Inhomogeneous Lévy processes in Lie groups and

homogeneous spaces

Ming Liao (Auburn University)

Summary We obtain a representation of an inhomogeneous Lévy process in a Lie group or a homogeneous space in terms of a drift, a matrix function and a measure function. Because the stochastic continuity is not assumed, our result generalizes the well known Lévy-Itô representation for stochastic continuous processes with independent increments in and its extension to Lie groups.

2000 Mathematics Subject Classification Primary 60J25, Secondary 58J65.

Key words and phrases Lévy processes, Lie groups, homogeneous spaces.

## 1 Introduction

Let , , be a process in with rcll paths (right continuous paths with left limits). It is said to have independent increments if for , is independent of (the -algebra generated by , ). The process is called a Lévy process if it also has stationary increments, that is, if has the same distribution as . It is well known that the class of Lévy processes in coincides with the class of rcll Markov processes with translation-invariant transition functions . The celebrated Lévy -Khinchin formula provides a useful representation of a Lévy process in by a triple of a drift vector, a covariance matrix and a Lévy measure.

More generally, a rcll process in with independent but possibly non-stationary increments may be called an inhomogeneous Lévy process. It is easy to show that such processes, also called additive processes in literature, coincide with rcll inhomogeneous Markov processes with translation invariant two-parameter transition functions (see [14]).

A Lévy process is always stochastically continuous, that is, a.s. (almost surely) for any fixed . An inhomogeneous Lévy process may not be stochastically continuous, but if it is, then the well known Lévy-Itô representation holds ([8, chapter 15]):

(1) |

where is a continuous path in with , called a drift, is a -dim continuous Gaussian process of zero mean and independent increments, is an independent Poisson random measure on with intensity being a Lévy measure function (to be defined later), and is the compensated form of .

The distribution of the Gaussian process is determined by its covariance matrix function and the distribution of the Poisson random measure is determined by its intensity measure . Thus, the distribution of a stochastically continuous inhomogeneous Lévy process in is completely determined by the time-dependent triple .

In general, an inhomogeneous Lévy process may not be a semimartingale (see [7, II]). By Itô’s formula, it can be shown that the process is a semi-martingale and for any smooth function on of compact support,

(2) |

is a martingale, where and denote respectively the first and second order partial derivatives of . This in fact provides a complete characterization for the distribution of a stochastically continuous inhomogeneous Lévy process in .

This martingale representation is extended to stochastically continuous inhomogeneous Lévy process in a general Lie group in [3], generalizing an earlier result in [13] for continuous processes. A different form of martingale representation in terms of the abstract Fourier analysis is obtained in [5], where the processes considered are also stochastically continuous.

A Lévy process in a Lie group is defined as a rcll process with independent and stationary (multiplicative) increments, that is, for , is independent of and has the same distribution as . Such a process may also be characterized as a rcll Markov process in whose transition function is invariant under left translations on . The classical triple representation of Lévy processes was extended to Lie groups in [6] in the form of a generator formula. A functional form of Lévy-Itô representation for Lévy processes in Lie groups was obtained in [1].

An inhomogeneous Lévy process in a Lie group is defined to be a rcll process that has independent but not necessarily stationary increments, which may also be characterized as an inhomogeneous rcll Markov process with a left invariant transition function (see Proposition 1). As mentioned earlier, the stochastically continuous inhomogeneous Lévy processes in may be represented by a martingale determined by a triple .

The notion of Lévy processes as invariant Markov processes, including inhomogeneous ones, may be extended to more general homogeneous spaces, such as a sphere. A homogeneous space may be regarded as a manifold under the transitive action of a Lie group with being the subgroup that fixes a point in . As on a Lie group , a Markov process in with a -invariant transition function will be called a Lévy process, and an inhomogeneous Markov process with a -invariant transition function will be called an inhomogeneous Lévy process. Although there is no natural product on , the increments of a process in may be properly defined, and a Lévy process in may be characterized by independent and stationary increments, with inhomogeneous ones just by independent increments, same as on . See section §7 for more details.

The purpose of this paper is to study the representation of inhomogeneous Lévy processes, not necessarily stochastically continuous, in a Lie group or a homogeneous space . We will show that such a process is represented by a triple with possibly discontinuous and . This is in contrast to the stochastically continuous case when the triple is continuous in . A non-stochastically-continuous process may have a fixed jump, that is, a time such that . It turns out to be quite non-trivial to handle fixed jumps which may form a countable dense subset of .

Our result applied to leads to a martingale representation of inhomogeneous Lévy processes in , for which the martingale (2) contains an extra term:

where is the distribution of the fixed jump at time with mean . This complements the Fourier transform representation on obtained in [7, II.5].

We will obtain the representation of inhomogeneous Lévy processes not only on a Lie group but also on a homogeneous space . For this purpose, we will formulate a product structure and develop certain invariance properties on so that the formulas obtained on , and their proofs, may be carried over to . We will also show that an inhomogeneous Lévy process in is the natural projection of an inhomogeneous Lévy process in . On an irreducible , such as a sphere, the representation takes a very simple form: there is no drift and the covariance matrix function for some function . This is even simpler than the representation on .

Our interest in non-stochastically-continuous inhomogeneous Lévy processes lies in the following application. Let be a Markov process in a manifold invariant under the action of a Lie group . It is shown in [10] that may be decomposed into a radial part , transversal to -orbits, and an angular part , in a standard -orbit . The can be an arbitrary Markov process in a transversal subspace, whereas given , the conditioned is an inhomogeneous Lévy process in the homogeneous space . For example, a Markov process in invariant under the group of orthogonal transformations is decomposed into a radial Markov process in a half line and an angular process in the unit sphere.

In [10], the representation of stochastically continuous inhomogeneous Lévy processes in is used to obtain a skew product decomposition of a -invariant continuous Markov process in which the angular part is a time changed Brownian motion in , generalizing the well known skew product of Brownian motion in .

When the -invariant Markov process is discontinuous, its conditioned angular part is typically not stochastically continuous. For example, a discontinuous -invariant Lévy process in is stochastically continuous, but its conditioned angular part is not. The present result will provide a useful tool in this situation.

We note that an important related problem, the weak convergence of convolution products of probability measures to a two-parameter convolution semigroup, which is the distribution of an inhomogeneous Lévy process, is not pursued in this paper. The stochastically continuous case is studied in [12] and [11].

Our paper is organized as follows. The main results on Lie groups are stated in the next section with proofs given in the four sections to follow. In §3, we establish the martingale representation, associated to a triple , for a given inhomogeneous Lévy process, under two technical assumptions (A) and (B). These assumptions are verified in §4. We then prove the uniqueness of the triple for a given process in §5, and the uniqueness and the existence of the process for a given triple in §6. The results on homogeneous spaces are presented in §7. We follow the basic ideas in [3], but with many changes, not only to deal with fixed jumps but also to clarify some obscure arguments in [3]. To save the space, we rely on some results proved in the first half of [3] (which is relatively easier than the second half), and have to omit some tedious computations after having stated the main technical points.

All processes are assumed to be defined on the infinite time interval , but it is clear that the results also hold on a finite time interval. For a manifold , let be the Borel -field on and let be the space of nonnegative Borel functions on . Let , and be respectively the spaces of continuous functions, bounded continuous functions, and smooth functions with compact supports on .

## 2 Inhomogeneous Lévy processes in Lie groups

Let be an inhomogeneous Lévy process in a Lie group . By definition, is a rcll process in with independent increments, that is, is independent of for . It becomes a Lévy process in if it also has stationary increments, that is, if the distribution of depends only on . Let be the distribution of . Then for ,

This shows that is an inhomogeneous Markov process with transition function given by .

Note that is left invariant in the sense that its transition function is left invariant, that is, for and , where is the left translation on . Conversely, if is a left invariant inhomogeneous Markov process in , then , where is the identity element of . This implies that has independent increments. We have proved the following result.

###### Proposition 1

A rcll process in is an inhomogeneous Lévy process if and only if it is a left invariant inhomogeneous Markov process.

Note that the proof of Proposition 1 may be slightly modified to show that a rcll process in is a Lévy process if and only it is a Markov process with a left invariant transition function (see also [9, Proposition 1.2]).

A measure function on is a family of -finite measures on , , such that for , and as . Here the limit is set-wise, that is, for . The left limit at , defined as the nondecreasing limit of measures as , exists and is .

A measure function may be regarded as a -finite measure on and may be written as , given by for and . Conversely, any measure on such that is a -finite measure on for any may be identified with the measure function .

A measure function is called continuous at if , and continuous if it is continuous at all . In general, the set : , of discontinuity times, is at most countable, and

(3) |

where is a continuous measure function, called the continuous part of , and

Recall is the identity element of . The jump intensity measure of an inhomogeneous Lévy process is the measure function on defined by

(4) |

the expected number of jumps in by time . The required -finiteness of will be clear from Proposition 7 later, and then the required right continuity, as , follows from (4). It is clear that the process is continuous if and only if .

Note that and , and for , and

(5) |

is the distribution of , so is continuous if and only if is stochastically continuous.

Let be a basis of the Lie algebra of . We will write for the exponential map on . There are such that for near , called coordinate functions associated to the basis of . Note that . The -truncated mean, or simply the mean, of a -valued random variable or its distribution is defined to be

(6) |

The distribution is called small if its mean has coordinates , that is,

(7) |

This is the case when is sufficiently concentrated near .

As defined in [3], a Lévy measure function on is a continuous measure function such that

(a) , and for any and neighborhood of .

(b) for any , where .

The notion of Lévy measure functions is now extended. A measure function on is called an extended Lévy measure function if (a) above and (b) below hold.

Note that and at a continuous point of , and hence the sum in (8) has at most countably many nonzero terms. If is continuous, then (b) becomes (b), and hence a continuous extended Lévy measure function is a Lévy measure function.

It can be shown directly that conditions (b) and (b) are independent of the choice for coordinate functions and basis . This is also a consequence of Theorem 2 below.

A continuous path in with will be called a drift. A symmetric matrix-valued function will be called a covariance matrix function if , (nonnegative definite) for , and is continuous. A triple of a drift , a covariance matrix function and a Lévy measure function will be called a Lévy triple on . For , the adjoint map : is the differential of the conjugation at . It is shown in [3] that if is a stochastically continuous inhomogeneous Lévy process in with , then there is a unique Lévy triple such that and

(9) | |||||

is a martingale under . Conversely, given a Lévy triple , there is a stochastically continuous inhomogeneous Lévy process with represented as above, unique in distribution.

Therefore, a stochastically continuous inhomogeneous Lévy process in is represented by a triple just like its counterpart in . The complicated form of the martingale in (9) with the presence of the drift , as compared with its counterpart (2) on , is caused by the non-commutativity of . This representation is extended to all inhomogeneous Lévy processes in , not necessarily stochastically continuous, in Theorem 2 below.

An extended drift on is a rcll path in with . A triple of an extended drift , a covariance matrix function and an extended Lévy measure function will be called an extended Lévy triple on if for any .

We note that for , our definition of an extended Lévy triple corresponds to the assumptions in [7, Theorem II 5.2]. In particular, (a) and (b) corresponds to (i) - (iii), and to (v), but (iv) in [7] is redundant as it is implied by the other conditions.

A rcll process in is said to be represented by an extended Lévy triple if with ,

(10) | |||||

is a martingale under the natural filtration of for any .

We note that in (10), the -integral is absolutely integrable and the sum converges absolutely a.s., and hence is a bounded random variable. This may be verified by (b) and Taylor’s expansions of at and at (see the computation in the proof of Lemma 15).

###### Theorem 2

Let be an inhomogeneous Lévy process in with . Then there is a unique extended Lévy triple on such that is represented by as defined above. Moreover, is the jump intensity measure of process given by (4). Consequently, is stochastically continuous if and only if is a Lévy triple.

Conversely, given an extended Lévy triple on , there is an inhomogeneous Lévy process in with , unique in distribution, that is represented by .

Remark 1 As the jump intensity measure, is clearly independent of the choice for the basis of and coordinate functions . By Lemma 11, is independent of and the operator is independent of .

In Theorem 2, the representation of process is given in the form of a martingale property of the shifted process . By Theorem 3 below, when the drift has a finite variation, a form of martingale property holds directly for .

A rcll path in a manifold is said to have a finite variation if for any , has a finite variation on any finite -interval. Let be a family of smooth vector fields on such that form a basis of the tangent space of at any . If is a continuous path in with a finite variation, then there are uniquely defined real valued continuous functions of finite variation, , with , such that

(11) |

Indeed, under local coordinates on , and , where , then are determined by . Conversely, given as above, a continuous path of finite variation satisfying (11) may be obtained by solving the integral equation

for by the usual successive approximation method.

The functions above will be called components of the path under the vector fields . When , these vector fields will be the basis of chosen before.

More generally, if is a rcll path in of finite variation, then there is a unique continuous path in of finite variation with such that, letting be the components of ,

(12) |

To prove this, cover the path by finitely many coordinate neighborhoods and then prove the claim on a Euclidean space. The path will be called the continuous part of .

###### Theorem 3

Let be an inhomogeneous Lévy process in with , represented by an extended Lévy triple . Assume is of finite variation. Then

(13) |

is a martingale under for any .

Conversely, given an extended Lévy triple with of finite variation, there is an inhomogeneous Lévy process in with , unique in distribution, such that (13) is a martingale under for .

The above theorem follows directly from Theorem 2 and the next lemma. Note that because has a finite variation and for all but finitely many , and hence . The absolute integrability of the -integral in (13) can be verified as in (10).

###### Lemma 4

Proof Let us assume (10) is a martingale. Let : (as ) be a sequence of partitions of with mesh as , and let . Then

where is a martingale with and

Let for . Let be the set of fixed jump times of . We may assume as in the sense that , for large . Then is a step function and as uniformly for bounded . It follows that for ,

where , , is a martingale, and is the right translation on . As , by the uniform convergence , , and by the continuity of and ,

and, by (12) and noting that is continuous in and as ,

where is a bounded martingale. It follows that converges boundedly to a martingale given by (13).