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

Authors: Wilfried Hazod
Journal: Journal of Mathematics
 (2013)

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## 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 toτ-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=τt denoting a continuous group of automorphisms acting as contracting mod. a compact subgroup.


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## Body

## 1. Introduction

Let𝔾 be a locally compact group, λ∈ℳ1(𝔾) an (w.l.o.g.) adapted probability measure. λ, more precisely, the random walk {λk}k≥0, is called nondissipating (also nonscattering) if for some compact subset C⊆𝔾 the (right) concentration functions fλk(C)∶=supx∈𝔾λk(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,{λk}k≥0 is relatively (right) shift compact, equivalently, {λk*λ~k}k≥0 is relatively compact [1].Furthermore, ifN=Nλ 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*εx with suppν⊆N.Denoting the restriction of the inner automorphismix:z↦xzx-1 to N by τ∶=ix|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 λk=ν(k)⊗εk, 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 of𝔾: it is shown in [1] that N=CK(τ), where K is a compact τ-invariant subgroup of N and C{e}(τ)=:C(τ)∶={x∈N:τn(x)→e} (for K={e}), respectively, CK(τ)∶={x∈N:τn(x)→emodK} denote the contractible, respectively, the K-contractible subgroups. As easily seen, for a random walk {λk=ν(k)⊗εk}⊆ℳ1(N⋊τℤ), ν(k) is representable as ν(k)=⋆j=0k-1τj(ν), k∈ℕ, (“⋆” denoting convolution on N) and hence satisfies the relation ν(k+ℓ)=ν(k)⋆τk(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 {λk} 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 {λk=ν(k)⊗εk} 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∶=limk→∞ν(k)⋆ωK exists and is τ-decomposable, that is, ρ=ν(k)*τk(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)⋆ωK=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 andτ-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 {λt}t∈ℝ+, the distributions of a Lévy process. At first (Propositions 12 and 13), we show that {λt}t∈ℝ+ is nondissipating and adapted iff some (hence all) skeleton random walk{λt0k=λt0k}k∈ℤ+ is nondissipating; it is adapted on 𝔾t0, the group generated by suppλt0. Furthermore, we show that the subgroup N∶=Nλt0 is independent of t0>0 and is a normal subgroup in 𝔾 (not only within the group 𝔾t0), and we observe 𝔾=⋃t‍𝔾t. 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=(τt∶=ix(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=CK(T)=C⋊βK, β∈Hom(K,Aut(C)).Theorems20 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, CK(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. 𝔘𝔾 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, λk the kth convolution power, λ0∶=εe, where εx 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=Nλ denotes the smallest closed normal subgroup containing suppλ*λ~. The concentration function of the random walk {λk}k∈ℤ+ is defined as ℤ+∋k↦fλk(C)∶=supx∈𝔾λk(Cx) for compact C⊆𝔾. The probability measure λ, or more precisely, the random walk {λk} is called nondissipating if fλk(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: CK(τ)∶={x∈N:τk(x)→emodK} and C(τ)=C{e}(τ)={x:τk(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){λk} is nondissipating iff {λk} is relatively (right) shift compact; that is, for some xk∈𝔾, {λk*εxk-1} is relatively compact. Equivalently, iff {λk*λ~k} is relatively compact. Indeed, then limkλk*λ~k exists.
In that case, if𝔾 is noncompact, the following assertions (b)–(f) hold:
(b)𝔾/N≅ℤ, hence the shifts can be chosen as xk=xk for some x∈𝔾∖N.
(c) The restriction toN of the inner automorphism τ∶=ix|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)=(gτk(h),k+ℓ) for g,h∈N,k,ℓ∈ℤ, where x=(e,1).)
(d) There exists a compact subgroupK⊆N such that N=CK(τ). If N is compact then N=K.
(e)λ and ν∶=λ*εx-1 are representable as λ=ν⊗ε1∈ℳ1(N⋊τℤ), ν=:ν(1) is identified with a probability of ℳ1(N). Hence by induction, λk=ν(k)⊗εk, k∈ℤ+, with ν(0)=εe, ν(1)=ν, ν(k)=★j=0k-1τj(ν)=:ν⋆τ(ν)⋆⋯⋆τk-1(ν)∈ℳ1(N), and x is identified with (e,1). Furthermore, {ν(k)}k∈ℤ+ is relatively compact. (Indeed, as λk*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⊗εk)*(ν⊗εℓ)=(μ⋆τk(ν))⊗εk+ℓ.)
(f) PutL∶=⋃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 λk*λ~k, see [26, 27].(b), (c) See [1, Theorem 3.4], [25]. (d) See [1, Theorems 3.5 and 3.9]. If N=CK(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 {λk=ν(k)⊗εk}, ν(k)∈ℳ1(N), is nondissipating.We have to recall some more notations and facts: letℍ 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 Uf,ϵ:={f<ϵ}∈𝔘ℍ. To any f we fix such an ϵ (and a neighbourhood U⊆Uf,ϵ).Letℍ be compactly generated, ℍ=⋃n≥0‍Vn for some symmetric compact V⊆𝔘ℍ. Then we define a subadditive function δV:x↦inf{n∈ℤ+:x∈Vn}. According to [28, Proposition 1], any gauge, in particular any subadditive function f, is dominated by δV, that is,
(1)f≤AδV+BforsomeconstantsA>0,B≥0.Definition 1.
(a) Letℍ=N=CK(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(τp(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].)
Letμ∈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=Uf,ϵ∈𝔘ℍ are defined previously.
(c) Analogously, we define for a groupℍ=N=CK(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=Uf,ϵ∈𝔘ℍ 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=Uf,ϵ (as above), they are finite for all U∈𝔘ℍ. And if the integrals are finite for some relatively compact U0∈𝔘ℍ, they are finite for all U∈𝔘ℍ.Remark 2.
The definition of logarithmic moments depends onτ-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)~δV(·,0) (cf. [1, Proposition 2.14].) If W is an other relatively compact symmetric neighbourhood generating 𝔾, then δV~δW (see [28, Remarque 3].) Hence λ∈ℳ1(𝔾) possesses finite first-order moments if δV, equivalently, δW is integrable. Hence λ=μ⊗εk possesses finite first-order moments iff ∫N‍δV|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=(τt) such that τk∈T (cf., e.g., [6, 2.8.14]), and therefore, Lemma A.2, respectively, Theorem 22 apply also for discrete times.Fact 2.
LetN be a locally compact group; τ∈Aut(N), N=CK(τ), and, as before, 𝔾=N⋊τℤ. Let ν(k)=★j=0k-1τj(ν(1)). (ωK 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, limkν(k)⋆ωK 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 groups𝔾 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/Γα, respectively, 𝔾=lim←(N/Γα)⋊ταℤ, where Γα are compact, τ-invariant subgroups Γα⊲N and τα∈Aut(N/Γα). Therefore in many investigations second countability may be assumed w.l.o.g. For example, ν(k)⋆ωK 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
(2)ν(0)=εe,ν(1)=:ν,ν(k+ℓ)=ν(k)⋆τk(ν(ℓ)),k,ℓ∈ℤ+
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 timesk; hence the definition of λ0=ν(0)⊗ε0 is of minor importance. We could replace λ0 by an idempotent ωH⊗εe such that λ0*λ*λ0=λ, that is, ωH⋆ν⋆τ(ωH)=ν. ωH 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)⋆τk(ρ). Cofρ(τk) denotes the set of cofactors of ρ with respect to τk. As immediately seen, one can choose ν(k)=★j=0k-1τj(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⋆ωK=ρ (for some compact subgroup K).
(d) For short,ρ is K-τ-decomposable if ρ is τ-decomposable and right K-invariant.Co
f
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]), Cofρ(τ) is compact in ℳ1(N). If {τk(ρ)} is relatively compact, in particular, if N=CK(τ), then, again by the shift-compactness theorem, ⋃k≥0‍Cofρ(τk) is relatively compact.Remark 5.
In the following, we distinguish carefully between adapted and nonadapted space-time random walks. Let{λk=ν(k)⊗εk} be a nondissipating space-time random walk on 𝔾=N⋊τℤ with relatively compact M-semigroup {ν(k)}⊆ℳ1(N). If the random walk {λk} is not adapted let 𝔾λ⊆𝔾 denote the closed subgroup generated by suppλ. Hence, on 𝔾λ, {λk} is nondissipating and adapted. Therefore there exists a closed subgroup Nλ in 𝔾, normal in 𝔾λ, containing the supports suppλk*λ~k, and y∈𝔾λ, such that with τ′∶=iy|Nλ∈Aut(Nλ) there exists a relatively compact M-semigroup (with respect to τ′) {ν′(k)}⊆ℳ1(Nλ), such that λk=ν′(k)*εyk for k∈ℤ+. Let y=(z,1); hence yk=(z(k),k) with z=z(1), z(k)∈N. Then yNλ=Nλy iff τ(Nλ)=iz-1(Nλ)=z-1Nλz. Note that τ′=iz∘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′⋆εz}⊗ε1. Hence ν=ν′⋆εz, analogously, ν(k)=ν′(k)⋆εz(k), k∈ℤ+. And thus suppν(k)⊆Nλ·z(k).
N
λ is the smallest closed normal subgroup of 𝔾λ containing suppλk*λ~k, and λk*λ~k=ν(k)⋆ν(k)~=ν′(k)⋆ν′(k)~; hence it follows that a nondissipating space-time random walk {λk=ν(k)⊗εk} is adapted on 𝔾 iff for any closed subgroup N′⊆N and z∈N, with (z,1) normalizing N′ and satisfying
(3)(1)N′⊇suppν(1)⋆ν(1)~,(2)τ(N′)=z-1N′z⟹N′=N.In this sense we may w.l.o.g. assume that a nondissipating random walk{λk=ν(k)⊗εk} 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 𝔾λ,N′, τ′=iz∘τ, ν′(k)=ν(k)⋆εz(k) as previously mentioned.)Relations between nondissipative random walks, M-semigroups, andτ-decomposable laws are collected in the following.Theorem 6.
(a) Let𝔾 be a locally compact group and {λk}⊆ℳ1(𝔾) a nondissipating adapted random walk. Then there exist a closed normal subgroup N and x∈𝔾, such that with τ∶=ix|N∈Aut(N), one has 𝔾≅N⋊τℤ, N=CK(τ) for a compact, τ-invariant subgroup K⊆N. Furthermore, the random walk is representable as {λk=ν(k)⊗εk}, where {ν(k)}⊆ℳ1(N) is a relatively compact M-semigroup (with respect to τ), and condition (3) is satisfied.
(b) Conversely, letN 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 {λk=ν(k)⊗εk}⊆ℳ1(𝔾) is nondissipating and adapted. (𝔾 denotes N⋊τℤ). Hence (a) applies. In particular, N=CK(τ) for some compact τ-invariant subgroup K⊆N.
(c) Non-dissipative random walks, respectively, relatively compact M-semigroups defineτ-decomposable laws.
In fact, in case (a) or (b) we obtains thatρ∶=limkν(k)⋆ωK∈ℳ1(N) exists and is K-τ-decomposable.
(d) Conversely,τ-decomposable laws define nondissipating random walks, respectively, relatively compact M-semigroups.
Precisely, assume thatN=CK(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)⋆ωK→ρ. In particular, {ν(k)} is relatively compact, and then and hence (a), respectively, (b) apply.
(e) Let𝔾=N⋊τℤ, N=CK(τ). An adapted space-time random walk {λk=ν(k)⊗εk}⊆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 Fact1 and the discussion in Remark 5.
(c) Let{ν(k)} be a relatively compact M-semigroup on N=CK(τ). Since (τk) is uniformly K-contracting on compact subsets, the accumulation points of {τk(ν(n))}n∈ℤ+,k→∞ are supported by K. Assume, for some subnets, that ν(kn)→α and ν(ℓn)→β and, w.l.o.g., kn≤ℓn for all n. Then ν(ℓn)⋆ωK→β⋆ωK on the one hand, ν(ℓn)⋆ωK=ν(kn)⋆τkn(ν(ℓn-kn))⋆ωK with limτkn(ν(ℓn-kn))⋆ωK=ωK on the other. Hence α⋆ωK=β⋆ωK follows. That is, ρ=limν(k)⋆ωK exists and is obviously right K-invariant.
Furthermore,ρ=limν(k+1)⋆ωK=limν(1)⋆τ(ν(k)⋆ωK)=ν(1)⋆τ(ρ) yields K-τ-decomposability of ρ.
(d) Conversely, assumeρ to be K-τ-decomposable. {τk(ρ)} is relatively compact as τ is K-contracting, and all accumulation points are supported by K. Right K-invariance of ρ implies that τk(ρ)→ωK. As afore mentioned, we can choose ν(k)=★j=0k-1τj(ν), ν=ν(1) in Cofρ(τ), hence as an M-semigroup of cofactors. Hence ρ=ν(k)⋆τk(ρ) and τk(ρ)→ωK yield (1) relative compactness of {ν(k)} and (2) ν(k)⋆ωK→ρ according to the shift-compactness theorem.
(e) Follows by Fact2, Remark 3, and [1, Corollary 2.15, resp., Proposition 4.3].More generally we have the following.Remark 7.
(a) LetN be a locally compact group, τ∈Aut(N), 𝔾=N⋊τℤ as previously mentioned. Let {ν(k)},  {λk=ν(k)⊗εk} 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) LetN 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)⋆τk(ρ), k∈ℤ+, yields that {ν(k)} is relatively shift compact and therefore (a) applies.
(c) If in (b) in addition{τk(ρ)} is assumed to be relatively compact (e.g., if N=CK(τ) as in Theorem 6), then, again by the shift-compactness theorem, the M-semigroup of cofactors is already relatively compact.Note that in Theorem6 and Remark 7, if N=CK(τ) and K≠{e}, limkν(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) LetΓ 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)⋆σk(μ(ℓ)). As σk{μ(ℓ):ℓ≥0}→k→∞{εe}, 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)⊗εx0. Then {ν(k)=μ(k)⊗εy(k)} is a relatively compact M-semigroup (with respect to τ) in ℳ1(N), where y(k)=∏j=0k-1ξj(x0). But y(k)=e and y(k)=x0, both infinitely often; hence ν(k) is not convergent. In fact, LIM{ν(k)}={ρ1⊗εe,ρ1⊗εx0}. But we have ν(k)⋆ωH→ρ1⊗ωH.
(b) A trivial case: letK(=N) be a monothetic compact group with dense subgroup {x(k)∶=x0k:k∈ℤ}≅ℤ (for some x0∈K). Put τ∶=id, obviously K-contracting; hence 𝔾=K⊗ℤ. Then {ν(k)∶=εx(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)}={εκ:κ∈K}. Here we have ν(k)⋆ωK≡ωK.In [12, Theorem 1], it is shown for a nondissipating random walk {λk=ν(k)⊗εk} that there exists a measurable dense τ-invariant subgroup N1⊆N on which a sequence τqn acts as contracting mod. a compact subgroup H, and λn=(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=CK(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=CH(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{ρt}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. {ρt} is (strictly) (τ,c)-semistable if for all t≥0, τ(ρt)=ρt·c. The idempotent ρ0=ωH is a normalized Haar measure on a compact τ-invariant subgroup H. If N is second countable, the contraction subgroups C(τ) and CH(τ) are Borel sets, and we have ρt(CH(τ))=1 for all t. Hence we assume that N=(CH(τ))-, and therefore N=CK(τ) (as mentioned previously).
Let nowρ∶=ρ1. Then ρ=ρ1-cn⋆ρcn=ρ1-cn⋆τn(ρ). Hence ρ is H-τ-decomposable with cofactors ν(n)∶=ρ1-cn∈Cofρ(τn). In particular, ρ1-cn (and hence all ρt) 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)⋆ωK=ρ⋆ωK).IfC(τ) is closed and CH(τ)=C(τ)·H, then CH(τ) is closed, hence H=K and N=CK(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 CH(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 CH(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=(CH(τ))-, τ is weakly contracting modH on N, and hence N=CK(τ) for some compact, τ-invariant subgroup K⊇H. Note that, if CH(τ) is not closed, additional conditions are needed to guarantee existence of limkν(k)⋆ωH (cf. Example 8 above).The following result will explain in more detail the interplay between limit behaviour of relatively compact M-semigroups andτ-decomposability.Proposition 10.
LetN=CK(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{τn(ν(kn)):n→∞,(kn)⊆ℤ+}⊆ℳ1(K).
Then one has the following(a)
𝒜⊆ρ⋆𝒮(for anyρ∈𝒜).(b)
τ(𝒮)=𝒮.(c)
If at least someρ0∈𝒜and𝒮belong to a commutative⋆-sub-semigroupℳofℳ1(N),   then anyρ∈𝒜isτ-decomposable withν(k)⋆αk∈Cofρ(τk)for someαk∈𝒮, k∈ℤ+.(d)
Ifν(k)(CH(τ))=1for some compactτ-invariant subgroupH⊆Kandν(k)⋆ωH=ν(k),   for allk≥1,   and if one assumes in addition that𝒮⊆ℳ1(H),   thenρ=limkν(k)exists, andρisH-τ-decomposable with cofactors ν(k)∈Cofρ(τk).Proof.
By assumption,(τk) is compactly contracting modK on N.(a)
Letρ,σ∈𝒜, ν(kn)→ρ and ν(ℓn)→σ. Assume that that w.l.o.g. ℓn≥kn for all n (else pass to subsequences). Then ν(ℓn)=ν(kn)⋆τkn(ν(ℓn-kn))→ρ⋆α, α∈𝒮. Hence σ=ρ⋆α∈ρ⋆𝒮, and analogously, ρ∈σ⋆𝒮 follows.(b)
Assume thatτℓn(ν(mn))→γ∈𝒮. Then τ(γ)=limτℓn+1(ν(mn)); hence τ(γ)∈𝒮, and analogously τ-1(γ)∈𝒮 follows.(c)
Letρ∈𝒜; hence ρ=ρ0⋆α for some α∈𝒮. Let ν(kn)→ρ. w.l.o.g. assume that ρ=ρ0, else replace in the following ρ by ρ0⋆α. Then, for all j∈ℕ, ν(kn)=ν(j)⋆τj(ν(kn-j)) and ν(kn-j)→ρ⋆βj (along a subsequence) for some βj∈𝒮. Hence ν(kn)→ν(j)⋆τj(ρ)⋆τj(βj)=(ν(j)⋆τj(βj))⋆τj(ρ). That is, ν(j)⋆τj(βj)∈Cofρ(τj). Whence the assertion with αj=τj(βj).(d)
Since by assumptionν(k) are right H-invariant, the accumulation points α∈𝒮 are right H-invariant and supported by H; thus 𝒮={ωH}. Whence 𝒜={ρ} and thus ν(k)→ρ.The following example will illustrate Proposition10.Example 11.
LetN=CK(τ) as before.
(a) Let{ρt}⊆ℳ1(N) be a (τ,α)-semistable continuous convolution semigroups (as in Example 9), and let ξ∈ℳ1(ℝ+) be Hα-decomposable with cofactor η, with Hα denoting the homothetic transformation x↦α·x on ℝ. Then the subordinated measure λ∶=∫ℝ+‍ρtdξ(t) is τ-decomposable with cofactor ∫ℝ+‍ρtdη(t). The set ℳ∶={∫ℝ+‍ρtdγ(t):γ∈ℳ1(ℝ+)} is a commutative sub-semigroup of ℳ1(N) containing λ and the cofactors (cf. Proposition 10(c)).
LetN′∶=CH(τ) for some compact τ-invariant subgroup H⊆K. Assume ρt(N′)=1 for all t, then also λ(N′)=1 and ν(N′)=1 for all ν∈ℳ.
(b) Letρ(i)∈ℳ1(N) be τ-decomposable with cofactors ν(i)(k), i=1,2. ⟦e.g., let {ρt(i)} be (τ,αi)-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 ρt(1),ρs(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.⟧
Ifρ(i)(CHi(τ))=1, for τ-invariant compact subgroups Hi⊆K, then ρ(CH1(τ)·CH2(τ))=1. If ωH1 and ωH2 commute, then H1·H2=:H3 is a compact τ-invariant subgroup and CH3(τ)⊇〈CH1(τ)·CH2(τ)〉, the subgroup generated by CHi(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 εx, x∈N we obtain: ρ¯=ρ(1)⋆εx is in general not τ-decomposable. In fact, without commutativity assumption, the left shifted measure, ρ∶=εx⋆ρ(1), is, as easily seen, τ-decomposable with cofactors ν(k)=εx⋆ν(1)(k)⋆ετk(x-1). Representing ρ¯ as εx⋆ix-1(ρ(1)) we see that ρ¯ is τ¯-decomposable, with τ¯∶=ix-1∘τ∘ix, but in general not τ-decomposable.
(d) To simplify notations we assume thatHi={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′τk(x-1))=1. Hence, with N′′∶=ix(N′) we have: ρ(N′′x)=1 and ν(k)(N′′xτk(x-1))=1. Furthermore, N′′=CH¯(τ¯), where τ¯:=ix∘τ∘ix-1, H¯∶=ix(H).
(e) In case (c), (d) we determine the sets of accumulation points𝒮 and 𝒜 (Proposition 10):
𝒮
=
LIM
{
τ
n
(
ν
(
k
n
)
)
}
=
LIM
{
ε
τ
n
(
x
)
⋆
τ
n
(
ν
(
1
)
(
k
n
)
)
⋆
ε
τ
n
+
k
n
(
x
-
1
)
}, hence 𝒮=LIM{ετn(x)τn+kn(x-1)}=𝒮0⋆𝒮0-1, where 𝒮0=LIM{ετn(x)}.  ⟦In fact, obviously 𝒮⊆𝒮0⋆𝒮0-1. On the other hand, assume τk(n)(x)→z and τℓ(n)(x)→u, w.l.o.g. ℓ(n)≥k(n). Then εzu-1∈𝒮0⋆𝒮0-1 and εzu-1=limετk(n)(x)⋆ετk(n)+(ℓ(n)-k(n))(x-1)∈𝒮.⟧
Furthermore,𝒜=LIM{ν(n)}=LIM{εx⋆ν(1)(n)⋆ετn(x-1)}. Thus 𝒜=εx⋆ρ(1)⋆𝒮0-1.


## 3. Continuous Time: Nondissipating Continuous Convolution Semigroups

Next we replace the random walk by a continuous convolution semigroup{λt}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=limt→0λt is uniquely determined.W.l.o.g. we assume that𝔾 is generated by {supp(λt):t≥0}. For short, {λt} is called adapted then. {λt}t∈ℝ+ is nondissipating if the concentration functions fλt(C)∶=supx∈𝔾λt(Cx) do not converge to 0 for some compact C (for t→∞).For anyt0>0 the random walk {λt0k=λt0k} 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 𝔾λt0=:𝔾t0 as smallest closed subgroup containing suppλt0. Hence {λt0k} is adapted on 𝔾t0. We define Nt0:=Nλt0 to be the smallest closed normal subgroup of 𝔾t0 containing suppλt0*λ~t0. Hence 𝔾t0=Nt0⋊τt0ℤ and λt0k=ν(t0k)⊗εt0k. (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 𝔾t0 are noncompact and the results of Section 1 apply. In fact, if for some t0>0, 𝔾t0 is compact, then {λt0k=λt0k:k∈ℤ+} is uniformly tight. But then {λs:s≥0}={λt0k:k∈ℤ+}⋆{λs:0≤s≤t0} is uniformly tight. Therefore 𝔾 (and all groups 𝔾t,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{λt}⊆ℳ1(𝔾) be an adapted continuous convolution semigroup. Then the following assertions are equivalent.(i)
{λt}   is nondissipating.(ii)
For all   [(ii′)     some]    t0>0,   the skeleton random walk   {λt0k=λkt0}   is nondissipating (and adapted on   𝔾t0).(iii)
{λt}t≥0   is relatively (right) shift compact.(iv)
{λt*λ~t}t≥0   is relatively compact. (Thenlimt→∞λt*λt~   exists.)(v)
For all   [(v′)   some]    t0>0, λt0  (considered as measure on   𝔾t0)   has finite first-order moments.(vi)
For all   [(vi′)   some]    t0>0, ν(t0)  (considered as measure onNt0)    has finite logarithmic moments. That is, for all regular   τt0-functions   f   on   Nt0   one has   ∫Nt0‍log(1+f)dν(t0)<∞.The precise formulations of conditions (v) (resp., (vi)) are: for all subadditive functionsf on 𝔾t0 (resp., for all regular τt0-functions g on Nt0) we have ∫𝔾t0‍fdλt0<∞ (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 𝔾t0, might have no subadditive extension to 𝔾. Analogously, for t0≠t1, the sets of τt0- and τt1-functions may be different (even if Nt0=Nt1, as will be shown in Proposition 13 later).Proof.
f
α
*
β
(
C
)
≤
f
α
(
C
) for all probabilities α,β, hence t↦fλt(C) is decreasing. Hence fλt(C)↛0 iff infλt(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 σt∶=λt*λt~ follows by Fact 1(a): For all t>0  limkσtk=: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↦〈σt,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 {λt0k}. Let xk∈𝔾 such that {λt0k*εxk-1} is relatively compact. Then, {λt} being a continuous convolution semigroup, the set {λr:0≤r<t0}*{λt0k*εxk-1:k∈ℤ+}=:{λt⋆εx(t)-1:t≥0} is relatively compact (where x(t)∶=xk, if t0k≤t<t0(k+1)). Whence (iii) follows.
“(iii)⇒(i)”. Let {λt*εx(t)-1} be relatively compact. Hence for any ϵ∈(0,1) there exists a compact C such that for all t∈ℝ+, λt(Cx(t))≥ϵ. Therefore, fλt(C) fails to converge to 0.A prioriNt0∶=Nλt0, the smallest closed normal subgroup of 𝔾t0 containing suppλt0*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{λt} be an adapted continuous convolution semigroup. Then one has Nt0=N1=:N for all t0>0, and N⊲𝔾.Proof.
Obviously,𝔾=(⋃t‍𝔾t)- and for all t>0, 𝔾t=⋃k∈ℤNtxtk for some (all) xt∈suppλt. First we consider dyadic numbers t and then proceed to real t applying continuity of {λt}.
Lett0=1. Then obviously 𝔾1⊆𝔾1/2 since λ1=λ1/22. For all x,x1/2∈suppλ1/2 it follows, xx1/2,x1/2x,x1/22∈suppλ1⊆𝔾1, hence
(4)x1/2xN1=N1x1/2x=xx1/2N1=N1xx1/2=N1x1/22=x1/22N1.
In fact,Nt⊲𝔾t for all t>0, and hence yNt=Ntz for all y,z∈suppλt. Therefore,
(5)x-1x1/22N1=x1/2N1,N1x1/22x-1=N1x1/2.
Put N1*∶=N1∩x1/2N1x1/2-1, and hence, by (5)(6)x1/2N1*=x1/2N1∩N1x1/2=N1*x1/2.
Claim 1.  We have
(7)λ1/2(N1*x1/2)=λ1/2(x1/2N1*)=1.
By (6) it suffices to show that λ1/2(x1/2N1)=λ1/2(N1x1/2)=1. According to (4) and (5) we have
(8)1=λ1(N1x1/22)=∫‍λ1/2(x-1N1x1/22)dλ1/2(x)=(4)∫‍λ1/2(x-1x1/22N1)dλ1/2(x)=(5)∫‍λ1/2(x1/2N1)dλ1/2(x)=λ1/2(x1/2N1).
Analogously we have 1=∫‍λ1/2(N1x1/22x-1)dλ1/2(x)=λ1/2(N1x1/2). Consequently, suppλ1/2⊆x1/2N1*=N1*x1/2 yields 𝔾1/2⊆⋃k∈ℤx1/2kN1*=⋃k∈ℤN1*x1/2k. Thus, as x1/2 normalizes N1*, we obtain N1*⊲𝔾1/2.
Claim 2.
N
1
/
2
⊆
N
1
*
⊆
N
1. N1*∩N1/2=:N* is a closed normal subgroup of 𝔾1/2 such that 1=λ1/2(x1/2N*)=λ1/2(N*x1/2). But N1/2 is minimal with this property. Whence N1/2⊆N*, thus N1/2⊆N1*⊆N1.
Claim 3.
N
1
*
=
N
*
=
N
1
/
2
=
N
1
=
:
N. ⟦N1/2⊲𝔾1/2 yields N1/2⊲𝔾1 (as G1⊆𝔾1/2). And λ1(x1/22N1/2)=∫‍λ1/2(x-1x1/22N1/2)dλ1/2(x)=∫‍λ1/2(x-1x1/2N1/2x1/2)dλ1/2(x); therefore λ1(x1/22N1/2)=λ1/2(N1/2x1/2)=1 (since x-1x1/2∈N1/2). Hence, according to the definition, N1⊆N1/2 follows. Together we obtain N1⊆N1/2⊆N1*⊆N1, whence the assertion follows.⟧
By induction, we obtainN=N1/2n for all n∈ℤ+, and N⊲⋃k∈ℤ,n≥0‍x1/2nkN1.
Claim 4.
N
t
=
N for all t>0.
At first we consider dyadict=k/2n.(1)
suppλk/2n⊆[(x1/2nN)k]-=x1/2nkN=Nx1/2nk, whence Nk/2n⊆N. Hence suppλk/2n⋆λ~k/2n⊆N.(2)
suppλk/2n⋆λ~k/2n⊇x1/2nk-1·(suppλ1/2n⋆λ~1/2n)·x1/2n-(k-1). Thus Nk/2n⊇x1/2nk-1Nx1/2n-(k-1)=N.
HenceNt=N for all dyadic t>0. Continuity of t↦λt⋆λ~t easily yields Nt⊆N1=N, and N⊲[⋃k,n≥0‍x1/2nkN]-.
Replacing1 by any real t0>0 we obtain analogously Nt0=Nt0k/2n for k,n≥0, and again Nt⊆Nt0 for all real t>0. In particular, N=N1⊆Nt0. Consequently, Nt=N for all t>0.
Now the assertions follow immediately, and the proof is complete.Consequently, for any nondissipating continuous convolution semigroup{λt} we obtain that {λt} is adapted iff 𝔾=⋃t≥0‍𝔾t=⋃s∈ℝ‍Nxs. Indeed, we have the following.Proposition 14.
Let{λt}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 suppλt⊆Nx(t),t≥0.
Let, fort∈ℝ, τt∶=ix(t)|N denote the restriction of the inner automorphism to N. Then T=(τt) is a continuous one-parameter group in Aut(N). And one has 𝔾≅N⋊Tℝ (with group operation (g,t)(h,s)=(gτt(h),t+s), g,h∈N,t,s∈ℝ). Furthermore, λt may be represented as λt=ν(t)⊗εt, 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=ωH⊗ε0 for some compact subgroup H⊆N.Proof.
As shown before, there existsxt∈𝔾 such that suppλt⊆Nxt (for all t∈ℝ+). Hence, π:𝔾→𝔾/N denoting the canonical homomorphism, and we obtain π(λt)=εz¯(t) with z¯(t)=xtN. Hence {εz¯(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, 𝔾t0/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 that𝔾 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.
Henceℍ∶=⋃t‍𝔾t=N⋊Tℝ is closed in 𝔾 containing ⋃t‍suppλt, whence 𝔾=ℍ.The existence of nondissipating adapted continuous convolution semigroups(λt) has strong influence on the structure of 𝔾.Corollary 15.
Assume, as in Proposition14, (λt) 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.⟦
If
N is totally disconnected then the continuous automorphism group T={ix(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 (λt) 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 thatN≅CK(τ). As mentioned in the previous proof, in the continuous time case, for all t>0, we have N=CK(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:τt(x)→t→∞e} and CK(T)={x∈N:τt(x)→t→∞emodK} for some compact, T-invariant subgroup K.Proposition 16.
Let, as before,𝔾=N⋊Tℝ.(a)With the notations of Proposition14 one has   N≅CK(T).
In fact, for any locally compact groupN, admitting a continuous one parameter group T=(τt)⊆Aut(N) and a T-invariant compact subgroup K one has the following.(b)For all   t>0, CK(T)=CK(τt), in particular, C(T)=C(τt).(c)C(T)  is closed,    T-invariant, connected, and isomorphic to a contractible Lie group (hence a homogeneous group), and   CK(T)   is a closed   T-invariant subgroup of   N.   If   N   is compact, then   CK(T)=K.(d)C(T)⊲CK(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   CK(T)=C(T)⋊βK. In particular,    CK(T)   is compactly generated.(e)The restrictions   S∶=T|C(T)=(σt)t∈ℝ⊆Aut(C(T))   and   Γ∶=T|K=(γt)t∈ℝ⊆Aut(K)   are continuous one-parameter groups, with   Γ   consisting of inner automorphisms of   K.(f)T=(τt=(σt,γt))   satisfies the following consistency conditions   σt∘β(κ)=β(γt(κ))∘σt, κ∈K, t∈ℝ.
Conversely, any groupT arises in that way.Proof.
(a) As mentioned in the previous proof, for anyt>0, we have N=CK(t)(τt), K(t) denoting a compact τt-invariant subgroup K(t).
It follows easily thatK(t)=K(s)=:K for all t,s>0. Indeed: (1)
Fort>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 τt/Nk-invariant for all k∈ℤ, in particular, K(t/N) is τt-invariant. As τt is compactly contracting modK(t), we obtain K(t/N)=τt/Nn·N(K(t/N))=τtn(K(t/N))→emodK(t), whence K(t/N)⊆K(t) follows.⟧(2)
Lett,s>0. Then there exists a sequence k(n)→∞ such that  n·s=k(n)·t+rn with 0≤rn<t. For x∈N=CK(t)(τt) we have τtn(x)=yntκnt with ynt→e and κnt∈K(t). Hence τsn(x)=τrn(ynt)τrn(κnt). The first terms converge to e, and LIM(τrn(κnt))⊆⋃0≤u≤t‍τu(K(t)). Hence LIM(τsn(x))⊆K(s) (since N=CK(s)(τs)), on the other hand, LIM(τsn(x))⊆⋃0≤u≤t‍τu(K(t)).
Replacingt by t/N, N∈ℕ, in view of Step (1). we obtain K(s)⊆⋃0≤u≤t/N‍τu(K(t)). Continuity of u↦τu 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 CK(τt)=CK(T). Thus (a) is proved.
(b) See [2], [6, Lemma 3.2.26]. (Note that the group {τt}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 Theorem22 and in Lemmas A.1–A.5) we mention the following.Proposition 17.
With the notations introduced afore (in Proposition16) 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 groupT¯∶=(τ¯t:=iy(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 Theorem20, 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.
LetN be a locally compact group, T=(τt)⊆Aut(N) as before.(a){ν(t)}t∈ℝ+⊆ℳ1(N) is a (continuous time) M-semigroup (with respect to T=(τt))—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:
(9)ν(t+s)=ν(t)⋆τt(ν(s))∀t,s∈ℝ+.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)⋆τt(ρ). Again, Cofρ(τt) denotes the set of cofactors. ρ∈ℳ1(N) is K-T-decomposable if in addition, ρ is right K-invariant.In analogy to Remark5 we note the following.Remark 19.
LetN be a locally compact group, T⊆Aut(N). Let {λt=ν(t)⊗εt} be a space-time continuous convolution semigroup  on 𝔾=N⋊Tℝ. If {λt} 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, λt=(ν′(t)⊗ε0)*ε(z(t),t)=(ν′(t)⋆εz(t))⊗εt where {ν′(t)} is a continuous M-semigroup with respect to T′=(τt′), with (τt′∶=iy(t)|N′=iz(t)∘τt).
Therefore,{λt} 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(10)(1)τt(N′)=iz(t)-1(N′),(2)suppν(t)⊆N′z(t)⟹N′=N.We have the continuous time analogues of Theorem6(a)–(d).Theorem 20.
Let, as before (Proposition16), 𝔾=N⋊Tℝ, λs=ν(s)⊗εs, s≥0, be probabilities on the on the space-time building 𝔾≅N⋊Tℝ. Obviously, {λt} is a continuous convolution semigroup iff {ν(t)} is a continuous M-semigroup with respect to T. λ0=limt→0λt is idempotent and λ0*λt*λ0=λt, for all t, hence ν(0)=ωH with ωH⋆ν(t)⋆τt(ωH)=ν(t), t≥0, and ν(0)=limt→0ν(t). (H denotes a compact subgroup of N.)(a)
If{λt} is nondissipating and adapted then {ν(t)} is relatively compact and satisfies condition (10). Furthermore, N=CK(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 {λt} is nondissipating and adapted. Hence (a) applies.(c)
In (a), respectively, (b),limt→∞ν(t)⋆ωK=:ρ∈ℳ1(N) exists and is T-K-decomposable with cofactors {ν(t)}.(d)
Conversely, letN=CK(T) and let ρ∈ℳ1(N) be K-T-decomposable. Then there exists a nondissipating continuous convolution semigroup {λt=ν(t)⊗εt}⊆ℳ1(𝔾), with a relatively compact continuous M-semigroup {ν(t)} in ℳ1(N) such that ν(t)∈Cofρ(τt) for all t, and furthermore, ρ=limt→∞ν(t)⋆ωK.Proof.
(a) is proved as in the discrete time case, Theorem6(d). (For aperiodic groups see [22, Proposition A], [23, Theorem 3.16]).
(b) and (c) are proved analogous to Theorem6(b) and (c).
(d) Letρ∈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 {νt(k)}k≥0 of cofactors (with respect to (τtk)), such that limkνt(k)⋆ωK=ρ. In fact, for any selection ν¯(t)∈Cofρ(τt), t≥0, it follows as in Theorem 6(d) that τt(ρ)→ωK and ν¯(t)⋆ωK→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 {λt=ν(t)⊗εt} 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 (λt=ν(t)⊗εt) such that ν(t)∈Cofρ(τt), 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∈Cofρ(τ1) with roots λ1/m=ν(1/m)⊗ε1/m where ν(1/m)∈Cofρ(τ1/m) and ν=★0m-1τj/m(ν(1/m)). Furthermore, λ is embeddable into a rational convolution semigroup {λr}r∈ℚ+. Moreover, the root sets ℛ1/m∶={σ∈ℳ1(𝔾):σm=λ} are compact and consist of measures σ=μ⊗ε1/m, with μ∈Cofρ(τ1/m).
⟦
Choose
, for n∈ℕ, ν¯(1/n)∈Cofρ(τ1/n), put ν(n)∶=★0n-1τj/n(ν¯(1/n)); hence ν(n)∈Cofρ(τ1) for all n. Put λ1/n(n)∶=ν¯(1/n)⊗ε1/n; hence λ(n)∶=(λ1/n(n))n=(ν¯(1/n)⊗ε1/n)n=ν(n)⊗ε1.
According to the shift-compactness theorem,Cofρ(τt) is compact for all t>0. For n≥m, put λ¯1/m(n)∶=(λ1/n!(n!))(n!/m)=:ν¯(n)(1/m)⊗ε1/m (with ν¯(n)(1/m)∈Cofρ(τ1/m)), and λ¯1(n)=λ(n).
By Tychonoff's theorem, there exists a subnet(n′)⊆ℕ satisfying ν¯(n)(1/m)→(n′)ν(1/m)∈Cofρ(τ1/m), equivalently, λ¯1/m(n)→(n′)λ1/m∶=ν(1/m)⊗ε1/m, for all m∈ℕ. Continuity of convolution yields λ1/mm=λ1=λ=ν(1)⊗ε1, ν(1)∈Cofρ(τ1) (for all m∈ℕ).
We have shown thatλ=λ1 is infinitely divisible, and by construction, the roots satisfy (λ1/m!)m=λ1/(m-1)!; hence, λ is embeddable into a rational convolution semigroup {λr}r∈ℚ+, (cf. [36, 44]), and the construction easily yields moreover λr=ν(r)⊗εr, with ν(r)∈Cofρ(τr), r≥0.⟧
Letσ∈ℳ1(𝔾) with σm=λ=ν⊗ε1. Then, as easily seen, σ=μ⊗ε1/m with ★0m-1τj/m(μ)=ν. Hence, μ∈Cofρ(τ1/m). Therefore, in particular, ℛ1/m is compact for all m.
In order to show that acontinuous convolution semigroup {λt′} can be selected, put ℛ∶=⋃n≥1‍{σnm:σnn=λ,0≤m≤n}. As mentioned previously, σn=μn⊗ε1/n and σnm=★0m-1τj/n(μn).
Claim 2. ℛ is relatively compact in ℳ1(𝔾) with all accumulation points of the form ν(t)⊗εt, ν(t)∈Cofρ(τt),0≤t≤1.
⟦
Let
t
k
∶
=
m
k
/
n
k
∈
[
0,1
] with mk,nk∈ℕ, tk→t∈[0,1]. Let σnk=μnk⊗ε1/nk∈ℛ, and put ν(tk)∶=⋆j=0mkτj/nk(μnk). Then τtk(ρ)→τt(ρ) and ρ=ν(tk)⋆τtk(ρ) yield that {ν(tk)} is relatively compact and LIM{ν(tk)}⊆Cofρ(τt).⟧
Let𝒜∶=LIM{λt:t∈ℚ+∩[0,1],t→0}. Therefore, σ∈𝒜 is representable as σ=u⊗ε0 for u∈Cofρ(τ0), with τ0=id; hence u⋆ρ=ρ. And, by definition of 𝒜, it follows that σ*λt=(u⋆ν(t))⊗εt, λt*σ=(ν(t)⋆τt(u))⊗εt 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 {λt′} of the form λt′=ν′(t)⊗εt for all t, with λt′=σt*λt=λt*σt where σt∈𝒜, hence σt=ut⊗ε0, ut∈Cofρ(τ0). In particular, ν′(t)=ν(t)⋆τt(ut). Hence, as immediately seen, ν′(t)∈Cofρ(τt).
Since{λt′} is a continuous convolution semigroup, {ν′(t)} is a continuous M-semigroup of cofactors of ρ. Whence the assertion.For the following Theorem22 we need some preparations.Letℍ be a locally compact group and {λt} a continuous convolution semigroup. The Lévy measure η is defined by ∫‍gdη=d+/dt|t=0∫‍gdλt, for g∈Cb(ℍ) vanishing in a neighbourhood of the unit. If as before, ℍ=𝔾=N⋊Tℝ and λt=ν(t)⊗εt, 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.
Letg≥1 be a submultiplicative function. Let ϵ>1, such that U=Ug,ϵ={g≤ϵ}∈𝔘ℍ. Then we have
(11)∫‍gdλt<∞forsome[all]t≥0iff∫𝔾∖U‍gdη<∞.
Thus for subadditive functions f≥0 we have with g∶=1+f: ∫‍1+fdλt<∞ iff ∫𝔾∖U‍1+fdη<∞. Equivalently, ∫‍fdλt<∞ iff ∫𝔾∖U‍fdη<∞.
As mentioned before, if the integral on the right side of (11) is finite for U=Ug,ϵ, then it is finite for any U∈𝔘ℍ. And, if it is finite for some relatively compact U0∈𝔘ℍ, it is finite for all U∈𝔘ℍ.The following Steps1–5 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=CK(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=(τt=ix(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 ∥δt(x)∥=t·∥x∥, where (δt) denotes a group of dilations defined by T (cf. for example, [6, 2.1.10, 2.7.26 (d)], [29]).Step 1 (cf. LemmaA.1).
There exists a homogeneous norm∥·∥ on C which is in addition β(K)-invariant.Step 2 (cf. LemmaA.2).
Define the functionϕ: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 anyt0>0 there exist constants 0<r≤R≤1,0<s≤S<1, satisfying for all γ∈N the growth conditions
(12)rnϕ(γ)≤ϕ(τt0n(γ))≤Rnϕ(γ),n≥0,Snϕ(γ)≤ϕ(τt0n(γ))≤snϕ(γ),n<0.
For large t0 one can put for example, R=∥τt0|C∥, r∶=∥τt0-1|C∥-1,S=R,s=r. The automorphism norm is defined as ∥σ∥∶=sup{∥σ(x)∥:x∈C,∥x∥=1}. In particular, ∥τs+t|C∥≤∥τt|C∥·∥τs|C∥, ∥τt|C∥→0 for t→∞ and ∥τ-t|C∥≥1 for t≥0.Step 3 (cf. LemmaA.3).
Letf:N→ℝ+ be a regular τt0-function for some t0>0, f(τt0p(γ))=αf(γ), 0<α<1. (See Definition 1.)
Then we obtainlog(1+f)~log(1+ϕ), (ϕ defined in Step 2.)Remark 21.
As mentioned before, in the situationN=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 τt-functions—they are δt-functions for dilations δt, as the proof shows—but these functions have similar properties (cf. (12)).Step 4 (cf. LemmaA.4).
With the notations of the preceding Steps we have: the functionsψ: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 𝔘𝔾 for all α>2.Step 5 (cf. LemmaA.5).
Let, as before,N=C⋊βK and 𝔾=N⋊Tℝ. N and 𝔾 are compactly generated for any neighbourhood U∈𝔘C, V∶=U·K generates N, and, for example, for any ϵ>0, W∶=V⊗[-ϵ,ϵ] generates 𝔾.
Letf be a regular τt0-function, for some t0>0. Let, for t>0, Wt∶=⋃0≤s≤t{V⊗{s}∪{V⊗{s}}-1} where V={f≤1}. Then Wt generates 𝔾, and for any symmetric relatively compact W∈𝔘𝔾 with W=WK, we have δWt~δW. (For the definition of δW cf. (1).) In particular, for all t1,t2>0, it follows that δWt1~δWt2.Next we characterize—in analogy to Theorem6(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 τt0.Theorem 22.
Let, as before,𝔾=N⋊Tℝ with N=CK(T)=C⋊βK, C=C(T) a homogeneous group. Let {λt=ν(t)⊗εt}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){λt} is nondissipating.(b)  For some (any)t0>0, λt0 has finite first-order moments (on 𝔾).(b*)  For some (any) t0>0, ∫𝔾‍δWdλt0<∞ for a compact K-invariant neighbourhood W∈𝔘𝔾.(c)η has finite first-order moments (on 𝔾), that is, ∫∁U‍fdη<∞ for all subadditive f:𝔾→ℝ+ and U∈𝔘𝔾.(d)  For some (any)t0>0, ν(t0) has finite logarithmic moments, that is, for any regular τt0-function f, one has ∫N‍log(1+f)dν(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 τt-functions, for all U∈𝔘N.(e*)  ∫N∖U‍log(1+ϕ)dη=∫N∖U‍log(1+∥x∥)dη(x,κ)<∞, for all U∈𝔘N.Proof.
According to Proposition12 and Theorem 20 we know that condition (a) is equivalent with shift compactness of a skeleton random walk {λt0k} for some  (hence any) t0>0. Hence (a) is equivalent to (b1) For some [any] t0>0, for any subadditive function f on 𝔾t0, ∫𝔾t0‍fdλt0<∞ (i.e., λt0 has finite first-order moments on 𝔾t0.)
Obviously, “(b1)⇒(b)”. By Fact 2, “(b1)⇔(d)”; furthermore, by Step 3 (cf. Lemma A.3), “(d)⇔(d*)” and “(e*)⇔(e)”.
According to Step5 (Lemma A.5), (b)⇔∫𝔾‍δWt0dλt0<∞, and therefore (b)⇔(b*) ∫𝔾‍δWdλt0<∞ for any t0>0, equivalently, for any t0,t1>0, ∫𝔾‍δWt1dλt0<∞, (since δW~δWt0~δWt1). ⟦Let us mention that (b1)⇒ ∫𝔾t0‍δVt0dλt0<∞, and since δWt0≤δVt0, we obtain again “(b1)⇒(b)”.⟧ Applying Siebert's result, Fact 3, to the subadditive function δWt0 (resp., to δW for any K-invariant relatively compact symmetric W∈𝔘𝔾) yields “(b*)⇔(c)”.
log
ψ is subadditive—ψ as in Step 4 (Lemma A.4)—hence again by Fact 3 we obtain (b)⇒∫𝔾‍logψdλt0<∞⇔∫∁U‍logψdη<∞ for all U∈𝔘𝔾.
But we have∫𝔾‍logψdλt0=∫N‍log(1+ϕ)dν(t0)<∞, hence (b)⇒(d*) and ∫𝔾∖U‍logψdη=∫N∖U∩N‍log(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 thatlimt→∞ν(t) does not need to exist if N=CK(T), K≠{e}. (For K={e}, i.e., N=C(T), limtν(t) always exists, cf. [22, Proposition A]).Example 23.
LetΓ be a contractible Lie group, hence simply connected  and nilpotent, with contracting one-parameter group S=(σt)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∶=(τt)⊆Aut(N) by τt=σt⊗id, t∈ℝ. Hence N=CK(T). Let {μ(t)} be a continuous M-semigroup (with respect to S) in ℳ1(Γ) such that ρ=limμ(t) exists. As ξt∶=τt|K≡id, continuous M-semigroups in ℳ1(K) (with respect to (ξt)) are just  continuous convolution semigroups; in particular, {εz(t)} is a M-semigroup. It is immediately verified that {ν(t)∶=μ(t)⊗εz(t)} is a relatively compact M-semigroup in ℳ1(N) (with respect to T), with nondissipating continuous convolution semigroup {λt∶=ν(t)⊗εt}.
As(z(t)) is dense in K, LIM{ν(t):t→∞}={ρ⊗εκ:κ∈K}. In particular, limt→∞ν(t) does not exist.In analogy to Example9, 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.
LetN be a simply connected nilpotent Lie group and S=(σt)⊆Aut(N) be a contracting continuous group with multiplicative parametrization, σtσs=σt·s for t,s>0, and limt→0σt(x)=e for all x∈N. A continuous convolution semigroup {ρt} in ℳ1(N) is called S-stable if σt(ρ1)=ρt, for all t>0, equivalently, σt(ρs)=ρt·s for t>0,s≥0. Put ρ∶=ρ1. We have ρ=ρ1-s⋆ρs=ρ1-s⋆σs(ρ), for all 0<s≤1; hence ρ is σs-decomposable with cofactors ρ1-s∈Cofρ(σs).
To obtain a continuous M-semigroup of cofactors we have to switch to additive parametrization:T∶=(τt∶=σe-t)t∈ℝ is a continuous one-parameter group satisfying τt+s=τtτs and limt→∞τt(x)=e for all x∈N. And with this notations we obtain ρ=ρ1-e-t⋆τt(ρ) for t≥0.
As immediately verified,{ν(t)∶=ρ1-e-t}t∈ℝ+ is a relatively compact continuous M-semigroup with ν(t)∈Cofρ(τt) (and ρ=limν(t)); hence it defines a nondissipating  continuous convolution semigroup {λt=ρ1-e-t⊗εt}⊆ℳ1(𝔾), 𝔾=N⋊Tℝ. Note that in this (trivial) case, the M-semigroup belongs to {ρs}s≥0, respectively, to the orbit {τt(ρ)}t≥0: ν(t)=ρ1-e-t=σ1-e-t(ρ)=τ-log(1-e-t)(ρ).
Analogously, stable laws with nontrivial idempotents onN=C⋊TK generate T-decomposable laws (see [6, § 3.5]).
A variety of examples of (self-) decomposable laws are obtained via subordination as in Example11: Let {ρt} be a continuous convolution semigroup which is stable with respect to T as above. Let ξ be a (He-t)-decomposable probability measure on ℝ+, where Hα:x↦α·x denotes the homothetic transformation on ℝ. Then, as immediately verified, the subordinated measure μ∶=∫0∞‍ρtdξ(t) is T-decomposable. In particular, if ξ is an exponential distribution, we obtain geometric stable laws (or geometric semistable laws, if {ρt} 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).



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