On the Abstract Subordinated Exit Equation

Abstract

Let ℙ=(𝑃_𝑡)_𝑡>0 be a 𝐶₀-contraction semigroup on a real Banach space ℬ. A ℙ-exit law is a ℬ-valued function 𝑡∈]0,∞[→𝜑_𝑡∈ℬ satisfying the functional equation: 𝑃_𝑡𝜑_𝑠=𝜑_𝑡+𝑠, 𝑠,𝑡>0. Let 𝛽 be a Bochner subordinator and let ℙ^𝛽 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 ℙ^𝛽-exit law is subordinated to a unique ℙ-exit law.

1. Introduction

Let ℙ:=(𝑃_𝑡)_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,∞[→𝜑_𝑡∈ℬ satisfying the functional equation: 𝑃_𝑡𝜑_𝑠=𝜑_𝑡+𝑠,𝑠,𝑡>0. (1.1) 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 a ℙ-exit law 𝜑 is Bochner integrable at 0 (shortly zero-integrable), this is equivalent to, ¹₀‖𝜑_𝑡‖𝑑𝑡<∞, (1.2) then 𝜑 is of the form 𝜑_𝑡=𝑞𝑃_𝑡𝑉_𝑞(𝜑)−𝐴𝑃_𝑡𝑉_𝑞(𝜑),𝑡,𝑞>0, (1.3) where 𝑉_𝑞(𝜑):=∫^∞₀𝑒^−𝑞𝑠𝜑_𝑠𝑑𝑠.

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, let 𝛽=(𝛽_𝑡)_𝑡>0 be a Bochner subordinator, that is, a vaguely continuous convolution semigroup of subprobability measures on [0,+∞[ and let ℙ^𝛽∶=(𝑃^𝛽_𝑡)_𝑡>0 be the subordinated 𝐶₀-semigroup of ℙ in the sense of Bochner by means of 𝛽, that is, 𝑃^𝛽_𝑡𝑓:=^∞₀𝑃_𝑠𝑓𝛽_𝑡(𝑑𝑠),𝑓∈ℬ,𝑡>0. (1.4) It can be seen that, for each exit law 𝜑=(𝜑_𝑡)_𝑡>0 for ℙ, the function 𝜑^𝛽 defined by 𝜑^𝛽_𝑡:=^∞₀𝜑_𝑠𝛽_𝑡(𝑑𝑠),𝑡>0, (1.5) is an exit law for ℙ^𝛽. The function 𝜑^𝛽 is said to be subordinated to 𝜑 by means of 𝛽.

Conversely, it is natural to ask if any ℙ^𝛽-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 ℙ^𝛽-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 ℙ^𝛽-exit laws in the abstract case. Namely, we prove the following.

Theorem 1.2. Let 𝜓:=(𝜓_𝑡)_𝑡>0 be a zero-integrable ℙ^𝛽-exit law satisfying the following conditions: There exist a constant 𝑞>0 such that: (𝑃_𝑡𝑉_𝑞(𝜓))_𝑡>0⊂𝐷(𝐴^𝛽),₁₀‖𝐴^𝛽𝑃_𝑠𝑉_𝑞(𝜓)‖𝛽_𝑡(𝑑𝑠)<∞,𝑡>0, (1.6)(1.7) where 𝑉_𝑞(𝜓)∶=∫^∞₀𝑒^−𝑞𝑠𝜓_𝑠𝑑𝑠 and (𝐴^𝛽,𝐷(𝐴^𝛽)) is the associated generator to ℙ^𝛽. Then, 𝜓 is subordinated to a unique ℙ-exit law 𝜑∶=(𝜑_𝑡)_𝑡>0. Moreover, 𝜑 is explicitly given by 𝜑_𝑡:=𝑞𝑃_𝑡𝑉_𝑞(𝜓)−𝐴^𝛽𝑃_𝑡𝑉_𝑞(𝜓). (1.8)

The conditions in Theorem 1.2 are fulfilled for the closed ℙ^𝛽-exit laws 𝜓. This is always the case for the zero-integrable ℙ^𝛽-exit laws in the bounded case.

As application, we consider the holomorphic case and we prove the following result:

Theorem 1.3. We suppose that ℙ is a 𝐶₀-contraction holomorphic semigroup on ℬ and 𝛽 be a Bochner subordinator satisfying ¹₀1𝑠𝛽_𝑡(𝑑𝑠)<∞,𝑡>0. (1.9) Then each zero-integrable ℙ^𝛽-exit law 𝜓 is subordinated to a unique ℙ-exit law 𝜑. Moreover, 𝜑 is given by 𝜑_𝑡=(𝑞+𝑎)𝑃_𝑡𝑉_𝑞(𝜓)−𝑏𝐴𝑃_𝑡𝑉_𝑞(𝜓)+^∞₀(𝑃_𝑠𝑉_𝑞(𝜓)−𝑃_𝑠+𝑡𝑉_𝑞(𝜓))𝜈(𝑑𝑠),𝑡>0, (1.10) 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 about 𝐶₀-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 {𝐴_𝑖∈𝒜∶𝜇(𝐴_𝑖)<∞}_{1≤𝑖≤𝑛} and 𝑋₁,…,𝑋_𝑛∈ℬ such that 𝑋(𝑡)=∑^𝑛_𝑖=1𝑋_𝑖1_𝐴𝑖(𝑡) for all 𝑡>0. A ℬ-valued function 𝑋∶]0,∞[→ℬ is also denoted by 𝑋:=(𝑋_𝑡)_𝑡>0.

In this paper, we consider the integral in Bochner sense for functions 𝑋∶]𝑎,𝑏[⊂]0,∞[→ℬ which are 𝜇-strongly measurable (i.e., there exists a sequence of simple functions 𝑋_𝑛∶]𝑎,𝑏[→ℬ satisfying lim_𝑛→∞‖𝑋_𝑛−𝑋‖=0, 𝜇.a.e.). For such functions 𝑋, it is known that 𝑋 is 𝜇-Bochner integrable if and only if ∫^𝑏_𝑎‖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 𝑇(^𝑏_𝑎𝑋(𝑠)𝜇(𝑑𝑠))=^𝑏_𝑎𝑇𝑋(𝑠)𝜇(𝑑𝑠). (2.1) In the sequel of this work, 𝜇 is omitted whenever it is the Lebesgue measure.

2.1. 𝐶₀-Contraction Semigroups

A 𝐶₀-contraction semigroup on ℬ is a family of linear operators ℙ:=(𝑃_𝑡)_𝑡≥0 on ℬ satisfying 𝑃₀=𝐼, 𝑃_𝑠+𝑡=𝑃_𝑠𝑃_𝑡 for all 𝑠,𝑡≥0, ‖𝑃_𝑡‖≤1 for all 𝑡≥0 and lim_𝑡→0‖𝑃_𝑡𝑓−𝑓‖=0 for all 𝑓∈ℬ.

Let ℙ be a 𝐶₀-contraction semigroup on ℬ. The associated generator 𝐴 of ℙ is defined by 𝐴𝑓:=lim_𝑡→0𝑃_𝑡𝑓−𝑓𝑡 (2.2) 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 resolvent ℛ_𝑞:=(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. Let ℙ be a 𝐶₀-contraction semigroup on ℬ with generator (𝐴,𝐷(𝐴)) and resolvent ℛ:=(ℛ_𝑞)_𝑞>0. (1)For 𝑓∈ℬ and 0≤𝑎<𝑏<∞, the function 𝑡→𝑃_𝑡𝑓 is strongly measurable and the Bochner integral ∫^𝑏_𝑎𝑃_𝑟𝑓𝑑𝑟 is well defined. (2) For each 𝑡≥0 and 𝑓∈𝐷(𝐴), we have 𝑃_𝑡𝑓∈𝐷(𝐴),𝐴𝑃_𝑡𝑓=𝑃_𝑡𝑑𝐴𝑓=𝑃𝑑𝑡^𝑡𝑃𝑓,_𝑡𝑓−𝑓=^𝑡₀𝐴(𝑃_𝑟𝑓)𝑑𝑟. (2.3)(2.4)(3)For each 𝑡,q>0, we have ℛ_𝑞=∫^∞₀𝑒^−𝑞𝑠𝑃_𝑠𝑑𝑠, 𝑃_𝑡ℛ_𝑞=ℛ_𝑞𝑃_𝑡, ℛ_𝑞(ℬ)⊂𝐷(𝐴) and lim_𝑞→∞𝑞ℛ_𝑞𝑢=𝑢,𝑢∈ℬ. (2.5)

Example 2.2. Let ℬ=𝒞_𝑏([0,∞[) be the Banach space of bounded uniformly continuous real-valued functions on [0,∞[ and let 𝑃_𝑡𝑓(𝑥):=𝑓(𝑥+𝑡),𝑡≥0,𝑥≥0,𝑓∈𝒞_𝑏([0,∞[). (2.6) Then ℙ:=(𝑃_𝑡)_𝑡≥0 is 𝐶₀-contraction semigroup with generator 𝐴𝑓=𝑓^′ where 𝐷(𝐴):={𝑓∈𝒞_𝑏([0,∞[)∶𝑓^′exist,𝑓^′∈𝒞_𝑏([0,∞[)}.

3. Exit Equation

3.1. Exit Laws

Definition 3.1. Let ℙ be a 𝐶₀-contraction semigroup on ℬ. A ℙ- exit law is a ℬ-valued function 𝑡∈]0,∞[→𝑢_𝑡∈ℬ which verifies the so-called exit equation: 𝑃_𝑡𝑢_𝑠=𝑢_𝑡+𝑠,𝑠,𝑡>0. (3.1) We point here that a ℙ-exit law 𝑡→𝑢_𝑡 may be also denoted by 𝑢∶=(𝑢_𝑡)_𝑡>0.

Proposition 3.2. Let ℙ be a 𝐶₀-contraction semigroup on ℬ with generator (𝐴,𝐷(𝐴)). (1)For each ℙ-exit law 𝜑∶=(𝜑_𝑡)_𝑡>0, the function 𝑠→𝜑_𝑠 is strongly measurable on ]0,∞[. (2)For each ℎ∈ℬ, the ℬ-valued function 𝑡→𝑃_𝑡ℎ is a ℙ-exit law. It is called a closed exit law. (3)Let ℎ∈ℬ such that (𝑃_𝑡ℎ)_𝑡>0⊂𝐷(𝐴), then 𝑡→𝐴𝑃_𝑡ℎ is a ℙ-exit law. It is said to be differentiable.

Proof. The function 𝑠→𝑃_𝑠ℎ is strongly measurable for each ℎ∈ℬ; then for each 𝑏>0, the function 𝑠→𝜑_𝑠+𝑏=𝑃_𝑠𝜑_𝑏 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

Let ℙ be a 𝐶₀-contraction semigroup on ℬ with generator (𝐴,𝐷(𝐴)). In the sequel, we consider ℙ-exit laws 𝑡→𝜑_𝑡 which are Bochner integrable at 0 (shortly zero-integrable). This is equivalent to ¹₀‖𝜑_𝑠‖𝑑𝑠<∞. (3.2)

Theorem 3.3. Let 𝜑 be a zero-integrable ℙ-exit law. Then 𝜑 is of the form 𝜑_𝑡=𝑞𝑃_𝑡𝑉_𝑞(𝜑)−𝐴𝑃_𝑡𝑉_𝑞(𝜑),𝑡>0, (3.3) where 𝑞>0 and 𝑉_𝑞(𝜑)∶=∫^∞₀𝑒^−𝑞𝑠𝜑_𝑠𝑑𝑠.

Proof. Let 𝑞>0 be fixed. Since (3.2) holds if and only if 𝑠→𝑒^𝜔𝑠𝜑_𝑠 is zero-integrable for all 𝜔∈ℝ, then using (3.1) and (3.2), we have ^∞₀𝑒^−𝑞𝑠‖𝜑_𝑠‖𝑑𝑠=¹₀𝑒^−𝑞𝑠‖𝜑_𝑠‖𝑑𝑠+^∞₁𝑒^−𝑞𝑠‖𝜑_𝑠=‖𝑑𝑠₁₀𝑒^−𝑞𝑠‖𝜑_𝑠‖𝑑𝑠+^∞₁𝑒^−𝑞𝑠‖𝑃_𝑠−1𝜑₁≤‖𝑑𝑠₁₀𝑒^−𝑞𝑠‖𝜑_𝑠‖𝑑𝑠+‖𝜑₁‖^∞₁𝑒^−𝑞𝑠𝑑𝑠<∞. (3.4) This implies that 𝑠→𝑒^−𝑞𝑠𝜑_𝑠 is Bochner integrable on ]0,∞[. Hence, 𝑉_𝑞(𝜑) is well defined and lies in 𝐵. Moreover, by (3.1), (2.1), and (2.3), we get 𝑃_𝑡𝑉_𝑞(𝜑)=^∞₀𝑒^−𝑞𝑠𝜑_𝑠+𝑡𝑑𝑠=^∞₀𝑒^−𝑞𝑠𝑃_𝑠𝜑_𝑡𝑑𝑠=ℛ_𝑞(𝜑_𝑡),𝑡>0. (3.5) Using (3.5) and (3.3) holds since 𝑞𝑃_𝑡𝑉_𝑞(𝜑)−𝐴𝑃_𝑡𝑉_𝑞(𝜑)=𝑞ℛ_𝑞(𝜑_𝑡)−𝐴ℛ_𝑞(𝜑_𝑡)=(𝑞𝐼−𝐴)ℛ_𝑞(𝜑_𝑡)=𝜑_𝑡. (3.6)

Corollary 3.4. Suppose that the generator 𝐴 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. If 𝜑_𝑡=𝑃_𝑡𝑓 for some 𝑓∈ℬ, then 𝑡→𝜑_𝑡 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 𝜑_𝑡=𝑞𝑃_𝑡𝑉_𝑞(𝜑)−𝐴𝑃_𝑡𝑉_𝑞(𝜑),𝑡>0, (3.7) where 𝑉_𝑞(𝜑):=∫^∞₀𝑒^−𝑞𝑠𝜑_𝑠𝑑𝑠 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 𝜑_𝑡=𝑞𝑃_𝑡𝑉_𝑞(𝜑)−𝐴𝑃_𝑡𝑉_𝑞(𝜑)=𝑞𝑃_𝑡𝑉_𝑞(𝜑)−𝑃_𝑡𝐴𝑉_𝑞(𝜑)=𝑃_𝑡(𝑞𝑉_𝑞(𝜑)−𝐴𝑉_𝑞(𝜑)). (3.8) Hence, 𝜑 is a closed exit law.

Remark 3.5. Results similar to Theorem 3.3 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 ∫¹₀‖𝐴𝑃_𝑡𝑢‖𝑑𝑡=∞.
We consider Example 2.2 and we define 𝜑^𝑎_𝑡1(𝑥):=(𝑥+𝑡)^𝑎,𝑎>0,𝑡>0,𝑥≥0. (3.9) It is proved in [10] that 𝜑^𝑎:=(𝜑^𝑎_𝑡)_𝑡>0 is a ℙ-exit law neither closed nor differentiable and ∫¹₀‖𝜑_𝑠‖𝑑𝑠<∞ if and only if 𝑎∈]0,1[.

4. Subordination of 𝐶₀-Contraction Semigroup

4.1. Bochner Subordinator

We consider ℝ endowed with its Borel 𝜎-field. We denote by 𝜀_𝑡 the Dirac measure at point 𝑡. Moreover, for each bounded measure 𝜇 on [0,∞[, ℒ denotes its Laplace transform, that is, ℒ(𝜇)(𝑟):=∫^∞₀exp(−𝑟𝑠)𝜇(𝑑𝑠) for 𝑟>0.

For the following classical notions, we refer the reader to [17–19].

A Bochner subordinator 𝛽∶=(𝛽_𝑡)_𝑡>0 is a vaguely continuous convolution semigroup of subprobability measures on [0,+∞[.

Let 𝛽 be a Bochner subordinator. The associated Bernstein function 𝑓 is defined by the Laplace transform ℒ(𝛽_𝑡)(𝑟)=exp(−𝑡𝑓(𝑟)),𝑟,𝑡>0. (4.1) In fact, (4.1) establishes a one-to-one correspondence between convolution semigroups 𝛽:=(𝛽_𝑡)_𝑡≥0 and Bernstein functions 𝑓 (cf. [18, Theorem 9.8]). In fact, 𝑓 admits the representation 𝑓(𝑟)=𝑎+𝑏𝑟+^∞₀(1−exp(−𝑟𝑠))𝜈(𝑑𝑠),𝑟>0, (4.2) 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. The fractional power subordinator 𝜂^𝛼:=(𝜂^𝛼_𝑡)_𝑡≥0 of index 𝛼∈]0,1[ is defined by its Laplace transform ℒ(𝜂^𝛼_𝑡)(𝑟)=exp(−𝑡𝑟^𝛼) for all 𝑟,𝑡>0.
The Γ-subordinator 𝛾∶=(𝛾_𝑡)_𝑡≥0 is defined by 𝛾_𝑡(𝑑𝑠):=1_]0,∞[1(𝑠)(Γ(𝑡))𝑠^𝑡−1exp(−𝑠)𝑑𝑠,𝑡≥0. (4.3)
The Poisson subordinator 𝜏∶=(𝜏_𝑡)_𝑡≥0 of jump 𝑐>0 is defined by 𝜏_𝑡:=exp(−𝑐𝑡)^∞_𝑛=0(𝑐𝑡)^𝑛𝜀𝑛!^𝑛,𝑡≥0. (4.4)
The Dirac subordinator 𝜀:=(𝜀_𝑡)_𝑡≥0.

4.2. Bochner Subordination

Let ℙ 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 𝑃^𝛽_𝑡𝑢:=^∞₀𝑃_𝑠𝑢𝛽_𝑡(𝑑𝑠). (4.5) Then ℙ^𝛽:=(𝑃^𝛽_𝑡)_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 ℙ^𝛽. In particular, 𝐴^𝛽 is the associated generator and ℛ^𝛽∶=(ℛ^𝛽_𝑞)_𝑞>0 its associated resolvent.

Let 𝐴^𝛽 be the generator of ℙ^𝛽. The following two remarks will be used throughout this paper: 𝐷(𝐴) is a subset of 𝐷(𝐴^𝛽) (cf. [17, page 299]) and 𝐴^𝛽𝑢=−𝑎𝑢+𝑏𝐴𝑢+^∞₀(𝑃_𝑡𝑢−𝑢)𝜈(𝑑𝑡),𝑢∈𝐷(𝐴), (4.6) where 𝑎,𝑏, and 𝜈 are given in (4.2).

Lemma 4.2. There exist some constants 𝐾₁,𝐾₂>0 such that ^∞₀‖𝑃_𝑠𝑢−𝑢‖𝜈(𝑑𝑠)≤𝐾₁‖𝑢‖+𝐾₂‖𝐴𝑢‖,𝑢∈𝐷(𝐴). (4.7)

Proof. Let 𝑢∈𝐷(𝐴). Using the semigroup property and (4.6), we have ^∞₀‖𝑃_𝑠𝑢−𝑢‖𝜈(𝑑𝑠)≤¹₀‖𝑃_𝑠𝑢−𝑢‖𝜈(𝑑𝑠)+^∞₁‖𝑃_𝑠≤𝑢−𝑢‖𝜈(𝑑𝑠)₁₀‖^𝑠₀𝑃_𝑟𝐴𝑢𝑑𝑟‖𝜈(𝑑𝑠)+2‖𝑢‖^∞₁𝜈(𝑑𝑠)≤(¹₀𝑠𝜈(𝑑𝑠))‖𝐴𝑢‖+2‖𝑢‖(^∞₁𝜈(𝑑𝑠)). (4.8) Hence, (4.7) holds for 𝐾₁:=2∫^∞₁𝜈(𝑑𝑠) and 𝐾₂:=∫¹₀𝑠𝜈(𝑑𝑠).

Proposition 4.3. Let ℙ be a 𝐶₀-contraction semigroup on ℬ, 𝛽 a Bochner subordinator, and ℙ^𝛽 be the subordinated to ℙ by means of 𝛽. Then 𝑃_𝑡𝐴^𝛽ℎ=𝐴^𝛽𝑃_𝑡ℎ,𝑡>0,ℎ∈𝐷(𝐴^𝛽). (4.9) In particular, we have 𝑃_𝑡(𝐷(𝐴^𝛽))⊂𝐷(𝐴^𝛽) for all 𝑡>0.

Proof. Step 1. First we suppose that ℎ∈𝐷(𝐴). From Lemma 4.2, ^∞₀‖𝑃_𝑠ℎ−ℎ‖𝜈(𝑑𝑠)<∞. (4.10) So by using (2.1), we get 𝑃_𝑡^∞₀(𝑃_𝑠ℎ−ℎ)𝜈(𝑑𝑠)=^∞₀(𝑃_𝑠𝑃_𝑡ℎ−𝑃_𝑡ℎ)𝜈(𝑑𝑠),𝑡>0. (4.11) Combining (2.3), (4.6), and (4.11), we have 𝑃_𝑡𝐴^𝛽ℎ=−𝑎𝑃_𝑡ℎ+𝑏𝑃_𝑡𝐴ℎ+𝑃_𝑡^∞₀(𝑃_𝑠ℎ−ℎ)𝜈(𝑑𝑠)=−𝑎𝑃_𝑡ℎ+𝑏𝐴𝑃_𝑡ℎ+^∞₀(𝑃_𝑠𝑃_𝑡ℎ−𝑃_𝑡ℎ)𝜈(𝑑𝑠)=𝐴^𝛽𝑃_𝑡ℎ. (4.12)Step 2. Now, we suppose that ℎ∈𝐷(𝐴^𝛽). Let ℛ∶=(ℛ_𝑞)_𝑞>0 be the associated resolvent to ℙ and let 𝑞,𝑡>0. Since ℛ_𝑞(ℬ)⊂𝐷(𝐴), then from Step 1 and Lemma 2.1, we have 𝑃_𝑡𝐴^𝛽ℛ_𝑞ℎ=𝐴^𝛽𝑃_𝑡ℛ_𝑞ℎ=𝐴^𝛽ℛ_𝑞𝑃_𝑡ℎ. (4.13) Hence, by the contraction property and (4.13), we get ‖𝑃_𝑡𝐴^𝛽ℎ−𝐴^𝛽𝑃_𝑡ℎ‖≤‖𝑃_𝑡𝐴^𝛽ℎ−𝑞𝑃_𝑡𝐴^𝛽ℛ_𝑞ℎ‖+‖𝑞𝐴^𝛽ℛ_𝑞𝑃_𝑡ℎ−𝐴^𝛽𝑃_𝑡ℎ‖≤‖𝐴^𝛽ℎ−𝐴^𝛽𝑞ℛ_𝑞ℎ‖+‖𝐴^𝛽𝑞ℛ_𝑞𝑃_𝑡ℎ−𝐴^𝛽𝑃_𝑡ℎ‖. (4.14) Finally, since 𝐴^𝛽 is closed then by (2.5) and by letting 𝑞↑∞, (4.9) holds.

5. Subordinated Exit Law

Let ℙ be a 𝐶₀-contraction semigroup with generator (𝐴,𝐷(𝐴)), let 𝛽 be a Bochner subordinator, and let ℙ^𝛽 be the subordinated to ℙ by means of 𝛽 with generator (𝐴^𝛽,𝐷(𝐴^𝛽)).

Definition 5.1. Let 𝜑 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 𝜑^𝛽∶=(𝜑^𝛽_𝑡)_𝑡>0 is a ℙ^𝛽-exit law which is said to be subordinated to 𝜑 in the Bochner sense by means of 𝛽. Notice that if 𝜑_𝑠=𝑃_𝑠𝑢 for some 𝑢∈ℬ, then (1.5) is just (4.5).

Remark 5.2. (1)Subordination problem: conversely, let 𝜓 be a ℙ^𝛽-exit law, does there exist a ℙ-exit law 𝜑 such that 𝜓 is subordinated to 𝜑? In this paper, we study this problem of ℙ^𝛽-exit laws which are Bochner integrable at 0. (2)The condition of zero-integrability is not necessary. Indeed, if we take 𝛽_𝑡=𝜀_𝑡, then ℙ^𝛽=ℙ and the subordination problem is solved for each ℙ^𝛽-exit law 𝜓 since 𝜓^𝛽=𝜓.

Example 5.3. Let ℬ=𝐿¹(𝜆¹) where 𝜆¹ is the Lebesgue measure on ℝ and let ℙ:=(𝑃_𝑡)_𝑡>0 be the left uniform translation on ℬ, that is, 𝑃_𝑡𝑢:=𝜀_𝑡∗𝑢 for 𝑡≥0 and 𝑢∈ℬ. It can be seen that each ℙ-exit law 𝜑 is closed, that is, of the form 𝜑_𝑡=𝑃_𝑡𝑢 for some fixed 𝑢∈ℬ.
On the other hand, let 𝜂^1/2:=(𝜂_𝑡^1/2)_𝑡>0 be the fractional powers subordinator of index 1/2. From [18, page 71], 𝜂_𝑡^1/2 is absolutely continuous with density 𝐺_𝑡(𝑠)=(1/√4𝜋)𝑡𝑠^−3/2exp(−𝑡²/4𝑠) for all 𝑡,𝑠>0. The extension of 𝐺_𝑡 by 0 on ℝ is denoted by 𝐺_𝑡. So, 𝑃^𝜂1/2_𝑡𝑢=𝐺_𝑡∗𝑢 for all 𝑡>0 and 𝑢∈ℬ. In particular, 𝑃^𝜂1/2_𝑡𝐺_𝑠(𝑥)=𝐺_𝑡∗𝐺_𝑠(𝑥)=𝐺_𝑠+𝑡(𝑥),𝑠,𝑡>0,𝑥∈ℝ, (5.1) by the convolution semigroup property of 𝜂^1/2. Therefore, the family 𝐺∶=(𝐺_𝑡)_𝑡>0 is a ℙ^𝜂1/2-exit law. Moreover, 𝐺 is zero-integrable (By using the change of variables 𝑦=𝑡⁻²𝑠, we have ‖𝐺_𝑡‖=‖𝐺₁‖ for all 𝑡>0). But there exists no 𝑢∈ℬ such that 𝐺_𝑡=𝐺_𝑡∗𝑢 for each 𝑡>0.
Hence, not every zero-integrable ℙ^𝜂1/2-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 ℙ^𝜂1/2-exit law.

Remark 5.4. Example 5.3 proves that we need to add some condition in order to solve the subordination problem. Next, we will suppose that 𝜓:=(𝜓_𝑡)_𝑡>0 satisfies the following conditions. (H):There exists a constant 𝑞>0 such as(1.6) and (1.7)where 𝑉_𝑞(𝜓)∶=∫^∞₀𝑒^−𝑞𝑠𝜓_𝑠𝑑𝑠.

Theorem 5.5. Let ℙ be a 𝐶₀-contraction semigroup on ℬ and let 𝛽 be a Bochner subordinator. Suppose that 𝜓 is a zero-integrable ℙ^𝛽-exit law satisfying (H). Then 𝜓 is subordinated to a unique ℙ-exit law 𝜑∶=(𝜑_𝑡)_𝑡>0. Moreover, 𝜑 is explicitly given by 𝜑_𝑡:=𝑞𝑃_𝑡𝑉_𝑞(𝜓)−𝐴^𝛽𝑃_𝑡𝑉_𝑞(𝜓),𝑡>0, (5.2) where 𝑞 and 𝑉_𝑞 are given by (H).

Proof. Since (𝑃_𝑡𝑉_𝑞(𝜓))_𝑡>0⊂𝐷(𝐴^𝛽), then the family 𝜑∶=(𝜑_𝑡)_𝑡>0 defined by (5.2) is well defined and lies in ℬ. Moreover, by (5.2) and (4.9), we get 𝑃_𝑠𝜑_𝑡=𝑃_𝑠(𝑞𝑃_𝑡𝑉_𝑞(𝜓)−𝐴^𝛽𝑃_𝑡𝑉_𝑞(𝜓))=𝑞𝑃_𝑠+𝑡𝑉_𝑞(𝜓)−𝐴^𝛽𝑃_𝑠+𝑡𝑉_𝑞(𝜓)=𝜑_𝑠+𝑡, (5.3) which implies that 𝜑∶=(𝜑_𝑡)_𝑡>0 is a ℙ-exit law. Now using (1.6) and (4.9), we have ^∞₁‖𝐴^𝛽𝑃_𝑠𝑉_𝑞(𝜓)‖𝛽_𝑡(𝑑𝑠)=^∞₁‖𝐴^𝛽𝑃_𝑠−1𝑃₁𝑉_𝑞(𝜓)‖𝛽_𝑡=(𝑑𝑠)_∞1‖𝑃_𝑠−1𝐴^𝛽𝑃₁𝑉_𝑞(𝜓)‖𝛽_𝑡≤(𝑑𝑠)_∞1‖𝐴^𝛽𝑃₁𝑉_𝑞(𝜓)‖𝛽_𝑡(𝑑𝑠)≤𝛽_𝑡([1,∞[)‖𝐴^𝛽𝑃₁𝑉_𝑞(𝜓)‖<∞, (5.4) and by (1.7), we conclude that ^∞₀‖𝐴^𝛽𝑃_𝑠𝑉_𝑞(𝜓)‖𝛽_𝑡(𝑑𝑠)<∞,𝑡>0. (5.5) Therefore, from (5.5), we have ^∞₀‖𝜑_𝑠‖𝛽_𝑡(𝑑𝑠)=^∞₀‖𝑞𝑃_𝑠𝑉_𝑞(𝜓)−𝐴^𝛽𝑃_𝑠𝑉_𝑞(𝜓)‖𝛽_𝑡(𝑑𝑠)≤𝑞^∞₀‖𝑃_𝑠𝑉_𝑞(𝜓)‖𝛽_𝑡(𝑑𝑠)+^∞₀‖𝐴^𝛽𝑃_𝑠𝑉_𝑞(𝜓)‖𝛽_𝑡(𝑑𝑠)≤𝑞𝛽_𝑡([0,∞[)‖𝑉_𝑞(𝜓)‖+^∞₀‖𝐴^𝛽𝑃_𝑠𝑉_𝑞(𝜓)‖𝛽_𝑡(𝑑𝑠)<∞. (5.6) Hence, the subordinated 𝜑^𝛽:=(𝜑^𝛽_𝑡)_𝑡>0 defined by (1.5) is well defined.
On the other hand, for all 𝑠,𝑡>0, we have 𝑃_𝑠𝜑=^𝛽_𝑡^(1.5)𝑃_𝑠^∞₀𝜑_𝑟𝛽_𝑡(𝑑𝑟)=^(2.1)∞₀𝑃_𝑠𝜑_𝑟𝛽_𝑡(𝑑𝑟)_(3.1)=^∞₀𝜑_𝑟+𝑡𝛽_𝑡(𝑑𝑟)=^∞₀𝑃_𝑟𝜑_𝑠𝛽_𝑡(𝑑𝑟)=𝑃^𝛽_𝑡𝜑=_𝑠⁽⁸⁾𝑃^𝛽_𝑡(𝑃_𝑠𝑉_𝑞(𝜓)−𝐴^𝛽𝑃_𝑠𝑉_𝑞(𝜓))_(2.3)=𝑃_𝑠𝑃^𝛽_𝑡𝑉_𝑞(𝜓)−𝐴^𝛽𝑃_𝑠𝑃^𝛽_𝑡𝑉_𝑞(𝜓)=⁽¹⁸⁾𝑃_𝑠(𝑃^𝛽_𝑡𝑉_𝑞(𝜓)−𝐴^𝛽𝑃^𝛽_𝑡𝑉_𝑞(𝜓))=𝑃_𝑠𝜓_𝑡 (5.7) by using Theorem 3.3 since 𝜓 is Bochner integrable at 0. Therefore, 𝜓_𝑡=𝑃^𝛽_𝑡/2𝜓_𝑡/2=^∞₀𝑃_𝑠𝜓_𝑡/2𝛽_𝑡/2(𝑑𝑠)=^∞₀𝑃_𝑠𝜑^𝛽_𝑡/2𝛽_𝑡/2(𝑑𝑠)=𝑃^𝛽_𝑡/2𝜑^𝛽_𝑡/2=𝜑^𝛽_𝑡, (5.8) which implies that 𝜓=𝜑^𝛽.
Finally, let us prove the uniqueness: Let 𝜙∶=(𝜙_𝑡)_𝑡>0 be a ℙ-exit law such that 𝜓=𝜙^𝛽. Since for all 𝑠,𝑡>0, we have 𝑃_𝑠𝜙^𝛽_𝑡=𝑃_𝑠^∞₀𝜙_𝑟𝛽_𝑡(𝑑𝑟)=^(2.1)∞₀𝑃_𝑠𝜙_𝑟𝛽_𝑡(𝑑𝑟)=^(3.1)∞₀𝜙_𝑟+𝑡𝛽_𝑡=(𝑑𝑟)_∞0𝑃_𝑟𝜙_𝑠𝛽_𝑡(𝑑𝑟)=^(4.4)𝑃^𝛽_𝑡𝜙_𝑠, (5.9) then for all 𝑡>0, 𝑃_𝑡𝑉_𝑞(𝜓)=^∞₀𝑒^−𝑠𝑞𝑃_𝑡𝜙^𝛽_𝑠𝑑𝑠=^∞₀𝑒^−𝑠𝑞𝑃^𝛽_𝑠𝜙_𝑡𝑑𝑠=𝑅^𝛽_𝑞(𝜙_𝑡). (5.10) Therefore, from (5.2), we have 𝜑_𝑡=𝑞𝑅^𝛽_𝑞(𝜙_𝑡)−𝐴^𝛽𝑅^𝛽_𝑞(𝜙_𝑡)=(𝑞𝐼−𝐴^𝛽)𝑅^𝛽_𝑞(𝜙_𝑡)=𝜙_𝑡,𝑡>0. (5.11)

Remark 5.6. In addition, if 𝑃_𝑡(𝑉_𝑞(𝜓))_𝑡>0⊂𝐷(𝐴), then from (4.6), 𝜑 is of the form 𝜑_𝑡=(𝑞+𝑎)𝑃_𝑡𝑉_𝑞(𝜓)−𝑏𝐴𝑃_𝑡𝑉_𝑞(𝜓)+^∞₀(𝑃_𝑠𝑉_𝑞(𝜓)−𝑃_𝑠+𝑡𝑉_𝑞(𝜓))𝜈(𝑑𝑠), (5.12) where 𝑎, 𝑏, and 𝜈 are the parameters associated to 𝛽.
In particular, this is the case of each 𝜓 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(𝐴^𝛽).
If 𝜓 satisfies (H) for some 𝑞>0, then it satisfies (H) for all 𝜀>0. Indeed, by Theorem 5.5, 𝜓 is subordinated to some ℙ-exit law 𝜑. Moreover, exactly as (5.10), we have 𝑃_𝑡𝑉_𝜀(𝜓)=^∞₀𝑒^−𝑠𝜀𝑃_𝑡𝜓_𝑠𝑑𝑠=^∞₀𝑒^−𝑠𝜀𝑃^𝛽_𝑠𝜑_𝑡𝑑𝑠=ℛ_𝜀(𝜑_𝑡),𝑡,𝜀>0. (5.13) So, 𝑃_𝑡𝑉_𝜀(𝜓)∈𝐷(𝐴^𝛽) and 𝐴^𝛽𝑃_𝑡𝑉_𝜀(𝜓)=𝐴^𝛽ℛ^𝛽_𝜀(𝜑_𝑡)=(𝜀𝐼−𝐴^𝛽)ℛ^𝛽_𝜀(𝜑_𝑡)−𝜀ℛ^𝛽_𝜀(𝜑_𝑡)=𝜑_𝑡−𝜀𝑃_𝑡𝑉_𝜀(𝜓). (5.14) Therefore, from the proof of Theorem 5.5, we have ¹₀‖𝐴^𝛽𝑃_𝑠𝑉_𝑞(𝜓)‖𝛽_𝑡(𝑑𝑠)=¹₀‖𝜑_𝑠−𝜀𝑃_𝑠𝑉_𝜀(𝜓)‖𝛽_𝑡≤(𝑑𝑠)₁₀(‖𝜑_𝑠‖+‖𝜀𝑃_𝑠𝑉_𝜀(𝜓)‖)𝛽_𝑡≤(𝑑𝑠)₁₀‖𝜑_𝑠‖𝛽_𝑡(𝑑𝑠)+𝜀‖𝑉_𝜀(𝜓)‖¹₀𝛽_𝑡(𝑑𝑠). (5.15) Hence, (1.6) and (1.7) hold for each 𝜀>0.
The conditions of Theorem 5.5 are fulfilled for the natural example of ℙ^𝛽-exit law. Indeed, we have the following result.

Corollary 5.7. Each closed ℙ^𝛽-exit law 𝜓, that is, 𝜓_𝑡=𝑃^𝛽_𝑡𝑢 for some 𝑢∈ℬ, is subordinated to unique exit law ℙ-exit law 𝜑∶=(𝜑_𝑡)_𝑡>0. Moreover, 𝜑 is explicitly given by 𝜑_𝑡=𝑃_𝑡𝑢 for all 𝑡>0.

Proof. Let 𝜓 be a closed ℙ^𝛽-exit law. It is easy to see that 𝜓 is zero-integrable. Moreover, for all 𝑞>0, we have 𝑉_𝑞(𝜓):=^∞₀𝑒^−𝑞𝑠𝜓_𝑠𝑑𝑠=^∞₀𝑒^−𝑞𝑠𝑃^𝛽_𝑠𝑢𝑑𝑠=ℛ^𝛽_𝑞𝑃(𝑢),_𝑡𝑉_𝑞(𝑢)=^∞₀𝑒^−𝑞𝑠𝑃^𝛽_𝑠𝑃_𝑡𝑢𝑑𝑠=ℛ^𝛽_𝑞(𝑃_𝑡𝑢),𝑡>0, (5.16) which implies that 𝑉_𝑞(𝜓)∈𝐷(𝐴^𝛽) and (𝑃_𝑡𝑉_𝑞(𝜓))_𝑡>0⊂𝐷(𝐴^𝛽). Moreover, ¹₀‖𝐴^𝛽𝑃_𝑠𝑉_𝑞(𝜓)‖𝛽_𝑡(𝑑𝑠)=¹₀‖𝑃_𝑠𝐴^𝛽𝑉_𝑞(𝜓)‖𝛽_𝑡≤(𝑑𝑠)₁₀‖𝐴^𝛽𝑉_𝑞(𝜓)‖𝛽_𝑡(𝑑𝑠)≤‖𝐴^𝛽𝑉_𝑞(𝜓)‖. (5.17) So, 𝜓 satisfies (H) and by Theorem 5.5, we get 𝜑_𝑡=(𝑞𝐼−𝐴^𝛽)𝑃_𝑡𝑉_𝑞(𝜓)=(𝑞𝐼−𝐴^𝛽)ℛ^𝛽(𝑃_𝑡𝑢))=𝑃_𝑡𝑢,𝑡>0. (5.18)

Corollary 5.8. Suppose that 𝐴 is bounded or 𝑓 is bounded, then Theorem 5.5 may be applied for each zero-integrable ℙ^𝛽-exit law.

Proof. According to [17, Theorem 4.3.8, page 303], 𝐴^𝛽 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. Let ℙ be a 𝐶₀-contraction semigroup on ℬ. ℙ is said to be holomorphic if there exists a holomorphic extension 𝑧→𝑃_𝑧 to 𝑆:={𝑧∈ℂ^∗∶|arg𝑧|<𝜃};0<𝜃<𝜋/2.

Remark 6.2 (Construction by Bochner subordination). Let 𝐹 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=(𝛽_𝑡)_𝑡>0 is said to be of type Carasso-Kato if:
The associated parameters 𝑎=𝑏=0 and the mapping 𝑡→𝛽_𝑡 is continuously differentiable from ]0,∞[ to 𝐹 such that ‖(𝜕/𝜕𝑡)𝛽_𝑡‖_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∶=ℚ^𝛽 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. Let ℙ be a 𝐶₀-contraction holomorphic semigroup on ℬ with generator (𝐴,𝐷(𝐴)). Then there exists a constant 𝑡>0 such that ‖𝐴𝑃_𝑡ℎ‖≤𝐶‖ℎ‖𝑡as𝑡→0. (6.1)
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 𝑃_𝑡 is contained in 𝐷(𝐴), hence also in 𝐷(𝐴^𝛽).

Proposition 6.4. Let 𝛽 be a Bochner subordinator such as (1.9)Then, ¹₀‖𝐴^𝛽𝑃_𝑠ℎ‖𝛽_𝑡(𝑑𝑠)<∞,𝑡>0,ℎ∈ℬ. (6.2)

Proof. Let 𝑡>0 and ℎ∈ℬ. Since 𝑃_𝑡ℎ∈𝐷(𝐴), then from (4.6), we have 𝐴^𝛽𝑃_𝑡ℎ=−𝑎𝑃_𝑡ℎ+𝑏𝐴𝑃_𝑡ℎ+^∞₀(𝑃_𝑠𝑃_𝑡ℎ−𝑃_𝑡ℎ)𝜈(𝑑𝑠), (6.3) where 𝑎,𝑏, and 𝜈 are given in (4.2). By using Lemma 4.2, we have ^∞₀‖𝑃_𝑠+𝑡ℎ−𝑃_𝑡ℎ‖𝜈(𝑑𝑠)≤𝐾₁‖𝑃_𝑡ℎ‖+𝐾₂‖𝐴𝑃_𝑡ℎ‖≤𝐾₁‖ℎ‖+𝐾₂‖𝐴𝑃_𝑡ℎ‖ (6.4) for some 𝐾₁,𝐾₂>0. Therefore, by (6.3), we conclude that ‖𝐴^𝛽𝑃_𝑡ℎ‖≤𝐾₃‖ℎ‖+𝐾₄‖𝐴𝑃_𝑡ℎ‖, (6.5) where 𝐾₃:=𝑎+𝐾₁ and 𝐾₄∶=𝑏+𝐾₂. Moreover, combining Lemma 6.3, (1.9), and (6.5), we have ¹₀‖𝐴^𝛽𝑃_𝑠ℎ‖𝛽_𝑡(𝑑𝑠)≤𝐾₃‖ℎ‖¹₀𝛽_𝑡(𝑑𝑠)+𝐾₄¹₀‖𝐴𝑃_𝑠ℎ‖𝛽_𝑡(𝑑𝑠)≤𝐾₃‖ℎ‖+𝐾₄𝐶‖ℎ‖¹₀1𝑠𝛽_𝑡(𝑑𝑠)<∞. (6.6) Hence, (6.2) holds.

Remark 6.5. The condition (1.9) holds as soon as the associated Bernstein function 𝑓 satisfies ^∞₀𝑒^{−𝑡𝑓(𝑟)}𝑑𝑟<∞,𝑡>0. (6.7) Indeed by using the Fubini's theorem, we have ¹₀1𝑠𝛽_𝑡(𝑑𝑠)≤^∞₀(^∞₀𝑒^−𝑠𝑟𝑑𝑟)𝛽_𝑡(𝑑𝑠)≤^∞₀(^∞₀𝑒^−𝑠𝑟𝛽_𝑡≤(𝑑𝑠))𝑑𝑟_∞0ℒ(𝛽_𝑡)(𝑟)𝑑𝑟≤^∞₀𝑒^{−𝑡𝑓(𝑟)}𝑑𝑟. (6.8) Hence, (1.9) holds for the Dirac and the fractional power subordinators.

Theorem 6.6. Let ℙ be a 𝐶₀-contraction holomorphic semigroup on ℬ and let 𝛽 be a Bochner subordinator satisfying (1.9). Then each zero-integrable ℙ^𝛽-exit law 𝜓 is subordinated to a unique ℙ-exit law 𝜑∶=(𝜑_𝑡)_𝑡>0. Moreover, 𝜑 is explicitly given by 𝜑_𝑡=(𝑞+𝑎)𝑃_𝑡𝑉_𝑞(𝜓)−𝑏𝐴𝑃_𝑡𝑉_𝑞(𝜓)+^∞₀(𝑃_𝑠𝑉_𝑞(𝜓)−𝑃_𝑠+𝑡𝑉_𝑞(𝜓))𝜈(𝑑𝑠),𝑡>0, (6.9) where 𝑎, 𝑏, and 𝜈 are the parameters of 𝛽 and 𝑉_𝑞(𝜓):=∫^∞₀𝑒^−𝑞𝑠𝜓_𝑠𝑑𝑠 for some 𝑞>0.

Proof. Let 𝑞>0. Since 𝑉_𝑞(𝜓):=∫^∞₀𝑒^−𝑞𝑠𝜓_𝑠𝑑𝑠∈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).

Acknowledgment

The authors want to thank professor Mohamed Hmissi for many helpful discussions on these and related topics.