Journal of Mathematics

Volume 2013 (2013), Article ID 540471, 15 pages

http://dx.doi.org/10.1155/2013/540471

Research Article

The Concentration Function Problem for Locally Compact Groups Revisited: Nondissipating Space-Time Random Walks, -Decomposable Laws, and Their Continuous Time Analogues

Wilfried Hazod

Faculty of Mathematics, TU Dortmund University, D-44221 Dortmund, Germany

Received 28 November 2012; Accepted 15 July 2013

Academic Editor: Ralf Meyer

Copyright © 2013 Wilfried Hazod. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The concentration function problem for locally compact groups is concerned with the structure of groups admitting adapted nondissipating random walks. It is closely related to discrete relatively compact M- or skew convolution semigroups and corresponding space-time random walks, and to -decomposable laws, respectively, where denotes an automorphism. Analogous results are obtained in the case of continuous time: nondissipating Lévy processes are related to relatively compact distributions of generalized Ornstein-Uhlenbeck processes and corresponding space-time processes and to -decomposable laws, respectively with denoting a continuous group of automorphisms acting as contracting mod. a compact subgroup.

1. Introduction

Let be a locally compact group, an (w.l.o.g.) adapted probability measure. , more precisely, the random walk , is called nondissipating (also nonscattering) if for some compact subset the (right) concentration functions fail to converge to , with time . To avoid trivialities, throughout is supposed to be noncompact, else any random walk would be nondissipating.

If the random walk is nondissipating, is relatively (right) shift compact, equivalently, is relatively compact [1].

Furthermore, if denotes the smallest closed normal subgroup of containing the support , then . Hence there exists such that with .

Denoting the restriction of the inner automorphism to by , we obtain , and is represented as , with hence the random walk is representable as , for all (with , .)

The existence of nondissipating adapted random walks has a strong impact on the structure of : it is shown in [1] that , where is a compact -invariant subgroup of and (for ), respectively, denote the contractible, respectively, the -contractible subgroups. As easily seen, for a random walk , is representable as , , (“” denoting convolution on ) and hence satisfies the relation , . We call a sequence satisfying such a relation “discrete time M-(or Mehler-)-semigroup” (in fact, in analogy to the continuous time case, the (discrete) semigroup of transition kernels is called discrete Mehler semigroup.) This M-(convolution)-semigroup is also called skew convolution semigroup or -semigroup. Moreover, if is nondissipating, the M-semigroup is relatively compact, a crucial property in the sequel. Conversely, discrete M-semigroups define space-time random walks on the space-time building   . So, nondissipating adapted random walks (on ) correspond in a 1-1-manner to relatively compact M-semigroups (on ), cf. Theorem 6 later. We furthermore show that exists and is -decomposable, that is, with relatively compact M-semigroups of cofactors   ; furthermore, is right -invariant. Conversely, for any such measure with cofactors , we have . Thus -decomposable measures , respectively, and the cofactors generate nondissipating random walks and vice versa.

The first part of the paper may be considered as an overview of more or less known results around the concentration function problem and its relations to M-semigroups and -decomposability. These results are needed as motivation and as a tool box for proofs in the second part which is concerned with the continuous time analogues. The random walk is replaced by a continuous convolution semigroup , the distributions of a Lévy process. At first (Propositions 12 and 13), we show that is nondissipating and adapted iff some (hence all) skeleton random walk   is nondissipating; it is adapted on , the group generated by . Furthermore, we show that the subgroup is independent of and is a normal subgroup in (not only within the group ), and we observe . Thus the results for discrete times apply easily to the continuous time setup. Furthermore, , and there exists a continuous one-parameter group such that with , . Moreover, in that case, there exists a homogeneous group (i.e., a contractible Lie group) and a compact -invariant subgroup , such that , .

Theorems 20 and 22 contain the continuous time analogues of Theorem 6, in particular, also a characterization of relative compactness of M-semigroups in terms of logarithmic, respectively, first order moments of the corresponding Lévy measure (on resp., .)

One of the aims of the paper is to point out that different branches of investigations lead to the same or equivalent objects: (1) nondissipating random walks, respectively, continuous convolution semigroups, (2) -decomposable laws and the subclasses of (semi-) stable and self-decomposable laws, and (3) (relatively compact) skew (discrete or continuous) convolution semigroups. While in the past in the cases (2) and (3) most investigations were concerned with continuous time models, the emphasis in the first case was on discrete time models.

All these investigations—for discrete and continuous times—are related to the structure of contractible, respectively, -contractible subgroups , respectively, of locally compact groups. See for example, [2–5], [6, Ch. III], [1, 7–15], and the literature mentioned there.

For the history of the concentration function problem for random walks on locally compact groups the reader is referred to the survey of Jaworski [1] showing previous developments and a recent state of investigations: beginning with the pioneer works [16–20] to the investigations [1, 11, 13].

Continuous time models (for the cases (2) and (3)) had been investigated in the past in different papers. Beginning with the pioneer work [21] (with slightly different representations) to [22, 23], [6, § 2.14]. See also [24] for Mehler hemigroups and embedding of discrete time models into continuous time ones. In [22] the reader will find some more hints to the literature, in particular also to vector spaces, beyond the locally compact group case.

2. Discrete Time: Nondissipating Random Walks

Recall the following notations: denotes a locally compact group with unit . denotes the filter basis of Borel neighbourhoods of the unit. For , the set of probabilities, let be the image of under the inverse mapping . Denotes convolution on , the th convolution power, , where denotes the point measure in . w.l.o.g. is supposed to be adapted; that is, is the closed group generated by the support . denotes the smallest closed normal subgroup containing . The concentration function of the random walk is defined as for compact . The probability measure , or more precisely, the random walk is called nondissipating if fails to converge to with , for some compact . Recall that we defined afore for and a compact -invariant subgroup : and . is contracting if . And if a contracting automorphism exists, is called contractible. Analogously we define -contracting automorphisms, respectively, -contractible groups, where denotes a compact -invariant subgroup. will throughout denote the set of accumulation points, and denotes the closure of a set .

We collect some properties of adapted random walks:

Fact 1. (a) is nondissipating iff is relatively (right) shift compact; that is, for some , is relatively compact. Equivalently, iff is relatively compact. Indeed, then exists.

In that case, if is noncompact, the following assertions (b)–(f) hold:

(b) , hence the shifts can be chosen as for some .

(c) The restriction to of the inner automorphism , , is considered as automorphism of , and hence has a canonical representation (with product for , where .)

(d) There exists a compact subgroup such that . If is compact then .

(e) and are representable as , is identified with a probability of . Hence by induction, , , with , , , and is identified with . Furthermore, is relatively compact. (Indeed, as is convergent, exists.) To distinguish, in the sequel “” will denote convolution on while convolution on is denoted by “”. (Thus we have .)

(f) Put . Then is the smallest closed -invariant normal subgroup of containing . (The closed group generated by may be smaller than .)

(a) See [1, Corollary 3.2] and the literature mentioned there. In particular, [25], [12, Theorem 1]. For the convergence of , see [26, 27].

(b), (c) See [1, Theorem 3.4], [25]. (d) See [1, Theorems 3.5 and 3.9]. If is compact, then, as is compactly contracting, follows.

(e) The representation of is immediately verified, and relative compactness follows by [1, Theorem 3.9].

(f) is adapted, hence is generated by . , hence is -invariant and . By definition, is the smallest closed normal subgroup with these properties. It is easily shown by examples that may be larger than the group generated by

As already mentioned, is always supposed to be noncompact, else any random walk would be nondissipating. If is compact then (by Fact 1(d)), and any “space-time” random walk , , is nondissipating.

We have to recall some more notations and facts: let be a locally compact group. A function is subadditive (resp., a “gauge”) if (resp., for some constant ). is submultiplicative if . As immediately seen, if is subadditive (and ≥0), then is submultiplicative (and ≥1), and if is submultiplicative (and ≥1), then is subadditive (and ≥0). Thus is subadditive and is submultiplicative. We will always assume tacitly that is Borel measurable. Therefore, is locally bounded (see [28, Proposition 1]), hence there exist such that . To any we fix such an (and a neighbourhood ).

Let be compactly generated, for some symmetric compact . Then we define a subadditive function . According to [28, Proposition 1], any gauge, in particular any subadditive function , is dominated by , that is,

Definition 1. (a) Let . A subadditive function is called -function (cf. [1, Definition 2.6]) if for some , , we have , . is a regular -function if in addition , is relatively compact for some (hence all) and iff . In fact, is continuous and -bi-invariant (see [1, Propositions 2.8 and 2.10].)

Let , and let denote a nonnegative measure on such that is a bounded measure outside any neighbourhood of the unit.

(b) , respectively, possess finite first-order moments if , respectively, for all subadditive Borel functions and are defined previously.

(c) Analogously, we define for a group : , respectively, possess finite logarithmic moments if , respectively, for all regular -functions , as in (c).

In fact, if the integrals in (b), respectively, in (c) are finite for some (as above), they are finite for all . And if the integrals are finite for some relatively compact , they are finite for all .

Remark 2. The definition of logarithmic moments depends on -functions and hence on and on the particular automorphism . If and is a regular -function, and if we define , , then is generated by and (cf. [1, Proposition 2.14].) If is an other relatively compact symmetric neighbourhood generating , then (see [28, Remarque 3].) Hence possesses finite first-order moments if , equivalently, is integrable. Hence possesses finite first-order moments iff (see [1, Proposition 2.14, Corollary 2.15].)

Note that for contractible groups logarithmic moments are frequently defined in terms of “group-norms” or “homogeneous norms” (e.g., in [29, 2.6, 2.11], [6, 2.7.26 (f)]), [30]), generalizing the situation of vector spaces. We show in Lemma A.2 and Theorem 22 (cf. Remark 21) that on homogeneous groups this definition is equivalent to Definition 1. In fact, if is a contracting automorphism, then there exist and a continuous group such that (cf., e.g., [6, 2.8.14]), and therefore, Lemma A.2, respectively, Theorem 22 apply also for discrete times.

Fact 2. Let be a locally compact group; , , and, as before, . Let . ( denotes the normalized Haar measure on the compact subgroup .) Then the following assertions are equivalent:

(i) is weakly convergent ; that is, exists.(ii) possesses finite logarithmic moments.(iii) possesses finite first-order moments.

The equivalence of the assertions “(i)-(ii)” is shown in the context of invariant metrics on , respectively, of regular -functions, in [1, Theorem 3.9, Propositions 4.2 and 4.3]. For “” see [1, Corollary 2.15].

The equivalence of the conditions “(i)-(ii)” (as well as equivalence with stochastic and almost sure convergence of corresponding random variables) is folklore for vector spaces (cf. [31], or e.g., the monograph [32, Lemma 3.6.5]), for homogeneous groups [29], [6, 2.14.24], for general contractible groups [30] (in the context of “group norms”.) As long as almost sure convergence of corresponding random variables is considered, Lévy's equivalence theorem is involved, and hence the underlying group is usually supposed to be second countable. In the situation here a superfluous condition as is metrizable, cf. [1, Theorems 2.5, Proof of Theorem 3.9]. Furthermore, in most cases, metrizability may be supposed without loss of generality.

Remark 3. The groups and , generated by the supports of the random walk, respectively, the M-semigroup (cf. Definition 4 below) are -compact. Therefore, and are representable as projective limits of second countable locally compact groups (cf. [33, page 101, Example 11] or [34, Theorem 5.2]): , respectively, , where are compact, -invariant subgroups and . Therefore in many investigations second countability may be assumed w.l.o.g. For example, is convergent, respectively, is relatively compact, respectively, satisfies (2) below iff for all the projections to the quotients share this property.

Definition 4. (a) A sequence in satisfying

is called discrete M-(convolution)-semigroup (also called Mehler semigroup, -semigroup, skew convolution semigroup, etc.)

Note that we are mainly interested in the behaviour of the random walk, respectively, the M-semigroup for large times ; hence the definition of is of minor importance. We could replace by an idempotent such that , that is, . denoting the Haar measure on a compact subgroup .

(b) is called -decomposable, if for some cofactor we have . Then by induction, . denotes the set of cofactors of with respect to . As immediately seen, one can choose for ; hence the cofactors can be chosen as (discrete) M-semigroup .

(c) is called right -invariant if (for some compact subgroup ).

(d) For short, is --decomposable if is -decomposable and right -invariant.

is closed, hence according to the shift-compactness theorem (cf. [35, III, Theorems 2.1 and 2.2] (for metrizable groups), [36, Theorem 1.2.21]), is compact in . If is relatively compact, in particular, if , then, again by the shift-compactness theorem, is relatively compact.

Remark 5. In the following, we distinguish carefully between adapted and nonadapted space-time random walks. Let be a nondissipating space-time random walk on with relatively compact M-semigroup . If the random walk is not adapted let denote the closed subgroup generated by . Hence, on , is nondissipating and adapted. Therefore there exists a closed subgroup in , normal in , containing the supports , and , such that with there exists a relatively compact M-semigroup (with respect to ) , such that for . Let ; hence with , . Then iff . Note that since .

Furthermore, , , and . Hence , analogously, , . And thus .

is the smallest closed normal subgroup of containing , and ; hence it follows that a nondissipating space-time random walk is adapted on iff for any closed subgroup and , with normalizing and satisfying

In this sense we may w.l.o.g. assume that a nondissipating random walk on a locally compact group is adapted, respectively, a relatively compact M-semigroup on a locally compact group (with respect to ) satisfies condition (3). (If necessary replacing by , , as previously mentioned.)

Relations between nondissipative random walks, M-semigroups, and -decomposable laws are collected in the following.

Theorem 6. (a) Let be a locally compact group and a nondissipating adapted random walk. Then there exist a closed normal subgroup and , such that with , one has , for a compact, -invariant subgroup . Furthermore, the random walk is representable as , where is a relatively compact M-semigroup (with respect to ), and condition (3) is satisfied.

(b) Conversely, let be a locally compact group, , and let be a relatively compact M-semigroup in , and assume condition (3) to be satisfied. Then the space-time random walk is nondissipating and adapted. ( denotes ). Hence (a) applies. In particular, for some compact -invariant subgroup .

(c) Non-dissipative random walks, respectively, relatively compact M-semigroups define -decomposable laws.

In fact, in case (a) or (b) we obtains that exists and is --decomposable.

(d) Conversely, -decomposable laws define nondissipating random walks, respectively, relatively compact M-semigroups.

Precisely, assume that . Let be --decomposable. Then there exists a M-semigroup of cofactors of such that . In particular, is relatively compact, and then and hence (a), respectively, (b) apply.

(e) Let , . An adapted space-time random walk is nondissipating, respectively, a M-semigroup satisfying (3) is relatively compact iff possesses finite logarithmic moments, respectively, iff possesses finite first-order moments.

Proof. For (a), (b) see Fact 1 and the discussion in Remark 5.

(c) Let be a relatively compact M-semigroup on . Since is uniformly -contracting on compact subsets, the accumulation points of are supported by . Assume, for some subnets, that and and, w.l.o.g., for all . Then on the one hand, with on the other. Hence follows. That is, exists and is obviously right -invariant.

Furthermore, yields --decomposability of .

(d) Conversely, assume to be --decomposable. is relatively compact as is -contracting, and all accumulation points are supported by . Right -invariance of implies that . As afore mentioned, we can choose , in , hence as an M-semigroup of cofactors. Hence and yield relative compactness of and (2) according to the shift-compactness theorem.

(e) Follows by Fact 2, Remark 3, and [1, Corollary 2.15, resp., Proposition 4.3].

More generally we have the following.

Remark 7. (a) Let be a locally compact group, , as previously mentioned. Let , be a M-semigroup, respectively, the corresponding space-time random walk. As immediately seen, if the M-semigroup is only supposed to be relatively shift compact, then the random walk is nondissipating. And if the random walk is not adapted, Remark 5 applies.

(b) Let be a locally compact group, , and let be -decomposable, with as M-semigroup of cofactors. Then , , yields that is relatively shift compact and therefore (a) applies.

(c) If in (b) in addition is assumed to be relatively compact (e.g., if as in Theorem 6), then, again by the shift-compactness theorem, the M-semigroup of cofactors is already relatively compact.

Note that in Theorem 6 and Remark 7, if and , will not in general exist. See for example, Example 3.16 in [1], with compact and an infinite number of accumulation points. Further types of examples are obtained in the following way.

Example 8. (a) Let be a contractible locally compact group, with contracting , hence . Let be a relatively compact M-semigroup, . As , exists (Theorem 6(c) with ). Let be a finite cyclic group, let generating with , and assume for some that (e.g., ). Put , define as , and put finally . Then is a relatively compact M-semigroup (with respect to ) in , where . But and , both infinitely often; hence is not convergent. In fact, . But we have .

(b) A trivial case: let be a monothetic compact group with dense subgroup (for some ). Put , obviously -contracting; hence . Then is a M-semigroup (with respect to ), which is relatively compact in . But is not convergent, in fact, . Here we have .

In [12, Theorem 1], it is shown for a nondissipating random walk that there exists a measurable dense -invariant subgroup on which a sequence acts as contracting mod. a compact subgroup , and are concentrated on cosets of . Hence is weakly contracting mod. on (cf. the definition in [12]), and thus , for some -invariant subgroup [12, Theorem 5]. In interesting examples, for example, in the context of (semi-) stable laws, we have for a compact -invariant subgroup . In fact, investigations of the structure of contractible and -contractible sub-groups had also been pushed forward in connection with investigations of (semi-) stable laws. See for example, [4–6, 10] and the literature mentioned there. However, concentration functions were not used as an essential tool (except in connection with random time substitutions and geometric (semi-) stability, cf. [6, 37]). Nevertheless it is worth to point out that semistable laws provide interesting examples of relatively compact M-semigroups, hence of nondissipating random walks.

Example 9 (cf., e.g., [6, § 3.5]). Let be a continuous convolution semigroup in , and let be a locally compact group. Let and . is (strictly) -semistable if for all , . The idempotent is a normalized Haar measure on a compact -invariant subgroup . If is second countable, the contraction subgroups and are Borel sets, and we have for all . Hence we assume that , and therefore (as mentioned previously).

Let now . Then . Hence is --decomposable with cofactors . In particular, (and hence all ) possess finite logarithmic moments. (This could also be proved by direct calculation.) In that example, are right -invariant and exists. (And thus trivially also ).

If is closed and , then is closed, hence and . is known to be closed if there exist contracting continuous one-parameter groups of automorphisms [5] moreover for -adic Lie groups [38] and more generally for totally disconnected groups if is a “tidy” automorphism [14]. And we have , at least in the case of totally disconnected groups or of Lie groups. See [11]; see also [13–15] for previous results. For Lie groups see [5], [6, Theorem 3.2.13]. For more information concerning the decomposition property , cf. [39, 40]. However, will not in general be closed, not even on a 2-dimensional torus (cf., e.g., [5], [6, Example ]). Then, as mentioned previously if w.l.o.g. , is weakly contracting on , and hence for some compact, -invariant subgroup . Note that, if is not closed, additional conditions are needed to guarantee existence of (cf. Example 8 above).

The following result will explain in more detail the interplay between limit behaviour of relatively compact M-semigroups and -decomposability.

Proposition 10. Let . Let be a relatively compact M-semigroup. To avoid measurability problems, is supposed to be second countable. Put and .

Then one has the following

(a)   (for any   ).(b).(c)If at least some   and     belong to a commutative  -sub-semigroup     of    ,   then any   is   -decomposable with    for some   , .(d)If      for some compact   -invariant subgroup    and   ,   for all   ,   and if one assumes in addition that   ,   then    exists, and    is   --decomposable with cofactors   .

Proof. By assumption, is compactly contracting on .

(a)Let , and . Assume that that w.l.o.g. for all (else pass to subsequences). Then , . Hence , and analogously, follows.(b)Assume that . Then ; hence , and analogously follows.(c)Let ; hence for some . Let . w.l.o.g. assume that , else replace in the following by . Then, for all , and (along a subsequence) for some . Hence . That is, . Whence the assertion with .(d)Since by assumption are right -invariant, the accumulation points are right -invariant and supported by ; thus . Whence and thus .

The following example will illustrate Proposition 10.

Example 11. Let as before.

(a) Let be a -semistable continuous convolution semigroups (as in Example 9), and let be -decomposable with cofactor , with denoting the homothetic transformation on . Then the subordinated measure is -decomposable with cofactor . The set is a commutative sub-semigroup of containing and the cofactors (cf. Proposition 10(c)).

Let for some compact -invariant subgroup . Assume for all , then also and for all .

(b) Let be -decomposable with cofactors , . e.g., let be -semistable continuous convolution semigroups and Assume that and commute. e.g., for semistable laws, assume that commute for all Then is -decomposable with cofactor . But note that if are semistable and if is in general not -semistable for

If , for -invariant compact subgroups , then . If and commute, then is a compact -invariant subgroup and , the subgroup generated by , .

(c) In general, without commutativity assumptions, in (b) will not in general be -decomposable. In particular, for point measures , we obtain: is in general not -decomposable. In fact, without commutativity assumption, the left shifted measure, , is, as easily seen, -decomposable with cofactors . Representing as we see that is -decomposable, with , but in general not -decomposable.

(d) To simplify notations we assume that . Let and assume that now . Then and (in (c)) are concentrated on cosets of compactly contractible subgroups of : and . Hence, with we have: and . Furthermore, , where , .

(e) In case (c), (d) we determine the sets of accumulation points and (Proposition 10):

, hence , where . fact, obviously . On the other hand, assume and , w.l.o.g. . Then and

Furthermore, . Thus .

3. Continuous Time: Nondissipating Continuous Convolution Semigroups

Next we replace the random walk by a continuous convolution semigroup (the distribution of a Lévy process on , if is metrizable). In contrast to the discrete time case, now the idempotent is uniquely determined.

W.l.o.g. we assume that is generated by . For short, is called adapted then. is nondissipating if the concentration functions do not converge to 0 for some compact (for ).

For any the random walk is called skeleton random walk. Note that a skeleton random walk do not need to be adapted on hence we introduce for the subgroups as smallest closed subgroup containing . Hence is adapted on . We define to be the smallest closed normal subgroup of containing . Hence and . (For later use we identify with .)

Again, to avoid trivialities, throughout in the sequel, is assumed to be noncompact (else any continuous convolution semigroup would be nondissipating). Hence also the closed subgroups are noncompact and the results of Section 1 apply. In fact, if for some , is compact, then is uniformly tight. But then is uniformly tight. Therefore (and all groups ) are compact.

First we compare the behaviour of concentration functions of continuous convolution semigroups and of their skeleton random walks.

Proposition 12. Let be an adapted continuous convolution semigroup. Then the following assertions are equivalent.

(i)   is nondissipating.(ii)For all   [( )     some]    ,   the skeleton random walk     is nondissipating (and adapted on   ).(iii)   is relatively (right) shift compact.(iv)   is relatively compact. (Then     exists.)(v)For all   [( )   some]    ,     (considered as measure on   )   has finite first-order moments.(vi)For all   [( )   some]    ,     (considered as measure on   )    has finite logarithmic moments. That is, for all regular   -functions     on     one has   .

The precise formulations of conditions (v) (resp., (vi)) are: for all subadditive functions on (resp., for all regular -functions on ) we have (resp., ). Note that for a subadditive function on the restriction is subadditive. However, the sets of subadditive functions defining moments on , might have no subadditive extension to . Analogously, for , the sets of - and -functions may be different (even if , as will be shown in Proposition 13 later).

Proof. for all probabilities , hence is decreasing. Hence iff . Thus, as immediately seen, “”. And obviously we have “” and “”, “”. The equivalences “” (for any fixed ) follow by Theorem 6, and hence we also have . Convergence of follows by Fact 1(a): For all    exists. Hence for commensurable , in particular, for all rational . Furthermore, for , is uniformly continuous. Whence for all real easily follows.

“”. By Fact 1(a), is equivalent to shift compactness of . Let such that is relatively compact. Then, being a continuous convolution semigroup, the set is relatively compact (where , if ). Whence follows.

“”. Let be relatively compact. Hence for any there exists a compact such that for all , . Therefore, fails to converge to .

A priori , the smallest closed normal subgroup of containing , might not be normal in and might depend on . In order to apply the results of Section 1 we have to overcome these difficulties.

Proposition 13. Let be an adapted continuous convolution semigroup. Then one has for all , and .

Proof. Obviously, and for all , for some (all) . First we consider dyadic numbers and then proceed to real applying continuity of .

Let . Then obviously since . For all it follows, , hence

In fact, for all , and hence for all . Therefore,

Put , and hence, by (5)

Claim 1. We have

By (6) it suffices to show that . According to (4) and (5) we have

Analogously we have . Consequently, yields . Thus, as normalizes , we obtain .

Claim 2.   . is a closed normal subgroup of such that . But is minimal with this property. Whence , thus .

Claim 3.    . yields (as ). And ; therefore (since ). Hence, according to the definition, follows. Together we obtain , whence the assertion follows

By induction, we obtain for all , and .

Claim 4.    for all .

At first we consider dyadic .

(1), whence . Hence .(2). Thus .

Hence for all dyadic . Continuity of easily yields , and .

Replacing by any real we obtain analogously for , and again for all real . In particular, . Consequently, for all .

Now the assertions follow immediately, and the proof is complete.

Consequently, for any nondissipating continuous convolution semigroup we obtain that is adapted iff . Indeed, we have the following.

Proposition 14. Let be nondissipating and adapted. Then, with the afore-introduced notations, there exists a continuous one-parameter group such that and .

Let, for , denote the restriction of the inner automorphism to . Then is a continuous one-parameter group in . And one has (with group operation , ). Furthermore, may be represented as , where is a continuous M-semigroup (with respect to ) (cf. Definition 18 later). Since it follows that for some compact subgroup .

Proof. As shown before, there exists such that (for all ). Hence, denoting the canonical homomorphism, and we obtain with . Hence and therefore are continuous one-parameter semigroups, extendible to groups, and thus . Indeed, if were compact, would be compact (hence finite) for any , a contradiction to Fact 1(b). Finally, there exists a continuous one-parameter group in with , , (cf., e.g., [41]), whence the assertion follows.

To show that splits as a semi-direct product, put , where . Assume to be nontrivial. Then is compact, hence is compact, in contradiction to , as shown previously.

Hence is closed in containing , whence .

The existence of nondissipating adapted continuous convolution semigroups has strong influence on the structure of .

Corollary 15. Assume, as in Proposition 14, to be adapted and nondissipating, and assume in addition to be totally disconnected. Then , is compact and the -semigroup is a continuous convolution semigroup. If moreover is assumed to be totally disconnected then , hence is compact, the trivial case excluded afore.

is totally disconnected then the continuous automorphism group is trivial; hence . Since is trivial, any continuous -semigroup (with respect to ) is a continuous convolution semigroup. Thus we obtain immediately that is compact if is nondissipating.

If moreover, is totally disconnected, then is trivial, whence is compact

In the discrete time case it was essentially used that . As mentioned in the previous proof, in the continuous time case, for all , we have . It turns out (cf. the next proposition) that is independent of . Furthermore, in the continuous time case the structure of is nicer. Put and for some compact, -invariant subgroup .

Proposition 16. Let, as before, .

 (a)  With the notations of Proposition 14 one has   .

In fact, for any locally compact group , admitting a continuous one parameter group and a -invariant compact subgroup one has the following.

 (b)  For all   , , in particular,  . (c)    is closed,    -invariant, connected, and isomorphic to a contractible Lie group (hence a homogeneous group), and     is a closed   -invariant subgroup of   .   If     is compact, then   . (d)  , and there exists a continuous homomorphism   , , , , such that   . In particular,      is compactly generated. (e)  The restrictions     and     are continuous one-parameter groups, with     consisting of inner automorphisms of   . (f)     satisfies the following consistency conditions   , , .

Conversely, any group arises in that way.

Proof. (a) As mentioned in the previous proof, for any , we have , denoting a compact -invariant subgroup .

It follows easily that for all . Indeed:

(1)For , , it follows that . . On the other hand, is compact and -invariant for all , in particular, is -invariant. As is compactly contracting , we obtain , whence follows(2)Let . Then there exists a sequence such that with . For we have with and . Hence . The first terms converge to , and . Hence (since ), on the other hand, .

Replacing by , , in view of Step . we obtain . Continuity of finally yields . And analogously, . Whence does not depend on . According to (b), we obtain . Thus (a) is proved.

(b) See [2], [6, Lemma 3.2.26]. (Note that the group is multiplicatively parametrized.) (c), (d) see [6, Theorem 3.2.32], [4]. For (e) see [5, Lemma 3.3], [6], and for (f) see [6, 3.3.4], [4].

For later use (in the proof of Theorem 22 and in Lemmas A.1–A.5) we mention the following.

Proposition 17. With the notations introduced afore (in Proposition 16) one has the following. There exists a continuous one-parameter subgroup such that for all for some —hence in particular, —and commutes element wise with .

Consequently, the automorphism group belongs to the centralizer .

Proof. is contracting mod. , is closed and is -invariant. Therefore, is a closed subgroup of , representable as semidirect product . According to [42, Proposition 9.4] or [43, Proposition 1.24], there exists a subgroup , , such that . That is, for some and for all and all .

In the sequel, in the proof of Theorem 20, respectively, in Lemmas A.1–A.5 we will always assume that and .

Now we define in analogy to the discrete time case the following.

Definition 18. Let be a locally compact group, as before.

 (a) is a (continuous time) M-semigroup (with respect to )—also called skew convolution semigroup, -semigroup, or distribution of a generalized Ornstein-Uhlenbeck process—if is continuous and if the following cocycle equation is satisfied:

 In the following we are interested in relatively compact M-semigroups. (b) is -decomposable (or -self-decomposable), if for all , . Again, denotes the set of cofactors. is --decomposable if in addition, is right -invariant.

In analogy to Remark 5 we note the following.

Remark 19. Let be a locally compact group, . Let be a space-time continuous convolution semigroup on . If is not adapted then there exists a closed subgroup on which it is adapted. is a closed subgroup and denotes a continuous one-parameter group in the normalizer of . Furthermore, where is a continuous M-semigroup with respect to , with .

Therefore, is adapted for all closed subgroups and all continuous one-parameter groups in the normalizer of , satisfying for all

We have the continuous time analogues of Theorem 6(a)–(d).

Theorem 20. Let, as before (Proposition 16), , , , be probabilities on the on the space-time building . Obviously, is a continuous convolution semigroup iff is a continuous M-semigroup with respect to . is idempotent and , for all , hence with , , and . ( denotes a compact subgroup of .)

(a)If is nondissipating and adapted then is relatively compact and satisfies condition (10). Furthermore, for some compact -invariant subgroup .(b)Conversely, if is relatively compact and satisfies condition (10) then the continuous convolution semigroup is nondissipating and adapted. Hence (a) applies.(c)In (a), respectively, (b), exists and is --decomposable with cofactors .(d)Conversely, let and let be --decomposable. Then there exists a nondissipating continuous convolution semigroup , with a relatively compact continuous M-semigroup in such that for all , and furthermore, .

Proof. (a) is proved as in the discrete time case, Theorem 6(d). (For aperiodic groups see [22, Proposition ], [23, Theorem 3.16]).

(b) and (c) are proved analogous to Theorem 6(b) and (c).

(d) Let be --decomposable. Again as in the case of discrete times, Theorem 6(d), for any , there exists a relatively compact (discrete) M-semigroup of cofactors (with respect to ), such that . In fact, for any selection , , it follows as in Theorem 6(d) that and . We have to show that the cofactors can be chosen as continuous M-semigroup in . For this purpose we need some tools from the embedding problem for infinitely divisible laws on groups. Recall that is a continuous M-semigroup iff is a continuous convolution semigroup (in ). So we have to prove the existence of , embeddable into a continuous convolution semigroup such that , (cf., for aperiodic groups, in particular simply connected nilpotent Lie groups, [23, Proposition 3.6]; see also [22, Theorem B]).

Claim 1. There exist infinitely divisible , with roots where and . Furthermore, is embeddable into a rational convolution semigroup . Moreover, the root sets are compact and consist of measures , with .

for , , put ; hence for all . Put ; hence .

According to the shift-compactness theorem, is compact for all . For , put (with ), and .

By Tychonoff's theorem, there exists a subnet satisfying , equivalently, , for all . Continuity of convolution yields , (for all ).

We have shown that is infinitely divisible, and by construction, the roots satisfy ; hence, is embeddable into a rational convolution semigroup , (cf. [36, 44]), and the construction easily yields moreover , with ,

Let with . Then, as easily seen, with . Hence, . Therefore, in particular, is compact for all .

In order to show that a continuous convolution semigroup can be selected, put . As mentioned previously, and .

Claim 2. is relatively compact in with all accumulation points of the form , .

with , . Let , and put . Then and yield that is relatively compact and

Let . Therefore, is representable as for , with ; hence . And, by definition of , it follows that , for all .

Siebert's embedding theorem (see [44, 5. Satz 1, 6. Satz 1], [36, Theorem 3.5.4]) shows the existence of a continuous convolution semigroup of the form for all , with where , hence , . In particular, . Hence, as immediately seen, .

Since is a continuous convolution semigroup, is a continuous M-semigroup of cofactors of . Whence the assertion.

For the following Theorem 22 we need some preparations.

Let be a locally compact group and a continuous convolution semigroup. The Lévy measure is defined by , for vanishing in a neighbourhood of the unit. If as before, and , , then, as easily seen, the Lévy measure is supported by and . Recall that Lévy measures are bounded outside neighbourhoods of . We recall a result of Siebert (see [45, Theorem 1], [46, Theorem 5]).

Fact 3. Let be a submultiplicative function. Let , such that . Then we have

Thus for subadditive functions we have with : iff . Equivalently, iff .

As mentioned before, if the integral on the right side of (11) is finite for , then it is finite for any . And, if it is finite for some relatively compact , it is finite for all .

The following Steps 1–5 are more or less folklore (though it seems to be hard to find precise references), respectively, they are easy generalizations of known results. Therefore we postpone the proofs to the Appendix (as Lemmas A.1 to A.5). Let throughout in the sequel with . Recall (cf. Proposition 16) that we have , where is a homogeneous group, that is, a contractible simply connected nilpotent Lie group, and . We will tacitly assume, according to Proposition 17, that and commute element wise. Recall that a homogeneous group norm (adapted to ) is a continuous subadditive function satisfying iff , and , where denotes a group of dilations defined by (cf. for example, [6, 2.1.10, 2.7.26 (d)], [29]).

Step 1 (cf. Lemma A.1). There exists a homogeneous norm on which is in addition -invariant.

Step 2 (cf. Lemma A.2). Define the function by , as in Step 1. Then is subadditive, continuous and satisfies , iff and furthermore, is -bi-invariant. (We call pseudo-group norm adapted to .)

Moreover, for any there exist constants , satisfying for all the growth conditions

For large one can put for example, , . The automorphism norm is defined as . In particular, , for and for .

Step 3 (cf. Lemma A.3). Let be a regular -function for some , , . (See Definition 1.)

Then we obtain , ( defined in Step 2.)

Remark 21. As mentioned before, in the situation , a contractible (nilpotent) Lie group, it follows that the definition of finite logarithmic moments via group norms (e.g., in [6, 29, 30]) and Definition 1(d) coincide. Note that and are not -functions—they are -functions for dilations , as the proof shows—but these functions have similar properties (cf. (12)).

Step 4 (cf. Lemma A.4). With the notations of the preceding Steps we have: the functions , is continuous, submultiplicative, ≥2 and satisfies iff . Therefore is subadditive and . Moreover is relatively compact and belongs to for all .

Step 5 (cf. Lemma A.5). Let, as before, and . and are compactly generated for any neighbourhood , generates , and, for example, for any , generates .

Let be a regular -function, for some . Let, for , where . Then generates , and for any symmetric relatively compact with , we have . (For the definition of cf. (1).) In particular, for all , it follows that .

Next we characterize—in analogy to Theorem 6(e)—nondissipativity by moment conditions. For the continuous time case we introduce additional equivalent conditions (), (), () which depend on but not on the particular automorphisms .

Theorem 22. Let, as before, with , a homogeneous group. Let be an adapted continuous convolution semigroup with Lévy measure . Let denote the pseudo-group norm on introduced in Step 2 via the -invariant homogeneous norm on (cf. Step 1). Then the following assertions are equivalent.

 (a)   is nondissipating. (b)  For some (any) , has finite first-order moments (on ). ()  For some (any) , for a compact -invariant neighbourhood . (c)   has finite first-order moments (on ), that is, for all subadditive and . (d)  For some (any) , has finite logarithmic moments, that is, for any regular -function , one has . ()  . (e)   (considered as measure on ) has finite logarithmic moments, that is, for all , for all regular -functions, for all . ()  , for all .

Proof. According to Proposition 12 and Theorem 20 we know that condition (a) is equivalent with shift compactness of a skeleton random walk for some (hence any) . Hence (a) is equivalent to (b1) For some [any] , for any subadditive function on , (i.e., has finite first-order moments on .)

Obviously, “(b1)(b)”. By Fact 2, “(b1)(d)”; furthermore, by Step 3 (cf. Lemma A.3), “(d)()” and “(e*)(e)”.

According to Step 5 (Lemma A.5), (b), and therefore (b)(b*) for any , equivalently, for any , , (since ). Let us mention that (b1) , and since , we obtain again “(b1)(b)” Applying Siebert's result, Fact 3, to the subadditive function (resp., to for any -invariant relatively compact symmetric ) yields “(b*)(c)”.

is subadditive— as in Step 4 (Lemma A.4)—hence again by Fact 3 we obtain (b) for all .

But we have , hence and . Hence the left integral is finite iff holds, whence .

The proof is complete: “(a)(b1)”, “(b1)(b)”, “(b1)(d)”, “(b)(b*)”, “(d)(d*)”, “(e)(e*)”, “(b*)(c)”, “(b)(d*)”, and “(d*)(e*)”.

We close with two examples in analogy with the case of discrete times. The first shows that does not need to exist if , . (For , i.e., , always exists, cf. [22, Proposition A]).

Example 23. Let be a contractible Lie group, hence simply connected and nilpotent, with contracting one-parameter group in , and let be a solenoidal compact group with dense one-parameter subgroup . Put and define by , . Hence . Let be a continuous M-semigroup (with respect to ) in such that exists. As , continuous M-semigroups in (with respect to ) are just continuous convolution semigroups; in particular, is a M-semigroup. It is immediately verified that is a relatively compact M-semigroup in (with respect to ), with nondissipating continuous convolution semigroup .

As is dense in , . In particular, does not exist.

In analogy to Example 9, the next example shows connections between stable laws and nondissipating continuous convolution semigroups (cf., e.g., [6, § 2.3]).

Example 24. Stable laws are particular self-decomposable laws. To show this we have to switch between additive and multiplicative parametrizations of continuous one-parameter groups.

Let be a simply connected nilpotent Lie group and be a contracting continuous group with multiplicative parametrization, for , and for all . A continuous convolution semigroup in is called -stable if , for all , equivalently, for . Put . We have , for all ; hence is -decomposable with cofactors .

To obtain a continuous M-semigroup of cofactors we have to switch to additive parametrization: is a continuous one-parameter group satisfying and for all . And with this notations we obtain for .

As immediately verified, is a relatively compact continuous M-semigroup with (and ); hence it defines a nondissipating continuous convolution semigroup , . Note that in this (trivial) case, the M-semigroup belongs to , respectively, to the orbit : .

Analogously, stable laws with nontrivial idempotents on generate -decomposable laws (see [6, 3.5]).

A variety of examples of (self-) decomposable laws are obtained via subordination as in Example 11: Let be a continuous convolution semigroup which is stable with respect to as above. Let be a -decomposable probability measure on , where denotes the homothetic transformation on . Then, as immediately verified, the subordinated measure is -decomposable. In particular, if is an exponential distribution, we obtain geometric stable laws (or geometric semistable laws, if is semi stable, cf. [6, § 2.13 resp., 2.14.36]. There the results are formulated for aperiodic groups, hence for M-semigroups with trivial idempotents. But the proofs do not depend on that assumption).

Appendix

Group Norms and -Functions

Let, as before, be a locally compact group, , a continuous one-parameter group such that .

Lemma A.1. There exists a homogeneous norm on which is in addition a -invariant.

Proof. Put, as before, . defines a group of dilations, that is, a multiplicative parametrized continuous group , , and a subadditive function such that , iff and for all , . Furthermore, commutes with all automorphisms which commute with all .

In fact, let denote the Lie algebra of . Then , and the correspondence is expressed by . can be defined via the Jordan decomposition of the exponent   , defined by (cf. [29, Proof of Proposition 2.4], [6, 2.1.13]). Whence the assertion follows

As by assumption (Proposition 17) it follows that .

Put , with denoting the Haar measure. As immediately seen, is symmetric, subadditive, with iff , compact in for , and is -invariant by construction. Furthermore, .

Lemma A.2. Define the function by . Then is subadditive and continuous, satisfies , iff , and furthermore, is -bi-invariant.

Moreover, for any there exist constants , , satisfying for all the growth conditions

For large one can put for example . The automorphism norm is defined as . In particular, one has , for and for .

Proof. Obviously, is continuous, iff , that is, iff . is subadditive: .

According to [29, Proposition 2.5(a) and (3.1)], [6, 2.10.15(b) and (2.14.11)], there exist satisfying the growth conditions (A.1) for and . (We have for large . If put for sufficiently large .)

Thus by the definition of the growth conditions (A.1) for follow.

Finally, for , shows -bi-invariance of the pseudo-group norm   .

Lemma A.3. Let be a regular -function for some , (for some ) (cf. Definition 1).

Then one obtains , ( defined in Lemma A.2).

Proof (similar to proof of Lemma 2.9 in [1]). With we have . Put , . Hence is increasing and . is a regular -function and hence , and thus all are compact neighbourhoods in , and by definition of , . is continuous and subadditive, hence locally bounded. Thus there exist constants , such that for all . According to Lemma A.2, we obtain for , hence for :

On the other hand, for , , equivalently, . Inserting this in (A.2) yields

and analogous estimates for .

Together, there exist constants such that , whence follows.

Lemma A.4. With the notations of the preceding lemmas define by .

Then is continuous, ≥2, submultiplicative and satisfies iff . Therefore, is subadditive and . Moreover is relatively compact and belongs to for all .

Proof (compare with Propositions 4.6 in [29], 2.14.28 in [6]). For short, we write instead of . We have .

The automorphism norm, restricted to , is submultiplicative, thus , satisfies , , and we have . Hence

Whence submultiplicativity of follows. Furthermore, , and iff and , that is, iff . The last assertion follows since .

Lemma A.5. Let, as before, and . is compactly generated: for any symmetric , generates , and, for example, for any , generates . Let be a regular -function, for some . Let, for , , where .

Then generates and for any symmetric relatively compact with , one has . In particular, for all it follows that .

Proof. Let again . By assumption, and commute element wise. We have , . Put , then is a subgroup of , such that . Hence , where the action is trivial on , that is, .

is connected, hence any neighbourhood generates . Thus generates .

The resting assertions follow immediately since , (since is -bi-invariant, cf. [1, Proposition 2.8]). Hence according to [28, Remarque 3], for any .

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