently, (47) is equivalent to ∀ T > ε > , lim n →∞ sup i ≥ P ( τ ε,ni +1 − τ ε,ni < δ ; τ ε,ni < T ) → δ → . (49)It is now clear that (49), and hence (47) and (45), are implied by (41) for any neighbor-hood U of e and T > 0. It remains to verify (44). It suffices to show that for any η > 0, thereare a compact K ⊂ G and integer m > P [ x nt ∈ K c for some t ≤ T ] ≤ η for all22 ≥ m . Because (41) implies (47), there are δ > m > n ≥ m , P ( A n ) < η/ 2, where A n = [min i ≥ { τ ε,ni +1 − τ ε,ni ; τ ε,ni < T } < δ ]. Let p = [ T /δ ], the integerpart of T /δ . Then on A cn , min i ≥ { τ ε,ni +1 − τ ε,ni ; τ ε,ni < T } ≥ δ , and hence x nt makes at most p displacements of size ε , before time T , between τ ε,n , τ ε,n , . . . , τ ε,np , with possible jumps atthese times. By (42), there is a compact H ⊂ G such that P ( B n,i ) ≤ η/ (2 p ) for all n ≥ m and1 ≤ i ≤ p , where B n,i = [( x nτ − ) − x nτ ∈ H c ; τ ≤ T ] with τ = τ ε,ni . Let U be the ε -ball around e and let K be a compact subset of G containing U HU H · · · U HU = { u h u h · · · u p h p u p +1 ; u i ∈ U and h i ∈ H } . Then for n ≥ m , P [ x nt ∈ K c for some t ≤ T ] ≤ P ( A n ) + p X i =1 P ( A cn ∩ B n,i ) ≤ η. ✷ We will now apply Lemma 16 to the processes z γ,nt . Because z γ,nt is constant for t ≥ σ ≤ f ∈ C ∞ c ( G ) be such that 0 ≤ f ≤ G , f ( e ) = 1 and f = 0 on U c . Write τ for the U c -displacement time for process z γ,nt from a stopping time σ . Let f σ = f ◦ ( z γ,nσ ) − . Then P ( τ < δ ) = E [ f σ ( z γ,nσ ) − f σ ( z γ,nσ + τ ); τ < δ ] ≤ E [ f σ ( z γ,nσ ) − f σ ( z γ,nσ + τ ∧ δ )] , noting f σ ( z γ,nσ ) = 1, f σ ( z γ,nσ + τ ) = 0 and τ = τ ∧ δ on [ τ < δ ], where a ∧ b = min( a, b ). Because M γ,nt f = f ( z γ,nt ) − X s ni ≤ t Z G [ f ( z γ,ns n i − b γ,ns n i − x ( b γni ) − ( b γ,ns n i − ) − ) − f ( z γ,ns n i − )] µ γni ( dx ) (50)is a martingale for any f ∈ C ∞ c ( G ), and σ and σ + τ ∧ δ are stopping times, E [ M γ,nσ f σ − M γ,nσ + τ ∧ δ f σ ] = E { E [ M γ,nσ f σ − M γ,nσ + τ ∧ δ f σ | F σ ] } = 0 . Writing z, b, b ′ , µ for z γ,ns n i − , b γ,ns n i − , b γni , µ γni , we obtain P ( τ < δ ) ≤ − E { X σ 1. If b γt has ajump of size r ( b γs − , b s ) > s , then b γ,nt would have a jump of size close to r ( b γs − , b s )at time t = λ − n ( s ), which is impossible because the jumps of b γ,nt are uniformly small when n is large. It follows that b γt is continuous in t and hence b γ,nt → b γt uniformly in t as n → ∞ .Then the convergence z γ,nt → z γt is also uniform in t .By (38), (39) and (52), the martingale M γ,nt f given by (50) converges to f ( z γt ) − Z t d X j,k =1 [Ad( b γs ) ξ j ][Ad( b γs ) ξ k ] f ( z γs ) dA γjk ( s ) (53)for any f ∈ C ∞ c ( G ). It follows that e = z γt b γt and for any f ∈ C ∞ c ( G ), (53) is a martingale.This provides a representation of the trivial L´evy process x t = e by the L´evy triple ( b γ , A γ , b γt = e and A γ ( t ) = 0. This is a contradiction because if (i) holds, then Trace A γ (1) = lim n q n ( t n , G ) ≥ ε ,and if (ii) holds, then b γ = lim n b nt n is at lease ε distance away from e . We now have provedthat (A) and (B) must hold.Because (A) and (B) have been verified, by Proposition 9 and Lemma 15, we have provedthat an inhomogeneous L´evy process x t is represented ( b, A, η ) with η ( t, · ) being its jumpintensity measure, and A ( t ) and b t given in Lemmas 11 and 14. This is the first part ofTheorem 2 except for the uniqueness of ( b, A, η ).24 Uniqueness of the triple Let ( b, A, η ) be a L´evy triple on G with b t of finite variation, and let ν = { ν t ; t ≥ } be afamily of probability measures on G such that ν t = δ e except for countably many t > 0, andfor any t > U of e , X u ≤ t ν u ( U c ) < ∞ , X u ≤ t d X j =1 | ν u ( φ j ) | < ∞ and X u ≤ t ν u ( k φ · k ) < ∞ . (54)A rcll process z t in G is said to have the martingale property under the quadruple( b, A, η, ν ) as described above, or the ( b, A, η, ν )-martingale property, if for f ∈ C ∞ c ( G ), f ( z t ) − Z t X i ξ i f ( z s ) db i ( s ) − Z t X j,k ξ j ξ k f ( z s ) dA jk ( s ) − Z t Z G [ f ( z s x ) − f ( z s ) − X i φ i ( x ) ξ i f ( z s )] η ( ds, dx ) − X u ≤ t Z G [ f ( z u − x ) − f ( z u − )] ν u ( dx ) (55)is a martingale under the natural filtration F zt of process z t , where b i ( t ) are the componentsof b t . In the sequel, a ( b, A, η, ν )-martingale property always refers to a L´evy triple ( b, A, η )with b t of finite variation and a family ν of probability measures ν t on G satisfying (54).Recall that [Ad( g )] is the matrix representing Ad( g ) under the basis { ξ , . . . , ξ d } of g .Let [Ad( g )] ′ be its transpose. Lemma 17 If x t is an inhomogeneous L´evy process in G represented by an extended L´evytriple ( b, A, η ) , and x t = z t b t , then z t has the (¯ b, ¯ A, ¯ η, ¯ ν ) -martingale property, where d ¯ A ( t ) =[Ad( b t )] dA ( t )[Ad( b t )] ′ , ¯ b t has components ¯ b i ( t ) = Z G { φ i ( b t xb − t ) − X p φ p ( x )[Ad( b t )] ip } η c ( t, dx ) , (56)¯ η ( t, f ) = R G f ( b t xb − t ) η c ( t, dx ) and ¯ ν t ( f ) = R G f ( b t − xh − t b − t − ) ν t ( dx ) , where η c is the continu-ous part of η and ν t is given in (5) with mean h t . Thus, ¯ ν t = δ e if η ( t, · ) is continuous attime t . Proof Formally this follows directly from (10), but we need to verify that ¯ b i ( t ) have finitevariation and (54) holds for ¯ ν t . Because e P j φ j ( b u xb − u ) ξ j = b u xb − u = e P j φ j ( x )Ad( b u ) ξ j for u ≤ t and x in a neighborhood U of e , the integrand in (56) vanishes in U . Noting η c ( t, U c ) < ∞ ,it is now easy to show that ¯ b i ( t ) have finite variation. The first inequality in (54) for ¯ ν , P u ≤ t ¯ ν u ( U c ) < ∞ , follows from P u ≤ t ν u ( U c ) < ∞ . The second and the third inequal-ities in (54), P u ≤ t | ¯ ν u ( φ j ) | = P u ≤ t | R φ j ( b u − xh − u b − u − ) ν u ( dx ) | < ∞ and P u ≤ t ¯ ν u ( k φ · k ) = P u ≤ t R k φ · ( b u − xh − u b − u − ) k ν u ( dx ) < ∞ , follow from a computation similar to the proof ofLemma 15 using Taylor expansion of φ j ( b u − xh − u b − u − ) = φ j ( b u − e P p φ p ( x ) ξ p h − u b − u − ) at x = h u . ✷ Lemma 18 Given a L´evy triple ( b, A, η ) with b t of finite variation, there is an inhomoge-neous L´evy process z t in G with z = e , unique in distribution, such that it has the ( b, A, η, -martingale property. Moreover, given a rcll process z t in G , there is at most one L´evy triple ( b, A, η ) with b t of finite variation such that the ( b, A, η, -martingale property holds. Lemma 19 If z t is a rcll process in G having a ( b, A, η, ν ) -martingale property, then thefixed jumps of z t are determined by ν , that is, the distribution of z − t − z t is ν t for all t > . Proof Let M t be the martingale (55). Then E [ M t − M t − | F zt − ] = 0. By (55), ∀ f ∈ C ∞ c ( G ) , E { f ( z t ) − f ( z t − ) − Z G [ f ( z t − x ) − f ( z t − )] ν t ( dx ) | F zt − } = 0 . Then E [ f ( z t ) | F zt − ] = R G f ( z t − x ) ν t ( dx ), and hence ν t is the distribution of z − t − z t . ✷ Corollary 20 If x t is a rcll process in G represented by an extended L´evy triple ( b, A, η ) ,then the fixed jumps of x t are determined by the discontinuous part of η , that is, x − t − x t hasdistribution ν t = η ( { t } × · ) + [1 − η ( { t } × G )] δ e for any t > . Proof Let x t = z t b t . By Lemma 17, z t has the (¯ b, ¯ A, ¯ η, ¯ ν )-martingale property, and byLemma 19, the fixed jumps of z t are determined by ¯ ν . Then E [ f ( x − t − x t )] = E [ f ( b − t − z − t − z t b t )] = Z G f ( b − t − xb t )¯ ν t ( dx ) = Z G f ( xh − t b − t − b t ) ν t ( dx ) = ν t ( f )for f ∈ B + ( G ), because b − t − b t = h t . ✷ A possible jump of a rcll process z t at time u may be removed to obtain a new process z ′ t defined by z ′ t = z t for t < u and z ′ t = z u − z − u z t for t ≥ u . Jumps may be successively removedat several time points and the resulting process is independent of the order at which theseoperations are performed. Lemma 21 Let z t be a rcll process in G . Then there is at most one quadruple ( b, A, η, ν ) such that the ( b, A, η, ν ) -martingale property holds for z t . Proof By Lemma 19, ν is determined by the process z t . Let J = { t > ν t = 0 } = { u , u , u , . . . } . We will write the martingale in (55) as M t f = f ( z t ) − Z t Hf ( z · ) − X u ≤ t Z G [ f ( z u − x ) − f ( z u − )] ν u ( dx ) , (57)26here R t Hf ( z · ) is the sum of integrals R t ( · · · ) db j ( s ), R t ( · · · ) dA jk ( s ) and R t R G ( · · · ) η ( ds, dx )with bounded integrands. Because b i ( t ), A jk ( t ) and η ( t, · ) are continuous in t , one can showthat R t Hf ( z · ) is a bounded continuous function of z · on D ( G ) under the Skorohod metric.Let z t be the process z t when its jump at time u ∈ J is removed. Then for t ≥ v ≥ u , M t f − M v f = f ( z t ) − f ( z v ) − Z tv Hf ( z · ) − X v 0, which verifies condition (41) in Lemma 16. Because the jumps of z nt are those of z t ,sup n sup σ ≤ T P [( z nσ − ) − z nσ ∈ K c ] ≤ sup σ ≤ T P [ z − σ − z σ ∈ K c ] ≤ P [ z − t − z t ∈ K c for some t ≤ T ] ↓ K ↑ G by the rcll property of z t . This verifies condition (42). Now Lemma 16may be applied to show the weak convergence of z nt in Skorohod metric to a rcll process z ′ t in G .Because R t Hf ( z · ) is a bounded continuous function on D ( G ), for any bounded contin-uous function F ( z · ) on D ( G ), E { [ R t Hf ( z n · )] F ( z n · ) } → E { R t Hf ( z ′· )] F ( z ′· ) } as n → ∞ . Inparticular, this holds when F is measureable under σ { z u ; u ≤ s } for s < t . Because M t f in(57) is a martingale when z t and J are replaced by z nt and J n , and the sum P u ≤ t in (57)with J replaced by J n converges to 0 as J n → ∅ , it follows that M t f is still a martingalewhen z t and J are replaced by z ′ t and ∅ . ✷ To prove the uniqueness of the extended L´evy triple ( b, A, η ) in Theorem 2, we will needa transformation rule for the martingale property (55) in the lemma below. Lemma 22 Let z t be a rcll process in G having the ( b, A, η, ν ) -martingale property. If u t isa drift of finite variation with components u i ( t ) , then z t u t has the ( b u , A u , η u , ν u ) -martingaleproperty, where dA u ( t ) = [Ad( u − t )] dA ( t )[Ad( u − t )] ′ , η u ( dt, f ) = R G f ( u − t xu t ) η ( dt, dx ) and ν ut ( f ) = R G f ( u − t − xu t − ) ν t ( dx ) for f ∈ B + ( G ) , and b ut is given in components by db ui ( t ) = X p [Ad( u − t )] ip db p ( t ) + du i ( t ) + Z G { φ i ( u − t xu t ) − X p [Ad( u − t )] ip φ p ( x ) } η ( dt, dx ) . Proof The proof is similar to the proof of Lemma 4, approximating u t by step functions u nt = u t ni , t ni ≤ t < t n i +1 and applying the ( b, A, η, ν )-martingale property for z t with thefunction f ni ( z ) = f ( zu t ni ), f ∈ C ∞ c ( G ). We omit the essentially repetitive details. ✷ Lemma 23 There is at most one extended L´evy triple ( b, A, η ) which represents a giveninhomogeneous L´evy process x t in G as in Theorem 2. Proof Suppose two extended L´evy triples ( b , A , η ) and ( b , A , η ) represent the sameinhomogeneous L´evy process x t in G . By Corollary 20, η and η have the same discontinuouspart, and hence b t and b t have the same jumps. Let x t = z t b t = z t b t . Then z t = z t u t ,where u t = b t ( b t ) − is a rcll path in G . Note that u t = ( z t ) − z t is a semimartingale, thatis, f ( u t ) is a real semimartingale for any f ∈ C ∞ c ( G ). Because u t is non-random, it thenfollows that u t has a finite variation. We now show u t is continuous and so is a drift. Let x = u − t − u t = b t − ( b t − ) − b t ( b t ) − , then ( b t ) − xb t = [( b t − ) − b t ] − [( b t − ) − b t ] = e because b t and b t have the same jumps. This implies x = e and hence u t is continuous.28y Lemma 17, z it has the (¯ b i , ¯ A i , ¯ η i , ¯ ν i )-martingale property for i = 1 , 2. By Lemma 22, z t also has the (¯ b u , ¯ A u , ¯ η u , ¯ ν u )-martingale property. By Lemma 21, ¯ b = ¯ b u , ¯ A = ¯ A u and ¯ η = ¯ η u . Then for any f ∈ C ∞ c ( G ), Z G f ( b t x ( b t ) − ) η c ( dt, dx ) = ¯ η ( dt, f ) = ¯ η u ( dt, f ) = Z G f ( u − t xu t )¯ η ( dt, dx )= Z G f ( u − t b t x ( b t ) − u t ) η c ( dt, dx ) = Z G f ( b t x ( b t ) − ) η c ( dt, dx ) . This implies η c = η c and hence η = η . Moreover, d ¯ b ui ( t ) = X j [ Ad ( u − t )] ij d ¯ b j ( t ) + du i ( t ) + Z G { φ i ( u − t xu t ) − X j φ j ( x )[Ad( u − t )] ij } ¯ η ( dt, dx )= Z G { X j [Ad( u − t )] ij φ j ( b t x ( b t ) − ) − X p φ p ( x )[Ad( u − t b t )] ip } η c ( dt, dx ) + du i ( t )+ Z G { φ i ( u − t b t x ( b t ) − u t ) − X j φ j ( b t x ( b t ) − )[Ad( u − t )] ij } η c ( dt, dx )= Z G { φ i ( u − t b t x ( b t ) − u t ) − X p φ p ( x )[Ad( u − t b t )] ip } η c ( dt, dx ) + du i ( t )= Z G { φ i ( b t x ( b t ) − ) − X p φ p ( x )[Ad( b t )] ip } η c ( dt, dx ) + du i ( t ) = d ¯ b i ( t ) + du i ( t ) . Because ¯ b = ¯ b u , this implies that u i ( t ) = 0 and hence b = b . Now ¯ A = ¯ A u = ¯ A , andwith b = b , it follows that A = A . The uniqueness of ( b, A, η ) is proved. ✷ Lemma 24 Let x nt be a sequence of rcll processes in G as in Lemma 16 and u n ∈ R + .For each n , let x n,mt be the process obtained from x nt when possible fixed jumps at times u , u , . . . , u m > are removed. If for any T > , η > and neighborhood U of e , there are δ > and an integer m ≥ such that lim n →∞ sup σ ≤ T P ( τ σ,n,mU < δ ) ≤ η, (59) where sup σ ≤ T is taken over all stopping times σ ≤ T and τ σ,n,mU is τ σU for process x n,mt , andif (42) also holds, then a subsequence of x nt converges weakly in D ( G ) . Proof Using the notation in the proof of Lemma 16, let τ ε,n,mi be the ε -displacement time τ εi for process x n,mt . By (48) and (59), for any η and ε > 0, there are δ and m such thatlim n →∞ P [min i ≥ { τ ε,n,mi +1 − τ ε,n,mi ; τ ε,n,mi < T } ≤ δ ] ≤ η. Then by (46), lim n →∞ P [ ω ′ ( x n,m , δ, T ) > ε ] ≤ η .29he computation of ω ′ ( x, δ, T ) in (43) is based on the oscillations over partitions thatcover [0 , T ] with spacing > δ , called δ -partitions. Let J m = { u , . . . , u m } . We may assume δ < min {| u | ∧ | u − v | ; u, v ∈ J m with u = v } , and either T ∈ J m or | T − u | > δ for u ∈ J m .Suppose there is a δ -partition with oscillation ≤ ε . Then any interval of this δ -partitioncontains at most one point in J m . Because adding more partition points will not increaseoscillation, adding J m to the δ -partition together with the midpoints of intervals that do notintercept J m , and then suitably combining intervals, we obtain a ( δ/ J m , with oscillation ≤ ε . Then lim n →∞ P [ ω ′ ( x n , δ/ , T ) > ε ] ≤ η . This verifies (45).It remains to verify (44). If ω ′ ( x, δ, T ) < ε , then there is a δ -partition { t i } with oscillation < ε . This implies that τ ε [0 , t ) and τ εi belong to different intervals of the partition. Thus ω ′ ( x, δ, T ) < ε = ⇒ min i ≥ { τ εi +1 − τ εi − ; τ εi < T } > δ (setting τ ε − = 0). (60)By (45) and (60), lim n →∞ P ( A n ) → δ → 0, where A n = [min i ≥ { τ ε,ni +1 − τ ε,ni − ; τ ε,ni < T } <δ ]. The rest of proof is very similar to the last part of the proof of Lemma 16. ✷ Let z t be an inhomogeneous L´evy process in G having the ( b, A, η, ν )-martingale propertyand let J = { u > ν u = 0 } = { u , u , . . . } . For p < q , let J qp = { u , . . . , u p , u q +1 , u q +2 , . . . } and let z p,qt be the process z t when its fixed jumps at u p +1 , . . . , u q are removed. As in theproof of Lemma 21, it can be shown that z p,qt has the ( b, A, η, ν p,q )-martingale property with ν p,q = { ν u ; u ∈ J qp } . Let J p = { u , u , . . . , u p } . Lemma 25 The family of processes, z p,qt for p < q , are weakly compact in D ( G ) . Moreover,along a subsequence of n → ∞ , z p,nt converges weakly to an inhomogeneous L´evy process z ′ pt that has the ( b, A, η, ν ′ p ) -martingale property with ν ′ p = { ν u ; u ∈ J p } . Proof Let z nt = z p n ,q n t , where p n < q n and q n → ∞ as n → ∞ . As in the proof ofLemma 21, but using Lemma 24 instead of Lemma 16, it can be shown that a subsequenceof z nt converges weakly in D ( G ) to some z ′ t . Note that the last term in (58) now takes theform P σ 0, choose φ ε ∈ C ( R ) such that 0 ≤ φ ε ≤ φ ε ( u ) = 1 and φ ε ( t ) = 0 for | t − u | > ε .For any g ∈ C ( G ) such that 0 ≤ g ≤ g = 0 near e , let H ε ( z · ) = P t> φ ε ( t ) g ( z − t − z t ).Then ˜ H ε ( z · ) = H ε ( z · ) ∧ D ( G ) and hence E [ ˜ H ε ( z p,n · )] → E [ ˜ H ε ( z ′ p · )] as n → ∞ . Because ˜ H ε ( z ) → g ( z − u − z u ) as ε → 0, ˜ H ε ( z ) ≤ P t ≤ T, | t − u |≤ ε g ( z − t − z t )and E [ P t ≤ T g ( z − t − z t )] ≤ P t ≤ T ν t ( g ) < ∞ , it follows that E [ ˜ H ε ( z p,n · )] → ν u ( g ) as ε → n . This implies that E [ g (( z ′ pu − ) − z ′ pu )] = lim ε → E [ ˜ H ε ( z ′ p · )] = ν u ( g ) and hencethe distribution of ( z ′ pu − ) − z ′ pu is ν u .We may assume F and f in (61) are bounded by 1 in absolute values. Let J ε ( z · ) = F ( z · ) P t> φ ε ( t )[ f ( z t ) − f ( z t − )] g ( z − t − z t ) and let ˜ J ε ( z · ) = [ J ε ( z · ) ∧ ∨ ( − a ∨ b =max( a, b ). Then ˜ J ε ( z · ) is a bounded continuous function on D ( G ) and hence E [ ˜ J ε ( z p,n · )] → E [ ˜ J ε ( z ′ p · )] as n → ∞ . Because ˜ J ε ( z · ) → F ( z · )[ f ( z u ) − f ( z u − )] g ( z − u − z u ) as ε → | ˜ J ε ( z · ) | ≤ P t ≤ T, | t − u |≤ ε g ( z − t − z t ), it follows that E [ ˜ J ε ( z p,n · )] → E { F ( z p,n · )[ f ( z p,nu ) − f ( z p,nu − )] g (( z p,nu − ) − z p,nu ) } as ε → n . Then E { F ( z ′ p · )[ f ( z ′ pu ) − f ( z ′ pu − )] g (( z ′ pu − ) − z ′ pu ) } = lim ε → E [ ˜ J ε ( z ′ p · )] =lim ε → lim n →∞ E [ ˜ J ε ( z p,n · )] = lim n →∞ lim ε → E [ ˜ J ε ( z p,n · )]. Letting g ↑ G −{ e } proves (61). ✷ Lemma 26 The distribution of an inhomogeneous L´evy process z t in G with z = e havingthe martingale property under a given quadruple ( b, A, η, ν ) is unique. Proof Because z ′ pt in Lemma 25 has only finitely many fixed jumps, after removing thesefixed jumps, it becomes stochastically continuous. By Lemma 18, the distribution of z ′ pt iscompletely determined by ( b, A, η, ν ′ p ), and hence also by ( b, A, η, ν ).By Lemma 25, the processes x p,qt , p < q , are weakly compact on D ( G ). Then by Re-mark 7.3 in [2, chapter 3], for any ε > 0, there is a compact K ε ⊂ G such that ∀ p < q, P ( z p,qs ∈ K ε for s ≤ t ) ≥ − ε. (62)Let f ∈ C ∞ c ( G ) with | f | ≤ 1. We will show that ∀ ε > ∃ constant c ε = c ε,f,t > ∀ p < q, | E [ f ( z t )] − E [ f ( z p,qt )] | ≤ c ε q X i = p +1 [ d X j =1 | ν u i ( φ j ) | + ν u i ( k φ · k ) + ν u i ( U c )] + 2 ε (63)for a small neighborhood U of e .Recall z p,qt is obtained from z t after the fixed jumps at times u p +1 , u p +2 , . . . , u q are re-moved. Let v < v < · · · < v q − p be the ordered values of these time points and let m ≤ q − p be the largest integer such that v m ≤ t . Let σ i = z − v i − z v i and for u < v , let z u,v = z − u z v − and z p,qu,v = ( z p,qu ) − z p,qv − . For x i ∈ G , let Q ni =1 x i = x x · · · x n . Set v = 0 and z u,u = z p,qu,u = e ,and let H = ∩ mi =1 [ z p,q ,v i ∈ K ε ]. By (62), P ( H ) ≥ − ε . For simplicity, assume z t = z t − . f ( z t ) − f ( z p,qt ) = f ( z p,q ,v σ z v ,t ) − f ( z p,q ,v m z v m ,t ) = m X i =1 [ f ( z p,q ,v i σ i z v i ,t ) − f ( z p,q ,v i z v i ,t )]31 m X i =1 [ f ( z ′ σz ) − f ( z ′ z )] ( z ′ = z p,q ,v i , z = z v i ,t and σ = σ i )= m X i =1 [ f ( z ′ e P j φ j ( σ ) ξ j z ) − f ( z ′ z )]1 [ σ ∈ U ] + m X i =1 [ f ( z ′ σz ) − f ( z ′ z )]1 [ σ ∈ U c ] = m X i =1 [ f ( e P j,k [Ad( z ′ )] kj φ j ( σ ) ξ k z ′ z ) − f ( z ′ z )]1 [ σ ∈ U ] + m X i =1 [ f ( z ′ σz ) − f ( z ′ z )]1 [ σ ∈ U c ] = m X i =1 { d X j,k =1 ξ rk f ( z ′ z )[Ad( z ′ )] kj φ j ( σ ) + O ( k φ · ( σ ) k ) } [ σ ∈ U ] + m X i =1 [ f ( z ′ σz ) − f ( z ′ z )]1 [ σ ∈ U c ] by Taylor’s formula, where ξ r f ( z ) = ddt f ( e tξ z ) | t =0 . Note that z p,qt and H are independent of σ i , and on H , z p,qs ∈ K ε and hence [Ad( z p,qs )] kj are bounded for all s ≤ t . It follows that | E [ f ( z t ) − f ( z p,qt )] | ≤ | E { [ f ( z t ) − f ( z p,qt )]1 H }| + 2 ε ≤ c ′ ε m X i =1 [ X j | ν v i ( φ j U ) | + O ( k φ · k ) + ν v i ( U c )] + 2 ε for some constant c ′ ε > 0. This proves (63) because | ν u ( φ j U ) | ≤ | ν u ( φ j ) | + (sup G | φ j | ) ν u ( U c ).By (54), letting first q → ∞ and then p → ∞ in (63) yields E [ f ( z t )] − E [ f ( z ′ pt )] → E [ f ( z ′ pt )] is determined by ( b, A, η, ν ), so is E [ f ( z t )]. This proves the uniqueness ofthe one-dimensional distribution of an inhomogeneous L´evy process z t having the ( b, A, η, ν )-martingale property. To complete the proof, one just need to use the standard argumentto derive the uniqueness of distribution from the uniqueness of one-dimensional distributionand the martingale property, see the proof of Theorem 4.2(a) in [2, chapter 4]. Note thesetups here and in [2] are not quite the same, but the argument is essentially the same. ✷ By Lemmas 17 and 26, we obtain the unique distribution for the process x t in Theorem 2. Corollary 27 The distribution of an inhomogeneous L´evy process x t in G with x = e represented by a given extended L´evy triple is unique. Proof of Theorem 2 It remains to prove that given an extended L´evy triple ( b, A, η ), thereis an inhomogeneous L´evy process x t in G represented by ( b, A, η ), that is, with x t = z t b t ,(10) is a martingale under the natural filtration F xt of x t for any f ∈ C ∞ c ( G ).Let (¯ b, ¯ A, ¯ η, ¯ ν ) be as in Lemma 17. Then (¯ b, ¯ A, ¯ η ) is a L´evy triple with ¯ b t of finite variation.By Lemma 18, there is a stochastically continuous inhomogeneous L´evy process z ′ t havingthe (¯ b, ¯ A, ¯ η, f ∈ C ∞ c ( G ), f ( z ′ t ) − R t ¯ Hf ( z ′· ) is amartingale under the natural filtration of z ′ t , where R t ¯ Hf ( z ′· ) is the expression R t Hf ( z · ) in(57) with b, A, η, z · replaced by ¯ b, ¯ A, ¯ η, z ′· .Let J = { u , u , u , . . . } be the set of discontinuity time points of η , and let { v u , u ∈ J } be a family of G -valued random variables with distributions ν u such that they are mutually32ndependent and independent of process z ′ t . For u = u , and let z t be the process z ′ t whenthe fixed jump ¯ v u = b u − v u h − u b − u − is added at time u , that is, z t = z ′ t for t < u and z t = z ′ u − ¯ v u z ′− u z ′ t for t ≥ u . Then it is easy to show that f ( z t ) − R t ¯ Hf ( z · ) − [ u ≤ t ] R G [ f ( z u − x ) − f ( z u − )]¯ ν u ( dx ) is a martingale under the natural filtration of z t . More generally, let z nt be theprocess z ′ t when the fixed jumps ¯ v u , ¯ v u , . . . , ¯ v u n are successively added at times u , u , . . . , u n .Then f ( z nt ) − Z t ¯ Hf ( z ns ) d ¯ b ( s ) − X u ≤ t, u ∈ J n Z G [ f ( z nu − x ) − f ( z nu − )]¯ ν u ( dx ) (64)is a martingale under the natural filtration of z nt , where J n = { u , u , . . . , u n } .As in the proof of Lemma 25, we may use Lemma 24 to show that a subsequence of z nt converge weakly in D ( G ) to a rcll process z t , and (64) is still a martingale when z nt and J n are replaced by z t and J . Therefore, with x t = z t b t , (10) is a martingale under F zt = F xt .It remains to show that x t is an inhomogeneous L´evy process in G . Because z nt isobtained by adding n independent fixed jumps to the inhomogeneous L´evy process z ′ t , so z nt is an inhomogeneous L´evy process. The same is true for z t , as the weak limit of z nt , andhence true also for x t = z t b t . ✷ Let K be a closed subgroup of a Lie group G . The space G/K of left cosets gK , g ∈ G , iscalled a homogeneous space. It is equipped with the unique manifold structure under whichthe natural G -action on G/K , ( g, g ′ K ) gg ′ K for g, g ′ ∈ G , is smooth (Theorem 4.2 in [4,chapter II]. Moreover, the natural projection π : G → G/K , g gK , is smooth and open.Let x t be a Markov process x t in G/K with rcll paths. It is called G -invariant if itstransition function P t is G -invariant in the sense that ∀ t ≥ , g ∈ G and f ∈ B + ( X ) , P t ( f ◦ g ) = ( P t f ) ◦ g. When K = { e } , such a process becomes a left invariant Markov process in G , that is, a L´evyprocess in G . Therefore, a rcll G -invariant Markov process in G/K will also be called a L´evyprocess. More generally, a rcll inhomogeneous Markov process in G/K with a G -invarianttransition function P s,t , will be called an inhomogeneous L´evy process.Let X be a manifold under the transitive action of a Lie group G and let K = { g ∈ G ; go = o } be the fix-point subgroup at a point o ∈ X . Then X may be identified with thehomogeneous space G/K via the map gK go (Theorem 3.2 and Proposition 4.3 in [4,chapter II]), under which the G -action on X is identified with the natural G -action on G/K .Although this identification is not unique, as a different o may be chosen, but (inhomoge-neous) L´evy processes in X , defined as G -invariant (inhomogeneous) Markov processes, areindependent of this identification. 33n the rest of this paper, let x t be an inhomogeneous L´evy process in X = G/K and let J be the countable set of fixed jump times of x t . We will assume K is compact.A measure µ on X = G/K is called K -invariant if kµ = µ for k ∈ K , where kµ isthe measure defined by kµ ( f ) = µ ( f ◦ k ) for f ∈ B + ( X ). Because P s,t is G -invariant, µ s,t = P s,t ( o, · ) is a K -invariant probability measure on X , where o = eK is the origin in X .A Borel measurable map S : X → G is called a section map if π ◦ S = id X (the identitymap on X ). Such a map is not unique and it may not be continuous on X . However, onemay always choose a section map that is smooth near any given point in X . Integrals like Z f ( xy ) µ ( dy ) = Z f ( S ( x ) y ) µ ( dy ) , Z f ( xyz ) µ ( dy ) ν ( dz ) = Z f ( S ( S ( x ) y ) z ) µ ( dy ) ν ( dz ) , are well defined for K -invariant measures µ and ν on X , independent of the choice for thesection map S . This is easily verified by the K -invariance of µ and ν , because if S ′ isanother section map, then for any x ∈ X , S ′ ( x ) = S ( x ) k x for some k x ∈ K . Note also that S ( S ( x ) y ) = S ( x ) S ( y ) k for some k = k ( x, y ) ∈ K , so the second integral above may also becomputed as R f ( S ( x ) S ( y ) z ) µ ( dy ) ν ( dz ).A random variable in X is called K -invariant if its distribution is so. It is clear thatif x and y are two independent random variables in X with y being K -invariant, then thedistribution of xy = S ( x ) y is independent of the choice for section map S .By the Markov property and the G -invariance of P s,t , E [ f ( x t , x t , . . . , x t n )] = Z µ ( dx ) µ ,t ( dx ) µ t ,t ( dx ) · · · µ t n − ,t n ( dx n ) f ( x x , x x x , . . . , x x x · · · x n ) (65)for t < t < · · · < t n and f ∈ B + ( G n ), where µ is the initial distribution. Proposition 28 A rcll process x t in X is an inhomogeneous L´evy process if and only if ithas independent increments in the sense that for s < t , x − s x t = S ( x s ) − x t is independent of F xs and has a K -invariant distribution µ s,t independent of the choice for the section map S . Proof From (65), it is easy to show that S ( x s ) − x t is independent of F xs and its distribution µ s,t is independent of S . Conversely, assume this property, then for s < t and f ∈ B + ( X ), E [ f ( x t ) | F xs ] = E [ f ( S ( x s ) S ( x s ) − x t ) | F xs ] = R X f ( S ( x s ) y ) µ s,t ( dy ). If µ s,t is K -invariant,then x t is an inhomogeneous Markov process in X with a G -invariant transition function P s,t f ( x ) = R X f ( S ( x ) y ) µ s,t ( dy ), and hence x t is an inhomogeneous L´evy process in X . ✷ The proof of Proposition 28 may be slightly modified to show that a rcll process x t in X is a L´evy process if and only if it has independent and stationary increments. Here thestationary increments mean the distribution of x − s x t = S ( x s ) − x t depends only on t − s .34 measure function η ( t, · ), t ≥ 0, on X is defined just as a measure function on G , thatis, as a nondecreasing and right continuous family of σ -finite measures on X . It is called K -invariant if η ( t, · ) is K -invariant for each t ≥ 0. As on G , a measure function η ( t, · ) on X may be regarded as a σ -finite measure on R + × X and it may be written as a sum of itscontinuous part η c ( t, · ) and its discontinuous part P s ≤ t η ( { s } × · ) as in (3).Let ∆ n : 0 = t n < t n < t n < · · · < t ni ↑ ∞ as i ↑ ∞ be a sequence of partitions of R + with mesh k ∆ n k → n → ∞ , and assume ∆ n contains J as n → ∞ as before. Let x t bean inhomogeneous L’evy process in X and let µ ni be the distribution of x − t n i − x t ni for i ≥ η n ( t, · ) = X t ni ≤ t µ ni , (66)setting η n (0 , · ) = 0. Then η n is a K -invariant measure function on X .The proof of Proposition 5 may be repeated on X , regarding x − s x t as S ( x s ) − x t withthe choice of a section map S . Note that with a G -invariant metric r on X , r ( x, y ) = r ( o, S − ( x ) y ), and hence S ( x s ) − x t ∈ U c for a neighborhood U of o if and only if r ( x s , x t ) > δ for some δ > 0. Then for any t > U of o , η n ( t, U c ) is bounded in n and η n ( t, U c ) ↓ U ↑ X uniformly in n. (67)For f ∈ C b ( X ), let ˆ f = R K dk ( f ◦ k ), where dk is the normalized Haar measure on K .Then ˆ f is K -invariant, that is, ˆ f ◦ k = ˆ f for k ∈ K , and for a K -invariant measure µ on X , µ ( f ) = µ ( ˆ f ). We note that to show the weak convergence of a sequence of K -invariantmeasures µ n on K , it suffices to show the convergence of µ n ( f ) for K -invariant f ∈ C b ( X ).Let f ∈ C b ( X ) be K -invariant and vanish near o . Then the function ( x, y ) f ( S ( x ) − y )is independent of the choice for the section map S and is continuous. The proof of Proposi-tion 7 may be repeated to show that as n → ∞ , η n ( t, f ) = X t ni ≤ t E [ f ( S ( x t n i − ) − x t ni )] → E [ X s ≤ t f ( S ( x s − ) − x s )] . This shows that η ( t, · ) defined by η (0 , · ) = 0, η ( t, { o } ) = 0 and η ( t, f ) = η ( t, ˆ f ) = E [ X s ≤ t ˆ f ( x − s − x s )] for f ∈ C b ( G ) vanishing near o, (68)is a well defined K -invariant measure function on X , independent of the choice of the sectionmap S to represented x − s − x s = S ( x s − ) − x s in (68), which will be called the jump intensitymeasure of process x t . It is clear that η = 0 if and only if the process x t is continuous, and η ( t, · ) is continuous in t if and only if x t is stochastically continuous. Moreover, for any t > f ∈ C b ( X ) vanishing near o , η n ( t, f ) → η ( t, f ) as n → ∞ .A point b ∈ X is called K -invariant if kb = b for all k ∈ K . For x ∈ X and K -invariant b , the product xb = S ( x ) b ∈ X is well defined, independent of choice for the section map S .35 roposition 29 Let N be the set of g ∈ G such that go = π ( g ) are K -invariant. Then N is the normalizer of K , that is, N = { g ∈ G ; gKg − = K } . Proof Note that g ∈ G with go being K -invariant is characterized by g − Kg ⊂ K . Thisimplies that g − Kg and K have the same identity component K = g − K g . Because K iscompact, its coset decomposition K = K ∪ [ ∪ pi =1 ( k i K )] is a finite union. Since g − Kg = K ∪ [ ∪ pi =1 ( g − k i gK )] is the coset decomposition of g − Kg , it follows that g − Kg = K . ✷ By Proposition 29, the set of K -invariant points in X has a natural group structure withproduct bb ′ = S ( b ) b ′ and inverse b − = S ( b ) − o , and the integral Z f ( xbyb ′ ) µ ( dy ) = Z f ( S ( x ) S ( b ) S ( y ) b ′ ) µ ( dy )makes sense for K -invariant measure µ on X and K -invariant points b, b ′ ∈ X , independentof the choice of the section map S .Let g and k be respectively the Lie algebras of G and K . Because K is compact, thereis a subspace p of g that is complementary to k and is Ad( K )-invariant in the sense thatAd( k ) p = p for k ∈ K . Choose a basis ξ , . . . , ξ n of p . There are φ , φ , . . . , φ n ∈ C ∞ c ( X )such that x = exp[ P ni =1 φ i ( x ) ξ i ] o for x near o . It then follows that ∀ k ∈ K, n X i =1 φ i ( x )[Ad( k ) ξ i ] = n X i =1 φ i ( kx ) ξ i (69)for x near o . Replacing φ i by φφ i for a K -invariant φ ∈ C ∞ c ( X ) with φ = 1 near o and φ = 0outside a small neighborhood of o , we may assume (69) holds for all x ∈ X . Then φ i ’s willbe called coordinate functions on X under the basis { ξ i } of p .Similar to the definitions given on G , the mean of a random variable x in X or itsdistribution µ is defined as b = e P ni =1 µ ( φ i ) ξ i o . By (69), if µ is K -invariant, then so is b . Wewill call x or µ small if b has coordinates µ ( φ i ), that is, φ i ( b ) = µ ( φ i ) for 1 ≤ i ≤ n .Any ξ ∈ g is a left invariant vector field on G . If ξ is Ad( K )-invariant, that is, ifAd( k ) ξ = ξ for k ∈ K , then e tξ o is K -invariant and ξ may also be regarded as a vector fieldon X given by ξf ( x ) = ddt f ( xe tξ o ) | t =0 for f ∈ C ∞ c ( X ) and x ∈ X , which is G -invariant inthe sense that ξ ( f ◦ g ) = ( ξf ) ◦ g for g ∈ G . In fact, any G -invariant vector field on X isgiven by a unique Ad( K )-invariant ξ ∈ p . Note that if ξ ∈ g is Ad( K )-invariant and b ∈ X is K -invariant, then Ad( b ) ξ = Ad( S ( b )) ξ is Ad( K )-invariant and is independent of the choiceof the section map S . By (69), for any K -invariant measure µ on X , R µ ( dx ) P i φ i ( x ) ξ i isAd( K )-invariant, and so is R µ ( dx ) P i φ i ( x )Ad( b ) ξ i for a K -invariant b ∈ X .Let ξ, η ∈ g . With a choice of section map S , ξη may be regarded as a second orderdifferential operator on X defined by ξηf ( x ) = ∂ ∂t ∂s f ( S ( x ) e tξ e sη o ) | t = s =0 . It is shown in [3,36art 2] that ξ i ξ j f ( x ) = ∂ ∂t ∂s f ( S ( x ) e tξ i + sξ j o ) | t = s =0 + P nk =1 ρ ijk ξ k f ( x ) with ρ ijk = − ρ jik . Thus,if a ij is an n × n symmetric matrix, then n X i,j =1 a ij ξ i ξ j f ( x ) = n X i,j =1 a ij ∂ ∂t i ∂t j f ( S ( x ) e P p t p ξ p o ) | t = ··· = t n =0 . (70)The matrix a ij is called Ad( K )-invariant if a ij = P p,q a pq [Ad( k )] ip [Ad( k )] jq for k ∈ K , where { [Ad( k )] ij } is the matrix representing Ad( k ), that is, Ad( k ) ξ j = P i [Ad( k )] ij ξ i . Then theoperator P i,j a ij ξ i ξ j is independent of section map S and is G -invariant. In fact, any secondorder G -invariant differential operator T on X with T G -invariant vector field. Note that 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 X for any b ∈ X (independent of S ) if a ij is Ad( K )-invariant.A drift or an extended drift b , a covariance matrix function A , and a L´evy measurefunction or an extended L´evy measure function η on X = G/K are defined just as on G , interms of the coordinate functions φ i on X , with the additional requirement that for each t , b t and η ( t, · ) are K -invariant, and A ( t ) is Ad( K )-invariant. With these modifications, L´evytriples and extended L´evy triples on X are defined exactly as on G . In particular, for anextended L´evy triple ( b, A, η ), b − t − b t = h t , where h t = exp[ P ni =1 ν t ( φ i ) ξ i ] o is the mean of the K -invariant probability measure ν t = η ( { t } × · ) + [1 − η ( { t } × X )] δ o .A rcll process x t in X = G/K is said to be represented by an extended L´evy triple( b, A, η ) on X if x t = z t b t and for any f ∈ C ∞ c ( X ), M t f = f ( z t ) − Z t n X j,k =1 [Ad( b s ) ξ j ][Ad( b s ) ξ k ] f ( z s ) dA jk ( s ) − Z t Z X { f ( z s b s xb − s ) − f ( z s ) − n X j =1 φ j ( x )[Ad( b s ) ξ j ] f ( z s ) } η c ( ds, dx ) − X u ≤ t Z X [ f ( z u − b u − xh − u b − u − ) − f ( z u − )] ν u ( dx ) (71)is a martingale under the natural filtration F xt of x t .By the K -invariance of b t and η ( t, · ), and the Ad( K )-invariance of A ( t ), the expressionin (71) makes sense. Moreover, by Taylor expansions of f ( z s b s xb − s ) = f ( z s e P i φ i ( x )Ad( b s ) ξ i )at x = o and f ( z u − b u − xh − u b − u − ) at x = h u , and the properties of η ( t, · ) as an extended L´evymeasure function, it can be shown that M t f given in (71) is a bounded random variable. Theorem 30 Let x t be an inhomogeneous L´evy process in X = G/K with x = o . Thenthere is a unique extended L´evy triple ( b, A, η ) on X such that x t is represented by ( b, A, η ) as defined above. Moreover, η ( t, · ) is the jump intensity measure of process x t given by (68).Consequently, x t is stochastically continuous if and only if ( b, A, η ) is a L´evy triple.Conversely, given an extended L´evy triple ( b, A, η ) on X , there is an inhomogeneous L´evyprocess x t in G with x = o , unique in distribution, such that x t is represented by ( b, A, η ) . roof The theorem may be proved by essentially repeating the proof of Theorem 2 forthe corresponding results on G , interpreting a product xy and an inverse x − on X = G/K as S ( x ) y and S ( x ) − by choosing a section map S as in the preceding discussion, andtaking various functions and sets to be K -invariant. As noted in [10], the results in [3] forstochastically continuous inhomogeneous L´evy processes in G hold also on X . In particular,Lemma 18, which summarizes part of [3], holds also on X . Note that because µ ni = µ t n i − ,t ni and ν t ( · ) are K -invariant, their means b ni and h t are K -invariant, and so is b nt . It then followsfrom (69) that A n ( t, f ) is Ad( K )-invariant for K -invariant f ∈ C b ( G ), and (22) holds on X with A ( t ) being a covariance matrix function on X . ✷ Remark 2 As in Remark 1 for the representation on G , η ( t, · ) in Theorem 30 is independentof the choice for the basis { ξ j } of p and coordinate functions φ j on X = G/K , A ( t ) isindependent of { φ j } and the G -invariant operator P nj,k =1 A ( t ) ξ j ξ k on X is independent of { ξ j } . However, η may depend on the choice for the origin o in X . Remark 3 We note that the representation of inhomogeneous L´evy processes in G by atriple ( b, A, η ) with b t of finite variation, as stated in Theorem 3, holds also on X = G/K with essentially the same proof, where b i ( t ) are components of b t under the basis { ξ i } of p .Because the action of K on X fixes o = eK , it induces a linear action on the tangentspace T o X at o . The homogeneous space X = G/K is called irreducible if the K -action on T o M is irreducible, that is, it has no nontrivial invariant subspace. By the identification of p and T o X via the differential of π : G → X , X is irreducible if and only if p has no nontrivialAd( K )-invariant subspace. Because π ◦ exp: g → X is diffeomorphic from a neighborhood V of 0 in p onto a neighborhood U of o in X , the only K -invariant point in U is o . Forexample, the n -dimensional sphere S n = O ( n + 1) /O ( n ) is irreducible.Let V above be convex. The coordinate functions φ i on X may be chosen so that P ni =1 φ i ( x ) ξ i ∈ V for all x ∈ X . Then the mean b = e P i µ ( φ i ) ξ i of any distribution µ on X belongs to U . Let X be irreducible. If µ is K -invariant, then so is b and hence b = o . Becausethe extended drift b t of an inhomogeneous L´evy process in X , represented by ( b, A, η ), is thelimit of b nt in (29). it follows that b t = o for all t ≥ X , up to a constant multiple, there is a unique K -invariant inner product on T o X . This implies that there is a unique Ad( K )-invariant innerproduct on p , and hence an Ad( K )-invariant matrix is a multiple of the identity matrix. Tosummarize, we record the following conclusion. Proposition 31 On an irreducible X = G/K , any covariance matrix function A ( t ) is givenby A ( t ) = a ( t ) I for some nondecreasing continuous function a ( t ) with a (0) = 0 , where I isthe identity matrix. Moreover, if the coordinate functions φ i are chosen as above, then the xtended drift b t of an inhomogeneous L´evy process in X is trivial, that is, b t = o for t ≥ . By (69), for K -invariant H ⊂ X and K -invariant measure µ on X , R H µ ( dx ) P nj =1 φ j ( x ) ξ j is a K -invariant vector in p and hence is 0 by the irreducibility of X = G/K . By Propo-sition 31, b t = o and A t = a ( t ) I , and hence z t = x t . The integral R t R X ( · · · ) η c ( ds, dx ) in(71) is the limit of R t R U c ( · · · ) η c ( ds, dx ) as a K -invariant neighborhood U of o decreases to o , and then the third term of the integrand may be removed. It follows that this integralcombined with the sum P u ≤ t in (71) may be written as R t R X [ f ( x s τ ) − f ( x s )] η ( ds, dτ ), wherethe integral is understood as the principal value, that is, as the limit of R t R U c [ · · · ] η ( ds, τ ) asa K -invariant neighborhood U of o shrinks to o . By Theorem 30, we obtain the followingsimple form of martingale representation on an irreducible X = G/K . Theorem 32 For an inhomogeneous L´evy process x t in an irreducible X = G/K with x = o , there is a unique pair ( a, η ) of a continuous nondecreasing function a ( t ) with a (0) = 0 andan extended L´evy measure function η ( t, · ) on X such that ∀ f ∈ C ∞ c ( X ) , f ( x t ) − Z t n X j =1 ξ j ξ j f ( x s ) da ( s ) − Z t Z X [ f ( x s τ ) − f ( x s )] η ( ds, dτ ) (72) is a martingale under F xt , where R t R X [ · · · ] η ( ds, dτ ) is the principal value as described aboveand is bounded. Moreover, η is the jump intensity measure of x t given by (68). Consequently, x t is stochastically continuous if and only if η is a L´evy measure function.Conversely, given a pair ( a, η ) as above, there is an inhomogeneous L´evy process x t in X with x = o , unique in distribution, such that (72) is a martingale under F xt . An inhomogeneous L´evy process g t in G is called K -conjugate invariant if its transitionfunction P s,t is K -conjugate invariant, that is, if P s,t ( f ◦ c k ) = ( P s,t f ) ◦ c k for f ∈ B + ( G ) and k ∈ K , where c g : x gxg − is the conjugation map on G . Note that when g = e , this isequivalent to saying that the process g t has the same distribution as kg t k − for any k ∈ K . Theorem 33 Let g t be a K -conjugate invariant inhomogeneous L´evy process in G with g = e . Then x t = g t o is an inhomogeneous L´evy process in X = G/K . Conversely, if x t is an inhomogeneous L´evy process in X with x = o , then there is a K -conjugate invariantinhomogeneous L´evy process g t in G such that processes x t and g t o are equal in distribution. Proof Let g t be a K -conjugate invariant inhomogeneous L´evy process in G with g = e andlet x t = g t o . Then for s < t and a section map S on X , S ( x s ) = g s k s for some F gs -measurable k s ∈ K , and S ( x s ) − x t = k − s g − s g t o d = g − s g t o (equal in distribution because of K -conjugateinvariance of g t ). This shows that S ( x s ) − x t is independent of F gs and its distribution isindependent of S . By Proposition 28, x t is an inhomogeneous L´evy process in X .39ow let x t be an inhomogeneous L´evy process in X with x = o , represented by anextended L´evy triple ( b, A, η ) on X . We will define an extended L´evy triple (¯ b, ¯ A, ¯ η ) on G from ( b, A, η ). Choose a section map S on X such that S ( x ) = e P ni =1 φ i ( x ) ξ i o for x containedin a K -invariant neighborhood U of o . By (69), ∀ k ∈ K and x ∈ U, kS ( x ) k − = S ( kx ) . (73)The basis { ξ , . . . , ξ n } of p is now extended to be a basis { ξ , . . . , ξ d } of g so that { ξ n +1 , . . . , ξ d } is a basis of k . Let ψ , . . . , ψ d ∈ C ∞ c ( G ) be the associated coordinate functionson G , that is, g = e P di =1 ψ i ( g ) ξ i for g near e . Because kgk − = e P ψ i ( g )Ad( k ) ξ i , ∀ k ∈ K, d X i =1 ( ψ i ◦ c k ) ξ i = d X i =1 ψ i [Ad( k ) ξ i ] (74)holds near e . Replacing ψ i by ψψ i for a K -conjugate invariant ψ ∈ C ∞ c ( G ) such that ψ = 1near e and ψ = 0 outside a small neighborhood of e , we may assume (74) holds on G .Because for x near o , e P ni =1 φ i ( x ) ξ i = S ( x ) = e P di =1 ψ i ( S ( x )) ξ i , it follows that ψ i ◦ S = φ i , ≤ i ≤ n, and ψ i ◦ S = 0 , n + 1 ≤ i ≤ d (75)holds near o . The ψ i may be modified outside a neighborhood of e so that (75) holds on X and(74) still holds on G . This may be accomplished by replacing ψ i by ( φ ◦ π ) ψ i +(1 − φ ◦ π )( φ i ◦ π )for i ≤ n and by ( φ ◦ π ) ψ i for i > n , where φ is a K -invariant smooth function on X suchthat φ = 1 near o and φ = 0 outside a small neighborhood of o .Let ¯ h t = e P j ν t ( φ j ) ξ j . Then π (¯ h t ) = b − t − b t . By (69), ¯ h t is K -conjugate invariant. Thereis a partition 0 = t < t < t < · · · < t n ↑ ∞ such that b − t i b t ∈ U for t ∈ [ t i , t i +1 ).Define ¯ b t = S ( b t ) for t < t and ¯ b t = ¯ b t − ¯ h t , and inductively let ¯ b t = ¯ b t i S ( b − t i b t ) for t i ≤ t < t i +1 and ¯ b t i +1 = ¯ b t i +1 − ¯ h t i +1 . Then ¯ b t is a K -conjugate invariant extended drift in G with π (¯ b t ) = b t . Let ¯ A ( t ) be the d × d matrix given by ¯ A ij ( t ) = A ij ( t ) for i, j ≤ n and¯ A ij = 0 otherwise. Then ¯ A ( t ) is a covariance matrix function on G that is Ad( K )-invariant,that is, [Ad( k )] ¯ A ( t )[Ad( k )] ′ = ¯ A ( t ) for k ∈ K , where [Ad( k )] is the matrix representing thelinear map Ad( k ): g → g under the basis { ξ , . . . , ξ d } . For f ∈ B + ( G ), let¯ η ( t, f ) = Z X Z K f ( kS ( x ) k − ) dkη ( t, dx ) . (76)Then ¯ η ( t, · ) is a K -conjugate invariant measure function. Its continuous part is ¯ η c ( t, f ) = R R f ( kS ( x ) k − ) dkη c ( t, dx ) and for each t > 0, ¯ ν t = ¯ η ( { t } , · ) + [1 − ¯ η ( { t } , G )] δ e is given by¯ ν t ( f ) = R R f ( kS ( x ) k − ) dkν t ( dx ).By (69) and (75) on X , and (74) on G , P dj =1 ¯ ν t ( ψ j ) ξ j = P dj =1 R X R K [ ψ j ( kS ( x ) k − ) ξ j ] dkν t ( dx ) = P dj =1 R X R K { ψ j ( S ( x ))[Ad( k )] ξ j } dkν t ( dx ) = P nj =1 R X { R K φ j ( x )[Ad( k )] ξ j } dkν t ( dx ) = P nj =1 ν t ( φ j ) ξ j .This shows that ¯ h t = e P j ν t ( φ j ) ξ j defined earlier is the mean of ¯ ν t .40e now show that ¯ η is an extended L´evy measure function on G by checking ¯ η c ( t, k ψ · k ) < ∞ and P u ≤ t ¯ ν u ( k ψ · − ψ · (¯ h u ) k ) < ∞ , where k ψ · k is the Euclidean norm of ψ · = ( ψ , . . . , ψ d ).Let V = π − ( U ). Then ¯ η ( t, V c ) = η ( t, U c ) < ∞ . By (73) and (75), the first conditionfollows from ¯ η c ( t, k ψ · k V ) = η c ( t, k φ · k U ) < ∞ . To verify the second condition, we mayassume all ¯ ν t are small, then ψ i (¯ h t ) = ¯ ν t ( ψ i ) and the second condition may be written as P u ≤ t R G ¯ ν u ( dx ) k R G ¯ ν u ( dy )[ ψ · ( x ) − ψ · ( y )] k < ∞ . Because P u ≤ t ¯ ν u ( V c ) = P u ≤ t ν u ( U c ) < ∞ ,this is equivalent to P u ≤ t R V ¯ ν u ( dx ) k R V ¯ ν u ( dy )[ ψ · ( x ) − ψ · ( y )] k < ∞ , but this is the same as P u ≤ t R U ν u ( dx ) k R U ν u ( dy )[ φ · ( x ) − φ · ( y )] k < ∞ , which holds by (8).Now let g t be an inhomogeneous L´evy process in G with g = e , represented by theextended L´evy triple (¯ b, ¯ A, ¯ η ). Write g t = ¯ z t ¯ b t and for f ∈ C ∞ c ( G ), let ¯ M t f (¯ z · ) be themartingale given in (10) with ( b, A, η ) and φ i replaced by (¯ b, ¯ A, ¯ η ) and ψ i . By the K -conjugate invariance of ¯ b and ¯ η , the Ad( K )-invariance of ¯ A , and (74), it is easy to showthat for k ∈ K , ¯ M t ( f ◦ c k )(¯ z · ) = ¯ M t f ( k ¯ z · k − ). This means that kg t k − = k ¯ z t k − ¯ b t is alsorepresented by (¯ b, ¯ A, ¯ η ). By the unique distribution in Theorem 2, g t and kg t k − have thesame distribution as processes, and hence g t is K -conjugate invariant.Let x ′ t = g t o . By the first part of Theorem 33 which has been proved, x ′ t is an inhomo-geneous L´evy process in X . We may write x ′ t = z t b t with z t = ¯ z t o . By the construction of(¯ b, ¯ A, ¯ η ) from ( b, A, η ), it can be shown that ¯ M t ( f ◦ π )(¯ z · ) is the martingale in (71). Thisshows that x ′ t is represented by ( b, A, η ). By the unique distribution in Theorem 30, x ′ t and x t are equal in distribution. ✷ Acknowledgment Helpful comments from David Applebaum and an anonymous refereehave led to a considerable improvement of the exposition. References [1] Applebaum, D. and Kunita, H., “L´evy flows on manifolds and L´evy processes on Liegroups”, J. Math. 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U of e . Let τ be the U c -displacement time of process z nt from a stoppingtime σ ≤ T . For f ∈ C ∞ c ( G ) with f ( e ) = 1, f = 0 on U c and 0 ≤ f ≤ G , let f σ ( z ) = f (( z nσ ) − z ). Then for δ > P ( τ < δ ) = E [ f σ ( z nσ ) − f σ ( z nσ + τ ); τ < δ ] ≤ E [ f σ ( z nσ ) − f σ ( z nσ + τ ∧ δ )]= E [ M nσ f σ − M nσ + τ ∧ δ f σ ] − E [ Z σ + τ ∧ δσ Hf σ ( z n · )] − E { X σ