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

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 𝜏-decomposable laws, respectively, where 𝜏 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 𝑇-decomposable laws, respectively with 𝑇=(𝜏_𝑡) denoting a continuous group of automorphisms acting as contracting mod. a compact subgroup.

1. Introduction

Let 𝔾 be a locally compact group, 𝜆∈ℳ¹(𝔾) an (w.l.o.g.) adapted probability measure. 𝜆, more precisely, the random walk {𝜆^𝑘}_𝑘≥0, is called nondissipating (also nonscattering) if for some compact subset 𝐶⊆𝔾 the (right) concentration functions 𝑓_𝜆^𝑘(𝐶)∶=sup_𝑥∈𝔾𝜆^𝑘(𝐶𝑥) fail to converge to 0, with time 𝑘→∞. To avoid trivialities, throughout 𝔾 is supposed to be noncompact, else any random walk would be nondissipating.

If the random walk is nondissipating, {𝜆^𝑘}_𝑘≥0 is relatively (right) shift compact, equivalently, {𝜆^𝑘∗𝜆^𝑘}_𝑘≥0 is relatively compact [1].

Furthermore, if 𝑁=𝑁_𝜆 denotes the smallest closed normal subgroup of 𝔾 containing the support supp𝜆∗𝜆, then 𝔾/𝑁≅ℤ. Hence there exists 𝑥∈𝔾 such that 𝜆=𝜈∗𝜀_𝑥 with supp𝜈⊆𝑁.

Denoting the restriction of the inner automorphism 𝑖_𝑥∶𝑧↦𝑥𝑧𝑥⁻¹ to 𝑁 by 𝜏∶=𝑖_𝑥|_𝑁, we obtain 𝐺≅𝑁⋊_𝜏ℤ, and 𝜆 is represented as 𝜆=𝜈(1)⊗𝜀₁, with 𝜈=𝜈(1)∈ℳ¹(𝑁) hence the random walk is representable as 𝜆^𝑘=𝜈(𝑘)⊗𝜀_𝑘, for all 𝑘∈ℤ₊ (with 𝜈(𝑘)∈ℳ¹(𝑁), 𝜈(0)∶=𝜀₀.)

The existence of nondissipating adapted random walks has a strong impact on the structure of 𝔾: it is shown in [1] that 𝑁=𝐶_𝐾(𝜏), where 𝐾 is a compact 𝜏-invariant subgroup of 𝑁 and 𝐶_{𝑒}(𝜏)=∶𝐶(𝜏)∶={𝑥∈𝑁∶𝜏^𝑛(𝑥)→𝑒} (for 𝐾={𝑒}), respectively, 𝐶_𝐾(𝜏)∶={𝑥∈𝑁∶𝜏^𝑛(𝑥)→𝑒mod𝐾} denote the contractible, respectively, the 𝐾-contractible subgroups. As easily seen, for a random walk {𝜆^𝑘=𝜈(𝑘)⊗𝜀_𝑘}⊆ℳ¹(𝑁⋊_𝜏ℤ), 𝜈(𝑘) is representable as 𝜈(𝑘)=⋆^𝑘−1_𝑗=0𝜏^𝑗(𝜈), 𝑘∈ℕ, (“⋆” denoting convolution on 𝑁) and hence satisfies the relation 𝜈(𝑘+ℓ)=𝜈(𝑘)⋆𝜏^𝑘(𝜈(ℓ)), 𝑘,ℓ∈ℤ₊. We call a sequence {𝜈(𝑘)}_𝑘∈ℤ+ 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 𝜏-semigroup. Moreover, if {𝜆^𝑘} is nondissipating, the M-semigroup {𝜈(𝑘)} is relatively compact, a crucial property in the sequel. Conversely, discrete M-semigroups {𝜈(𝑘)}⊆ℳ¹(𝑁) define space-time random walks {𝜆^𝑘=𝜈(𝑘)⊗𝜀_𝑘} on the space-time building   𝔾=𝑁⋊_𝜏ℤ. So, nondissipating adapted random walks (on 𝔾) correspond in a 1-1-manner to relatively compact M-semigroups (on 𝑁), cf. Theorem 6 later. We furthermore show that 𝜌∶=lim_𝑘→∞𝜈(𝑘)⋆𝜔_𝐾 exists and is 𝜏-decomposable, that is, 𝜌=𝜈(𝑘)∗𝜏^𝑘(𝜌) with relatively compact M-semigroups of cofactors   {𝜈(𝑘)}_𝑘∈ℤ+; furthermore, 𝜌 is right 𝐾-invariant. Conversely, for any such measure 𝜌 with cofactors {𝜈(𝑘)}, we have lim𝜈(𝑘)⋆𝜔_𝐾=𝜌. Thus 𝜏-decomposable measures 𝜌, respectively, and the cofactors {𝜈(𝑘)} 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 𝜏-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 {𝜆_𝑡}_𝑡∈ℝ+, the distributions of a Lévy process. At first (Propositions 12 and 13), we show that {𝜆_𝑡}_𝑡∈ℝ+ is nondissipating and adapted iff some (hence all) skeleton random walk  {𝜆^𝑘_𝑡0=𝜆_𝑡0𝑘}_𝑘∈ℤ+ is nondissipating; it is adapted on 𝔾_𝑡0, the group generated by supp𝜆_𝑡0. Furthermore, we show that the subgroup 𝑁∶=𝑁_𝜆𝑡0 is independent of 𝑡₀>0 and is a normal subgroup in 𝔾 (not only within the group 𝔾_𝑡0), and we observe 𝔾=⋃_𝑡𝔾_𝑡. Thus the results for discrete times apply easily to the continuous time setup. Furthermore, 𝔾/𝑁≅ℝ, and there exists a continuous one-parameter group (𝑥(𝑡))_𝑡∈ℝ such that with 𝑇=(𝜏_𝑡∶=𝑖_𝑥(𝑡)|_𝑁)⊆Aut(𝑁), 𝔾≅𝑁⋊_𝑇ℝ. Moreover, in that case, there exists a homogeneous group (i.e., a contractible Lie group) 𝐶 and a compact 𝑇-invariant subgroup 𝐾, such that 𝑁=𝐶_𝐾(𝑇)=𝐶⋊_𝛽𝐾, 𝛽∈Hom(𝐾,Aut(𝐶)).

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 𝑁 resp., 𝔾.)

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) 𝜏-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, 𝐾-contractible subgroups 𝐶(𝜏), respectively, 𝐶_𝐾(𝜏) 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: 𝔾 denotes a locally compact group with unit 𝑒. 𝔘_𝔾 denotes the filter basis of Borel neighbourhoods of the unit. For 𝜆∈ℳ¹(𝔾), the set of probabilities, let 𝜆 be the image of 𝜆 under the inverse mapping 𝑥↦𝑥⁻¹. ∗ Denotes convolution on 𝔾, 𝜆^𝑘 the 𝑘th convolution power, 𝜆⁰∶=𝜀_𝑒, where 𝜀_𝑥 denotes the point measure in 𝑥∈𝔾. w.l.o.g. 𝜆 is supposed to be adapted; that is, 𝔾 is the closed group generated by the support supp𝜆. 𝑁=𝑁_𝜆 denotes the smallest closed normal subgroup containing supp𝜆∗𝜆. The concentration function of the random walk {𝜆^𝑘}_𝑘∈ℤ+ is defined as ℤ₊∋𝑘↦𝑓_𝜆^𝑘(𝐶)∶=sup_𝑥∈𝔾𝜆^𝑘(𝐶𝑥) for compact 𝐶⊆𝔾. The probability measure 𝜆, or more precisely, the random walk {𝜆^𝑘} is called nondissipating if 𝑓_𝜆^𝑘(𝐶) fails to converge to 0 with 𝑘→∞, for some compact 𝐶. Recall that we defined afore for 𝜏∈Aut(𝑁) and a compact 𝜏-invariant subgroup 𝐾: 𝐶_𝐾(𝜏)∶={𝑥∈𝑁∶𝜏^𝑘(𝑥)→𝑒mod𝐾} and 𝐶(𝜏)=𝐶_{𝑒}(𝜏)={𝑥∶𝜏^𝑘(𝑥)→𝑒}. 𝜏∈Aut(𝔾) is contracting if 𝐶(𝜏)=𝔾. And if a contracting automorphism exists, 𝔾 is called contractible. Analogously we define 𝐾-contracting automorphisms, respectively, 𝐾-contractible groups, where 𝐾 denotes a compact 𝜏-invariant subgroup. LIM will throughout denote the set of accumulation points, and 𝐴⁻ denotes the closure of a set 𝐴.

We collect some properties of adapted random walks:

Fact 1. (a) {𝜆^𝑘} is nondissipating iff {𝜆^𝑘} is relatively (right) shift compact; that is, for some 𝑥_𝑘∈𝔾, {𝜆^𝑘∗𝜀_𝑥𝑘⁻¹} is relatively compact. Equivalently, iff {𝜆^𝑘∗𝜆^𝑘} is relatively compact. Indeed, then lim_𝑘𝜆^𝑘∗𝜆^𝑘 exists.
In that case, if 𝔾 is noncompact, the following assertions (b)–(f) hold:
(b) 𝔾/𝑁≅ℤ, hence the shifts can be chosen as 𝑥_𝑘=𝑥^𝑘 for some 𝑥∈𝔾⧵𝑁.
(c) The restriction to 𝑁 of the inner automorphism 𝜏∶=𝑖_𝑥|_𝑁, 𝑧↦𝑥𝑧𝑥⁻¹, is considered as automorphism of 𝑁, and hence 𝔾 has a canonical representation 𝔾=𝑁⋊_𝜏ℤ (with product (𝑔,𝑘)(ℎ,ℓ)=(𝑔𝜏^𝑘(ℎ),𝑘+ℓ) for 𝑔,ℎ∈𝑁,𝑘,ℓ∈ℤ, where 𝑥=(𝑒,1).)
(d) There exists a compact subgroup 𝐾⊆𝑁 such that 𝑁=𝐶_𝐾(𝜏). If 𝑁 is compact then 𝑁=𝐾.
(e) 𝜆 and 𝜈∶=𝜆∗𝜀_𝑥−1 are representable as 𝜆=𝜈⊗𝜀₁∈ℳ¹(𝑁⋊_𝜏ℤ), 𝜈=∶𝜈(1) is identified with a probability of ℳ¹(𝑁). Hence by induction, 𝜆^𝑘=𝜈(𝑘)⊗𝜀_𝑘, 𝑘∈ℤ₊, with 𝜈(0)=𝜀_𝑒, 𝜈(1)=𝜈, 𝜈(𝑘)=★^𝑘−1_𝑗=0𝜏^𝑗(𝜈)=∶𝜈⋆𝜏(𝜈)⋆⋯⋆𝜏^𝑘−1(𝜈)∈ℳ¹(𝑁), and 𝑥 is identified with (𝑒,1). Furthermore, {𝜈(𝑘)}_𝑘∈ℤ+ is relatively compact. (Indeed, as 𝜆^𝑘∗𝜆^𝑘 is convergent, lim𝜈(𝑘)⋆𝜈(𝑘) exists.) To distinguish, in the sequel “∗” will denote convolution on 𝔾 while convolution on 𝑁 is denoted by “⋆”. (Thus we have (𝜇⊗𝜀_𝑘)∗(𝜈⊗𝜀_ℓ)=(𝜇⋆𝜏^𝑘(𝜈))⊗𝜀_𝑘+ℓ.)
(f) Put 𝐿∶=⋃_𝑘∈ℤ+supp𝜈(𝑘). Then 𝑁 is the smallest closed 𝜏-invariant normal subgroup of 𝔾 containing 𝐿. (The closed group generated by 𝐿 may be smaller than 𝑁.)

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

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

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

(f) 𝜆 is adapted, hence 𝔾 is generated by supp𝜈(1)⊗{1}. 𝑁⊲𝔾, hence 𝑁 is 𝜏-invariant and 𝑁⊇𝐿. By definition, 𝑁 is the smallest closed normal subgroup with these properties. It is easily shown by examples that 𝑁 may be larger than the group generated by 𝐿.⟧

As already mentioned, 𝔾 is always supposed to be noncompact, else any random walk would be nondissipating. If 𝑁 is compact then (by Fact 1(d)), 𝑁=𝐾 and any “space-time” random walk {𝜆^𝑘=𝜈(𝑘)⊗𝜀_𝑘}, 𝜈(𝑘)∈ℳ¹(𝑁), is nondissipating.

We have to recall some more notations and facts: let ℍ be a locally compact group. A function 𝑓∶ℍ→ℝ₊ is subadditive (resp., a “gauge”) if 𝑓(𝑥𝑦)≤𝑓(𝑥)+𝑓(𝑦) (resp., ≤𝐶+𝑓(𝑥)+𝑓(𝑦) for some constant 𝐶≥0). 𝑓 is submultiplicative if 𝑓(𝑥𝑦)≤𝑓(𝑥)𝑓(𝑦). As immediately seen, if 𝑓 is subadditive (and ≥0), then 𝑔∶=1+𝑓 is submultiplicative (and ≥1), and if ℎ is submultiplicative (and ≥1), then log(ℎ) is subadditive (and ≥0). Thus log(1+𝑓) is subadditive and 1+log(1+𝑓) is submultiplicative. We will always assume tacitly that 𝑓 is Borel measurable. Therefore, 𝑓 is locally bounded (see [28, Proposition 1]), hence there exist 𝜖>0 such that 𝑈_𝑓,𝜖∶={𝑓<𝜖}∈𝔘_ℍ. To any 𝑓 we fix such an 𝜖 (and a neighbourhood 𝑈⊆𝑈_𝑓,𝜖).

Let ℍ be compactly generated, ℍ=⋃_𝑛≥0𝑉^𝑛 for some symmetric compact 𝑉⊆𝔘_ℍ. Then we define a subadditive function 𝛿_𝑉∶𝑥↦inf{𝑛∈ℤ₊∶𝑥∈𝑉^𝑛}. According to [28, Proposition 1], any gauge, in particular any subadditive function 𝑓, is dominated by 𝛿_𝑉, that is, 𝑓≤𝐴𝛿_𝑉+𝐵forsomeconstants𝐴>0,𝐵≥0. (1)

Definition 1. (a) Let ℍ=𝑁=𝐶_𝐾(𝜏). A subadditive function 𝑓≥0 is called 𝜏-function (cf. [1, Definition 2.6]) if for some 𝑝∈ℕ, 0<𝛿<1, we have 𝑓(𝜏^𝑝(𝑥))=𝛿⋅𝑓(𝑥), 𝑥∈ℍ. 𝑓 is a regular 𝜏-function if in addition 𝑓(𝑥⁻¹)=𝑓(𝑥), {𝑓≤𝛼} is relatively compact for some (hence all) 𝛼>0 and 𝑓(𝑥)=0 iff 𝑥∈𝐾. In fact, 𝑓 is continuous and 𝐾-bi-invariant (see [1, Propositions 2.8 and 2.10].)
Let 𝜇∈ℳ¹(ℍ), and let 𝜂 denote a nonnegative measure on ℍ⧵{𝑒} such that 𝜂 is a bounded measure outside any neighbourhood of the unit.
(b) 𝜇, respectively, 𝜂 possess finite first-order moments if ∫_ℍ𝑓d𝜇<∞, respectively, ∫_ℍ⧵𝑈𝑓d𝜂<∞ for all subadditive Borel functions 𝑓∶ℍ→ℝ₊ and 𝑈=𝑈_𝑓,𝜖∈𝔘_ℍ are defined previously.
(c) Analogously, we define for a group ℍ=𝑁=𝐶_𝐾(𝜏): 𝜇, respectively, 𝜂 possess finite logarithmic moments if ∫_𝑁log(1+𝑓)d𝜇<∞, respectively, ∫_ℍ⧵𝑈log(1+𝑓)d𝜂<∞ for all regular 𝜏-functions 𝑓, 𝑈=𝑈_𝑓,𝜖∈𝔘_ℍ as in (c).
In fact, if the integrals ∫_ℍ⧵𝑈𝑓d𝜂 in (b), respectively, ∫_ℍ⧵𝑈log(1+𝑓)d𝜂 in (c) are finite for some 𝑈=𝑈_𝑓,𝜖 (as above), they are finite for all 𝑈∈𝔘_ℍ. And if the integrals are finite for some relatively compact 𝑈₀∈𝔘_ℍ, they are finite for all 𝑈∈𝔘_ℍ.

Remark 2. The definition of logarithmic moments depends on 𝜏-functions and hence on 𝐾 and on the particular automorphism 𝜏. If ℍ=𝔾=𝑁⋊_𝜏ℤ and 𝑓 is a regular 𝜏-function, and if we define 𝑈={𝑓≤1}, 𝑉∶=𝑈⊗{1}∪𝑈⊗{0}∪{𝑈⊗{1}}⁻¹, then 𝔾 is generated by 𝑉 and log(1+𝑓)∼𝛿_𝑉(⋅,0) (cf. [1, Proposition 2.14].) If 𝑊 is an other relatively compact symmetric neighbourhood generating 𝔾, then 𝛿_𝑉∼𝛿_𝑊 (see [28, Remarque 3].) Hence 𝜆∈ℳ¹(𝔾) possesses finite first-order moments if 𝛿_𝑉, equivalently, 𝛿_𝑊 is integrable. Hence 𝜆=𝜇⊗𝜀_𝑘 possesses finite first-order moments iff ∫_𝑁𝛿_𝑉|_𝑁d𝜇<∞ (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 𝜏 is a contracting automorphism, then there exist 𝑘∈𝑁 and a continuous group 𝑇=(𝜏_𝑡) such that 𝜏^𝑘∈𝑇 (cf., e.g., [6, 2.8.14]), and therefore, Lemma A.2, respectively, Theorem 22 apply also for discrete times.

Fact 2. Let 𝑁 be a locally compact group; 𝜏∈Aut(𝑁), 𝑁=𝐶_𝐾(𝜏), and, as before, 𝔾=𝑁⋊_𝜏ℤ. Let 𝜈(𝑘)=★^𝑘−1_𝑗=0𝜏^𝑗(𝜈(1)). (𝜔_𝐾 denotes the normalized Haar measure on the compact subgroup 𝐾⊆𝑁.) Then the following assertions are equivalent:(i)𝜈(𝑘) is weakly convergent mod𝐾; that is, lim_𝑘𝜈(𝑘)⋆𝜔_𝐾 exists.(ii)𝜈(1) possesses finite logarithmic moments.(iii)𝜆=𝜈⊗𝜀₁ possesses finite first-order moments.

⟦The equivalence of the assertions “(i)-(ii)” is shown in the context of invariant metrics on 𝑁/𝐾, respectively, of regular 𝜏-functions, in [1, Theorem 3.9, Propositions 4.2 and 4.3]. For “(ii)⇔(iii)” 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 𝑁/𝐾 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 𝔾 and 𝑁, generated by the supports of the random walk, respectively, the M-semigroup (cf. Definition 4 below) are 𝜎-compact. Therefore, 𝔾 and 𝑁 are representable as projective limits of second countable locally compact groups (cf. [33, page 101, Example 11] or [34, Theorem 5.2]): 𝑁=lim_←𝑁/Γ_𝛼, respectively, 𝔾=lim_←(𝑁/Γ_𝛼)⋊_𝜏𝛼ℤ, where Γ_𝛼 are compact, 𝜏-invariant subgroups Γ_𝛼⊲𝑁 and 𝜏_𝛼∈Aut(𝑁/Γ_𝛼). Therefore in many investigations second countability may be assumed w.l.o.g. For example, 𝜈(𝑘)⋆𝜔_𝐾 is convergent, respectively, {𝜈(𝑘)} is relatively compact, respectively, satisfies (2) below iff for all 𝛼 the projections to the quotients share this property.

Definition 4. (a) A sequence {𝜈(𝑘)}_𝑘∈ℤ+ in ℳ¹(𝑁) satisfying 𝜈(0)=𝜀_𝑒(1)𝜈()(𝑘),𝜈=∶𝜈,𝑘+ℓ=𝜈⋆𝜏^𝑘(𝜈(ℓ)),𝑘,ℓ∈ℤ₊ (2) is called discrete M-(convolution)-semigroup (also called Mehler semigroup, 𝜏-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 𝑘; hence the definition of 𝜆⁰=𝜈(0)⊗𝜀₀ is of minor importance. We could replace 𝜆⁰ by an idempotent 𝜔_𝐻⊗𝜀_𝑒 such that 𝜆⁰∗𝜆∗𝜆⁰=𝜆, that is, 𝜔_𝐻⋆𝜈⋆𝜏(𝜔_𝐻)=𝜈. 𝜔_𝐻 denoting the Haar measure on a compact subgroup 𝐻⊆𝑁.
(b) 𝜌∈ℳ¹(𝑁) is called 𝜏-decomposable, if for some cofactor 𝜈=𝜈(1)∈ℳ¹(𝑁) we have 𝜌=𝜈⋆𝜏(𝜌). Then by induction, 𝜌=𝜈(𝑘)⋆𝜏^𝑘(𝜌). Cof_𝜌(𝜏^𝑘) denotes the set of cofactors of 𝜌 with respect to 𝜏^𝑘. As immediately seen, one can choose 𝜈(𝑘)=★^𝑘−1_𝑗=0𝜏^𝑗(𝜈(1)) for 𝑘≥2; hence the cofactors can be chosen as (discrete) M-semigroup {𝜈(𝑘)}.
(c) 𝜌 is called right 𝐾-invariant if 𝜌⋆𝜔_𝐾=𝜌 (for some compact subgroup 𝐾).
(d) For short, 𝜌 is 𝐾-𝜏-decomposable if 𝜌 is 𝜏-decomposable and right 𝐾-invariant.

Cof_𝜌(𝜏) 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]), Cof_𝜌(𝜏) is compact in ℳ¹(𝑁). If {𝜏^𝑘(𝜌)} is relatively compact, in particular, if 𝑁=𝐶_𝐾(𝜏), then, again by the shift-compactness theorem, ⋃_𝑘≥0Cof_𝜌(𝜏^𝑘) is relatively compact.

Remark 5. In the following, we distinguish carefully between adapted and nonadapted space-time random walks. Let {𝜆^𝑘=𝜈(𝑘)⊗𝜀_𝑘} be a nondissipating space-time random walk on 𝔾=𝑁⋊_𝜏ℤ with relatively compact M-semigroup {𝜈(𝑘)}⊆ℳ¹(𝑁). If the random walk {𝜆^𝑘} is not adapted let 𝔾_𝜆⊆𝔾 denote the closed subgroup generated by supp𝜆. Hence, on 𝔾_𝜆, {𝜆^𝑘} is nondissipating and adapted. Therefore there exists a closed subgroup 𝑁_𝜆 in 𝔾, normal in 𝔾_𝜆, containing the supports supp𝜆^𝑘∗𝜆^𝑘, and 𝑦∈𝔾_𝜆, such that with 𝜏^′∶=𝑖_𝑦|_𝑁𝜆∈Aut(𝑁_𝜆) there exists a relatively compact M-semigroup (with respect to 𝜏^′) {𝜈^′(𝑘)}⊆ℳ¹(𝑁_𝜆), such that 𝜆^𝑘=𝜈^′(𝑘)∗𝜀_𝑦𝑘 for 𝑘∈ℤ₊. Let 𝑦=(𝑧,1); hence 𝑦^𝑘=(𝑧(𝑘),𝑘) with 𝑧=𝑧(1), 𝑧(𝑘)∈𝑁. Then 𝑦𝑁_𝜆=𝑁_𝜆𝑦 iff 𝜏(𝑁_𝜆)=𝑖_𝑧−1(𝑁_𝜆)=𝑧⁻¹𝑁_𝜆𝑧. Note that 𝜏^′=𝑖_𝑧∘𝜏 since 𝑦=(𝑧,0)(𝑒,1).
Furthermore, 𝜆=𝜈⊗𝜀₁=𝜈∗𝜀_(𝑒,1), 𝜈=𝜈(1), and 𝜆=𝜈^′∗𝜀_(𝑧,1)={𝜈^′⋆𝜀_𝑧}⊗𝜀₁. Hence 𝜈=𝜈^′⋆𝜀_𝑧, analogously, 𝜈(𝑘)=𝜈^′(𝑘)⋆𝜀_𝑧(𝑘), 𝑘∈ℤ₊. And thus supp𝜈(𝑘)⊆𝑁_𝜆⋅𝑧(𝑘).
𝑁_𝜆 is the smallest closed normal subgroup of 𝔾_𝜆 containing supp𝜆^𝑘∗𝜆^𝑘, and 𝜆^𝑘∗𝜆^𝑘=𝜈(𝑘)⋆𝜈(𝑘)=𝜈^′(𝑘)⋆𝜈^′(𝑘); hence it follows that a nondissipating space-time random walk {𝜆^𝑘=𝜈(𝑘)⊗𝜀_𝑘} is adapted on 𝔾 iff for any closed subgroup 𝑁^′⊆𝑁 and 𝑧∈𝑁, with (𝑧,1) normalizing 𝑁^′ and satisfying (1)𝑁^′(1)⋆𝜈(1),(2)𝜏𝑁⊇supp𝜈_′=𝑧⁻¹𝑁^′𝑧⟹𝑁^′=𝑁. (3)

In this sense we may w.l.o.g. assume that a nondissipating random walk {𝜆^𝑘=𝜈(𝑘)⊗𝜀_𝑘} on a locally compact group 𝔾 is adapted, respectively, a relatively compact M-semigroup {𝜈(𝑘)} on a locally compact group 𝑁 (with respect to 𝜏∈Aut(𝑁)) satisfies condition (3). (If necessary replacing 𝔾,𝑁,𝜏,𝜈(𝑘) by 𝔾_𝜆,𝑁^′, 𝜏^′=𝑖_𝑧∘𝜏, 𝜈^′(𝑘)=𝜈(𝑘)⋆𝜀_𝑧(𝑘) as previously mentioned.)

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

Theorem 6. (a) Let 𝔾 be a locally compact group and {𝜆^𝑘}⊆ℳ¹(𝔾) a nondissipating adapted random walk. Then there exist a closed normal subgroup 𝑁 and 𝑥∈𝔾, such that with 𝜏∶=𝑖_𝑥|_𝑁∈Aut(𝑁), one has 𝔾≅𝑁⋊_𝜏ℤ, 𝑁=𝐶_𝐾(𝜏) for a compact, 𝜏-invariant subgroup 𝐾⊆𝑁. Furthermore, the random walk is representable as {𝜆^𝑘=𝜈(𝑘)⊗𝜀_𝑘}, where {𝜈(𝑘)}⊆ℳ¹(𝑁) is a relatively compact M-semigroup (with respect to 𝜏), and condition (3) is satisfied.
(b) Conversely, let 𝑁 be a locally compact group, 𝜏∈Aut(𝑁), and let {𝜈(𝑘)} be a relatively compact M-semigroup in 𝑀¹(𝑁), and assume condition (3) to be satisfied. Then the space-time random walk {𝜆^𝑘=𝜈(𝑘)⊗𝜀_𝑘}⊆ℳ¹(𝔾) is nondissipating and adapted. (𝔾 denotes 𝑁⋊_𝜏ℤ). Hence (a) applies. In particular, 𝑁=𝐶_𝐾(𝜏) for some compact 𝜏-invariant subgroup 𝐾⊆𝑁.
(c) Non-dissipative random walks, respectively, relatively compact M-semigroups define 𝜏-decomposable laws.
In fact, in case (a) or (b) we obtains that 𝜌∶=lim_𝑘𝜈(𝑘)⋆𝜔_𝐾∈ℳ¹(𝑁) exists and is 𝐾-𝜏-decomposable.
(d) Conversely, 𝜏-decomposable laws define nondissipating random walks, respectively, relatively compact M-semigroups.
Precisely, assume that 𝑁=𝐶_𝐾(𝜏). Let 𝜌∈ℳ¹(𝑁) be 𝐾-𝜏-decomposable. Then there exists a M-semigroup {𝜈(𝑘)} of cofactors of 𝜌 such that 𝜈(𝑘)⋆𝜔_𝐾→𝜌. In particular, {𝜈(𝑘)} is relatively compact, and then and hence (a), respectively, (b) apply.
(e) Let 𝔾=𝑁⋊_𝜏ℤ, 𝑁=𝐶_𝐾(𝜏). An adapted space-time random walk {𝜆^𝑘=𝜈(𝑘)⊗𝜀_𝑘}⊆ℳ¹(𝔾) is nondissipating, respectively, a M-semigroup {𝜈(𝑘)}⊆ℳ¹(𝑁) satisfying (3) is relatively compact iff 𝜈=𝜈(1) possesses finite logarithmic moments, respectively, iff 𝜆 possesses finite first-order moments.

Proof. For (a), (b) see Fact 1 and the discussion in Remark 5.
(c) Let {𝜈(𝑘)} be a relatively compact M-semigroup on 𝑁=𝐶_𝐾(𝜏). Since (𝜏^𝑘) is uniformly 𝐾-contracting on compact subsets, the accumulation points of {𝜏^𝑘(𝜈(𝑛))}_{𝑛∈ℤ+,𝑘→∞} are supported by 𝐾. Assume, for some subnets, that 𝜈(𝑘_𝑛)→𝛼 and 𝜈(ℓ_𝑛)→𝛽 and, w.l.o.g., 𝑘_𝑛≤ℓ_𝑛 for all 𝑛. Then 𝜈(ℓ_𝑛)⋆𝜔_𝐾→𝛽⋆𝜔_𝐾 on the one hand, 𝜈(ℓ_𝑛)⋆𝜔_𝐾=𝜈(𝑘_𝑛)⋆𝜏^𝑘𝑛(𝜈(ℓ_𝑛−𝑘_𝑛))⋆𝜔_𝐾 with lim𝜏^𝑘𝑛(𝜈(ℓ_𝑛−𝑘_𝑛))⋆𝜔_𝐾=𝜔_𝐾 on the other. Hence 𝛼⋆𝜔_𝐾=𝛽⋆𝜔_𝐾 follows. That is, 𝜌=lim𝜈(𝑘)⋆𝜔_𝐾 exists and is obviously right 𝐾-invariant.
Furthermore, 𝜌=lim𝜈(𝑘+1)⋆𝜔_𝐾=lim𝜈(1)⋆𝜏(𝜈(𝑘)⋆𝜔_𝐾)=𝜈(1)⋆𝜏(𝜌) yields 𝐾-𝜏-decomposability of 𝜌.
(d) Conversely, assume 𝜌 to be 𝐾-𝜏-decomposable. {𝜏^𝑘(𝜌)} is relatively compact as 𝜏 is 𝐾-contracting, and all accumulation points are supported by 𝐾. Right 𝐾-invariance of 𝜌 implies that 𝜏^𝑘(𝜌)→𝜔_𝐾. As afore mentioned, we can choose 𝜈(𝑘)=★^𝑘−1_𝑗=0𝜏^𝑗(𝜈), 𝜈=𝜈(1) in Cof_𝜌(𝜏), hence as an M-semigroup of cofactors. Hence 𝜌=𝜈(𝑘)⋆𝜏^𝑘(𝜌) and 𝜏^𝑘(𝜌)→𝜔_𝐾 yield (1) relative compactness of {𝜈(𝑘)} and (2) 𝜈(𝑘)⋆𝜔_𝐾→𝜌 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 𝑁 be a locally compact group, 𝜏∈Aut(𝑁), 𝔾=𝑁⋊_𝜏ℤ as previously mentioned. Let {𝜈(𝑘)}, {𝜆^𝑘=𝜈(𝑘)⊗𝜀_𝑘} 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 𝑁 be a locally compact group, 𝜏∈Aut(𝑁), and let 𝜌∈ℳ¹(𝑁) be 𝜏-decomposable, with {𝜈(𝑘)} as M-semigroup of cofactors. Then 𝜌=𝜈(𝑘)⋆𝜏^𝑘(𝜌), 𝑘∈ℤ₊, yields that {𝜈(𝑘)} is relatively shift compact and therefore (a) applies.
(c) If in (b) in addition {𝜏^𝑘(𝜌)} is assumed to be relatively compact (e.g., if 𝑁=𝐶_𝐾(𝜏) 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 𝑁=𝐶_𝐾(𝜏) and 𝐾≠{𝑒}, lim_𝑘𝜈(𝑘) will not in general exist. See for example, Example 3.16 in [1], with compact 𝑁=𝕋² and an infinite number of accumulation points. Further types of examples are obtained in the following way.

Example 8. (a) Let Γ be a contractible locally compact group, with contracting 𝜎∈Aut(Γ), hence Γ=𝐶(𝜎). Let {𝜇(𝑘)}⊆ℳ¹(Γ) be a relatively compact M-semigroup, 𝜇(𝑘+ℓ)=𝜇(𝑘)⋆𝜎^𝑘(𝜇(ℓ)). As 𝜎^𝑘{𝜇(ℓ)∶ℓ≥0}−−−−−^𝑘→∞→{𝜀_𝑒}, lim𝜇(𝑘)=∶𝜌₁ exists (Theorem 6(c) with 𝐾={𝑒}). Let 𝐻 be a finite cyclic group, let 𝑥₀∈𝐻 generating 𝐻 with ord(𝑥₀)>2, and assume for some 𝜉∈Aut(𝐻) that 𝜉(𝑥₀)=𝑥₀⁻¹ (e.g., 𝜉∶𝑥↦𝑥⁻¹). Put 𝑁∶=Γ⊗𝐻, define 𝜏∈Aut(𝑁) as 𝜏=𝜎⊗𝜉, and put finally 𝜈=𝜈(1)∶=𝜇(1)⊗𝜀_𝑥0. Then {𝜈(𝑘)=𝜇(𝑘)⊗𝜀_𝑦(𝑘)} is a relatively compact M-semigroup (with respect to 𝜏) in ℳ¹(𝑁), where 𝑦(𝑘)=∏^𝑘−1_𝑗=0𝜉^𝑗(𝑥₀). But 𝑦(𝑘)=𝑒 and 𝑦(𝑘)=𝑥₀, both infinitely often; hence 𝜈(𝑘) is not convergent. In fact, LIM{𝜈(𝑘)}={𝜌₁⊗𝜀_𝑒,𝜌₁⊗𝜀_𝑥0}. But we have 𝜈(𝑘)⋆𝜔_𝐻→𝜌₁⊗𝜔_𝐻.
(b) A trivial case: let 𝐾(=𝑁) be a monothetic compact group with dense subgroup {𝑥(𝑘)∶=𝑥^𝑘₀∶𝑘∈ℤ}≅ℤ (for some 𝑥₀∈𝐾). Put 𝜏∶=𝑖𝑑, obviously 𝐾-contracting; hence 𝔾=𝐾⊗ℤ. Then {𝜈(𝑘)∶=𝜀_𝑥(𝑘)}_𝑘∈ℤ+ is a M-semigroup (with respect to 𝜏), which is relatively compact in ℳ¹(𝐾). But 𝜈(𝑘) is not convergent, in fact, LIM{𝜈(𝑘)}={𝜀_𝜅∶𝜅∈𝐾}. Here we have 𝜈(𝑘)⋆𝜔_𝐾≡𝜔_𝐾.

In [12, Theorem 1], it is shown for a nondissipating random walk {𝜆^𝑘=𝜈(𝑘)⊗𝜀_𝑘} that there exists a measurable dense 𝜏-invariant subgroup 𝑁₁⊆𝑁 on which a sequence 𝜏^𝑞𝑛 acts as contracting mod. a compact subgroup 𝐻, and 𝜆^𝑛=(𝜈⊗𝜀₁)^𝑛 are concentrated on cosets of 𝑁₁. Hence 𝜏 is weakly contracting mod. 𝐻 on 𝑁 (cf. the definition in [12]), and thus 𝑁=𝐶_𝐾(𝜏), for some 𝜏-invariant subgroup 𝐾⊇𝐻 [12, Theorem 5]. In interesting examples, for example, in the context of (semi-) stable laws, we have 𝑁₁=𝐶_𝐻(𝜏) for a compact 𝜏-invariant subgroup 𝐻⊆𝐾. In fact, investigations of the structure of contractible and 𝐾-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 {𝜌_𝑡}_𝑡≥0 be a continuous convolution semigroup in ℳ¹(𝑁), and let 𝑁 be a locally compact group. Let 𝜏∈Aut(𝑁) and 0<𝑐<1. {𝜌_𝑡} is (strictly) (𝜏,𝑐)-semistable if for all 𝑡≥0, 𝜏(𝜌_𝑡)=𝜌_𝑡⋅𝑐. The idempotent 𝜌₀=𝜔_𝐻 is a normalized Haar measure on a compact 𝜏-invariant subgroup 𝐻. If 𝑁 is second countable, the contraction subgroups 𝐶(𝜏) and 𝐶_𝐻(𝜏) are Borel sets, and we have 𝜌_𝑡(𝐶_𝐻(𝜏))=1 for all 𝑡. Hence we assume that 𝑁=(𝐶_𝐻(𝜏))⁻, and therefore 𝑁=𝐶_𝐾(𝜏) (as mentioned previously).
Let now 𝜌∶=𝜌₁. Then 𝜌=𝜌_1−𝑐𝑛⋆𝜌_𝑐𝑛=𝜌_1−𝑐𝑛⋆𝜏^𝑛(𝜌). Hence 𝜌 is 𝐻-𝜏-decomposable with cofactors 𝜈(𝑛)∶=𝜌_1−𝑐𝑛∈Cof_𝜌(𝜏^𝑛). In particular, 𝜌_1−𝑐𝑛 (and hence all 𝜌_𝑡) possess finite logarithmic moments. (This could also be proved by direct calculation.) In that example, 𝜈(𝑛) are right 𝐻-invariant and lim𝜈(𝑛)=𝜌 exists. (And thus trivially also lim𝜈(𝑛)⋆𝜔_𝐾=𝜌⋆𝜔_𝐾).

If 𝐶(𝜏) is closed and 𝐶_𝐻(𝜏)=𝐶(𝜏)⋅𝐻, then 𝐶_𝐻(𝜏) is closed, hence 𝐻=𝐾 and 𝑁=𝐶_𝐾(𝜏). 𝐶(𝜏) is known to be closed if there exist contracting continuous one-parameter groups of automorphisms [5] moreover for 𝑝-adic Lie groups [38] and more generally for totally disconnected groups if 𝜏 is a “tidy” automorphism [14]. And we have 𝐶_𝐻(𝜏)=𝐶(𝜏)⋅𝐻, 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 𝐶_𝐻(𝜏)=𝐶(𝜏)⋅𝐻, cf. [39, 40]. However, 𝐶(𝜏) will not in general be closed, not even on a 2-dimensional torus (cf., e.g., [5], [6, Example 3.12.5]). Then, as mentioned previously if w.l.o.g. 𝑁=(𝐶_𝐻(𝜏))⁻, 𝜏 is weakly contracting mod𝐻 on 𝑁, and hence 𝑁=𝐶_𝐾(𝜏) for some compact, 𝜏-invariant subgroup 𝐾⊇𝐻. Note that, if 𝐶_𝐻(𝜏) is not closed, additional conditions are needed to guarantee existence of lim_𝑘𝜈(𝑘)⋆𝜔_𝐻 (cf. Example 8 above).

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

Proposition 10. Let 𝑁=𝐶_𝐾(𝜏). Let {𝜈(𝑘)} be a relatively compact M-semigroup. To avoid measurability problems, 𝑁 is supposed to be second countable. Put 𝒜∶=LIM{𝜈(𝑘)∶𝑘→∞} and 𝒮∶=LIM{𝜏^𝑛(𝜈(𝑘_𝑛))∶𝑛→∞,(𝑘_𝑛)⊆ℤ₊}⊆ℳ¹(𝐾).
Then one has the following(a)𝒜⊆𝜌⋆𝒮   (for any  𝜌∈𝒜).(b)𝜏(𝒮)=𝒮.(c)If at least some  𝜌₀∈𝒜and  𝒮  belong to a commutative  ⋆-sub-semigroup  ℳ  of   ℳ¹(𝑁),   then any  𝜌∈𝒜  is  𝜏-decomposable with  𝜈(𝑘)⋆𝛼_𝑘∈Cof_𝜌(𝜏^𝑘)  for some  𝛼_𝑘∈𝒮, 𝑘∈ℤ₊.(d)If   𝜈(𝑘)(𝐶_𝐻(𝜏))=1  for some compact  𝜏-invariant subgroup  𝐻⊆𝐾  and  𝜈(𝑘)⋆𝜔_𝐻=𝜈(𝑘),   for all  𝑘≥1,   and if one assumes in addition that  𝒮⊆ℳ¹(𝐻),   then  𝜌=lim_𝑘𝜈(𝑘)  exists, and  𝜌  is  𝐻-𝜏-decomposable with cofactors   𝜈(𝑘)∈Cof_𝜌(𝜏^𝑘).

Proof. By assumption, (𝜏^𝑘) is compactly contracting mod𝐾 on 𝑁.(a)Let 𝜌,𝜎∈𝒜, 𝜈(𝑘_𝑛)→𝜌 and 𝜈(ℓ_𝑛)→𝜎. Assume that that w.l.o.g. ℓ_𝑛≥𝑘_𝑛 for all 𝑛 (else pass to subsequences). Then 𝜈(ℓ_𝑛)=𝜈(𝑘_𝑛)⋆𝜏^𝑘𝑛(𝜈(ℓ_𝑛−𝑘_𝑛))→𝜌⋆𝛼, 𝛼∈𝒮. Hence 𝜎=𝜌⋆𝛼∈𝜌⋆𝒮, and analogously, 𝜌∈𝜎⋆𝒮 follows.(b)Assume that 𝜏^ℓ𝑛(𝜈(𝑚_𝑛))→𝛾∈𝒮. Then 𝜏(𝛾)=lim𝜏^ℓ𝑛+1(𝜈(𝑚_𝑛)); hence 𝜏(𝛾)∈𝒮, and analogously 𝜏⁻¹(𝛾)∈𝒮 follows.(c)Let 𝜌∈𝒜; hence 𝜌=𝜌₀⋆𝛼 for some 𝛼∈𝒮. Let 𝜈(𝑘_𝑛)→𝜌. w.l.o.g. assume that 𝜌=𝜌₀, else replace in the following 𝜌 by 𝜌₀⋆𝛼. Then, for all 𝑗∈ℕ, 𝜈(𝑘_𝑛)=𝜈(𝑗)⋆𝜏^𝑗(𝜈(𝑘_𝑛−𝑗)) and 𝜈(𝑘_𝑛−𝑗)→𝜌⋆𝛽_𝑗 (along a subsequence) for some 𝛽_𝑗∈𝒮. Hence 𝜈(𝑘_𝑛)→𝜈(𝑗)⋆𝜏^𝑗(𝜌)⋆𝜏^𝑗(𝛽_𝑗)=(𝜈(𝑗)⋆𝜏^𝑗(𝛽_𝑗))⋆𝜏^𝑗(𝜌). That is, 𝜈(𝑗)⋆𝜏^𝑗(𝛽_𝑗)∈Cof_𝜌(𝜏^𝑗). Whence the assertion with 𝛼_𝑗=𝜏^𝑗(𝛽_𝑗).(d)Since by assumption 𝜈(𝑘) are right 𝐻-invariant, the accumulation points 𝛼∈𝒮 are right 𝐻-invariant and supported by 𝐻; thus 𝒮={𝜔_𝐻}. Whence 𝒜={𝜌} and thus 𝜈(𝑘)→𝜌.

The following example will illustrate Proposition 10.

Example 11. Let 𝑁=𝐶_𝐾(𝜏) as before.
(a) Let {𝜌_𝑡}⊆ℳ¹(𝑁) be a (𝜏,𝛼)-semistable continuous convolution semigroups (as in Example 9), and let 𝜉∈ℳ¹(ℝ₊) be 𝐻_𝛼-decomposable with cofactor 𝜂, with 𝐻_𝛼 denoting the homothetic transformation 𝑥↦𝛼⋅𝑥 on ℝ. Then the subordinated measure 𝜆∶=∫_ℝ+𝜌_𝑡d𝜉(𝑡) is 𝜏-decomposable with cofactor ∫_ℝ+𝜌_𝑡d𝜂(𝑡). The set ℳ∶={∫_ℝ+𝜌_𝑡d𝛾(𝑡)∶𝛾∈ℳ¹(ℝ₊)} is a commutative sub-semigroup of ℳ¹(𝑁) containing 𝜆 and the cofactors (cf. Proposition 10(c)).
Let 𝑁^′∶=𝐶_𝐻(𝜏) for some compact 𝜏-invariant subgroup 𝐻⊆𝐾. Assume 𝜌_𝑡(𝑁^′)=1 for all 𝑡, then also 𝜆(𝑁^′)=1 and 𝜈(𝑁^′)=1 for all 𝜈∈ℳ.
(b) Let 𝜌^(𝑖)∈ℳ¹(𝑁) be 𝜏-decomposable with cofactors 𝜈^(𝑖)(𝑘), 𝑖=1,2. ⟦e.g., let {𝜌_𝑡^(𝑖)} be (𝜏,𝛼_𝑖)-semistable continuous convolution semigroups and 𝜌^(𝑖)=𝜌₁^(𝑖).⟧ Assume that 𝜈⁽²⁾(1) and 𝜏(𝜌⁽¹⁾) commute. ⟦e.g., for semistable laws, assume that 𝜌_𝑡⁽¹⁾,𝜌_𝑠⁽²⁾ commute for all 𝑠,𝑡.⟧ Then 𝜌∶=𝜌⁽¹⁾⋆𝜌⁽²⁾ is 𝜏-decomposable with cofactor 𝜈⁽¹⁾(1)⋆𝜈⁽²⁾(1). ⟦But note that if 𝜌^(𝑖) are semistable and if 𝛼₁≠𝛼₂−𝜌₁⁽¹⁾⋆𝜌₁⁽²⁾ is in general not (𝜏,𝛽)-semistable for 0<𝛽<1.⟧
If 𝜌^(𝑖)(𝐶_𝐻𝑖(𝜏))=1, for 𝜏-invariant compact subgroups 𝐻_𝑖⊆𝐾, then 𝜌(𝐶_𝐻1(𝜏)⋅𝐶_𝐻2(𝜏))=1. If 𝜔_𝐻1 and 𝜔_𝐻2 commute, then 𝐻₁⋅𝐻₂=∶𝐻₃ is a compact 𝜏-invariant subgroup and 𝐶_𝐻3(𝜏)⊇⟨𝐶_𝐻1(𝜏)⋅𝐶_𝐻2(𝜏)⟩, the subgroup generated by 𝐶_𝐻𝑖(𝜏), 𝑖=1,2.
(c) In general, without commutativity assumptions, 𝜌 in (b) will not in general be 𝜏-decomposable. In particular, for point measures 𝜀_𝑥, 𝑥∈𝑁 we obtain: 𝜌=𝜌⁽¹⁾⋆𝜀_𝑥 is in general not 𝜏-decomposable. In fact, without commutativity assumption, the left shifted measure, 𝜌∶=𝜀_𝑥⋆𝜌⁽¹⁾, is, as easily seen, 𝜏-decomposable with cofactors 𝜈(𝑘)=𝜀_𝑥⋆𝜈⁽¹⁾(𝑘)⋆𝜀_{𝜏𝑘(𝑥−1)}. Representing 𝜌 as 𝜀_𝑥⋆𝑖_𝑥−1(𝜌⁽¹⁾) we see that 𝜌 is 𝜏-decomposable, with 𝜏∶=𝑖_𝑥−1∘𝜏∘𝑖_𝑥, but in general not 𝜏-decomposable.
(d) To simplify notations we assume that 𝐻_𝑖={𝑒}. Let 𝑁^′∶=𝐶(𝜏) and assume that now 𝜌⁽¹⁾(𝑁^′)=𝜈⁽¹⁾(𝑘)(𝑁^′)=1. Then 𝜌 and 𝜈(𝑘) (in (c)) are concentrated on cosets of compactly contractible subgroups of 𝑁: 𝜌(𝑥𝑁^′)=1 and 𝜈(𝑘)(𝑥𝑁^′𝜏^𝑘(𝑥⁻¹))=1. Hence, with 𝑁^′′∶=𝑖_𝑥(𝑁^′) we have: 𝜌(𝑁^′′𝑥)=1 and 𝜈(𝑘)(𝑁^′′𝑥𝜏^𝑘(𝑥⁻¹))=1. Furthermore, 𝑁^′′=𝐶_𝐻(𝜏), where 𝜏∶=𝑖_𝑥∘𝜏∘𝑖_𝑥−1, 𝐻∶=𝑖_𝑥(𝐻).
(e) In case (c), (d) we determine the sets of accumulation points 𝒮 and 𝒜 (Proposition 10):
𝒮=LIM{𝜏^𝑛(𝜈(𝑘_𝑛))}=LIM{𝜀_{𝜏𝑛(𝑥)}⋆𝜏^𝑛(𝜈⁽¹⁾(𝑘_𝑛))⋆𝜀_{𝜏𝑛𝑛+𝑘(𝑥−1)}}, hence 𝒮=LIM{𝜀_{𝜏𝑛(𝑥)𝜏𝑛𝑛+𝑘(𝑥−1)}}=𝒮₀⋆𝒮₀⁻¹, where 𝒮₀=LIM{𝜀_{𝜏𝑛(𝑥)}}. ⟦In fact, obviously 𝒮⊆𝒮₀⋆𝒮₀⁻¹. On the other hand, assume 𝜏^𝑘(𝑛)(𝑥)→𝑧 and 𝜏^ℓ(𝑛)(𝑥)→𝑢, w.l.o.g. ℓ(𝑛)≥𝑘(𝑛). Then 𝜀_𝑧𝑢−1∈𝒮₀⋆𝒮₀⁻¹ and 𝜀_𝑧𝑢−1=lim𝜀_{𝜏𝑘(𝑛)(𝑥)}⋆𝜀_{𝜏𝑘(𝑛)+(ℓ(𝑛)−𝑘(𝑛))(𝑥−1)}∈𝒮.⟧
Furthermore, 𝒜=LIM{𝜈(𝑛)}=LIM{𝜀_𝑥⋆𝜈⁽¹⁾(𝑛)⋆𝜀_{𝜏𝑛(𝑥−1)}}. Thus 𝒜=𝜀_𝑥⋆𝜌⁽¹⁾⋆𝒮₀⁻¹.

3. Continuous Time: Nondissipating Continuous Convolution Semigroups

Next we replace the random walk by a continuous convolution semigroup {𝜆_𝑡}_𝑡∈ℝ+ (the distribution of a Lévy process on 𝔾, if 𝔾 is metrizable). In contrast to the discrete time case, now the idempotent 𝜆₀=lim_𝑡→0𝜆_𝑡 is uniquely determined.

W.l.o.g. we assume that 𝔾 is generated by {supp(𝜆_𝑡)∶𝑡≥0}. For short, {𝜆_𝑡} is called adapted then. {𝜆_𝑡}_𝑡∈ℝ+ is nondissipating if the concentration functions 𝑓_𝜆𝑡(𝐶)∶=sup_𝑥∈𝔾𝜆_𝑡(𝐶𝑥) do not converge to 0 for some compact 𝐶 (for 𝑡→∞).

For any 𝑡₀>0 the random walk {𝜆^𝑘_𝑡0=𝜆_𝑡0𝑘} is called skeleton random walk. Note that a skeleton random walk do not need to be adapted on 𝔾 hence we introduce for 𝑡₀>0 the subgroups 𝔾_𝜆𝑡0=∶𝔾_𝑡0 as smallest closed subgroup containing supp𝜆_𝑡0. Hence {𝜆^𝑘_𝑡0} is adapted on 𝔾_𝑡0. We define 𝑁_𝑡0∶=𝑁_𝜆𝑡0 to be the smallest closed normal subgroup of 𝔾_𝑡0 containing supp𝜆_𝑡0∗𝜆_𝑡0. Hence 𝔾_𝑡0=𝑁_𝑡0⋊_𝜏𝑡0ℤ and 𝜆_𝑡0𝑘=𝜈(𝑡₀𝑘)⊗𝜀_𝑡0𝑘. (For later use we identify ℤ with 𝑡₀⋅ℤ.)

Again, to avoid trivialities, throughout in the sequel, 𝔾 is assumed to be noncompact (else any continuous convolution semigroup would be nondissipating). Hence also the closed subgroups 𝔾_𝑡0 are noncompact and the results of Section 1 apply. In fact, if for some 𝑡₀>0, 𝔾_𝑡0 is compact, then {𝜆^𝑘_𝑡0=𝜆_𝑡0𝑘∶𝑘∈ℤ₊} is uniformly tight. But then {𝜆_𝑠∶𝑠≥0}={𝜆_𝑡0𝑘∶𝑘∈ℤ₊}⋆{𝜆_𝑠∶0≤𝑠≤𝑡₀} is uniformly tight. Therefore 𝔾 (and all groups 𝔾_𝑡,𝑡>0) are compact.

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

Proposition 12. Let {𝜆_𝑡}⊆ℳ¹(𝔾) be an adapted continuous convolution semigroup. Then the following assertions are equivalent.(i){𝜆_𝑡}   is nondissipating.(ii)For all   [(𝑖𝑖^′)     some]    𝑡₀>0,   the skeleton random walk   {𝜆^𝑘_𝑡0=𝜆_𝑘𝑡0}   is nondissipating (and adapted on   𝔾_𝑡0).(iii){𝜆_𝑡}_𝑡≥0   is relatively (right) shift compact.(iv){𝜆_𝑡∗𝜆_𝑡}_𝑡≥0   is relatively compact. (Then  lim_𝑡→∞𝜆_𝑡∗𝜆_𝑡   exists.)(v)For all   [(𝑣^′)   some]    𝑡₀>0, 𝜆_𝑡0     (considered as measure on   𝔾_𝑡0)   has finite first-order moments.(vi)For all   [(𝑣𝑖^′)   some]    𝑡₀>0, 𝜈(𝑡₀)     (considered as measure on  𝑁_𝑡0)    has finite logarithmic moments. That is, for all regular   𝜏_𝑡0-functions   𝑓   on   𝑁_𝑡0   one has   ∫_𝑁𝑡0log(1+𝑓)d𝜈(𝑡₀)<∞.

The precise formulations of conditions (v) (resp., (vi)) are: for all subadditive functions 𝑓 on 𝔾_𝑡0 (resp., for all regular 𝜏_𝑡0-functions 𝑔 on 𝑁_𝑡0) we have ∫_𝔾𝑡0𝑓d𝜆_𝑡0<∞ (resp., ∫_𝑁𝑡0log(1+𝑔)d𝜈(𝑡₀)<∞). Note that for a subadditive function 𝑓 on 𝔾 the restriction 𝑓|_𝔾𝑡0 is subadditive. However, the sets of subadditive functions defining moments on 𝔾_𝑡0, might have no subadditive extension to 𝔾. Analogously, for 𝑡₀≠𝑡₁, the sets of 𝜏_𝑡0- and 𝜏_𝑡1-functions may be different (even if 𝑁_𝑡0=𝑁_𝑡1, as will be shown in Proposition 13 later).

Proof. 𝑓_𝛼∗𝛽(𝐶)≤𝑓_𝛼(𝐶) for all probabilities 𝛼,𝛽, hence 𝑡↦𝑓_𝜆𝑡(𝐶) is decreasing. Hence 𝑓_𝜆𝑡(𝐶)↛0 iff inf𝜆_𝑡(𝐶)>0. Thus, as immediately seen, “(i)⇔(ii)⇔(ii^′)”. And obviously we have “(iii)⇔(iv)” and “(v)⇒(v^′)”, “(vi)⇒(vi^′)”. The equivalences “(ii^′)⇔(v^′)⇔(vi^′)” (for any fixed 𝑡₀>0) follow by Theorem 6, and hence we also have (ii)⇔(v)⇔(vi). Convergence of 𝜎_𝑡∶=𝜆_𝑡∗𝜆_𝑡 follows by Fact 1(a): For all 𝑡>0  lim_𝑘𝜎_𝑡𝑘=∶𝜎(𝑡) exists. Hence 𝜎(𝑡)=𝜎(𝑡^′) for commensurable (𝑡,𝑡^′), in particular, 𝜎(𝑡)=𝜎(1) for all rational 𝑡. Furthermore, for 𝑓∈𝐶₀(𝔾), 𝑡↦⟨𝜎_𝑡,𝑓⟩ is uniformly continuous. Whence 𝜎(𝑡)=𝜎(1) for all real 𝑡>0 easily follows.
“(ii^′)⇒(iii)”. By Fact 1(a), (ii^′) is equivalent to shift compactness of {𝜆^𝑘_𝑡0}. Let 𝑥_𝑘∈𝔾 such that {𝜆_𝑡0𝑘∗𝜀_𝑥𝑘⁻¹} is relatively compact. Then, {𝜆_𝑡} being a continuous convolution semigroup, the set {𝜆_𝑟∶0≤𝑟<𝑡₀}∗{𝜆_𝑡0𝑘∗𝜀_𝑥𝑘⁻¹∶𝑘∈ℤ₊}=∶{𝜆_𝑡⋆𝜀_𝑥(𝑡)⁻¹∶𝑡≥0} is relatively compact (where 𝑥(𝑡)∶=𝑥_𝑘, if 𝑡₀𝑘≤𝑡<𝑡₀(𝑘+1)). Whence (iii) follows.
“(iii)⇒(i)”. Let {𝜆_𝑡∗𝜀_𝑥(𝑡)⁻¹} be relatively compact. Hence for any 𝜖∈(0,1) there exists a compact 𝐶 such that for all 𝑡∈ℝ₊, 𝜆_𝑡(𝐶𝑥(𝑡))≥𝜖. Therefore, 𝑓_𝜆𝑡(𝐶) fails to converge to 0.

A priori 𝑁_𝑡0∶=𝑁_𝜆𝑡0, the smallest closed normal subgroup of 𝔾_𝑡0 containing supp𝜆_𝑡0∗𝜆_𝑡0, might not be normal in 𝔾 and might depend on 𝑡₀. In order to apply the results of Section 1 we have to overcome these difficulties.

Proposition 13. Let {𝜆_𝑡} be an adapted continuous convolution semigroup. Then one has 𝑁_𝑡0=𝑁₁=∶𝑁 for all 𝑡₀>0, and 𝑁⊲𝔾.

Proof. Obviously, 𝔾=(⋃_𝑡𝔾_𝑡)⁻ and for all 𝑡>0, 𝔾_𝑡=⋃_𝑘∈ℤ𝑁_𝑡𝑥^𝑘_𝑡 for some (all) 𝑥_𝑡∈supp𝜆_𝑡. First we consider dyadic numbers 𝑡 and then proceed to real 𝑡 applying continuity of {𝜆_𝑡}.
Let 𝑡₀=1. Then obviously 𝔾₁⊆𝔾_1/2 since 𝜆₁=𝜆²_1/2. For all 𝑥,𝑥_1/2∈supp𝜆_1/2 it follows, 𝑥𝑥_1/2,𝑥_1/2𝑥,𝑥²_1/2∈supp𝜆₁⊆𝔾₁, hence 𝑥_1/2𝑥𝑁₁=𝑁₁𝑥_1/2𝑥=𝑥𝑥_1/2𝑁₁=𝑁₁𝑥𝑥_1/2=𝑁₁𝑥²_1/2=𝑥²_1/2𝑁₁. (4)
In fact, 𝑁_𝑡⊲𝔾_𝑡 for all 𝑡>0, and hence 𝑦𝑁_𝑡=𝑁_𝑡𝑧 for all 𝑦,𝑧∈supp𝜆_𝑡. Therefore, 𝑥⁻¹𝑥²_1/2𝑁₁=𝑥_1/2𝑁₁,𝑁₁𝑥²_1/2𝑥⁻¹=𝑁₁𝑥_1/2. (5) Put 𝑁^∗₁∶=𝑁₁∩𝑥_1/2𝑁₁𝑥⁻¹_1/2, and hence, by (5)𝑥_1/2𝑁^∗₁=𝑥_1/2𝑁₁∩𝑁₁𝑥_1/2=𝑁^∗₁𝑥_1/2. (6)
Claim 1. We have 𝜆_1/2𝑁^∗₁𝑥_1/2=𝜆_1/2𝑥_1/2𝑁^∗₁=1. (7)
By (6) it suffices to show that 𝜆_1/2(𝑥_1/2𝑁₁)=𝜆_1/2(𝑁₁𝑥_1/2)=1. According to (4) and (5) we have 1=𝜆₁𝑁₁𝑥²_1/2=𝜆_1/2𝑥⁻¹𝑁₁𝑥²_1/2d𝜆_1/2(𝑥)₍₄₎=𝜆_1/2𝑥⁻¹𝑥²_1/2𝑁₁d𝜆_1/2(𝑥)₍₅₎=𝜆_1/2𝑥_1/2𝑁₁d𝜆_1/2(𝑥)=𝜆_1/2𝑥_1/2𝑁₁. (8) Analogously we have 1=∫𝜆_1/2(𝑁₁𝑥²_1/2𝑥⁻¹)d𝜆_1/2(𝑥)=𝜆_1/2(𝑁₁𝑥_1/2). Consequently, supp𝜆_1/2⊆𝑥_1/2𝑁^∗₁=𝑁^∗₁𝑥_1/2 yields 𝔾_1/2⊆⋃_𝑘∈ℤ𝑥^𝑘_1/2𝑁^∗₁=⋃_𝑘∈ℤ𝑁^∗₁𝑥^𝑘_1/2. Thus, as 𝑥_1/2 normalizes 𝑁^∗₁, we obtain 𝑁^∗₁⊲𝔾_1/2.
Claim 2.  𝑁_1/2⊆𝑁^∗₁⊆𝑁₁. 𝑁^∗₁∩𝑁_1/2=∶𝑁^∗ is a closed normal subgroup of 𝔾_1/2 such that 1=𝜆_1/2(𝑥_1/2𝑁^∗)=𝜆_1/2(𝑁^∗𝑥_1/2). But 𝑁_1/2 is minimal with this property. Whence 𝑁_1/2⊆𝑁^∗, thus 𝑁_1/2⊆𝑁^∗₁⊆𝑁₁.
Claim 3.   𝑁^∗₁=𝑁^∗=𝑁_1/2=𝑁₁=∶𝑁. ⟦𝑁_1/2⊲𝔾_1/2 yields 𝑁_1/2⊲𝔾₁ (as 𝐺₁⊆𝔾_1/2). And 𝜆₁(𝑥²_1/2𝑁_1/2)=∫𝜆_1/2(𝑥⁻¹𝑥²_1/2𝑁_1/2)d𝜆_1/2(𝑥)=∫𝜆_1/2(𝑥⁻¹𝑥_1/2𝑁_1/2𝑥_1/2)d𝜆_1/2(𝑥); therefore 𝜆₁(𝑥²_1/2𝑁_1/2)=𝜆_1/2(𝑁_1/2𝑥_1/2)=1 (since 𝑥⁻¹𝑥_1/2∈𝑁_1/2). Hence, according to the definition, 𝑁₁⊆𝑁_1/2 follows. Together we obtain 𝑁₁⊆𝑁_1/2⊆𝑁^∗₁⊆𝑁₁, whence the assertion follows.⟧
By induction, we obtain 𝑁=𝑁_1/2^𝑛 for all 𝑛∈ℤ₊, and 𝑁⊲⋃_{𝑘∈ℤ,𝑛≥0}𝑥^𝑘_1/2^𝑛𝑁₁.
Claim 4.   𝑁_𝑡=𝑁 for all 𝑡>0.
At first we consider dyadic 𝑡=𝑘/2^𝑛.(1)supp𝜆_𝑘/2^𝑛⊆[(𝑥_1/2^𝑛𝑁)^𝑘]⁻=𝑥^𝑘_1/2^𝑛𝑁=𝑁𝑥^𝑘_1/2^𝑛, whence 𝑁_𝑘/2^𝑛⊆𝑁. Hence supp𝜆_𝑘/2^𝑛⋆𝜆_𝑘/2^𝑛⊆𝑁.(2)supp𝜆_𝑘/2^𝑛⋆𝜆_𝑘/2^𝑛⊇𝑥^𝑘−1_1/2^𝑛⋅(supp𝜆_1/2^𝑛⋆𝜆_1/2^𝑛)⋅𝑥^{−(𝑘−1)}_1/2𝑛. Thus 𝑁_𝑘/2^𝑛⊇𝑥^𝑘−1_1/2^𝑛𝑁𝑥^{−(𝑘−1)}_1/2𝑛=𝑁.
Hence 𝑁_𝑡=𝑁 for all dyadic 𝑡>0. Continuity of 𝑡↦𝜆_𝑡⋆𝜆_𝑡 easily yields 𝑁_𝑡⊆𝑁₁=𝑁, and 𝑁⊲[⋃_{𝑘,𝑛≥0}𝑥^𝑘_1/2^𝑛𝑁]⁻.
Replacing 1 by any real 𝑡₀>0 we obtain analogously 𝑁_𝑡0=𝑁_𝑡0𝑘/2^𝑛 for 𝑘,𝑛≥0, and again 𝑁_𝑡⊆𝑁_𝑡0 for all real 𝑡>0. In particular, 𝑁=𝑁₁⊆𝑁_𝑡0. Consequently, 𝑁_𝑡=𝑁 for all 𝑡>0.
Now the assertions follow immediately, and the proof is complete.

Consequently, for any nondissipating continuous convolution semigroup {𝜆_𝑡} we obtain that {𝜆_𝑡} is adapted iff 𝔾=⋃_𝑡≥0𝔾_𝑡=⋃_𝑠∈ℝ𝑁𝑥_𝑠. Indeed, we have the following.

Proposition 14. Let {𝜆_𝑡}_𝑡≥0 be nondissipating and adapted. Then, with the afore-introduced notations, there exists a continuous one-parameter group {𝑥(𝑡)}_𝑡∈ℝ⊆𝔾 such that 𝔾/𝑁≅{𝑥(𝑡)∶𝑡∈ℝ}≅ℝ and supp𝜆_𝑡⊆𝑁𝑥(𝑡),𝑡≥0.
Let, for 𝑡∈ℝ, 𝜏_𝑡∶=𝑖_𝑥(𝑡)|_𝑁 denote the restriction of the inner automorphism to 𝑁. Then 𝑇=(𝜏_𝑡) is a continuous one-parameter group in Aut(𝑁). And one has 𝔾≅𝑁⋊_𝑇ℝ (with group operation (𝑔,𝑡)(ℎ,𝑠)=(𝑔𝜏_𝑡(ℎ),𝑡+𝑠), 𝑔,ℎ∈𝑁,𝑡,𝑠∈ℝ). Furthermore, 𝜆_𝑡 may be represented as 𝜆_𝑡=𝜈(𝑡)⊗𝜀_𝑡, where 𝑡↦𝜈(𝑡)∈ℳ¹(𝑁) is a continuous M-semigroup (with respect to 𝑇) (cf. Definition 18 later). Since 𝑥(0)=𝑒 it follows that 𝜆₀=𝜔_𝐻⊗𝜀₀ for some compact subgroup 𝐻⊆𝑁.

Proof. As shown before, there exists 𝑥_𝑡∈𝔾 such that supp𝜆_𝑡⊆𝑁𝑥_𝑡 (for all 𝑡∈ℝ₊). Hence, 𝜋∶𝔾→𝔾/𝑁 denoting the canonical homomorphism, and we obtain 𝜋(𝜆_𝑡)=𝜀_𝑧(𝑡) with 𝑧(𝑡)=𝑥_𝑡𝑁. Hence {𝜀_𝑧(𝑡)}_𝑡∈ℝ+ and therefore (𝑧(𝑡))_𝑡∈ℝ+ are continuous one-parameter semigroups, extendible to groups, and thus 𝔾/𝑁≅ℝ. ⟦Indeed, if 𝔾/𝑁 were compact, 𝔾_𝑡0/𝑁 would be compact (hence finite) for any 𝑡₀, a contradiction to Fact 1(b).⟧ Finally, there exists a continuous one-parameter group (𝑥(𝑡))_𝑡∈ℝ in 𝔾 with 𝜋(𝑥(𝑡))=𝑧(𝑡), 𝑡≥0, (cf., e.g., [41]), whence the assertion follows.
To show that 𝔾 splits as a semi-direct product, put 𝑁∩𝑋=∶𝐿, where 𝑋∶={𝑥(𝑡)}_𝑡∈ℝ. Assume 𝐿 to be nontrivial. Then 𝑋/𝐿 is compact, hence 𝔾/𝑁=(𝑁⋅𝑋)/𝑁 is compact, in contradiction to 𝔾/𝑁≅ℝ, as shown previously.
Hence ℍ∶=⋃_𝑡𝔾_𝑡=𝑁⋊_𝑇ℝ is closed in 𝔾 containing ⋃_𝑡supp𝜆_𝑡, whence 𝔾=ℍ.

The existence of nondissipating adapted continuous convolution semigroups (𝜆_𝑡) has strong influence on the structure of 𝔾.

Corollary 15. Assume, as in Proposition 14, (𝜆_𝑡) to be adapted and nondissipating, and assume in addition 𝑁 to be totally disconnected. Then 𝔾=𝑁⊗ℝ, 𝑁 is compact and the 𝑀-semigroup (𝜈(𝑡)) is a continuous convolution semigroup. If moreover 𝔾 is assumed to be totally disconnected then 𝔾=𝑁, hence 𝔾 is compact, the trivial case excluded afore.

⟦If𝑁 is totally disconnected then the continuous automorphism group 𝑇={𝑖_𝑥(𝑡)|_𝑁} is trivial; hence 𝑁⋊_𝑇ℝ=𝑁⊗ℝ. Since 𝑇 is trivial, any continuous 𝑀-semigroup (with respect to 𝑇) is a continuous convolution semigroup. Thus we obtain immediately that 𝑁 is compact if (𝜆_𝑡) is nondissipating.

If moreover, 𝔾 is totally disconnected, then {𝑥(𝑡)} is trivial, whence 𝔾=𝑁 is compact.⟧

In the discrete time case it was essentially used that 𝑁≅𝐶_𝐾(𝜏). As mentioned in the previous proof, in the continuous time case, for all 𝑡>0, we have 𝑁=𝐶_𝐾(𝑡). It turns out (cf. the next proposition) that 𝐾(𝑡) is independent of 𝑡>0. Furthermore, in the continuous time case the structure of 𝑁 is nicer. Put 𝐶∶=𝐶(𝑇)={𝑥∈𝑁∶𝜏_𝑡(𝑥)−−−−−^𝑡→∞→𝑒} and 𝐶_𝐾(𝑇)={𝑥∈𝑁∶𝜏_𝑡(𝑥)−−−−−^𝑡→∞→𝑒mod𝐾} for some compact, 𝑇-invariant subgroup 𝐾.

Proposition 16. Let, as before, 𝔾=𝑁⋊_𝑇ℝ. (a)  With the notations of Proposition 14 one has   𝑁≅𝐶_𝐾(𝑇).
In fact, for any locally compact group 𝑁, admitting a continuous one parameter group 𝑇=(𝜏_𝑡)⊆Aut(𝑁) and a 𝑇-invariant compact subgroup 𝐾 one has the following. (b)  For all   𝑡>0, 𝐶_𝐾(𝑇)=𝐶_𝐾(𝜏_𝑡), in particular,  𝐶(𝑇)=𝐶(𝜏_𝑡). (c)  𝐶(𝑇)  is closed,    𝑇-invariant, connected, and isomorphic to a contractible Lie group (hence a homogeneous group), and   𝐶_𝐾(𝑇)   is a closed   𝑇-invariant subgroup of   𝑁.   If   𝑁   is compact, then   𝐶_𝐾(𝑇)=𝐾. (d)  𝐶(𝑇)⊲𝐶_𝐾(𝑇), and there exists a continuous homomorphism   𝛽∶Aut(𝐾)→Aut(𝐶(𝑇)), 𝛽(𝜅)(𝑔)∶=𝜅𝑔𝜅⁻¹, 𝜅∈𝐾, 𝑔∈𝐶(𝑇), such that   𝐶_𝐾(𝑇)=𝐶(𝑇)⋊_𝛽𝐾. In particular,    𝐶_𝐾(𝑇)   is compactly generated. (e)  The restrictions   𝑆∶=𝑇|_𝐶(𝑇)=(𝜎_𝑡)_𝑡∈ℝ⊆Aut(𝐶(𝑇))   and   Γ∶=𝑇|_𝐾=(𝛾_𝑡)_𝑡∈ℝ⊆Aut(𝐾)   are continuous one-parameter groups, with   Γ   consisting of inner automorphisms of   𝐾. (f)  𝑇=(𝜏_𝑡=(𝜎_𝑡,𝛾_𝑡))   satisfies the following consistency conditions   𝜎_𝑡∘𝛽(𝜅)=𝛽(𝛾_𝑡(𝜅))∘𝜎_𝑡, 𝜅∈𝐾, 𝑡∈ℝ.
Conversely, any group 𝑇 arises in that way.

Proof. (a) As mentioned in the previous proof, for any 𝑡>0, we have 𝑁=𝐶_𝐾(𝑡)(𝜏_𝑡), 𝐾(𝑡) denoting a compact 𝜏_𝑡-invariant subgroup 𝐾(𝑡).
It follows easily that 𝐾(𝑡)=𝐾(𝑠)=∶𝐾 for all 𝑡,𝑠>0. Indeed: (1)For 𝑡>0, 𝑁∈ℕ, it follows that 𝐾(𝑡/𝑁)=𝐾(𝑡). ⟦Obviously,𝐾(𝑡)⊆𝐾(𝑡/𝑁). On the other hand, 𝐾(𝑡/𝑁) is compact and 𝜏^𝑘_𝑡/𝑁-invariant for all 𝑘∈ℤ, in particular, 𝐾(𝑡/𝑁) is 𝜏_𝑡-invariant. As 𝜏_𝑡 is compactly contracting mod𝐾(𝑡), we obtain 𝐾(𝑡/𝑁)=𝜏^𝑛⋅𝑁_𝑡/𝑁(𝐾(𝑡/𝑁))=𝜏^𝑛_𝑡(𝐾(𝑡/𝑁))→𝑒mod𝐾(𝑡), whence 𝐾(𝑡/𝑁)⊆𝐾(𝑡) follows.⟧(2)Let 𝑡,𝑠>0. Then there exists a sequence 𝑘(𝑛)→∞ such that 𝑛⋅𝑠=𝑘(𝑛)⋅𝑡+𝑟_𝑛 with 0≤𝑟_𝑛<𝑡. For 𝑥∈𝑁=𝐶_𝐾(𝑡)(𝜏_𝑡) we have 𝜏^𝑛_𝑡(𝑥)=𝑦^𝑡_𝑛𝜅^𝑡_𝑛 with 𝑦^𝑡_𝑛→𝑒 and 𝜅^𝑡_𝑛∈𝐾(𝑡). Hence 𝜏^𝑛_𝑠(𝑥)=𝜏_𝑟𝑛(𝑦^𝑡_𝑛)𝜏_𝑟𝑛(𝜅^𝑡_𝑛). The first terms converge to 𝑒, and LIM(𝜏_𝑟𝑛(𝜅^𝑡_𝑛))⊆⋃_{0≤𝑢≤𝑡}𝜏_𝑢(𝐾(𝑡)). Hence LIM(𝜏^𝑛_𝑠(𝑥))⊆𝐾(𝑠) (since 𝑁=𝐶_𝐾(𝑠)(𝜏_𝑠)), on the other hand, LIM(𝜏^𝑛_𝑠(𝑥))⊆⋃_{0≤𝑢≤𝑡}𝜏_𝑢(𝐾(𝑡)).
Replacing 𝑡 by 𝑡/𝑁, 𝑁∈ℕ, in view of Step (1). we obtain 𝐾(𝑠)⊆⋃_{0≤𝑢≤𝑡/𝑁}𝜏_𝑢(𝐾(𝑡)). Continuity of 𝑢↦𝜏_𝑢 finally yields 𝐾(𝑠)⊆𝐾(𝑡). And analogously, 𝐾(𝑡)⊆𝐾(𝑠). Whence 𝐾(𝑡)=∶𝐾 does not depend on 𝑡>0. According to (b), we obtain 𝐶_𝐾(𝜏_𝑡)=𝐶_𝐾(𝑇). Thus (a) is proved.
(b) See [2], [6, Lemma 3.2.26]. (Note that the group {𝜏_𝑡}_𝑡>0 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 𝑌=(𝑦(𝑡))_𝑡∈ℝ such that for all 𝑡,𝑦(𝑡)=𝜅(𝑡)𝑥(𝑡) for some 𝜅(𝑡)∈𝐾—hence in particular, 𝑁𝑥(𝑡)=𝑁𝑦(𝑡)—and 𝑌 commutes element wise with 𝐾.
Consequently, the automorphism group 𝑇∶=(𝜏_𝑡∶=𝑖_𝑦(𝑡)|_𝑁) belongs to the centralizer Cent(𝛽(𝐾),Aut(𝑁)).

Proof. 𝑇 is contracting mod. 𝐾, 𝑋=∶(𝑥(𝑡))≅ℝ is closed and 𝐾 is 𝑇-invariant. Therefore, Γ∶=𝐾⋅𝑋 is a closed subgroup of 𝔾, representable as semidirect product Γ=𝐾⋊_𝑇ℝ. According to [42, Proposition 9.4] or [43, Proposition 1.24], there exists a subgroup 𝑌=(𝑦(𝑡))≅𝑋(≅ℝ), 𝑌⊲Γ, such that Γ=𝐾⊗𝑌. That is, 𝑦(𝑡)=𝜅(𝑡)𝑥(𝑡) for some 𝜅(𝑡)∈𝐾 and 𝑦(𝑡)𝜅=𝜅𝑦(𝑡) for all 𝑡 and all 𝜅∈𝐾.

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

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

Definition 18. Let 𝑁 be a locally compact group, 𝑇=(𝜏_𝑡)⊆Aut(𝑁) as before. (a) {𝜈(𝑡)}_𝑡∈ℝ+⊆ℳ¹(𝑁) is a (continuous time) M-semigroup (with respect to 𝑇=(𝜏_𝑡))—also called skew convolution semigroup, 𝑇-semigroup, or distribution of a generalized Ornstein-Uhlenbeck process—if 𝑡↦𝜈(𝑡) is continuous and if the following cocycle equation is satisfied: 𝜈(𝑡+𝑠)=𝜈(𝑡)⋆𝜏_𝑡(𝜈(𝑠))∀𝑡,𝑠∈ℝ₊. (9) In the following we are interested in relatively compact M-semigroups. (b) 𝜌∈ℳ¹(𝑁) is 𝑇-decomposable (or 𝑇-self-decomposable), if for all 𝑡∈ℝ₊, 𝜌=𝜈(𝑡)⋆𝜏_𝑡(𝜌). Again, Cof_𝜌(𝜏_𝑡) denotes the set of cofactors. 𝜌∈ℳ¹(𝑁) is 𝐾-𝑇-decomposable if in addition, 𝜌 is right 𝐾-invariant.

In analogy to Remark 5 we note the following.

Remark 19. Let 𝑁 be a locally compact group, 𝑇⊆Aut(𝑁). Let {𝜆_𝑡=𝜈(𝑡)⊗𝜀_𝑡} be a space-time continuous convolution semigroup on 𝔾=𝑁⋊_𝑇ℝ. If {𝜆_𝑡} is not adapted then there exists a closed subgroup 𝔾^′=⋃_𝑡𝑁^′⋅(𝑧(𝑡),𝑡) on which it is adapted. 𝑁^′⊆𝑁 is a closed subgroup and (𝑦(𝑡)=(𝑧(𝑡),𝑡)) denotes a continuous one-parameter group in the normalizer of 𝑁^′. Furthermore, 𝜆_𝑡=(𝜈^′(𝑡)⊗𝜀₀)∗𝜀_{(𝑧(𝑡),𝑡)}=(𝜈^′(𝑡)⋆𝜀_𝑧(𝑡))⊗𝜀_𝑡 where {𝜈^′(𝑡)} is a continuous M-semigroup with respect to 𝑇^′=(𝜏^′_𝑡), with (𝜏^′_𝑡∶=𝑖_𝑦(𝑡)|_𝑁′=𝑖_𝑧(𝑡)∘𝜏_𝑡).
Therefore, {𝜆_𝑡} is adapted for all closed subgroups 𝑁^′⊆𝑁 and all continuous one-parameter groups ((𝑧(𝑡),𝑡)) in the normalizer of 𝑁^′, satisfying for all 𝑡∈ℝ(1)𝜏_𝑡𝑁^′=𝑖_𝑧(𝑡)⁻¹𝑁^′,(2)supp𝜈(𝑡)⊆𝑁^′𝑧(𝑡)⟹𝑁^′=𝑁. (10)

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

Theorem 20. Let, as before (Proposition 16), 𝔾=𝑁⋊_𝑇ℝ, 𝜆_𝑠=𝜈(𝑠)⊗𝜀_𝑠, 𝑠≥0, be probabilities on the on the space-time building 𝔾≅𝑁⋊_𝑇ℝ. Obviously, {𝜆_𝑡} is a continuous convolution semigroup iff {𝜈(𝑡)} is a continuous M-semigroup with respect to 𝑇. 𝜆₀=lim_𝑡→0𝜆_𝑡 is idempotent and 𝜆₀∗𝜆_𝑡∗𝜆₀=𝜆_𝑡, for all 𝑡, hence 𝜈(0)=𝜔_𝐻 with 𝜔_𝐻⋆𝜈(𝑡)⋆𝜏_𝑡(𝜔_𝐻)=𝜈(𝑡), 𝑡≥0, and 𝜈(0)=lim_𝑡→0𝜈(𝑡). (𝐻 denotes a compact subgroup of 𝑁.)(a)If {𝜆_𝑡} is nondissipating and adapted then {𝜈(𝑡)} is relatively compact and satisfies condition (10). Furthermore, 𝑁=𝐶_𝐾(𝑇) for some compact 𝑇-invariant subgroup 𝐾⊆𝑁.(b)Conversely, if {𝜈(𝑡)} is relatively compact and satisfies condition (10) then the continuous convolution semigroup {𝜆_𝑡} is nondissipating and adapted. Hence (a) applies.(c)In (a), respectively, (b), lim_𝑡→∞𝜈(𝑡)⋆𝜔_𝐾=∶𝜌∈ℳ¹(𝑁) exists and is 𝑇-𝐾-decomposable with cofactors {𝜈(𝑡)}.(d)Conversely, let 𝑁=𝐶_𝐾(𝑇) and let 𝜌∈ℳ¹(𝑁) be 𝐾-𝑇-decomposable. Then there exists a nondissipating continuous convolution semigroup {𝜆_𝑡=𝜈(𝑡)⊗𝜀_𝑡}⊆ℳ¹(𝔾), with a relatively compact continuous M-semigroup {𝜈(𝑡)} in ℳ¹(𝑁) such that 𝜈(𝑡)∈Cof_𝜌(𝜏_𝑡) for all 𝑡, and furthermore, 𝜌=lim_𝑡→∞𝜈(𝑡)⋆𝜔_𝐾.

Proof. (a) is proved as in the discrete time case, Theorem 6(d). (For aperiodic groups see [22, Proposition A], [23, Theorem 3.16]).
(b) and (c) are proved analogous to Theorem 6(b) and (c).
(d) Let 𝜌∈ℳ¹(𝑁) be 𝐾-𝑇-decomposable. Again as in the case of discrete times, Theorem 6(d), for any 𝑡>0, there exists a relatively compact (discrete) M-semigroup {𝜈_𝑡(𝑘)}_𝑘≥0 of cofactors (with respect to (𝜏^𝑘_𝑡)), such that lim_𝑘𝜈_𝑡(𝑘)⋆𝜔_𝐾=𝜌. In fact, for any selection 𝜈(𝑡)∈Cof_𝜌(𝜏_𝑡), 𝑡≥0, it follows as in Theorem 6(d) that 𝜏_𝑡(𝜌)→𝜔_𝐾 and 𝜈(𝑡)⋆𝜔_𝐾→𝜌. We have to show that the cofactors {𝜈(𝑡)} can be chosen as continuous M-semigroup in ℳ¹(𝑁). For this purpose we need some tools from the embedding problem for infinitely divisible laws on groups. Recall that {𝜈(𝑡)} is a continuous M-semigroup iff {𝜆_𝑡=𝜈(𝑡)⊗𝜀_𝑡} is a continuous convolution semigroup (in ℳ¹(𝔾)). So we have to prove the existence of 𝜆=𝜆₁∈ℳ¹(𝔾), embeddable into a continuous convolution semigroup (𝜆_𝑡=𝜈(𝑡)⊗𝜀_𝑡) such that 𝜈(𝑡)∈Cof_𝜌(𝜏_𝑡), 𝑡≥0 (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 𝜆=𝜆₁=𝜈⊗𝜀₁, 𝜈∈Cof_𝜌(𝜏₁) with roots 𝜆_1/𝑚=𝜈(1/𝑚)⊗𝜀_1/𝑚 where 𝜈(1/𝑚)∈Cof_𝜌(𝜏_1/𝑚) and 𝜈=★₀^𝑚−1𝜏_𝑗/𝑚(𝜈(1/𝑚)). Furthermore, 𝜆 is embeddable into a rational convolution semigroup {𝜆_𝑟}_𝑟∈ℚ+. Moreover, the root sets ℛ_1/𝑚∶={𝜎∈ℳ¹(𝔾)∶𝜎^𝑚=𝜆} are compact and consist of measures 𝜎=𝜇⊗𝜀_1/𝑚, with 𝜇∈Cof_𝜌(𝜏_1/𝑚).
⟦Choose, for 𝑛∈ℕ, 𝜈(1/𝑛)∈Cof_𝜌(𝜏_1/𝑛), put 𝜈^(𝑛)∶=★₀^𝑛−1𝜏_𝑗/𝑛(𝜈(1/𝑛)); hence 𝜈^(𝑛)∈Cof_𝜌(𝜏₁) for all 𝑛. Put 𝜆^(𝑛)_1/𝑛∶=𝜈(1/𝑛)⊗𝜀_1/𝑛; hence 𝜆^(𝑛)∶=(𝜆^(𝑛)_1/𝑛)^𝑛=(𝜈(1/𝑛)⊗𝜀_1/𝑛)^𝑛=𝜈^(𝑛)⊗𝜀₁.
According to the shift-compactness theorem, Cof_𝜌(𝜏_𝑡) is compact for all 𝑡>0. For 𝑛≥𝑚, put 𝜆^(𝑛)_1/𝑚∶=(𝜆^(𝑛!)_1/𝑛!)^(𝑛!/𝑚)=∶𝜈^(𝑛)(1/𝑚)⊗𝜀_1/𝑚 (with 𝜈^(𝑛)(1/𝑚)∈Cof_𝜌(𝜏_1/𝑚)), and 𝜆₁^(𝑛)=𝜆^(𝑛).
By Tychonoff's theorem, there exists a subnet (𝑛^′)⊆ℕ satisfying 𝜈^(𝑛)(1/𝑚)−−−→^(𝑛′)𝜈(1/𝑚)∈Cof_𝜌(𝜏_1/𝑚), equivalently, 𝜆−−−→^(𝑛)_1/𝑚^(𝑛′)𝜆_1/𝑚∶=𝜈(1/𝑚)⊗𝜀_1/𝑚, for all 𝑚∈ℕ. Continuity of convolution yields 𝜆^𝑚_1/𝑚=𝜆₁=𝜆=𝜈(1)⊗𝜀₁, 𝜈(1)∈Cof_𝜌(𝜏₁) (for all 𝑚∈ℕ).
We have shown that 𝜆=𝜆₁ is infinitely divisible, and by construction, the roots satisfy (𝜆_1/𝑚!)^𝑚=𝜆_{1/(𝑚−1)!}; hence, 𝜆 is embeddable into a rational convolution semigroup {𝜆_𝑟}_𝑟∈ℚ+, (cf. [36, 44]), and the construction easily yields moreover 𝜆_𝑟=𝜈(𝑟)⊗𝜀_𝑟, with 𝜈(𝑟)∈Cof_𝜌(𝜏_𝑟), 𝑟≥0.⟧
Let 𝜎∈ℳ¹(𝔾) with 𝜎^𝑚=𝜆=𝜈⊗𝜀₁. Then, as easily seen, 𝜎=𝜇⊗𝜀_1/𝑚 with ★₀^𝑚−1𝜏_𝑗/𝑚(𝜇)=𝜈. Hence, 𝜇∈Cof_𝜌(𝜏_1/𝑚). Therefore, in particular, ℛ_1/𝑚 is compact for all 𝑚.
In order to show that a continuous convolution semigroup {𝜆^′_t} can be selected, put ℛ∶=⋃_𝑛≥1{𝜎^𝑚_𝑛∶𝜎^𝑛_𝑛=𝜆,0≤𝑚≤𝑛}. As mentioned previously, 𝜎_𝑛=𝜇_𝑛⊗𝜀_1/𝑛 and 𝜎^𝑚_𝑛=★₀^𝑚−1𝜏_𝑗/𝑛(𝜇_𝑛).
Claim 2. ℛ is relatively compact in ℳ¹(𝔾) with all accumulation points of the form 𝜈(𝑡)⊗𝜀_𝑡, 𝜈(𝑡)∈Cof_𝜌(𝜏_𝑡),0≤𝑡≤1.
⟦Let𝑡_𝑘∶=𝑚_𝑘/𝑛_𝑘∈[0,1] with 𝑚_𝑘,𝑛_𝑘∈ℕ, 𝑡_𝑘→𝑡∈[0,1]. Let 𝜎_𝑛𝑘=𝜇_𝑛𝑘⊗𝜀_1/𝑛𝑘∈ℛ, and put 𝜈(𝑡_𝑘)∶=⋆^𝑚𝑘_𝑗=0𝜏_{𝑗/𝑛𝑘}(𝜇_𝑛𝑘). Then 𝜏_𝑡𝑘(𝜌)→𝜏_𝑡(𝜌) and 𝜌=𝜈(𝑡_𝑘)⋆𝜏_𝑡𝑘(𝜌) yield that {𝜈(𝑡_𝑘)} is relatively compact and LIM{𝜈(𝑡_𝑘)}⊆Cof_𝜌(𝜏_𝑡).⟧
Let 𝒜∶=LIM{𝜆_𝑡∶𝑡∈ℚ₊∩[0,1],𝑡→0}. Therefore, 𝜎∈𝒜 is representable as 𝜎=𝑢⊗𝜀₀ for 𝑢∈Cof_𝜌(𝜏₀), with 𝜏₀=𝑖𝑑; hence 𝑢⋆𝜌=𝜌. And, by definition of 𝒜, it follows that 𝜎∗𝜆_𝑡=(𝑢⋆𝜈(𝑡))⊗𝜀_𝑡, 𝜆_𝑡∗𝜎=(𝜈(𝑡)⋆𝜏_𝑡(𝑢))⊗𝜀_𝑡 for all 𝑡≥0.
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 {𝜆^′_𝑡} of the form 𝜆^′_𝑡=𝜈^′(𝑡)⊗𝜀_𝑡 for all 𝑡, with 𝜆^′_𝑡=𝜎_𝑡∗𝜆_𝑡=𝜆_𝑡∗𝜎_𝑡 where 𝜎_𝑡∈𝒜, hence 𝜎_𝑡=𝑢_𝑡⊗𝜀₀, 𝑢_𝑡∈Cof_𝜌(𝜏₀). In particular, 𝜈^′(𝑡)=𝜈(𝑡)⋆𝜏_𝑡(𝑢_𝑡). Hence, as immediately seen, 𝜈^′(𝑡)∈Cof_𝜌(𝜏_𝑡).
Since {𝜆^′_𝑡} is a continuous convolution semigroup, {𝜈^′(𝑡)} is a continuous M-semigroup of cofactors of 𝜌. Whence the assertion.

For the following Theorem 22 we need some preparations.

Let ℍ be a locally compact group and {𝜆_𝑡} a continuous convolution semigroup. The Lévy measure 𝜂 is defined by ∫𝑔d𝜂=d⁺/d𝑡|_𝑡=0∫𝑔d𝜆_𝑡, for 𝑔∈𝐶^𝑏(ℍ) vanishing in a neighbourhood of the unit. If as before, ℍ=𝔾=𝑁⋊_𝑇ℝ and 𝜆_𝑡=𝜈(𝑡)⊗𝜀_𝑡, 𝑡≥0, then, as easily seen, the Lévy measure 𝜂 is supported by 𝑁 and ∫_𝑁𝑔d𝜂=d⁺/d𝑡|_𝑡=0∫_𝑁𝑔d𝜈(𝑡). Recall that Lévy measures are bounded outside neighbourhoods of 𝑒. We recall a result of Siebert (see [45, Theorem 1], [46, Theorem 5]).

Fact 3. Let 𝑔≥1 be a submultiplicative function. Let 𝜖>1, such that 𝑈=𝑈_𝑔,𝜖={𝑔≤𝜖}∈𝔘_ℍ. Then we have 𝑔d𝜆_𝑡<∞forsome[all]𝑡≥0iff_𝔾⧵𝑈𝑔d𝜂<∞. (11) Thus for subadditive functions 𝑓≥0 we have with 𝑔∶=1+𝑓: ∫1+𝑓d𝜆_𝑡<∞ iff ∫_𝔾⧵𝑈1+𝑓d𝜂<∞. Equivalently, ∫𝑓d𝜆_𝑡<∞ iff ∫_𝔾⧵𝑈𝑓d𝜂<∞.
As mentioned before, if the integral on the right side of (11) is finite for 𝑈=𝑈_𝑔,𝜖, then it is finite for any 𝑈∈𝔘_ℍ. And, if it is finite for some relatively compact 𝑈₀∈𝔘_ℍ, it is finite for all 𝑈∈𝔘_ℍ.

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 𝔾=𝑁⋊_𝑇ℝ with 𝑁=𝐶_𝐾(𝑇). Recall (cf. Proposition 16) that we have 𝑁=𝐶⋊_𝛽𝐾, where 𝐶=𝐶(𝑇) is a homogeneous group, that is, a contractible simply connected nilpotent Lie group, and 𝑇=(𝜏_𝑡=𝑖_𝑥(𝑡)|_𝑁)_𝑡∈ℝ. We will tacitly assume, according to Proposition 17, that {𝑥(𝑡)}_𝑡∈ℝ and 𝐾 commute element wise. Recall that a homogeneous group norm (adapted to 𝑇) is a continuous subadditive function 𝐶(𝑇)→ℝ₊ satisfying ‖𝑥‖=0 iff 𝑥=𝑒, ‖𝑥‖=‖𝑥⁻¹‖ and ‖𝛿_𝑡(𝑥)‖=𝑡⋅‖𝑥‖, where (𝛿_𝑡) denotes a group of dilations defined by 𝑇 (cf. for example, [6, 2.1.10, 2.7.26 (d)], [29]).

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

Step 2 (cf. Lemma A.2). Define the function 𝜙∶𝑁=𝐶⋊_𝛽𝐾→ℝ₊ by 𝜙(𝑥,𝜅)∶=‖𝑥‖, ‖⋅‖ as in Step 1. Then 𝜙 is subadditive, continuous and satisfies 𝜙(𝛾)=𝜙(𝛾⁻¹), 𝜙(𝛾)=0 iff 𝛾=(𝑒,𝜅)∈𝐾 and furthermore, 𝜙 is 𝐾-bi-invariant. (We call 𝜙 pseudo-group norm adapted to 𝑇.)
Moreover, for any 𝑡₀>0 there exist constants 0<𝑟≤𝑅≤1,0<𝑠≤𝑆<1, satisfying for all 𝛾∈𝑁 the growth conditions 𝑟^𝑛𝜙𝛾≤𝜙𝜏^𝑛_𝑡0𝛾≤𝑅^𝑛𝜙𝛾𝑆,𝑛≥0,_𝑛𝜙𝛾≤𝜙𝜏^𝑛_𝑡0𝛾≤𝑠^𝑛𝜙𝛾,𝑛<0. (12) For large 𝑡₀ one can put for example, 𝑅=‖𝜏_𝑡0|_𝐶‖, 𝑟∶=‖𝜏_𝑡⁻¹₀|_𝐶‖⁻¹,𝑆=𝑅,𝑠=𝑟. The automorphism norm is defined as ‖𝜎‖∶=sup{‖𝜎(𝑥)‖∶𝑥∈𝐶,‖𝑥‖=1}. In particular, ‖𝜏_𝑠+𝑡|_𝐶‖≤‖𝜏_𝑡|_𝐶‖⋅‖𝜏_𝑠|_𝐶‖, ‖𝜏_𝑡|_𝐶‖→0 for 𝑡→∞ and ‖𝜏_−𝑡|_𝐶‖≥1 for 𝑡≥0.

Step 3 (cf. Lemma A.3). Let 𝑓∶𝑁→ℝ₊ be a regular 𝜏_𝑡0-function for some 𝑡₀>0, 𝑓(𝜏^𝑝_𝑡0(𝛾))=𝛼𝑓(𝛾), 0<𝛼<1. (See Definition 1.)
Then we obtain log(1+𝑓)∼log(1+𝜙), (𝜙 defined in Step 2.)

Remark 21. As mentioned before, in the situation 𝑁=𝐶, 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 𝜙 are not 𝜏_𝑡-functions—they are 𝛿_𝑡-functions for dilations 𝛿_𝑡, 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 𝜓∶𝔾=𝑁⋊_𝑇ℝ→ℝ₊, 𝜓(𝛾,𝑡)∶=1+𝜙(𝛾)+‖𝜏_−|𝑡||_𝐶‖ is continuous, submultiplicative, ≥2 and satisfies 𝜓(𝛾,𝑡)=2 iff 𝑡=0,𝛾∈𝐾. Therefore log𝜓 is subadditive and >0. Moreover {𝜓<𝛼} is relatively compact and belongs to 𝔘_𝔾 for all 𝛼>2.

Step 5 (cf. Lemma A.5). Let, as before, 𝑁=𝐶⋊_𝛽𝐾 and 𝔾=𝑁⋊_𝑇ℝ. 𝑁 and 𝔾 are compactly generated for any neighbourhood 𝑈∈𝔘_𝐶, 𝑉∶=𝑈⋅𝐾 generates 𝑁, and, for example, for any 𝜖>0, 𝑊∶=𝑉⊗[−𝜖,𝜖] generates 𝔾.
Let 𝑓 be a regular 𝜏_𝑡0-function, for some 𝑡₀>0. Let, for 𝑡>0, 𝑊_𝑡∶=⋃_{0≤𝑠≤𝑡}{𝑉⊗{𝑠}∪{𝑉⊗{𝑠}}⁻¹} where 𝑉={𝑓≤1}. Then 𝑊_𝑡 generates 𝔾, and for any symmetric relatively compact 𝑊∈𝔘_𝔾 with 𝑊=𝑊𝐾, we have 𝛿_𝑊𝑡∼𝛿_𝑊. (For the definition of 𝛿_𝑊 cf. (1).) In particular, for all 𝑡₁,𝑡₂>0, it follows that 𝛿_𝑊𝑡1∼𝛿_𝑊𝑡2.

Next we characterize—in analogy to Theorem 6(e)—nondissipativity by moment conditions. For the continuous time case we introduce additional equivalent conditions (b^∗), (d^∗), (e^∗) which depend on 𝑇 but not on the particular automorphisms 𝜏_𝑡0.

Theorem 22. Let, as before, 𝔾=𝑁⋊_𝑇ℝ with 𝑁=𝐶_𝐾(𝑇)=𝐶⋊_𝛽𝐾, 𝐶=𝐶(𝑇) a homogeneous group. Let {𝜆_𝑡=𝜈(𝑡)⊗𝜀_𝑡}_𝑡≥0 be an adapted continuous convolution semigroup with Lévy measure 𝜂. Let 𝜙 denote the pseudo-group norm on 𝑁 introduced in Step 2 via the 𝐾-invariant homogeneous norm ‖⋅‖ on 𝐶=𝐶(𝑇) (cf. Step 1). Then the following assertions are equivalent. (a)  {𝜆_𝑡} is nondissipating. (b)  For some (any) 𝑡₀>0, 𝜆_𝑡0 has finite first-order moments (on 𝔾). (b^∗)  For some (any) 𝑡₀>0, ∫_𝔾𝛿_𝑊d𝜆_𝑡0<∞ for a compact 𝐾-invariant neighbourhood 𝑊∈𝔘_𝔾. (c)  𝜂 has finite first-order moments (on 𝔾), that is, ∫_∁𝑈𝑓d𝜂<∞ for all subadditive 𝑓∶𝔾→ℝ₊ and 𝑈∈𝔘_𝔾. (d)  For some (any) 𝑡₀>0, 𝜈(𝑡₀) has finite logarithmic moments, that is, for any regular 𝜏_𝑡0-function 𝑓, one has ∫_𝑁log(1+𝑓)d𝜈(𝑡₀)<∞. (d^∗)  ∫_𝑁log(1+𝜙)d𝜈(𝑡₀)=∫_𝑁log(1+‖𝑥‖)d𝜈(𝑡₀)(𝑥,𝜅)<∞. (e)  𝜂 (considered as measure on 𝑁) has finite logarithmic moments, that is, for all 𝑡, ∫_𝑁⧵𝑈log(1+𝑓)d𝜂<∞ for all regular 𝜏_𝑡-functions, for all 𝑈∈𝔘_𝑁. (e^∗)  ∫_𝑁⧵𝑈log(1+𝜙)d𝜂=∫_𝑁⧵𝑈log(1+‖𝑥‖)d𝜂(𝑥,𝜅)<∞, for all 𝑈∈𝔘_𝑁.

Proof. According to Proposition 12 and Theorem 20 we know that condition (a) is equivalent with shift compactness of a skeleton random walk {𝜆^𝑘_𝑡0} for some (hence any) 𝑡₀>0. Hence (a) is equivalent to (b1) For some [any] 𝑡₀>0, for any subadditive function 𝑓 on 𝔾_𝑡0, ∫_𝔾𝑡0𝑓d𝜆_𝑡0<∞ (i.e., 𝜆_𝑡0 has finite first-order moments on 𝔾_𝑡0.)
Obviously, “(b1)⇒(b)”. By Fact 2, “(b1)⇔(d)”; furthermore, by Step 3 (cf. Lemma A.3), “(d)⇔(d^∗)” and “(e*)⇔(e)”.
According to Step 5 (Lemma A.5), (b)⇔∫_𝔾𝛿_𝑊𝑡0d𝜆_𝑡0<∞, and therefore (b)⇔(b*) ∫_𝔾𝛿_𝑊d𝜆_𝑡0<∞ for any 𝑡₀>0, equivalently, for any 𝑡₀,𝑡₁>0, ∫_𝔾𝛿_𝑊𝑡1d𝜆_𝑡0<∞, (since 𝛿_𝑊∼𝛿_𝑊𝑡0∼𝛿_𝑊𝑡1). ⟦Let us mention that (b1)⇒ ∫_𝔾𝑡0𝛿_𝑉𝑡0d𝜆_𝑡0<∞, and since 𝛿_𝑊𝑡0≤𝛿_𝑉𝑡0, we obtain again “(b1)⇒(b)”.⟧ Applying Siebert's result, Fact 3, to the subadditive function 𝛿_𝑊𝑡0 (resp., to 𝛿_𝑊 for any 𝐾-invariant relatively compact symmetric 𝑊∈𝔘_𝔾) yields “(b*)⇔(c)”.
log𝜓 is subadditive—𝜓 as in Step 4 (Lemma A.4)—hence again by Fact 3 we obtain (b)⇒∫_𝔾log𝜓d𝜆_𝑡0<∞⇔∫_∁𝑈log𝜓d𝜂<∞ for all 𝑈∈𝔘_𝔾.
But we have ∫_𝔾log𝜓d𝜆_𝑡0=∫_𝑁log(1+𝜙)d𝜈(𝑡₀)<∞, hence (b)⇒(d^∗) and ∫_𝔾⧵𝑈log𝜓d𝜂=∫_{𝑁⧵𝑈∩𝑁}log(1+𝜙)d𝜂. Hence the left integral is finite iff (e^∗) holds, whence (d^∗)⇔(e^∗).
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 lim_𝑡→∞𝜈(𝑡) does not need to exist if 𝑁=𝐶_𝐾(𝑇), 𝐾≠{𝑒}. (For 𝐾={𝑒}, i.e., 𝑁=𝐶(𝑇), lim_𝑡𝜈(𝑡) always exists, cf. [22, Proposition A]).

Example 23. Let Γ be a contractible Lie group, hence simply connected and nilpotent, with contracting one-parameter group 𝑆=(𝜎_𝑡)_𝑡>0 in Aut(Γ), and let 𝐾 be a solenoidal compact group with dense one-parameter subgroup (𝑧(𝑡))_𝑡∈ℝ. Put 𝑁∶=Γ⊗𝐾 and define 𝑇∶=(𝜏_𝑡)⊆Aut(𝑁) by 𝜏_𝑡=𝜎_𝑡⊗𝑖𝑑, 𝑡∈ℝ. Hence 𝑁=𝐶_𝐾(𝑇). Let {𝜇(𝑡)} be a continuous M-semigroup (with respect to 𝑆) in ℳ¹(Γ) such that 𝜌=lim𝜇(𝑡) exists. As 𝜉_𝑡∶=𝜏_𝑡|_𝐾≡𝑖𝑑, continuous M-semigroups in ℳ¹(𝐾) (with respect to (𝜉_𝑡)) are just continuous convolution semigroups; in particular, {𝜀_𝑧(𝑡)} is a M-semigroup. It is immediately verified that {𝜈(𝑡)∶=𝜇(𝑡)⊗𝜀_𝑧(𝑡)} is a relatively compact M-semigroup in ℳ¹(𝑁) (with respect to 𝑇), with nondissipating continuous convolution semigroup {𝜆_𝑡∶=𝜈(𝑡)⊗𝜀_𝑡}.
As (𝑧(𝑡)) is dense in 𝐾, LIM{𝜈(𝑡)∶𝑡→∞}={𝜌⊗𝜀_𝜅∶𝜅∈𝐾}. In particular, lim_𝑡→∞𝜈(𝑡) 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 𝑁 be a simply connected nilpotent Lie group and 𝑆=(𝜎_𝑡)⊆Aut(𝑁) be a contracting continuous group with multiplicative parametrization, 𝜎_𝑡𝜎_𝑠=𝜎_𝑡⋅𝑠 for 𝑡,𝑠>0, and lim_𝑡→0𝜎_𝑡(𝑥)=𝑒 for all 𝑥∈𝑁. A continuous convolution semigroup {𝜌_𝑡} in ℳ¹(𝑁) is called 𝑆-stable if 𝜎_𝑡(𝜌₁)=𝜌_𝑡, for all 𝑡>0, equivalently, 𝜎_𝑡(𝜌_𝑠)=𝜌_𝑡⋅𝑠 for 𝑡>0,𝑠≥0. Put 𝜌∶=𝜌₁. We have 𝜌=𝜌_1−𝑠⋆𝜌_𝑠=𝜌_1−𝑠⋆𝜎_𝑠(𝜌), for all 0<𝑠≤1; hence 𝜌 is 𝜎_𝑠-decomposable with cofactors 𝜌_1−𝑠∈Cof_𝜌(𝜎_𝑠).
To obtain a continuous M-semigroup of cofactors we have to switch to additive parametrization: 𝑇∶=(𝜏_𝑡∶=𝜎_e−𝑡)_𝑡∈ℝ is a continuous one-parameter group satisfying 𝜏_𝑡+𝑠=𝜏_𝑡𝜏_𝑠 and lim_𝑡→∞𝜏_𝑡(𝑥)=𝑒 for all 𝑥∈𝑁. And with this notations we obtain 𝜌=𝜌_1−e−𝑡⋆𝜏_𝑡(𝜌) for 𝑡≥0.
As immediately verified, {𝜈(𝑡)∶=𝜌_{1−𝑒−𝑡}}_𝑡∈ℝ+ is a relatively compact continuous M-semigroup with 𝜈(𝑡)∈Cof_𝜌(𝜏_𝑡) (and 𝜌=lim𝜈(𝑡)); hence it defines a nondissipating continuous convolution semigroup {𝜆_𝑡=𝜌_{1−𝑒−𝑡}⊗𝜀_𝑡}⊆ℳ¹(𝔾), 𝔾=𝑁⋊_𝑇ℝ. Note that in this (trivial) case, the M-semigroup belongs to {𝜌_𝑠}_𝑠≥0, respectively, to the orbit {𝜏_𝑡(𝜌)}_𝑡≥0: 𝜈(𝑡)=𝜌_{1−𝑒−𝑡}=𝜎_{1−𝑒−𝑡}(𝜌)=𝜏_{−log(1−𝑒−𝑡)}(𝜌).
Analogously, stable laws with nontrivial idempotents on 𝑁=𝐶⋊_𝑇𝐾 generate 𝑇-decomposable laws (see [6, § 3.5]).
A variety of examples of (self-) decomposable laws are obtained via subordination as in Example 11: Let {𝜌_𝑡} be a continuous convolution semigroup which is stable with respect to 𝑇 as above. Let 𝜉 be a (𝐻_𝑒−𝑡)-decomposable probability measure on ℝ₊, where 𝐻_𝛼∶𝑥↦𝛼⋅𝑥 denotes the homothetic transformation on ℝ. Then, as immediately verified, the subordinated measure 𝜇∶=∫^∞₀𝜌_𝑡d𝜉(𝑡) is 𝑇-decomposable. In particular, if 𝜉 is an exponential distribution, we obtain geometric stable laws (or geometric semistable laws, if {𝜌_𝑡} 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 𝜏_𝑡-Functions

Let, as before, 𝑁 be a locally compact group, 𝑇=(𝜏_𝑡)⊆Aut(𝑁), a continuous one-parameter group such that 𝑁=𝐶_𝐾(𝑇).

Lemma A.1. There exists a homogeneous norm ‖⋅‖ on 𝐶(𝑇) which is in addition a 𝛽(𝐾)-invariant.

Proof. Put, as before, 𝐶=𝐶(𝑇). 𝑇|_𝐶=(𝜏_𝑡|_𝐶)_𝑡∈ℝ defines a group of dilations, that is, a multiplicative parametrized continuous group 𝐷=(𝛿_𝑠)_𝑠>0⊆Aut(𝐶), 𝛿_𝑠𝛿_𝑟=𝛿_𝑠𝑟, and a subadditive function ‖⋅‖₁∶𝐶→ℝ₊ such that ‖𝑥⁻¹‖₁=‖𝑥‖₁, ‖𝑥‖₁=0 iff 𝑥=𝑒 and ‖𝛿_𝑠(𝑥)‖₁=𝑠‖𝑥‖₁ for all 𝑠>0, 𝑥∈𝐶. Furthermore, 𝛿_𝑠 commutes with all automorphisms 𝛼∈Aut(𝐶) which commute with all 𝜏_𝑡.
⟦In fact, let 𝕍 denote the Lie algebra of 𝐶. Then Aut(𝐶)≅Aut(𝕍)⊆GL(𝕍), and the correspondence is expressed by Aut(𝐶)∋𝛼↔𝛼^∘∈Aut(𝕍). 𝛿^∘_𝑠 can be defined via the Jordan decomposition of the exponent  𝐸, defined by 𝜏^∘_𝑡=𝑒^𝑡𝐸 (cf. [29, Proof of Proposition 2.4], [6, 2.1.13]). Whence the assertion follows.⟧
As by assumption (𝜏_𝑡|_𝐶)⊆Cent(𝛽(𝐾),Aut(𝐶)) (Proposition 17) it follows that (𝛿_𝑠)⊆Cent(𝛽(𝐾),Aut(𝐶)).
Put ‖𝑥‖∶=∫_𝐾‖𝛽(𝜅)(𝑥)‖₁d𝜔_𝐾(𝜅), with 𝜔_𝐾 denoting the Haar measure. As immediately seen, ‖⋅‖ is symmetric, subadditive, with ‖𝑥‖=0 iff 𝑥=𝑒, {‖𝑥‖≤𝛼} compact in 𝔘_𝐶 for 𝛼>0, and ‖⋅‖ is 𝛽(𝐾)-invariant by construction. Furthermore, ‖𝛿_𝑠(𝑥)‖=∫_𝐾‖𝛽(𝜅)(𝛿_𝑠(𝑥))‖₁d𝜔_𝐾(𝜅)=∫_𝐾‖𝛿_𝑠(𝛽(𝜅)(𝑥))‖₁d𝜔_𝐾(𝜅)=𝑠∫_𝐾‖𝛽(𝜅)(𝑥)‖₁d𝜔_𝐾(𝑥)=𝑠‖𝑥‖.

Lemma A.2. Define the function 𝜙∶𝑁=𝐶⋊_𝛽𝐾→ℝ₊ by 𝜙(𝑥,𝜅)∶=‖𝑥‖. Then 𝜙 is subadditive and continuous, satisfies 𝜙(𝛾)=𝜙(𝛾⁻¹), 𝜙(𝛾)=0 iff 𝛾=(𝑒,𝜅)∈𝐾, and furthermore, 𝜙 is 𝐾-bi-invariant.
Moreover, for any 𝑡₀>0 there exist constants 0<𝑟≤𝑅<1, 0<𝑠≤𝑆<1, satisfying for all 𝛾∈𝑁 the growth conditions 𝑟^𝑛𝜙𝛾≤𝜙𝜏^𝑛_𝑡0𝛾≤𝑅^𝑛𝜙𝛾𝑆,𝑛≥0,_𝑛𝜙𝛾≤𝜙𝜏^𝑛_𝑡0𝛾≤𝑠^𝑛𝜙𝛾,𝑛<0. (A.1) For large 𝑡₀ one can put for example 𝑅∶=‖𝜏_𝑡0|_𝐶‖,𝑟∶=‖𝜏_𝑡⁻¹₀|_𝐶‖,𝑆=𝑅,𝑠=𝑟. The automorphism norm is defined as ‖𝜎‖∶=sup{‖𝜎(𝑥)‖∶𝑥∈𝐶,‖𝑥‖=1}. In particular, one has ‖𝜏_𝑠+𝑡|_𝐶‖≤‖𝜏_𝑡|_𝐶‖⋅‖𝜏_𝑠|_𝐶‖, ‖𝜏_𝑠|_𝐶‖→0 for 𝑡→∞ and ‖𝜏_−𝑡|_𝐶‖≥1 for 𝑡≥0.

Proof. Obviously, 𝜙 is continuous, 𝜙≥0,𝜙(𝑥,𝜅)=0 iff 𝑥=𝑒, that is, iff (𝑥,𝜅)∈𝐾. 𝜙 is subadditive: 𝜙((𝑥,𝜅)(𝑦,𝜅^′))=𝜙(𝑥𝛽(𝜅)(𝑦),𝜅𝜅^′)=‖𝑥𝛽(𝜅)(𝑦)‖≤‖𝑥‖+‖𝛽(𝜅)(𝑦)‖=‖𝑥‖+‖𝑦‖=𝜙(𝑥,𝜅)+𝜙(𝑦,𝜅^′).
According to [29, Proposition 2.5(a) and (3.1)], [6, 2.10.15(b) and (2.14.11)], there exist 𝑠,𝑟,𝑆,𝑅 satisfying the growth conditions (A.1) for 𝐶,‖⋅‖ and 𝜏_𝑡|_𝐶∈Aut(𝐶). (We have 𝑅∶=‖𝜏_𝑡0|_𝐶‖<1 for large 𝑡₀. If ‖𝜏_𝑡0|_𝐶‖≥1 put 𝑅∶=‖𝜏_𝑚𝑡0|_𝐶‖^1/𝑚<1 for sufficiently large 𝑚.)
Thus by the definition of 𝜙 the growth conditions (A.1) for 𝑁 follow.
Finally, 𝜙((𝑒,𝜅₁)(𝑥,𝜅₀)(𝑒,𝜅₂))=𝜙(𝛽(𝜅₁)(𝑥),𝜅₁𝜅₀𝜅₂)=𝜙(𝑥,𝜅₀) for 𝑥∈𝑁, 𝜅_𝑖∈𝐾 shows 𝐾-bi-invariance of the pseudo-group norm  𝜙.

Lemma A.3. Let 𝑓∶𝑁→ℝ₊ be a regular 𝜏_𝑡0-function for some 𝑡₀>0, 𝑓(𝜏^𝑝_𝑡0(𝛾))=𝛼𝑓(𝛾) (for some 𝛼>0) (cf. Definition 1).
Then one obtains log(1+𝑓)∼log(1+𝜙), (𝜙 defined in Lemma A.2).

Proof (similar to proof of Lemma 2.9 in [1]). With 𝑡₁=𝑝𝑡₀>0 we have 𝑓(𝜏_𝑡1(𝛾))=𝛼𝑓(𝛾). Put 𝐶₀∶={𝑓≤1}, 𝐶_𝑛∶=𝜏_𝑡^−𝑛₁(𝐶₀),𝑛∈ℤ. Hence 𝑛↦𝐶_𝑛 is increasing and 𝑁=𝐾∪⋃_𝑛∈ℤ(𝐶_𝑛+1⧵𝐶_𝑛). 𝑓 is a regular 𝜏_𝑡1-function and hence 𝐶₀, and thus all 𝐶_𝑛 are compact neighbourhoods in 𝔘_𝑁, and by definition of 𝑓, 𝐶_𝑛={𝑓≤𝛼^−𝑛}. 𝜙 is continuous and subadditive, hence locally bounded. Thus there exist constants 𝑚,𝑀, such that 0≤𝑚≤𝜙(𝛾)≤𝑀<∞ for all 𝛾∈𝐶₀⧵𝐶₋₁. According to Lemma A.2, we obtain for 𝛾∈𝐶_𝑛⧵𝐶_𝑛−1, hence for 𝜏^𝑛_𝑡1(𝛾)∈𝐶₀⧵𝐶₋₁: 𝑆^−𝑛𝑚≤𝜙𝛾=𝜙𝜏_𝑡^−(𝑛+1)₁𝜏_𝑡^𝑛+1₁𝛾≤𝑠^−𝑛𝑟𝑀,𝑛≥0,_−𝑛𝑚≤𝜙𝛾=𝜙𝜏_𝑡^−(𝑛+1)₁𝜏_𝑡^𝑛+1₁𝛾≤𝑅^−𝑛𝑀,𝑛<0. (A.2) On the other hand, for 𝑛≥0, 𝛼^−𝑛≤𝑓(𝛾)≤𝛼^−(𝑛+1), equivalently, 𝑛+1≥log𝑓(𝛾)^{−1/log(𝛼)}≥𝑛. Inserting this in (A.2) yields 𝑚𝑓𝛾^{log𝑆/log𝛼}≤𝜙𝛾≤𝑀𝑓𝛾^{log𝑠/log𝛼},𝑛≥0 (A.3) and analogous estimates for 𝑛<0.
Together, there exist constants 𝑢,𝑣,𝑈,𝑉 such that 𝑢𝑓^𝑣≤𝜙≤𝑈𝑓^𝑉, whence log(1+𝜙)∼log(1+𝑓) follows.

Lemma A.4. With the notations of the preceding lemmas define 𝜓∶𝔾=𝑁⋊_𝑇ℝ→ℝ₊ by 𝜓(𝛾,𝑡)∶=1+𝜙(𝛾)+‖𝜏_−|𝑡||_𝐶‖.
Then 𝜓 is continuous, ≥2, submultiplicative and satisfies 𝜓(𝛾,𝑡)=2 iff 𝑡=0,𝛾∈𝐾. Therefore, log𝜓 is subadditive and >0. Moreover {𝜓<𝛼} is relatively compact and belongs to 𝔘_𝔾 for all 𝛼>2.

Proof (compare with Propositions 4.6 in [29], 2.14.28 in [6]). For short, we write ‖𝜏_𝑡‖ instead of ‖𝜏_𝑡|_𝐶‖. We have ‖𝜏_𝑡‖−−−−−^𝑡→∞→0.
The automorphism norm, restricted to Cent(Aut(𝐶),(𝛿_𝑡)), is submultiplicative, thus ‖𝜏_𝑡+𝑠‖≤‖𝜏_𝑡‖⋅‖𝜏_𝑠‖, satisfies ‖𝜏_𝑡‖≤‖𝜏_−|𝑡|‖, ‖𝜏_−|𝑡|‖≥1,‖𝜏₀‖=1, and we have 𝜙(𝜏_𝑡(𝛾))≤‖𝜏_−|𝑡|‖⋅𝜙(𝛾). Hence 𝜓𝛾,𝑡𝛾^′,𝑠=𝜓𝛾𝜏_𝑡𝛾^′,𝑡+𝑠=1+𝜙𝛾𝜏_𝑡𝛾^′+𝜏_{−|𝑡+𝑠|}≤1+𝜙𝛾+𝜙𝜏_𝑡𝛾^′+𝜏_−|𝑡|⋅𝜏_−|𝑠|≤1+𝜙𝛾+𝜏_−|𝑡|⋅𝜙𝛾^′+𝜏_−|𝑡|⋅𝜏_−|𝑠|≤1+𝜙𝛾+𝜏_−|𝑡|⋅1+𝜙𝛾^′+𝜏_−|𝑠|. (A.4) Whence submultiplicativity of 𝜓 follows. Furthermore, 𝜓(𝛾,𝑡)=1+𝜙(𝛾)+‖𝜏_−|𝑡|‖≥2+𝜙(𝛾)≥2, and 𝜓(𝛾,𝑡)=2 iff 𝑡=0 and 𝜙(𝛾)=0, that is, iff (𝛾,𝑡)∈𝐾. The last assertion follows since 𝜓(𝛾,𝑡)<𝛼⇔𝜙(𝛾)+‖𝜏_−|𝑡|‖<𝛼−1.

Lemma A.5. Let, as before, 𝑁=𝐶⋊_𝛽𝐾 and 𝔾=𝑁⋊_𝑇ℝ. 𝑁 is compactly generated: for any symmetric 𝑈∈𝔘_𝐶, 𝑉∶=𝑈⋅𝐾 generates 𝑁, and, for example, for any 𝜖>0, 𝑊∶=𝑉⊗[−𝜖,𝜖] generates 𝔾. Let 𝑓 be a regular 𝜏_𝑡0-function, for some 𝑡₀>0. Let, for 𝑡>0, 𝑊_𝑡∶=⋃_{0≤𝑠≤𝑡}{𝑉⊗{𝑠}∪{𝑉⊗{𝑠}}⁻¹}, where 𝑉={𝑓≤1}.
Then 𝑊_𝑡 generates 𝔾 and for any symmetric relatively compact 𝑊∈𝔘_𝔾 with 𝑊=𝑊𝐾, one has 𝛿_𝑊𝑡∼𝛿_𝑊. In particular, for all 𝑡₁,𝑡₂>0 it follows that 𝛿_𝑊𝑡1∼𝛿_𝑊𝑡2.

Proof. Let again 𝑇=(𝜏_𝑡=𝑖_𝑥(𝑡)|_𝑁). By assumption, (𝑥(𝑡)) and 𝐾 commute element wise. We have 𝑁=𝐶⋅𝐾, 𝐶⊲𝑁,𝐾⊆𝑁,𝐶∩𝐾={𝑒}. Put 𝑋∶=(𝑥(𝑡))⊆𝔾, then Γ∶=𝐶⋅𝑋≅𝐶⋊_𝑇|𝐶ℝ is a subgroup of 𝔾, such that Γ𝐾=𝐶𝑋𝐾=𝐶𝐾𝑋=𝔾. Hence 𝔾≅𝑁⋊_𝑇ℝ≅(𝐶⋊_𝑇|𝐶ℝ)⋊_𝛽𝐾, where the action 𝛽 is trivial on ℝ, that is, 𝛽(𝜅)(𝑐,𝑡)=(𝛽(𝜅)(𝑐),𝑡).
Γ is connected, hence any neighbourhood 𝐵∈𝔘_Γ generates Γ. Thus 𝐵⋅𝐾 generates 𝔾.
The resting assertions follow immediately since 𝑊_𝑡0∈𝔘_𝔾, 𝑊_𝑡0𝐾=𝑊_𝑡0 (since 𝑓 is 𝐾-bi-invariant, cf. [1, Proposition 2.8]). Hence according to [28, Remarque 3], 𝛿_𝑊𝑡0∼𝛿_𝐵∼𝛿_𝑊𝑡1 for any 𝑡₀,𝑡₁>0.