# On the Abstract Subordinated Exit Equation

Authors: Hassen Mejri; Ezzedine Mliki
Journal: Abstract and Applied Analysis
 (2010)

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## Abstract

Letℙ=(Pt)t>0 be a C0-contraction semigroup on a real Banach space ℬ. A ℙ-exit law is a ℬ-valued function t∈]0,∞[→φt∈ℬ satisfying the functional equation: Ptφs=φt+s, s,t>0. Let β be a Bochner subordinator and let ℙβ be the subordinated semigroup of ℙ (in the Bochner sense) by means of β. Under some regularity assumption, it is proved in this paper that each ℙβ-exit law is subordinated to a unique ℙ-exit law.


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## Body

## 1. Introduction

Letℙ:=(Pt)t≥0 be a C0-contraction semigroup on a real Banach space ℬ with generator (A,D(A)). A ℙ-exit law is a ℬ-valued function t∈]0,∞[→φt∈ℬ satisfying the functional equation:  (1.1)Ptφs=φt+s,s,t>0.
Exit laws are introduced by Dynkin (cf. [1]). They play an important role in the framework of potential theory without Green function. Indeed, they allow in this case, an integral representation of potentials and explicit energy formulas. Moreover, this notion was investigated in many papers (cf. [2–13]).In particular, the following theorem is proved in our paper[10].Theorem 1.1.
If aℙ-exit law φ is Bochner integrable at 0 (shortly zero-integrable), this is equivalent to,
(1.2)∫01∥φt∥dt<∞,
then φ is of the form
(1.3)φt=qPtVq(φ)-APtVq(φ),t,q>0,
where Vq(φ):=∫0∞e-qsφsds.The present paper is devoted to investigate the subordinated abstract case where we study the zero-integrable solution of the exit equation (1.1) after Bochner subordination.More precisely, letβ=(βt)t>0 be a Bochner subordinator, that is, a vaguely continuous convolution semigroup of subprobability measures on [0,+∞[ and let ℙβ:=(Ptβ)t>0 be the subordinated C0-semigroup of ℙ in the sense of Bochner by means of β, that is,  (1.4)Ptβf:=∫0∞Psfβt(ds),f∈ℬ,t>0.
It can be seen that, for each exit law φ=(φt)t>0 for ℙ, the function φβ defined by  (1.5)φtβ:=∫0∞φsβt(ds),t>0,
is an exit law for ℙβ. The function φβ is said to be subordinated to φ by means of β.Conversely, it is natural to ask if anyℙβ-exit law is subordinated to some ℙ-exit law. In general, we do not have a positive answer (see Example 5.3 below or [2, page 1922]). However, this problem was solved (cf. [2, 4–6]) for ℬ=L2(m) and positive ℙβ-exit laws ψ, and under some regularity assumptions on ℙ, β, and ψ. Basing on our paper[10, Theorem 1],we consider in this paper the zero-integrable ℙβ-exit laws in the abstract case. Namely, we prove the following.Theorem 1.2.
Letψ:=(ψt)t>0 be a zero-integrable ℙβ-exit law satisfying the following conditions: There exist a constant q>0 such that:
(1.6)(PtVq(ψ))t>0⊂D(Aβ),(1.7)∫01∥AβPsVq(ψ)∥βt(ds)<∞,t>0,
where Vq(ψ):=∫0∞e-qsψsds and (Aβ,D(Aβ)) is the associated generator to ℙβ. Then, ψ is subordinated to a unique ℙ-exit law φ:=(φt)t>0. Moreover, φ is explicitly given by
(1.8)φt:=qPtVq(ψ)-AβPtVq(ψ).The conditions in Theorem1.2 are fulfilled for the closed ℙβ-exit laws ψ. This is always the case for the zero-integrable ℙβ-exit laws in the bounded case.As application, we consider the holomorphic case and we prove the following result:Theorem 1.3.
We suppose thatℙ is a C0-contraction holomorphic semigroup on ℬ and β be a Bochner subordinator satisfying
(1.9)∫011sβt(ds)<∞,t>0.
Then each zero-integrable ℙβ-exit law ψ is subordinated to a unique ℙ-exit law φ. Moreover, φ is given by
(1.10)φt=(q+a)PtVq(ψ)-bAPtVq(ψ)+∫0∞(PsVq(ψ)-Ps+tVq(ψ))ν(ds),t>0,
where a, b, and ν are the parameters of β.The condition (1.9) is fulfilled for the fractional power subordinator and the Dirac subordinator.


## 2.C0-Contraction Semigroup

For the following notions and properties aboutC0-contraction semigroups, we will refer essentially to [14, 15] (cf. also [16, 17]).Let(ℬ,∥·∥) be a real Banach space and let I be the identity operator on ℬ. For a linear operator T:ℬ→ℬ, we denote also by ∥T∥:=sup∥f∥≤1∥Tf∥ the norm of T. If ∥T∥<∞, T is said to be bounded.We consider[0,∞[ endowed with its Borel field 𝒜 and a measure μ on ([0,∞[,𝒜). We say that a property holds μ.a.e. if the set for which this property fails is μ-negligible. A ℬ-valued function X : ]0,∞[→ℬ is said simple if there exists a disjoint sequence {Ai∈𝒜:μ(Ai)<∞}1≤i≤n and X1,…,Xn∈ℬ such that X(t)=∑i=1nXi1Ai(t) for all t>0. A ℬ-valued function X:]0,∞[→ℬ is also denoted by X:=(Xt)t>0.In this paper, we consider the integral in Bochner sense for functionsX:]a,b[⊂]0,∞[→ℬ which are μ-strongly measurable (i.e., there exists a sequence of simple functions Xn:]a,b[→ℬ satisfying limn→∞∥Xn-X∥=0, μ.a.e.). For such functions X, it is known that X is μ-Bochner integrable if and only if ∫ab∥X(s)∥μ(ds)<∞ (cf. [15, page 133]). For such functions X, it is also known that for each bounded linear operator T:ℬ→ℬ, we have  (2.1)T(∫abX(s)μ(ds))=∫abTX(s)μ(ds).
In the sequel of this work, μ is omitted whenever it is the Lebesgue measure.


### 2.1.C0-Contraction Semigroups

AC0-contraction semigroup on ℬ is a family of linear operators ℙ:=(Pt)t≥0 on ℬ satisfying P0=I, Ps+t=PsPt for all s,t≥0, ∥Pt∥≤1 for all t≥0 and limt→0∥Ptf-f∥=0 for all f∈ℬ.Letℙ be a C0-contraction semigroup on ℬ. The associated generatorA of ℙ is defined by  (2.2)Af:=limt→0Ptf-ft
on its domain D(A):={f∈ℬ:  the limit in (2.2) exists inℬ}. It is known that    (1)
A:D(A)→ℬ is a closed linear operator;(2)
D(A) is dense in the Banach space ℬ;(3)
the resolventℛq:=(qI-A)-1 of A exists for each q>0.The proof of the following useful classical properties can be found in [14, pages 4 and 108] and in [15, pages 233–240].Lemma 2.1.
Letℙ be a C0-contraction semigroup on ℬ with generator (A,D(A)) and resolvent ℛ:=(ℛq)q>0.    (1)
Forf∈ℬ and 0≤a<b<∞, the function t→Ptf is strongly measurable and the Bochner integral ∫abPrfdr is well defined.(2)
For eacht≥0 and f∈D(A), we have Ptf∈D(A),(2.3)APtf=PtAf=ddtPtf,(2.4)Ptf-f=∫0tA(Prf)dr.(3)
For eacht,q>0, we have ℛq=∫0∞e-qsPsds, Ptℛq=ℛqPt, ℛq(ℬ)⊂D(A) and
(2.5)limq→∞qℛqu=u,u∈ℬ.Example 2.2.
Letℬ=𝒞b([0,∞[) be the Banach space of bounded uniformly continuous real-valued functions on [0,∞[ and let
(2.6)Ptf(x):=f(x+t),t≥0,x≥0,f∈𝒞b([0,∞[).
Then ℙ:=(Pt)t≥0 is C0-contraction semigroup with generator Af=f' where D(A):={f∈𝒞b([0,∞[):f′exist,f′∈𝒞b([0,∞[)}.


## 2.1.C0-Contraction Semigroups

AC0-contraction semigroup on ℬ is a family of linear operators ℙ:=(Pt)t≥0 on ℬ satisfying P0=I, Ps+t=PsPt for all s,t≥0, ∥Pt∥≤1 for all t≥0 and limt→0∥Ptf-f∥=0 for all f∈ℬ.Letℙ be a C0-contraction semigroup on ℬ. The associated generatorA of ℙ is defined by  (2.2)Af:=limt→0Ptf-ft
on its domain D(A):={f∈ℬ:  the limit in (2.2) exists inℬ}. It is known that    (1)
A:D(A)→ℬ is a closed linear operator;(2)
D(A) is dense in the Banach space ℬ;(3)
the resolventℛq:=(qI-A)-1 of A exists for each q>0.The proof of the following useful classical properties can be found in [14, pages 4 and 108] and in [15, pages 233–240].Lemma 2.1.
Letℙ be a C0-contraction semigroup on ℬ with generator (A,D(A)) and resolvent ℛ:=(ℛq)q>0.    (1)
Forf∈ℬ and 0≤a<b<∞, the function t→Ptf is strongly measurable and the Bochner integral ∫abPrfdr is well defined.(2)
For eacht≥0 and f∈D(A), we have Ptf∈D(A),(2.3)APtf=PtAf=ddtPtf,(2.4)Ptf-f=∫0tA(Prf)dr.(3)
For eacht,q>0, we have ℛq=∫0∞e-qsPsds, Ptℛq=ℛqPt, ℛq(ℬ)⊂D(A) and
(2.5)limq→∞qℛqu=u,u∈ℬ.Example 2.2.
Letℬ=𝒞b([0,∞[) be the Banach space of bounded uniformly continuous real-valued functions on [0,∞[ and let
(2.6)Ptf(x):=f(x+t),t≥0,x≥0,f∈𝒞b([0,∞[).
Then ℙ:=(Pt)t≥0 is C0-contraction semigroup with generator Af=f' where D(A):={f∈𝒞b([0,∞[):f′exist,f′∈𝒞b([0,∞[)}.


## 3. Exit Equation

### 3.1. Exit Laws

Definition 3.1.
Letℙ be a C0-contraction semigroup on ℬ. A ℙ-exit law is a ℬ-valued function t∈]0,∞[→ut∈ℬ which verifies the so-called exit equation: (3.1)Ptus=ut+s,s,t>0.
We point here that a ℙ-exit law t→ut may be also denoted by u:=(ut)t>0.Proposition 3.2.
Letℙ be a C0-contraction semigroup on ℬ with generator (A,D(A)).    (1)
For eachℙ-exit law φ:=(φt)t>0, the function s→φs is strongly measurable on ]0,∞[.(2)
For eachh∈ℬ, the ℬ-valued function t→Pth is a ℙ-exit law. It is called a closed exit law.(3)
Leth∈ℬ such that (Pth)t>0⊂D(A), then t→APth is a ℙ-exit law. It is said to be differentiable.Proof.
The functions→Psh is strongly measurable for each h∈ℬ; then for each b>0, the function s→φs+b=Psφb is strongly measurable on [b,∞[. Since b>0 is arbitrary, then u is strongly measurable on ]0,∞[.
It is immediate from the semigroup property.
It is a consequence of the semigroup property and (2.3).


### 3.2. Integrable Exit Laws

Letℙ be a C0-contraction semigroup on ℬ with generator (A,D(A)). In the sequel, we consider ℙ-exit laws t→φt which are Bochner integrable at 0 (shortly zero-integrable). This is equivalent to (3.2)∫01∥φs∥ds<∞.Theorem 3.3.
Letφ be a zero-integrable ℙ-exit law. Then φ is of the form
(3.3)φt=qPtVq(φ)-APtVq(φ),t>0,
where q>0 and Vq(φ):=∫0∞e-qsφsds.Proof.
Letq>0 be fixed. Since (3.2) holds if and only if s→eωsφs is zero-integrable for all ω∈ℝ, then using (3.1) and (3.2), we have
(3.4)∫0∞e-qs∥φs∥ds=∫01e-qs∥φs∥ds+∫1∞e-qs∥φs∥ds=∫01e-qs∥φs∥ds+∫1∞e-qs∥Ps-1φ1∥ds≤∫01e-qs∥φs∥ds+∥φ1∥∫1∞e-qsds<∞.
This implies that s→e-qsφs is Bochner integrable on ]0,∞[. Hence, Vq(φ) is well defined and lies in B. Moreover, by (3.1), (2.1), and (2.3), we get
(3.5)PtVq(φ)=∫0∞e-qsφs+tds=∫0∞e-qsPsφtds=ℛq(φt),t>0.
Using (3.5) and (3.3) holds since
(3.6)qPtVq(φ)-APtVq(φ)=qℛq(φt)-Aℛq(φt)=(qI-A)ℛq(φt)=φt.Corollary 3.4.
Suppose that the generatorA of ℙ is bounded and let φ be a ℙ-exit law. Then φ is zero-integrable if and only if φ is closed.Proof.
Ifφt=Ptf for some f∈ℬ, then t→φt is zero-integrable by Lemma 2.1. Conversely, let φ be a ℙ-exit law satisfying (3.2). Theorem 3.3 may be applied: φ is of the form
(3.7)φt=qPtVq(φ)-APtVq(φ),t>0,
where Vq(φ):=∫0∞e-qsφsds for some q>0. Moreover, since A is bounded then D(A)=ℬ (cf. [16, Corollary 1.5] and therefore by (2.3), we get
(3.8)φt=qPtVq(φ)-APtVq(φ)=qPtVq(φ)-PtAVq(φ)=Pt(qVq(φ)-AVq(φ)).
Hence, φ is a closed exit law.Remark 3.5.
Results similar to Theorem3.3 are proved in our paper [10]. Indeed, the proof given in [10] depends fundamentally on the properties of the rescaled C0-semigroup; however, in this paper, it is based on the resolvent properties of C0-contraction semigroup.
For closed exit laws, the condition (3.2) is satisfied. However, this not the case, for differentiable exit laws. Indeed, consider again Example 2.2 and let u(x)=xsin(1/x). Then u∈ℬ and ∫01∥APtu∥dt=∞.
We consider Example2.2 and we define
(3.9)φta(x)≔1(x+t)a,a>0,t>0,x≥0.
It is proved in [10] that φa:=(φta)t>0 is a ℙ-exit law neither closed nor differentiable and ∫01∥φs∥ds<∞ if and only if a∈]0,1[.


## 3.1. Exit Laws

Definition 3.1.
Letℙ be a C0-contraction semigroup on ℬ. A ℙ-exit law is a ℬ-valued function t∈]0,∞[→ut∈ℬ which verifies the so-called exit equation: (3.1)Ptus=ut+s,s,t>0.
We point here that a ℙ-exit law t→ut may be also denoted by u:=(ut)t>0.Proposition 3.2.
Letℙ be a C0-contraction semigroup on ℬ with generator (A,D(A)).    (1)
For eachℙ-exit law φ:=(φt)t>0, the function s→φs is strongly measurable on ]0,∞[.(2)
For eachh∈ℬ, the ℬ-valued function t→Pth is a ℙ-exit law. It is called a closed exit law.(3)
Leth∈ℬ such that (Pth)t>0⊂D(A), then t→APth is a ℙ-exit law. It is said to be differentiable.Proof.
The functions→Psh is strongly measurable for each h∈ℬ; then for each b>0, the function s→φs+b=Psφb is strongly measurable on [b,∞[. Since b>0 is arbitrary, then u is strongly measurable on ]0,∞[.
It is immediate from the semigroup property.
It is a consequence of the semigroup property and (2.3).


## 3.2. Integrable Exit Laws

Letℙ be a C0-contraction semigroup on ℬ with generator (A,D(A)). In the sequel, we consider ℙ-exit laws t→φt which are Bochner integrable at 0 (shortly zero-integrable). This is equivalent to (3.2)∫01∥φs∥ds<∞.Theorem 3.3.
Letφ be a zero-integrable ℙ-exit law. Then φ is of the form
(3.3)φt=qPtVq(φ)-APtVq(φ),t>0,
where q>0 and Vq(φ):=∫0∞e-qsφsds.Proof.
Letq>0 be fixed. Since (3.2) holds if and only if s→eωsφs is zero-integrable for all ω∈ℝ, then using (3.1) and (3.2), we have
(3.4)∫0∞e-qs∥φs∥ds=∫01e-qs∥φs∥ds+∫1∞e-qs∥φs∥ds=∫01e-qs∥φs∥ds+∫1∞e-qs∥Ps-1φ1∥ds≤∫01e-qs∥φs∥ds+∥φ1∥∫1∞e-qsds<∞.
This implies that s→e-qsφs is Bochner integrable on ]0,∞[. Hence, Vq(φ) is well defined and lies in B. Moreover, by (3.1), (2.1), and (2.3), we get
(3.5)PtVq(φ)=∫0∞e-qsφs+tds=∫0∞e-qsPsφtds=ℛq(φt),t>0.
Using (3.5) and (3.3) holds since
(3.6)qPtVq(φ)-APtVq(φ)=qℛq(φt)-Aℛq(φt)=(qI-A)ℛq(φt)=φt.Corollary 3.4.
Suppose that the generatorA of ℙ is bounded and let φ be a ℙ-exit law. Then φ is zero-integrable if and only if φ is closed.Proof.
Ifφt=Ptf for some f∈ℬ, then t→φt is zero-integrable by Lemma 2.1. Conversely, let φ be a ℙ-exit law satisfying (3.2). Theorem 3.3 may be applied: φ is of the form
(3.7)φt=qPtVq(φ)-APtVq(φ),t>0,
where Vq(φ):=∫0∞e-qsφsds for some q>0. Moreover, since A is bounded then D(A)=ℬ (cf. [16, Corollary 1.5] and therefore by (2.3), we get
(3.8)φt=qPtVq(φ)-APtVq(φ)=qPtVq(φ)-PtAVq(φ)=Pt(qVq(φ)-AVq(φ)).
Hence, φ is a closed exit law.Remark 3.5.
Results similar to Theorem3.3 are proved in our paper [10]. Indeed, the proof given in [10] depends fundamentally on the properties of the rescaled C0-semigroup; however, in this paper, it is based on the resolvent properties of C0-contraction semigroup.
For closed exit laws, the condition (3.2) is satisfied. However, this not the case, for differentiable exit laws. Indeed, consider again Example 2.2 and let u(x)=xsin(1/x). Then u∈ℬ and ∫01∥APtu∥dt=∞.
We consider Example2.2 and we define
(3.9)φta(x)≔1(x+t)a,a>0,t>0,x≥0.
It is proved in [10] that φa:=(φta)t>0 is a ℙ-exit law neither closed nor differentiable and ∫01∥φs∥ds<∞ if and only if a∈]0,1[.


## 4. Subordination ofC0-Contraction Semigroup

### 4.1. Bochner Subordinator

We considerℝ endowed with its Borel σ-field. We denote by ɛt the Dirac measure at point t. Moreover, for each bounded measure μ on [0,∞[, ℒ denotes its Laplace transform, that is, ℒ(μ)(r):=∫0∞exp(-rs)μ(ds) for r>0.For the following classical notions, we refer the reader to [17–19].ABochner subordinatorβ:=(βt)t>0 is a vaguely continuous convolution semigroup of subprobability measures on [0,+∞[.Letβ be a Bochner subordinator. The associated Bernstein functionf is defined by the Laplace transform  (4.1)ℒ(βt)(r)=exp(-tf(r)),r,t>0.
In fact, (4.1) establishes a one-to-one correspondence between convolution semigroups β:=(βt)t≥0 and Bernstein functions f (cf. [18, Theorem 9.8]). In fact, f admits the representation  (4.2)f(r)=a+br+∫0∞(1-exp(-rs))ν(ds),r>0,
where a,b≥0 and ν is a measure on ]0,∞[ verifying ∫0∞(s/(s+1))ν(ds)<∞. They are called parameters of β or of f.Example 4.1.
Thefractional power subordinatorηα:=(ηtα)t≥0of indexα∈]0,1[ is defined by its Laplace transform ℒ(ηtα)(r)=exp(-trα) for all r,t>0.
TheΓ-subordinatorγ:=(γt)t≥0 is defined by
(4.3)γt(ds)≔1]0,∞[(s)(1Γ(t))st-1exp(-s)ds,t≥0.
ThePoisson subordinatorτ:=(τt)t≥0of jumpc>0 is defined by
(4.4)τt≔exp(-ct)∑n=0∞(ct)nn!ɛn,t≥0.
The  Dirac subordinatorɛ:=(ɛt)t≥0.


### 4.2. Bochner Subordination

Letℙ be a C0-contraction semigroup on ℬ and let β be a Bochner subordinator. For every t>0 and for every u∈ℬ, we may define (4.5)Ptβu:=∫0∞Psuβt(ds).
Then ℙβ:=(Ptβ)t>0 is a C0-contraction semigroup on ℬ (see, e.g., [17, Theorem 4.3.1]). It is said to be subordinated to ℙ in the sense of Bochner by means of β. In what follows, we index by “β” all entities associated to ℙβ. In particular, Aβ is the associated generator and ℛβ:=(ℛqβ)q>0 its associated resolvent.LetAβ be the generator of ℙβ. The following two remarks will be used throughout this paper: D(A) is a subset of D(Aβ) (cf. [17, page 299]) and  (4.6)Aβu=-au+bAu+∫0∞(Ptu-u)ν(dt),u∈D(A),
where a,b, and ν are given in (4.2).Lemma 4.2.
There exist some constantsK1,K2>0 such that
(4.7)∫0∞∥Psu-u∥ν(ds)≤K1∥u∥+K2∥Au∥,u∈D(A).Proof.
Letu∈D(A). Using the semigroup property and (4.6), we have
(4.8)∫0∞∥Psu-u∥ν(ds)≤∫01∥Psu-u∥ν(ds)+∫1∞∥Psu-u∥ν(ds)≤∫01∥∫0sPrAudr∥ν(ds)+2∥u∥∫1∞ν(ds)≤(∫01sν(ds))∥Au∥+2∥u∥(∫1∞ν(ds)).
Hence, (4.7) holds for K1:=2∫1∞ν(ds) and K2:=∫01sν(ds).Proposition 4.3.
Letℙ be a C0-contraction semigroup on ℬ, β a Bochner subordinator, and ℙβ be the subordinated to ℙ by means of β. Then
(4.9)PtAβh=AβPth,t>0,h∈D(Aβ).
In particular, we have Pt(D(Aβ))⊂D(Aβ) for all t>0.Proof.
Step 1.
First we suppose thath∈D(A). From Lemma 4.2,
(4.10)∫0∞∥Psh-h∥ν(ds)<∞.
So by using (2.1), we get
(4.11)Pt∫0∞(Psh-h)ν(ds)=∫0∞(PsPth-Pth)ν(ds),t>0.
Combining (2.3), (4.6), and (4.11), we have
(4.12)PtAβh=-aPth+bPtAh+Pt∫0∞(Psh-h)ν(ds)=-aPth+bAPth+∫0∞(PsPth-Pth)ν(ds)=AβPth.Step 2.
Now, we suppose thath∈D(Aβ). Let ℛ:=(ℛq)q>0 be the associated resolvent to ℙ and let q,t>0. Since ℛq(ℬ)⊂D(A), then from Step 1 and Lemma 2.1, we have
(4.13)PtAβℛqh=AβPtℛqh=AβℛqPth.
Hence, by the contraction property and (4.13), we get
(4.14)∥PtAβh-AβPth∥≤∥PtAβh-qPtAβℛqh∥+∥qAβℛqPth-AβPth∥≤∥Aβh-Aβqℛqh∥+∥AβqℛqPth-AβPth∥.
Finally, since Aβ is closed then by (2.5) and by letting q↑∞, (4.9) holds.


## 4.1. Bochner Subordinator

We considerℝ endowed with its Borel σ-field. We denote by ɛt the Dirac measure at point t. Moreover, for each bounded measure μ on [0,∞[, ℒ denotes its Laplace transform, that is, ℒ(μ)(r):=∫0∞exp(-rs)μ(ds) for r>0.For the following classical notions, we refer the reader to [17–19].ABochner subordinatorβ:=(βt)t>0 is a vaguely continuous convolution semigroup of subprobability measures on [0,+∞[.Letβ be a Bochner subordinator. The associated Bernstein functionf is defined by the Laplace transform  (4.1)ℒ(βt)(r)=exp(-tf(r)),r,t>0.
In fact, (4.1) establishes a one-to-one correspondence between convolution semigroups β:=(βt)t≥0 and Bernstein functions f (cf. [18, Theorem 9.8]). In fact, f admits the representation  (4.2)f(r)=a+br+∫0∞(1-exp(-rs))ν(ds),r>0,
where a,b≥0 and ν is a measure on ]0,∞[ verifying ∫0∞(s/(s+1))ν(ds)<∞. They are called parameters of β or of f.Example 4.1.
Thefractional power subordinatorηα:=(ηtα)t≥0of indexα∈]0,1[ is defined by its Laplace transform ℒ(ηtα)(r)=exp(-trα) for all r,t>0.
TheΓ-subordinatorγ:=(γt)t≥0 is defined by
(4.3)γt(ds)≔1]0,∞[(s)(1Γ(t))st-1exp(-s)ds,t≥0.
ThePoisson subordinatorτ:=(τt)t≥0of jumpc>0 is defined by
(4.4)τt≔exp(-ct)∑n=0∞(ct)nn!ɛn,t≥0.
The  Dirac subordinatorɛ:=(ɛt)t≥0.


## 4.2. Bochner Subordination

Letℙ be a C0-contraction semigroup on ℬ and let β be a Bochner subordinator. For every t>0 and for every u∈ℬ, we may define (4.5)Ptβu:=∫0∞Psuβt(ds).
Then ℙβ:=(Ptβ)t>0 is a C0-contraction semigroup on ℬ (see, e.g., [17, Theorem 4.3.1]). It is said to be subordinated to ℙ in the sense of Bochner by means of β. In what follows, we index by “β” all entities associated to ℙβ. In particular, Aβ is the associated generator and ℛβ:=(ℛqβ)q>0 its associated resolvent.LetAβ be the generator of ℙβ. The following two remarks will be used throughout this paper: D(A) is a subset of D(Aβ) (cf. [17, page 299]) and  (4.6)Aβu=-au+bAu+∫0∞(Ptu-u)ν(dt),u∈D(A),
where a,b, and ν are given in (4.2).Lemma 4.2.
There exist some constantsK1,K2>0 such that
(4.7)∫0∞∥Psu-u∥ν(ds)≤K1∥u∥+K2∥Au∥,u∈D(A).Proof.
Letu∈D(A). Using the semigroup property and (4.6), we have
(4.8)∫0∞∥Psu-u∥ν(ds)≤∫01∥Psu-u∥ν(ds)+∫1∞∥Psu-u∥ν(ds)≤∫01∥∫0sPrAudr∥ν(ds)+2∥u∥∫1∞ν(ds)≤(∫01sν(ds))∥Au∥+2∥u∥(∫1∞ν(ds)).
Hence, (4.7) holds for K1:=2∫1∞ν(ds) and K2:=∫01sν(ds).Proposition 4.3.
Letℙ be a C0-contraction semigroup on ℬ, β a Bochner subordinator, and ℙβ be the subordinated to ℙ by means of β. Then
(4.9)PtAβh=AβPth,t>0,h∈D(Aβ).
In particular, we have Pt(D(Aβ))⊂D(Aβ) for all t>0.Proof.
Step 1.
First we suppose thath∈D(A). From Lemma 4.2,
(4.10)∫0∞∥Psh-h∥ν(ds)<∞.
So by using (2.1), we get
(4.11)Pt∫0∞(Psh-h)ν(ds)=∫0∞(PsPth-Pth)ν(ds),t>0.
Combining (2.3), (4.6), and (4.11), we have
(4.12)PtAβh=-aPth+bPtAh+Pt∫0∞(Psh-h)ν(ds)=-aPth+bAPth+∫0∞(PsPth-Pth)ν(ds)=AβPth.Step 2.
Now, we suppose thath∈D(Aβ). Let ℛ:=(ℛq)q>0 be the associated resolvent to ℙ and let q,t>0. Since ℛq(ℬ)⊂D(A), then from Step 1 and Lemma 2.1, we have
(4.13)PtAβℛqh=AβPtℛqh=AβℛqPth.
Hence, by the contraction property and (4.13), we get
(4.14)∥PtAβh-AβPth∥≤∥PtAβh-qPtAβℛqh∥+∥qAβℛqPth-AβPth∥≤∥Aβh-Aβqℛqh∥+∥AβqℛqPth-AβPth∥.
Finally, since Aβ is closed then by (2.5) and by letting q↑∞, (4.9) holds.


## 5. Subordinated Exit Law

Letℙ be a C0-contraction semigroup with generator (A,D(A)), let β be a Bochner subordinator, and let ℙβ be the subordinated to ℙ by means of β with generator (Aβ,D(Aβ)).Definition 5.1.
Letφ be a ℙ-exit law and define(1.5).If the family of Bochner integrals (1.5) is well defined, it easy to verify that φβ:=(φtβ)t>0 is a ℙβ-exit law which is said to be subordinated toφin the Bochner sense by means ofβ. Notice that if φs=Psu for some u∈ℬ, then (1.5) is just (4.5).Remark 5.2.
(1)
Subordination problem: conversely, let ψ be a ℙβ-exit law, does there exist a ℙ-exit law φ such that ψ is subordinated to φ?In this paper, we study this problem ofℙβ-exit laws which are Bochner integrable at 0.(2)
The condition of zero-integrability is not necessary. Indeed, if we takeβt=ɛt, then ℙβ=ℙ and the subordination problem is solved for each ℙβ-exit law ψ since ψβ=ψ.Example 5.3.
Letℬ=L1(λ1) where λ1 is the Lebesgue measure on ℝ and let ℙ:=(Pt)t>0 be the left uniform translation on ℬ, that is, Ptu:=ɛt*u for t≥0 and u∈ℬ. It can be seen that each ℙ-exit law φ is closed, that is, of the form φt=Ptu for some fixed u∈ℬ.
On the other hand, letη1/2:=(ηt1/2)t>0 be the fractional powers subordinator of index 1/2. From [18,  page 71], ηt1/2 is absolutely continuous with density Gt(s)=(1/4π)ts-3/2exp(-t2/4s) for all t,s>0. The extension of Gt by 0 on ℝ is denoted by Gt. So, Ptη1/2u=Gt*u for all t>0 and u∈ℬ. In particular,
(5.1)Ptη1/2Gs(x)=Gt*Gs(x)=Gs+t(x),s,t>0,x∈ℝ,
by the convolution semigroup property of η1/2. Therefore, the family G:=(Gt)t>0 is a ℙη1/2-exit law. Moreover, G is zero-integrable (By using the change of variables y=t-2s, we have ∥Gt∥=∥G1∥ for all t>0). But there exists no u∈ℬ such that Gt=Gt*u for each t>0.
Hence, not every zero-integrableℙη1/2-exit law is subordinated to a ℙ-exit law, because each ℙ-exit law is closed, while this is not the case for all zero-integrable ℙη1/2-exit law.Remark 5.4.
Example5.3 proves that we need to add some condition in order to solve the subordination problem. Next, we will suppose that ψ:=(ψt)t>0 satisfies the following conditions.   (H):
There exists a constantq>0 such as(1.6) and (1.7)where Vq(ψ):=∫0∞e-qsψsds.Theorem 5.5.
Letℙ be a C0-contraction semigroup on ℬ and let β be a Bochner subordinator. Suppose that ψ is a zero-integrable ℙβ-exit law satisfying (H). Then ψ is subordinated to a unique ℙ-exit law φ:=(φt)t>0. Moreover, φ is explicitly given by
(5.2)φt:=qPtVq(ψ)-AβPtVq(ψ),t>0,
where q and Vq are given by (H).Proof.
Since(PtVq(ψ))t>0⊂D(Aβ), then the family φ:=(φt)t>0 defined by (5.2) is well defined and lies in ℬ. Moreover, by (5.2) and (4.9), we get
(5.3)Psφt=Ps(qPtVq(ψ)-AβPtVq(ψ))=qPs+tVq(ψ)-AβPs+tVq(ψ)=φs+t,
which implies that φ:=(φt)t>0 is a ℙ-exit law. Now using (1.6) and (4.9), we have
(5.4)∫1∞∥AβPsVq(ψ)∥βt(ds)=∫1∞∥AβPs-1P1Vq(ψ)∥βt(ds)=∫1∞∥Ps-1AβP1Vq(ψ)∥βt(ds)≤∫1∞∥AβP1Vq(ψ)∥βt(ds)≤βt([1,∞[)∥AβP1Vq(ψ)∥<∞,
and by (1.7), we conclude that
(5.5)∫0∞∥AβPsVq(ψ)∥βt(ds)<∞,t>0.
Therefore, from (5.5), we have
(5.6)∫0∞∥φs∥βt(ds)=∫0∞∥qPsVq(ψ)-AβPsVq(ψ)∥βt(ds)≤q∫0∞∥PsVq(ψ)∥βt(ds)+∫0∞∥AβPsVq(ψ)∥βt(ds)≤qβt([0,∞[)∥Vq(ψ)∥+∫0∞∥AβPsVq(ψ)∥βt(ds)<∞.
Hence, the subordinated φβ:=(φtβ)t>0 defined by (1.5) is well defined.
On the other hand, for alls,t>0, we have
(5.7)Psφtβ=(1.5)Ps∫0∞φrβt(dr)=(2.1)∫0∞Psφrβt(dr)=(3.1)∫0∞φr+tβt(dr)=∫0∞Prφsβt(dr)=Ptβφs=(8)Ptβ(PsVq(ψ)-AβPsVq(ψ))=(2.3)PsPtβVq(ψ)-AβPsPtβVq(ψ)=(18)Ps(PtβVq(ψ)-AβPtβVq(ψ))=Psψt
by using Theorem 3.3 since ψ is Bochner integrable at 0. Therefore,
(5.8)ψt=Pt/2βψt/2=∫0∞Psψt/2βt/2(ds)=∫0∞Psφt/2ββt/2(ds)=Pt/2βφt/2β=φtβ,
which implies that ψ=φβ.
Finally, let us prove the uniqueness: Letϕ:=(ϕt)t>0 be a ℙ-exit law such that ψ=ϕβ. Since for all s,t>0, we have
(5.9)Psϕtβ=Ps∫0∞ϕrβt(dr)=(2.1)∫0∞Psϕrβt(dr)=(3.1)∫0∞ϕr+tβt(dr)=∫0∞Prϕsβt(dr)=(4.4)Ptβϕs,
then for all t>0,
(5.10)PtVq(ψ)=∫0∞e-sqPtϕsβds=∫0∞e-sqPsβϕtds=Rqβ(ϕt).
Therefore, from (5.2), we have
(5.11)φt=qRqβ(ϕt)-AβRqβ(ϕt)=(qI-Aβ)Rqβ(ϕt)=ϕt,t>0.Remark 5.6.
In addition, ifPt(Vq(ψ))t>0⊂D(A), then from (4.6), φ is of the form
(5.12)φt=(q+a)PtVq(ψ)-bAPtVq(ψ)+∫0∞(PsVq(ψ)-Ps+tVq(ψ))ν(ds),
where a,  b, and ν are the parameters associated to β.
In particular, this is the case of eachψ satisfying (H) whenever the parameter b of β is not zero or A is bounded. Indeed, from [17, Theorem 5.3.8], we have D(A)=D(Aβ).
Ifψ satisfies (H) for some q>0, then it satisfies (H) for all ɛ>0. Indeed, by Theorem 5.5, ψ is subordinated to some ℙ-exit law φ. Moreover, exactly as (5.10), we have
(5.13)PtVɛ(ψ)=∫0∞e-sɛPtψsds=∫0∞e-sɛPsβφtds=ℛɛ(φt),t,ɛ>0.
So, PtVɛ(ψ)∈D(Aβ) and
(5.14)AβPtVɛ(ψ)=Aβℛɛβ(φt)=(ɛI-Aβ)ℛɛβ(φt)-ɛℛɛβ(φt)=φt-ɛPtVɛ(ψ).
Therefore, from the proof of Theorem 5.5, we have
(5.15)∫01∥AβPsVq(ψ)∥βt(ds)=∫01∥φs-ɛPsVɛ(ψ)∥βt(ds)≤∫01(∥φs∥+∥ɛPsVɛ(ψ)∥)βt(ds)≤∫01∥φs∥βt(ds)+ɛ∥Vɛ(ψ)∥∫01βt(ds).
Hence, (1.6) and (1.7) hold for each ɛ>0.
The conditions of Theorem5.5 are fulfilled for the natural example of ℙβ-exit law. Indeed, we have the following result.Corollary 5.7.
Each closedℙβ-exit law ψ, that is, ψt=Ptβu for some u∈ℬ, is subordinated to unique exit law ℙ-exit law φ:=(φt)t>0. Moreover, φ is explicitly given by φt=Ptu for all t>0.Proof.
Letψ be a closed ℙβ-exit law. It is easy to see that ψ is zero-integrable. Moreover, for all q>0, we have
(5.16)Vq(ψ):=∫0∞e-qsψsds=∫0∞e-qsPsβuds=ℛqβ(u),PtVq(u)=∫0∞e-qsPsβPtuds=ℛqβ(Ptu),t>0,
which implies that Vq(ψ)∈D(Aβ) and (PtVq(ψ))t>0⊂D(Aβ). Moreover,
(5.17)∫01∥AβPsVq(ψ)∥βt(ds)=∫01∥PsAβVq(ψ)∥βt(ds)≤∫01∥AβVq(ψ)∥βt(ds)≤∥AβVq(ψ)∥.
So, ψ satisfies (H) and by Theorem 5.5, we get
(5.18)φt=(qI-Aβ)PtVq(ψ)=(qI-Aβ)ℛβ(Ptu))=Ptu,t>0.Corollary 5.8.
Suppose thatA is bounded or f is bounded, then Theorem 5.5 may be applied for each zero-integrable ℙβ-exit law.Proof.
According to [17, Theorem 4.3.8, page 303], Aβ is bounded if and only if A is bounded or f is bounded. So the proof is an immediate consequence of Corollaries 3.4 and 5.7.


## 6. Application to Holomorphic Case

Definition 6.1.
Letℙ be a C0-contraction semigroup on ℬ. ℙ is said to be holomorphic if there exists a holomorphic extension z→Pz to S:={z∈ℂ*:|argz|<θ};0<θ<π/2.Remark 6.2 (Construction by Bochner subordination).
LetF be the Banach algebra of complex Borel measures on [0,∞[, with convolution as multiplication, and normed by the total variation ∥·∥F. A Bochner subordinator β=(βt)t>0 is said to be of type Carasso-Kato if:
The associated parametersa=b=0 and the mapping t→βt is continuously differentiable from ]0,∞[ to F such that ∥(∂/∂t)βt∥F≤c/t as t→0 and for some constant c>0.
It is proved in [19] that for each C0-contraction semigroup ℚ and each subordinator β of type Carasso-Kato, the subordinated ℙ:=ℚβ is a C0-contraction holomorphic semigroup.
Note that the fractional power subordinator,Γ-subordinator, and Poisson subordinator are of type Carasso-Kato.The proof of the following useful classical properties can be found in [14, pages 4 and 108] and in [15, pages 233–240].Lemma 6.3.
Letℙ be a C0-contraction holomorphic semigroup on ℬ with generator (A,D(A)). Then there exists a constant t>0 such that
(6.1)∥APth∥≤C∥h∥tast→0.
Note that the condition (1.6) from (H) is fulfilled for all C0-contraction holomorphic semigroup ℙ. Indeed, using the above Lemma, the range of Pt is contained in D(A), hence also in D(Aβ).Proposition 6.4.
Letβ be a Bochner subordinator such as (1.9)Then,
(6.2)∫01∥AβPsh∥βt(ds)<∞,t>0,h∈ℬ.Proof.
Lett>0 and h∈ℬ. Since Pth∈D(A), then from (4.6), we have
(6.3)AβPth=-aPth+bAPth+∫0∞(PsPth-Pth)ν(ds),
where a,b, and ν are given in (4.2). By using Lemma 4.2, we have
(6.4)∫0∞∥Ps+th-Pth∥ν(ds)≤K1∥Pth∥+K2∥APth∥≤K1∥h∥+K2∥APth∥
for some K1,K2>0. Therefore, by (6.3), we conclude that
(6.5)∥AβPth∥≤K3∥h∥+K4∥APth∥,
where K3:=a+K1 and K4:=b+K2. Moreover, combining Lemma 6.3, (1.9), and (6.5), we have
(6.6)∫01∥AβPsh∥βt(ds)≤K3∥h∥∫01βt(ds)+K4∫01∥APsh∥βt(ds)≤K3∥h∥+K4C∥h∥∫011sβt(ds)<∞.
Hence, (6.2) holds.Remark 6.5.
The condition (1.9) holds as soon as the associated Bernstein function f satisfies
(6.7)∫0∞e-tf(r)dr<∞,t>0.
Indeed by using the Fubini's theorem, we have
(6.8)∫011sβt(ds)≤∫0∞(∫0∞e-srdr)βt(ds)≤∫0∞(∫0∞e-srβt(ds))dr≤∫0∞ℒ(βt)(r)dr≤∫0∞e-tf(r)dr.
Hence, (1.9) holds for the Dirac and the fractional power subordinators.Theorem 6.6.
Letℙ be a C0-contraction holomorphic semigroup on ℬ and let β be a Bochner subordinator satisfying (1.9). Then each zero-integrable ℙβ-exit law ψ is subordinated to a unique ℙ-exit law φ:=(φt)t>0. Moreover, φ is explicitly given by
(6.9)φt=(q+a)PtVq(ψ)-bAPtVq(ψ)+∫0∞(PsVq(ψ)-Ps+tVq(ψ))ν(ds),t>0,
where a, b, and ν are the parameters of β and Vq(ψ):=∫0∞e-qsψsds for some q>0.Proof.
Letq>0. Since Vq(ψ):=∫0∞e-qsψsds∈ℬ, then from Proposition 6.4, (1.6) and (1.7) hold. Therefore, ψ satisfies (H). So the proof is an immediate consequence of Theorem 5.5 and (4.6).



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