ON S-SUBORDINATION AND APPLICATIONS TO ENTRANCE LAWS
MOHAMED HMISSI and KLAUS JANSSEN
Let U and V be two sub-Markovian resolvents of kernels such that V is S- subordinated toU, i.e. eachU-excessive function isV-excessive. Based on results of J. Steens, we prove that the energy functional of Vis partially induced by the one of Uby some sort of projection. As application, we solve the so called Bochner subordination problem for entrance laws in complete generality.
AMS 2010 Subject Classication: Primary: 60J35. Secondary: 60J40, 60J45, 31D05.
Key words: resolvent of kernels, semigroup of kernels, excessive function, exces- sive measure, energy functional, S-subordination, Bochner subordi- nation, entrance law, exit law.
1. INTRODUCTION
Let U := (Up) and V:= (Vp) be two transient sub-Markovian resolvents of kernels on a Lusin measurable space (E,E). Vis said to be S-subordinated to U if each U-excessive function is V-excessive. This notion, introduced in [22], generalizes classical subordinations: Subordination by killing (Kac subor- dination), subordination in the sense of Bochner, time change, etc. For Levy processes, S-subordination is equivalent to the subordination in the wide sense introduced in [31].
The aim of this paper is to study some consequences of theS-subordination for the energy functional. As application, we solve the so-called Bochner sub- ordination problem for entrance laws.
Let L(resp. M) be the energy functional dened by U(resp. V). Under general hypotheses onU, we prove rst that, ifVisS-subordinated toUthen, for each suciently nite V-excessive measure m, there exists a unique U- excessive measure l:= Υ(m) such that
M(m, u) = L(l, u)
REV. ROUMAINE MATH. PURES APPL. 59 (2014), 1, 105121
for each U-excessive function u (c.f. Theorem 6). This result is in fact a consequence of a representation Theorem of J. Steens [34]. Moreover, if the cone ofU-excessive functions is rich enough and is stable byVp, p >0, we prove the injectivity of the operatorΥ(cf. Proposition 9).
Suppose thatUis the resolvent of a measurable sub-Markovian semigroup P. A P-entrance law is a family (νt)t>0 of σ-nite measures on E satisfying νtPs=νs+t. LetPβ be the subordinated ofP by means of a Bochner subordi- nator β, i.e. Ptβ =R∞
0 Psβt(ds). As application we prove, under some natural niteness assumptions that each Pβ-entrance law (µt)t>0 is subordinated to a unique P-entrance law(νt)t>0, by means of β, i.e.
µt= Z ∞
0
νsβt(ds), t >0
(cf. Theorem 12). Using strong duality, we deduce a similar result for Pβ- exit laws whenever P is absolutely continuous with respect to a xed σ-nite measure (cf. Corollary 15).
2. REPRESENTATION BY THE ENERGY FUNCTIONAL In the following, we refer to [5, 9, 13, 16, 30] for notations and results in potential theory associated with resolvents and semigroups.
2.1. Excessive structure
In the sequel,E denotes a Lusin measurable space equipped with its Borel σ-eldE. We denote bypEthe cone of positive numerical measurable functions on E and by Mea the cone of σ-nite positive measures on E. For µ ∈ Mea and f ∈pE, we use sometimes the notationµ(f) :=R
f(x)µ(dx).
A kernel onEis a mappingK :E×E →[0,∞]such that (a)x→K(x, B) is measurable for each B ∈ E, (b)B →K(x, B) is a (positive) measure on E, for eachx∈E. In this case,Kacts to the right onpE and to the left on Mea by Kf(x) := R
f(y)K(x, dy) for f ∈ pE, x∈ E and mK(B) := R
K(x, B)m(dx) for m ∈ Mea, B ∈ E (note that mK is not necessarily σ-nite). Denote by S(K) :={u∈pE :Ku≤u}. K is said to be sub-Markovian if 1∈ S(K).
A (sub-Markovian) resolvent on E is a family U:= (Up)p≥0 of kernels on (E,E)such thatpUpis sub-Markovian,UpUq=UqUpandUp =Uq+(q−p)UpUq
for q > p≥0. We denote by U :=U0 := supUp the potential kernel ofU.
In this paper, we will suppose mostly that the kernel U is proper, i.e.
there exists a strictly positive function f0 ∈pE such thatU f0<∞.
Remember that a set N ∈ E is called of potential zero if U1N = 0. We say that some property holds U-a.e. if this property holds except on a set of potential zero.
Let Ube a resolvent onE with proper potential kernel U.
A U-a.e. nite function u ∈ pE is called U-excessive if pUpu ↑ u as p ↑ ∞. If in addition, pUpu ↓ 0, U-a.e. as p ↓ 0, u is said to be U-purely excessive. If pUpu =u for each p > 0 thenu is calledU-invariant. Note that for f ∈pE, the functionU f is U-purely excessive whenever it is U-a.e. nite.
We denote respectively by S(U),S0(U),I(U) the convex cone of U-excessive, U-purely excessive, U-invariant functions.
In the same way, a σ-nite measure m ∈ Mea is called U-excessive if pmUp ↑ m as p ↑ ∞. If in addition, pmUp(f) ↓ 0 for some strictly positive f ∈pE asp↓0,mis said to beU-purely excessive. IfpmUp =mfor eachp >0 thenmis calledU-invariant. We denote by Exc(U),Pur(U),Inv(U) the convex cone of U-excessive,U-purely excessive,U-invariant measures, respectively.
Given µ∈Mea, the measure µU is U-purely excessive if it isσ-nite. In this case,µU is said to be the potential of the measureµ. We denote by Pot(U) the convex cone of potential measures.
Let us recall the Riesz decomposition (cf. [13], XII, 37)
(1) Exc(U) = Pur(U) M
Inv(U) and Hunt's approximation theorem (cf. [13], XII, 38) namely, (2)
Exc(U) ={supµnU ∈Mea: (µn)⊂Mea, µn bounded, (µnU) is increasing} Remark 1. Often (but not always) a resolvent U on E is dened from a measurable semigroup Pon E, i.e.
(3) Up =
Z ∞
0
exp(−pt)Ptdt, p≥0,
where P := (Pt)t>0 is a family of sub-Markovian kernels on (E,E) such that PsPt=Ps+t for s, t >0 and (t, x)→Ptf(x) is measurable for eachf ∈pE.
The present paper is based on a representation result of J. Steens [34], which we present in the following.
2.2. Representation theorem of Steens
For a resolventUwith proper potential kernelU, we recall rst the notion of energy functional introduced by Meyer (cf. [13] XII, 39). Namely the function L: Exc(U)× S(U)→[0,∞]dened by
(4) L(l, u) := sup{µ(u) :µU ∈Pot(U), µU ≤l}·
We use freely the basic properties of this functional as developed in [13]
(cf. also [5, 16]).
The following fundamental result is proved in ([34], Corollary 2.5) Theorem 2. Let U be a resolvent on E, with proper kernel U and such that U1>0.
1. For each m∈Exc(U), the mapping v →L(m, v) is a positive increasing linear functional on S(U) which is continuous from below and satises L(m, v0)<∞ for some strictly positivev0 ∈ S(P).
2. Conversely, every positive increasing linear functional ϕ:S(U)→[0,∞]
which is continuous from below and satises ϕ(v0)<∞ for some strictly positive v0 ∈ S(U), is given by L(m, .) for some unique m∈Exc(U).
3. STABILITY BY S-SUBORDINATION
Let U and V be two resolvents on E. We say that V is S-subordinated to U if each U-excessive function is V-excessive, i.e. S(U) ⊂ S(V). This notion is introduced recently in [22]. As a rst fundamental application, it is proved in [22] that S-subordination preserves unicity of charges and solidness of potentials. Combining this with a characterization of J. Steens [33], it is deduced in [22] that, if VisS-subordinated to a resolventU of a right process thenV itself is associated with a right process (cf. more comments about this result at the end of the following Example 4).
3.1. Examples
1. A trivial example: Let V:= (Vp) be the resolvent given by
(5) Vp:= c
1 +pcI, p≥0,
for the identity kernel I on E and an arbitrary strictly positive and nite function c ∈pE. Then V is S-subordinated to every resolvent U with proper potential kernel onE.
2. Another trivial example: Let U be a resolvent on E, let p0 ∈]0,∞[
and let V := (Vp) be the resolvent dened by Vp := Up+p0, p ≥0. Then V is S-subordinated to U.
3. Subordination by killing: Let U and V be two resolvents on E such that Vp ≤Up for p ≥0, U1>0 and V1>0. Then V is S-subordinated toU (cf. [5, 9, 15, 30]).
a) If U is associated with a right process (Xt), then it is known that V is associated with a right process (Yt) obtained by killing (Xt) by some nice multiplicative functional.
It is well known, that under some regularity assumptions we haveVp ≤Up for allp≥0if and only if we haveU =V+P U for some subordination operator P (cf. [5, 10]).
A special case is the Kac-subordination whereP =Uh(h.)for some su- ciently nite strictly positiveh and V =Uh (c.f. [2, 5, 10]).
If there is a nice Markov process (Xt) associated with U, then the sub- ordination operator P is determined by some multiplicative functional (Mt) (cf. [5, 9, 30]), and the associated semigroup is given byQtf(x) =Ex(f◦XtMt) for t≥0, x∈E and f ∈pE (E denotes the usual expectation).
In the special case of Kac-subordination we have Mt = exp(−Rt
0h ◦ Xsds), t≥0. Example 2 is the special case where h is constant.
b) It may happen that Vis associated with a right process, butUis only associated with a right process on a bigger state spaceEˆ such thatEˆ\E is of potential zero forU. Such examples appear in the context of full-superharmonic structures (cf. [26, 27, 29]). Typical examples for this are (killed) Brownian motion and (killed) reected Brownian motion.
4. Subordination in the sense of Bochner: Let β := (βt)t≥0 be a Bochner subordinator, that is β is a convolution semigroup of subprobability measures on R+ such thatβt→ε0 weakly fort→0.
For the following notions, we will refer to [3] and [19]. βis said to be of (K)- type if the associated potential measureκ:=R∞
0 βtdtis absolutely continuous.
In this case we may write κ(dt) = g(t)·dt where g :]0,∞[→ R is completely monotone (i.e. g is a C∞-function and (−1)ng(n) ≥0 for all integers n∈N).
Moreover,g is integrable at 0 and (6) κp(dt) :=
Z ∞
0
exp(−pt)βtdt=gp(t)·dt, p >0,
where gp is also a completely monotone and integrable function on]0,∞[, for each p >0. Therefore, gp is the Laplace transform of a positive measureρp on ]0,∞[such that
(7) ρp({0}) = 0andZ ∞ 0
1
sρp(ds) ≤ 1/p, p >0.
Now, letU be a resolvent onE. The familyUβ := (Upβ)p>0 dened by
(8) Upβ :=
Z ∞
0
Usρp(ds), p >0
is a resolvent on E with proper potential kernel Uβ (cf. [19]). Uβ is the
subordinated resolvent to Uby means ofβ in the Bochner sense. Moreover, it can be veried using Fatou's Lemma thatUβ isS-subordinated to U.
Suppose thatUis the resolvent of a semigroupP onE. For any Bochner subordinatorβ (not necessary of (K)-type) we may denePβ the subordinated semigroup of Pby means ofβ in the Bochner sense, namely
(9) Ptβf :=
Z ∞
0
Psf βt(ds), t≥0, f ∈pE
(cf. [3] and related references therein). In this case Uβ is always dened (as resolvent of Pβ).
If U is associated with a right process (Xt) and if τ := (τt) is an in- dependent right process associated with β on R+, then (Xτt) is a right pro- cess associated with Uβ under some regularity assumptions according to [11]
and [20].
This result is recently proved in [28] (Theorem 3.3) for general Bochner subordination without any regularity assumption. In fact, for Bochner sub- ordination, Theorem 3.3 of [28] improves and completes Theorem 3 of [22] by using essentially the same arguments namely Steens's characterization of right processes in [33].
Notice that the rst and second examples are particular cases of subordi- nation in the Bochner sense.
Repeating the above subordination procedures gives new S-subordinated semigroups provided the corresponding potential kernels are proper. Moreover, one may piece together dierent subordinations in dierent subsets of E to obtain newS-subordinations.
5. Time change: Let U and V be two resolvents on E with proper po- tential kernels associated with right processes (Xt) and (Yt). IfS(U) =S(V), then hitting distributions for both processes coincide, and consequently (Xt) is obtained from (Yt) by a time change determined by some strict continuous additive functional (cf. [9]).
In particular, if U is a resolvent and if 0 < h < ∞ in pE is suciently nite, then V : f → U(hf) is the potential kernel of a resolvent V such that S(U) =S(V).
Notice that time changes of Levy processes arise in many interesting ap- plications in option pricing (cf. [12]).
6. Subordination in the wide sense for Levy processes: LetUandVbe two resolvents onE =Rdwith proper potential kernels associated with (transient) Levy processes (Xt) and (Yt). Let (µt)t≥0 and (νt)t≥0 be the associated con- volution semigroups and let µ:=R∞
0 µtdtand ν :=R∞
0 νtdtbe the associated potential measures. Moreover, let S and T be the Levy generator of (µt)t≥0
and (νt)t≥0. Following [31], S(U) ⊂ S(V) if and only if one of the following equivalent assertions holds
1. Exc(U)⊂Exc(V).
2. µis a quotient of ν, i.e. there existsκ∈Mea such thatµ=ν∗κ. 3. S is a quotient ofT, i.e. there existsκ∈Mea which tends to0at innity
such thatS =T∗κ.
4. (Yt) is subordinated to (Xt) in the wide sense (cf. [31] for the detailed denition).
7. Subordinated killed and killed subordinated Brownian motions. For this example, we refer to [32] and related references therein.
Let (Xt) be a d-dimensional Brownian motion in Rd, let(τt) be an α/2- stable independent subordinator starting at zero, 0 < α < 2 and let D be a domain in Rd. Let (Yt) be the subordinated process (in the Bochner sense) to (Xt) with respect to (τt), that is Yt = Xτt, t≥ 0. Then the process (YtD) obtained by killing (Yt) upon exiting D, is the so-called killed subordinated Brownian motion in D.
On the other hand, the subordinated killed Brownian motion (ZtD) in D is obtained as follows: We rst kill the Brownian motion (Xt) at the rst exit time of(Xt)fromDto obtain(XtD)and then we subordinate the process(XtD) using the α/2-stable subordinator (τt).
Let (t, x, y)7→q(t, x, y) (resp. (t, x, y)7→r(t, x, y)) the transition density of the process(YtD) (resp. (ZtD)). Following [32],
(10) r(t, x, y)≤q(t, x, y), t >0, x, y∈D
In particular, (ZtD) is S-subordinated to(YtD). This is a special case of Exemple 3.
8. A dierent example: We have no idea if there is a general description of S-subordination. As an example which is completely dierent from the above examples (which are all related to some killing and time change) we mention the following resolvents onE := [0,1[: Vis the resolvent of uniform motion to the right on E, andU is the resolvent associated with Brownian motion onE, reected at zero and killed at the exit at 1. Then S(V) is the convex cone of all l.s.c. decreasing and positive functions on E whereas S(U) = {u ∈ S(V) : u is concave}.
3.2. Two useful results
LetUbe a resolvent on(E,E) with proper kernelU. The resolventU(or its potential kernel U) is said to satisfy the principle of unicity of charges, if for some µ, ν ∈Mea,µU =νU ∈Pot(U) thenµ=ν.
The following result is proved in [22] (cf. Theorem 1).
Lemma 3. Let Uand V be two resolvent on (E,E) with proper kernelsU andVand such that VisS-subordinated to U. Then the principle of unicity of charges holds for V when it holds for U.
Remark 4. Let U be a resolvent on (E,E) and let u be a xed strictly positive nite U-excessive function. Forf ∈pE and p≥0, let
(11) Up/uf := 1
uUp(uf).
Then U/u := (Up/u)p≥0 is a resolvent on E such that 1 is U/u-excessive.
Moreover, the associated potential kernel is given by
(12) U/uf = 1
uU(uf), f ∈pE
whereU is the potential kernel ofU. U/uis calledu-transform ofUin the Doob sense.
The proof of the following useful Lemma is obvious since S(U/u) = {vu : v∈ S(U)}.
Lemma 5. Let U and V be two resolvents on (E,E) such that V is S- subordinated to U. Then for each strictly positive nite functionu∈ S(U), the resolvent V/u isS-subordinated toU/u.
3.3. The main result
LetU andVbe two resolvents on E with proper potential kernels U and V, respectively. We denote by L and M the energy functional dened by U and V, respectively. In this paragraph, we suppose thatV is S-subordinated to U.
Let
(13) Dom(V,U) :={m∈Exc(V) :M(m, u)<∞ for someu >0 inS(U)}.
Theorem 6. For each m ∈ Dom(V,U), there exits a unique Υ(m) ∈ Exc(U) such that
(14) M(m, u) =L(Υ(m), u), u∈ S(U).
Moreover:
1. If for(µn)⊂Mea, we have µnV ↑m then µnU ↑Υ(m). In particular, if µV ∈Dom(V,U) then
(15) M(µV, u) =L(µU, u), u∈ S(U), i.e. Υ(µV) =µU.
2. IfK is a kernel onE which commutes with U and V and S(U)⊂ S(K), then mK ∈Dom(V,U) and
(16) M(mK, u) =L(Υ(m)K, u), u∈ S(U),
Proof. Let m ∈Dom(V,U). Inserting a suitable strictly positive density into the potential kernels and making a suitable Doob transformation (cf. Re- mark 4 and Lemma 5), we may assume that U1 ≤ 1, V1 ≤ 1, M(m,1) < ∞ and 1∈ S(U).
By Theorem 2 the mapping
(17) ϕ:u→M(m, u), u∈ S(V)
is a positive increasing linear functional which is continuous from below. Con- sequently, the restriction ψof ϕtoS(U)also has these properties, and ψ(1) = ϕ(1) < ∞. Again, by Theorem 2 we conclude the existence of a unique l:= Υ(m)∈Exc(U)such that
(18) ψ(u) =L(l, u), u∈ S(U).
Moreover, by (17) and (18) we have for u∈ S(U) (19) M(m, u) =ϕ(u) =ψ(u) =L(l, u).
1. Let(µn)⊂Mea such that µnV ↑m. Then, for eachu∈ S(U) we have (20) µn(u) =M(µnV, u)↑M(m, u) =L(l, u).
Hence, µnU ↑lby taking in (20) U f instead ofu for all f ∈pE.
If we apply the preceding result for µn=µandm=µV, we deduce that Υ(µV) =µU.
2. Let K be a kernel on E such that U K = KU, V K = KV and S(U)⊂ S(K). There exists by Hunt's approximation (2) a sequence of bounded measures (µn)⊂Mea such thatm= supµnV. Thus,
(21) mK = (supµnV)K= supµnV K = supµnKV.
However, since 1 ∈ S(U) we have K1 ≤ 1 and therefore (µnK) is a sequence of bounded measures. Hence, mK ∈Exc(V) by (2) again.
Now, letm∈Dom(V,U)and letu∈ S(U), u >0such thatM(m, u)<∞. Since Ku≤u, we get
∞> M(m, u) =M(supµnV, u) = supµn(u)≥supµn(Ku)
= supM(µnKV, u) = supM(µnV K, u) =M(mK, u).
Consequently,mK ∈Dom(V,U). Using similar arguments, for u∈ S(U), we get
M(mK, u) = supµnK(u) = supL(µnKU, u)
= supL(µnU K, u) =L(Υ(m)K, u), which proves (16).
Remark 7. In general, Υ is not dened on the whole cone Exc(V). For example: Let U (resp. P) be the resolvent (resp. semigroup) of a transient convolution semigroup on Rd (cf. [3] for details). Letp0 >0, Vp := Up+p0 for p ≥ 0 and V := (Vp) (cf. Example 2.1.2). Since the Lebesgue measure m is U-invariant then m=p0mUp0 =p0mV =µV for µ=p0m.
Now, if Υ(m) is dened then Υ(m) = Υ(µV) = µU. However, m is P-invariant, hence,
(22) µU(f) =p0
Z
mPtfdt=∞ ·m(f).
Therefore, µU is notσ-nite.
Example 8. Let U be a resolvent on E and p0 > 0; let Vp := Up+p0 for p ≥ 0. From Theorem 6 we know that for m0 ∈ Dom(V,U) there exists a unique m∈Exc(U) such that
(23) L0(m0, u) =L(m, u), u∈ S(U),
whereL0 andLdenote the energy functionals associated withVandU, respec- tively. We have
(24) m (i)= m0+p0mUp0
(ii)= m0+p0m0U.
More generally, ifVis obtained fromUby general subordination by killing by a subordination operator P, then form0 ∈Dom(V,U)there exists a unique m ∈Exc(U) such that (23) holds and we havem =m0+mP. In the special case of Kac-killing by h ∈ pE this reads m = m0 +mUh(h.). To obtain the equation corresponding to (ii) we need the assumptionU =QV forQ= P
n≥0
Pn and then we havem=m0Qcorresponding to (ii).
For the applications, we need to prove the injectivity of the operator Υ dened by Theorem 6. For this aim, some additional (but natural) assumptions are necessary. In fact, the proof of the following result is based on remarks in [4] and the results in [7], p. 858. In [7] the reader may also nd some details concerning Ray-compactications, c.f. also [5, 6, 15].
Proposition 9. Suppose that
(i) S(U) is inf-stable and generates the σ-algebraE. (ii) For u∈ S(U) we have Vpu∈ S(U) for p >0.
Let m1, m2∈Dom(V,U) such that
(25) M(m1, u) =M(m2, u) u∈ S(U).
Then m1 =m2.
In particular, the operator Υ :Dom(V,U)→Exc(U) is injective.
Proof. Let m1, m2 ∈ Dom(V,U) such that (25) holds. By putting suit- able densities into the potentials kernels and measures, and by taking suit- able Doob's transformations we may assume 1 ∈ S(U), U1 +V1 is bounded, m1(1) +m2(1)<∞, and M(m1,1)<1.
Let R0 be the Ray cone with respect to Uof bounded U-excessive func- tions. We augment it to a Ray cone R with respect to V by iterating the following procedure: apply nitely many of the kernels Vpi, take nite inma and generate convex cones. Due to assumption (ii) we see thatRis a common Ray cone for Uand Vconsisting of U-excessive functions.
If we denote by Eˆ the Ray compactication of E with respect to R then the energy functionals M(mi, .) (for i = 1,2) are represented on R by the integral with respect to a unique measure on the non-branch points ofE, hence,ˆ m1=m2.
Now, let m1, m2 ∈Dom(V,U) such thatΥ(m1) = Υ(m2). Then by (14) we get
(26) M(m1, u) =L(Υ(m1), u) =L(Υ(m2), u) =M(m2, u) u∈ S(U).
Hence, (25) is satised and therefore m1 = m2 by the rst part of the Proposition.
4. APPLICATIONS TO EXIT AND ENTRANCE LAWS From now, we consider a resolvent Udened by a measurable semigroup P on(E,E) such that
1. The associated potential kernelU is proper and satises the principle of unicity of charges.
2. The cone S(U) ofU-excessive functions is inf-stable and generates E. 4.1. An application to entrance laws
A P-entrance law is a family(νt)t>0 in Mea satisfying (27) νsPt=νs+t, s, t >0.
Let us denote by Ent(P) the cone of P-entrance laws (νt) such that R∞
0 νtdt is σ-nite. It can be easily seen that, if ν ∈Ent(P) then R∞
0 νtdt∈ P ur(U). Conversely, we have the following result.
Theorem 10. For each l∈Pur(U), there exists a unique P-entrance law ν ∈Ent(P) such that
(28) l =
Z ∞
0
νtdt.
This integral representation was proved in [25] (cf. also [14, 18] for other versions under some regularity assumptions).
In fact, l ∈ Pur(U) admits a integral representation by entrance laws if and only if there exists a family(νt) of Mea such that
(29) l·Pt = νtU, t >0.
The proof is based on the unicity of charges principle (cf. [25], Proposi- tion).
Let β := (βt) be a Bochner subordinator. We denote by Pβ the subordi- nated semigroup of P by means of β (cf. relation (9)). Moreover, U and Uβ denote of the resolvent ofPandPβ andU andUβ the associated initial kernels, respectively.
If (νt) is aP-entrance law then the family(νtβ) dened by
(30) νtβ :=
Z ∞
0
νsβt(ds), t >0
is a Pβ-entrance law whenever it is σ-nite. It is also said to be subordinated to (νt) by means ofβ.
The Bochner subordination problem for entrance laws consists of the con- verse: Is a given Pβ-entrance law subordinated to some P-entrance law?
In order to prove the converse, we need rst the following characterization.
The proof is obtained by routine calculus.
Proposition 11. Let (µt) ∈ Ent(Pβ) a Pβ-entrance law and let m :=
R∞
0 µtdt. Then (µt) is subordinated to a (unique) P-entrance law by means of β if and only if there exists a family (νt) of Mea such that
(31) m·Pt = νt·Uβ, t >0
Proof. Suppose that (µt) is subordinated to aP-entrance law(νt) and let t >0. Then (30) and (27) yield
m·Pt = ( Z ∞
0
µsds)·Pt= ( Z ∞
0
Z ∞
0
νrβs(dr)ds)·Pt
= Z ∞
0
Z ∞
0
νrPtβs(dr)ds= Z ∞
0
Z ∞
0
νtPrβs(dr)ds
= νt·( Z ∞
0
Z ∞
0
Prβs(dr)ds) =νt·( Z ∞
0
Psβds) =νt·Uβ. Conversely, lets, t >0. By (31) and the semigroup property we have (32) νs+tUβ =mPs+t= (mPs)·Pt=νsUβPt= (νsPt)Uβ,
since Pt commutes with Uβ. Hence, (νt) satises the entrance laws equation (27) by the unicity of charges principle (Lemma 3). Moreover, from (29), (9)
and (31) we deduce fort >0 that (33)
µtUβ =mPtβ = Z ∞
0
mPsβt(ds) =Z ∞ 0
νsUββt(ds) = (Z ∞ 0
νsβt(ds))·Uβ. Therefore, (µt) is subordinated to(νt) by Lemma 3 again.
LetLandLβ be the energy functionals dened byUandUβ, respectively.
We denote by
(34) Ent(Pβ)∗ :={(µt)∈Ent(Pβ) : Z ∞
0
µtdt∈ Dom(Uβ,U)}.
Theorem 12. Any Pβ-entrance law in Ent(Pβ)∗ is subordinated to a unique P-entrance law in Ent(P) by means of β.
Proof. Let (µt) ∈ Ent(Pβ)∗ and let m := R∞
0 µtdt. There exists by Theorem 6, a uniquel∈Pur(U) such thatl= Υ(m) and
(35) Lβ(m, u) =L(l, u), u∈ S(U).
Then l=R∞
0 νtdt for some uniqueP-entrance law (νt).
Since each Pt commute withUβ andU, we have by (16) and (35) that (36) Lβ(mPt, u) =L(lPt, u), u∈ S(U), t >0.
Using successively (15), (29) and (36) we get
(37) Lβ(νtUβ, u) =L(νtU, u) =L(lPt, u) =Lβ(mPt, u), u∈ S(U), t >0.
Since the resolvents Uand Uβ commute, Proposition 9 implies that (38) νtUβ =mPt, t >0.
We conclude by Proposition 11 that(µt)is subordinated to(νt)by means of β.
Remark 13. Since each Upβ commute with Uβ and withU, (16) and (35) give
(39) Lβ(m(pUpβ), u) =L(l(pUpβ), u), u∈ S(U), p >0.
We deduce easily from (39) that l = Υ(m) ∈ Pur(U) whenever m ∈ Pur(Uβ). Hence, by the Riesz decomposition (1) and the additivity of Lβ, we can see that l= Υ(m)∈Inv(U)if m∈Inv(Uβ). The two last conclusions hold for general S-subordination ifVp commutes with U for all p >0.
4.2. An application to exit laws
LetPandPˆbe two measurable semigroups onE, which are in strong dual- ity with respect to aσ-nite xed measurem∈Mea, and absolutely continuous with respect to m. That is
(40)
Z
E
gPtfdm= Z
E
fPˆtgdm, t >0, f, g∈pE
and all measures Pt(x, .),Pˆt(x, .) :t >0, x∈E are absolutely continuous with respect to m.
A P-exit law is a family (ft)t>0 ofm-a.e. nite inpE satisfying (41) Ptfs=fs+t, s, t >0
We denote byE(P) the cone of allP-exit laws(ft)for whichR∞
0 ftdtism-a.e.
nite. Obviously, R∞
0 ftdt∈ S0(U) if it isU-a.e. nite.
For absolutely continuous semigroups we have the following useful result.
Lemma 14. The map
Θ : E(P) → Ent(ˆP) (ft) → (ft·m) is one-to-one.
Proof. The proof uses standard arguments and is adapted from [23] (cf.
also related references therein). By the strong duality relation (40), it is obvious that (ft·m) ∈ Ent(ˆP) whenever (ft) ∈ E(P). Conversely, let (νt) be a P- entrance laws such that R∞
0 νtdt is σ-nite. Since each kernel Pˆt, t > 0 is absolutely continuous with respect tom then so is each measureνsPˆt, s, t >0. We deduce from (27) that all measuresνt, t >0are absolutely continuous with respect to m. Let (ht) ⊂ pE such that νt = ht·m, t > 0. Using (40), (27) becomes (m-a.e. means almost everywhere with respectm)
(42) hs+t=Psht m−a.e., s, t >0 Fix (tk)↓0. For 0< tk< tl< twe have
(43) ftk :=Pt−tkhtk =Pt−tl+tl−tkhtk =Pt−tlPtl−tkhtk =Pt−tlhtl=ftl. This is immediate from (42) since the semigroupPis absolutely continuous with respect to m.
Hence, ft:=ftk is well dened,ft=htm-a.e (hence,ftis a density ofνt) and (ft) is a P-exit law (in fact the unique one such that νt = ft·m, t > 0).
Finally,R∞
0 ftdtisU-a.e. nite since it ism-a.e nite (as density of theσ-nite measure R∞
0 νtdt) andU is absolutely continuous with respect to m.
Let β := (βt) be a Bochner subordinator and let Pβ (resp. Pˆβ) be the subordinated semigroup ofP(resp. P) by means ofˆ β. Let(ft) be aP-exit law and dene (ftβ) by
(44) ftβ :=
Z ∞
0
fsβt(ds), t >0.
Then (ftβ) is a Pβ-exit law. As for entrance laws, (ftβ) is said to be subordinated to(ft) by means ofβ. Conversely, denote
(45) E(Pβ)∗ :={(gt)∈E(Pβ) : Z ∞
0
gt·mdt∈ Dom(Uβ,U)}.
Corollary 15. Any Pβ-exit law in E(Pβ)∗ is subordinated to a unique P-exit law by means of β.
Proof. Let (gt) be a Pβ-exit law and let (gt·m) be the associated Pˆβ- entrance law by Lemma 14. There exists, by Theorem 12, a unique P-entranceˆ law (νt) such that
(46) gt·m=νtβ, t >0.
On the other hand, there exists by Lemma 14, a unique P-exit law (ft) such that
(47) νs=fs·m, s >0.
After integrating (47) with respect to βt, t >0 we get (48) νtβ =ftβ·m, t >0.
Let t >0. It follows from (46) and (48) that (49) gt/2(x) =ft/2β (x), x∈At for some At∈ E withm(At) = 0.
Now, since P is absolutely continuous with respect to m then so is Pβ. Thus, we conclude by (49) and (27) that
(50) gt=Pt/2gt/2 =Pt/2ft/2β =ftβ.
Remark 16. 1. By Corollary 15, we solve the Bochner subordination problem for absolutely continuous semigroups. This result is proved in [21] under some integrability hypothesis on the common density ofPand P. Note that Theorem 12 and Corollary 15 seem to be new even for killedˆ Brownian motions.
2. For general semigroups, some additional assumptions are needed in order to obtain a positive response for this problem. We refer the reader to [1]
where non-trivial examples and counter examples are given and to [24]
where a general characterization is proved.
Acknowledgments. We thank L. Beznea for his remarks [4] which helped us to prove the injectivity of the operator Υ(i.e. Proposition 9), cf. also [6].
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Received 8 Novembeber 2013 Universite de Tunis Elmanar, Departement de Mathematiques,
Faculte des Sciences de Tunis, TN-2092 Elmanar,
Tunis, Tunisia Med.Hmissi@fst.rnu.tn Universitat Dusseldorf, Mathematisches Institut,
Mathematisch-Naturwissenschaftliche Fakultat, Universitatsstrasse 1,
D-40225 Dusseldorf, Germany
janssenk@uni-duesseldorf.de
