A CLASS OF SUBORDINATION OPERATORS ON A DIRECT SUM

on the occasion of his 70th birthday

A CLASS OF SUBORDINATION OPERATORS ON A DIRECT SUM

LUCIAN BEZNEA and ANDREI-GEORGE OPRINA

We present an example of a subordination operator (in the sense of G. Moko- bodzki) which occurs in a potential theoretical view on the discrete branching processes, introduced by N. Ikeda, S. Watanabe, and M. Nagasawa. It turns out that it is an operator which is neither of potential type nor of hitting distribution type. Notice that the state space of the branching processes is the set of all finite sums of Dirac measures on a given spaceE, identified with the direct sum of the symmetricn-powers ofE.

AMS 2000 Subject Classification: 60J45, 60J57, 60J40, 60J80, 31D05.

Key words: subordinate resolvent, subordination operator, right process, branch- ing process.

1. INTRODUCTION

The subordination operators have been considered in potential theory by G. Mokobodzki as an analytic tool in the study of the perturbations of a Markov process by a right continuous multiplicative functional; see (3.1) below for the precise relations between the analytic and the probabilistic notions, more details may be found in [2] and [5]; see also [6] for further results on the uniqueness problem for the subordination operators.

There are two basic examples of subordination operators. The first one is the hitting distribution kernel and corresponds to the killing of the process at the first entry time in the complement of an open set. The second one is the so called “potential type” subordination operator and corresponds to the zero order perturbation of the infinitesimal generator of the process, or equiva- lently to the perturbation given by a Feynman-Kac formula with a continuous additive functional of the process; see Example 3.2 for a complete presentation.

In this paper we derive an example of subordination operator useful in a potential theoretical view on the discrete branching processes as introduced by N. Ikeda, S. Watanabe, and M. Nagasawa in the pioneering paper [7].

We work in a general frame given by a Borel right (Markov) process on a

MATH. REPORTS12(62),2 (2010), 119–126

Lusin topological space; several preliminary results are given in Section 2.

Theorem 4.1 and Remark 4.2 ii) below present this example (which is neither of potential type nor of hitting distribution type) in its natural frame given by a direct sum of measurable spaces; for the motivation see assertion iii) of the Remark 4.2. In fact, we prove in Proposition 3.3 a more general result on the summation of a sequence of subordination operators.

Our approach was inspired by an analytic method proposed by K. Janssen (cf. [8]) in the frame of the balayage spaces (on metrizable locally compact spaces).

2. PRELIMINARIES

Let U = (U_α)_α>0 be a proper sub-Markovian resolvent of kernels on a Lusin measurable space (E,B). We assume further that:

(2.1) There exists a Lusin topologyT on E such that B is precisely the Borelσ-algebra onEand there exists a Borel right processXwith state space E having U as associated resolvent, i.e.,

U_αf(x) =E^x Z ∞

0

e^−αtf◦X_tdt for all f ∈pB and x∈E.

We recall now some basic notions and results on the excessive functions associated with U; see, e.g., [2] for details.

An universally B-measurable function u : E → R+ is called U-super- median if αU_αu ≤ u for all α > 0. If in addition sup

α>0

αU_αu(x) = u(x) for all x ∈E, then u is named U-excessive. If u is U-supermedian then one can consider the U-excessive regularizationbuofu, defined asu(x) = supb

α>0

αU_αu(x), x ∈ E.We denote by E(U) the set of all B-measurableU-excessive functions on E.

Aσ-finite measureξon (E,B) is calledU-excessiveif for allα >0 we have ξ(αU_αf)≤ξ(f) and sup

α>0

ξ(αU_αf) =ξ(f), f ∈pB;pB (resp.bpB) denotes the set of all numerical positive (resp. bounded positive) B-measurable functions on E. We denote by Exc(U) the set of all U-excessive measures.

If µis a positive measure on (E,B) thenµ◦U is U-excessive, provided it is σ-finite; such an U-excessive measure is calledpotential.

Thefine topologyis the topology onE generated by allU-excessive func- tions.

A Luzin topology on E having B as Borel σ-algebra is called natural (with respect to U) provided that it is smaller than the fine topology.

(2.2) Assertion (2.1) is equivalent to each of the following ones:

(2.3) For all u, v ∈ E(U) we have inf(u, v) ∈ E(U), 1 ∈ E(U), B is generated by E(U), and every U-excessive measure dominated by a potential is also a potential.

(2.4) For every natural topology on E there exists a Borel right process with state space E, havingU as associated resolvent.

3. SUBORDINATION OPERATORS

A kernel P on (E,B^u) is called subordination operator with respect to U provided that P u ≤ u and the function inf(u, P u+v−P v+P f) is U- excessive for all u, v∈ E(U) withv <∞ and f ∈pB; we have denoted by B^u the universal completion of B.

Remark 3.1. The subordination operators considered here correspond to the exact subordination operators from [2] and notice that for the applica- tion we intend to give, we need to consider only B-measurable subordination operators (compare with [2]).

Recall that a second sub-Markovian resolvent of kernels W = (Wα)α>0

on (E,B) is called subordinate toU provided thatW_α ≤U_α for allα > 0. If in addition U f−W f isU-excessive for everyf ∈bpB withU f <∞, then we say that W is exactly subordinate toU.

(3.1) We collect some results on subordination operators and subordinate resolvents (see [2, Section 5.1]).

i) LetP be a subordination operator with respect toU. Then there exists a sub-Markovian resolvent W = (Wα)α>0 on (E,B^u) such that W is exactly subordinate to U and W f =U f −P U f for all f ∈ pB^u with U f <∞. The sub-Markovian resolvent W is called generatedbyP.

ii) Let W = (Wα)α>0 be a sub-Markovian resolvent on (E,B^u) which is exactly subordinate to U. Then there exists a uniquely determined sub- ordination operator P with respect to U such that W is generated by P; in particular, we have P U f =U f−W f for all f ∈pB withU f <∞.

iii) If P is a subordination operator with respect toU we put EP ={x∈E : there existsu∈ E(U) withP u(x)< u(x)}.

We assume for simplicity that P is a kernel on (E,B); see Remark 3.1.

By [2, Theorem 5.1.6] it follows that the restriction toE_P of the subor- dinate resolvent generated by P (resolvent also denoted byW) satisfies (2.3) and therefore (2.4). We get also that the trace of the topology T on E_P is natural for the resolvent W on EP and we conclude by (2.4) that W is the resolvent of a Borel right process Y with state space E_P. Notice thatP f is a W-excessive function (onEP) for all f ∈pB.

iv) The probabilistic interpretation is as follows (for details see [5], [10]

and [2, Section 5.3]). For every exact subordination operatorP there exists an exact (right continuous) multiplicative functionalM = (Mt)t≥0ofX such that

P f(x) =−E^x Z ∞

0

f(Xt)dMt, f ∈pB.

The set E_P becomes the set of all permanent points with respect to M and the process Y on EP is obtained from X by killing withM.

Example 3.2. i) For each u ∈ E(U) and A ⊂E we consider the reduced functionof u on A,

R^Au= inf{v∈ E(U) : v≥u onA}.

If A ∈ B and u ∈ E(U) then R^Au, the reduced function of u on A, is a B- universally measurable U-supermedian function. Assume that R^Au ∈ pB for all u ∈ E(U). Then Rc^A (called the balayage of u on A) is a subordination operator with respect to U. The maps u 7→ R^Au and u 7→ Rc^Au extend to kernels on E. Note that by Hunt’s Theorem we haveR^Af(x) =E^x(f◦XD_A) and Rc^Af(x) =E^x(f ◦X_TA), where D_A is the entry time of A,D_A= inf{t≥ 0 : X_t∈A}, and T_A is thehitting time of A,T_A= inf{t >0 : X_t∈A}.

ii) Let v < ∞ be a regular U-excessive function. Then there exists a regular U-excessive kernel V⁰ on (E,B) such that v = V⁰1 and let V be the kernel on E defined by V f = V⁰(gf) for all f ∈ pB, where g ∈pB is a fixed real valued function. Notice thatV possesses the complete maximum principle.

Assume that there exists a sub-Markovian resolvent of kernels V = (Vα)α>0

having V as initial kernel (V = V0). We recall that, by [2, Theorem 1.1.12 (Kondo-Taylor-Hirsch)], this is the case whenevergis bounded sinceV1<∞.

Then V1 is a subordination operator with respect to U and EV1 = E. In particular, for all α > 0 the kernel αU_α is a subordination operator with respect to U. These subordination operators are called of potential type.

LetW = (W_α)_α>0 be the subordinate resolvent generated by the subor- dination operatorV1. HenceW =U−V1U and recall that we have in addition U =

∞

P

n=0

V₁ⁿW.

The probabilistic counterpart is as follows; cf. [5] and [2]. There exists a continuous additive functional (At)t≥0 of X such that for all α≥0 we have Vαf(x) = E^xR∞

0 e^−αAtf◦XtdAt. The subordinate resolvent W = (Wα)α>0

is given by the Feynman-Kac formula Wαf(x) =E^x

Z ∞

0

e^−At−αtf ◦Xtdt.

For further analytic and probabilistic developments on the potential type sub- ordination operators see [3] and [4].

Proposition 3.3. Let (Pn)n≥1 be a family of subordination operators with respect to a sub-Markovian resolvent of kernels U on a Lusin measurable space (E,B). In addition, we suppose that EPn∩EPm =∅ for all n, m∈N^∗, n6=m, and S

n≥1E_Pn =E. Then the kernel P on (E,B) defined as P f :=X

n≥1

1E_PnPnf , f ∈pB,

is a subordination operator with respect to the resolventU and we haveEP =E.

Proof. Clearly,P is a kernel on (E,B) and P u≤ufor all u∈ E(U). We show that for all u∈ E(U) we have

(3.2) P u= inf

n≥1P_nu.

Indeed, if m ≥ 1, n 6= m, and x ∈ E_Pm then P_nu(x) = u(x). Therefore, P u(x) =Pmu(x)≤u(x) =Pnu(x) and thus

P u(x) = inf

n≥1P_nu(x).

We consider u, v∈ E(U), v <∞ and f ∈pBand define the functions h := inf (u, P u+v−P v+P f),

h_n:= inf (u, P_nu+v−P_nv+P_nf) for all n≥1.

We haveh=hnon EPn for alln≥1, henceh=P

n≥11E_Pnhn. We show that

(3.3) h= inf

n≥1h_n.

Indeed, let m∈N^∗ and x∈EPm. Since Png(x) =g(x) for all g∈pB,n6=m, we get

n6=minf hn(x) = inf

n6=minf(u(x), Pnu(x) +v(x)−Pnv(x) +Pnf(x)) =

= inf(u(x), u(x) +f(x)) =u(x)≥h_m(x).

Consequently,

n≥1inf hn=hm=h on EPm.

BecauseP_n is a subordination operator with respect toU we have h_n∈ E(U) for alln≥1, and therefore by (3.3) we deduce thath isU-supermedian.

But h = h_n on E_Pn and h_n ∈ E(U), therefore h is finely continuous on each finely open set EPn and thush∈ E(U).

BecauseP u≤P_nu(cf. (3.2)), we getE_Pn ⊂E_P for alln∈N^∗, therefore EP =E.

Remark.The hypothesisS

n≥1EPn=Eis not necessary in Proposition 3.3.

However, without this hypothesis we have to define the kernelP on (E,B) as P f =X

n≥1

1_EPnP_nf+ 1_E\^S

n≥1E_Pnf,

and we can deduce that h is a strongly supermedian function, hence P is a subordination operator in the general sense considered in [2] and one can show that E_P =S

n≥1E_Pn.

4. SUBORDINATION OPERATORS ON A DIRECT SUM

In this section we consider the resolvent U = (Uα)α>0 on the Lusin topological space E as in Section 2. Let I be a countable set, card(I) ≥ 2, and assume that (E,B) is a countable sum

E:=L

i∈I

E_i, E_i∩E_j =∅for all i, j∈I, i6=j, (E_i)i∈I ⊂ B.

Hence (E_i)i∈I is a countable family of Lusin measurable spaces, every E_i is endowed with the trace topology of E and B_i is the σ-algebra on Ei. We suppose thatE_i is a finely open subset of E, for alli∈I.

A function f : E → R is uniquely identified by its restrictions to every set E_i,i∈I. We write f = (f_i)i∈I, wheref_i:=f_|Ei.

A kernelN :pB →pB on the measurable space (E,B) can be identified through a matrix of kernels (Nij)i,j∈I, where Nij : pB_i → pB_j is a kernel from (E_i,B_i) to (E_j,B_j), defined as N_ijf_i := (Nfe_i)_j, f_i ∈ pB, fe_i being the extension of fi from Ei to E with 0 on E\Ei. Conversely, given a family of kernels (Nij)i,j∈I, Nij : pB_i → pB_j, we define a kernel N on (E,B) as N f := P

j∈IN_jif_j

i∈I, f ∈pB.

A kernelN on (E,B) is calleddiagonal ifN_ij = 0 for alli, j∈I,i6=j.A resolvent of kernelsV = (Vα)α>0on (E,B) is calleddiagonalifVαis a diagonal kernel for every α >0.

Let M and N be two kernels on the measurable space (E,B). If in addition N is diagonal andM ≤N (i.e.,M f ≤N f for allf ∈pB) thenM is also diagonal.

We define the kernel P on (E,B) as

(P f)_i := (R^E\Eif)_i, i∈I.

From the above considerations we can identify the kernel P with the family of kernels (P_ij)i,j∈I,P_ij :pB_i →pB_j, by

Pijfi := (Pfei)j = (R^E\Ejfei)j.

Theorem 4.1. The following assertions hold for W =U−P U.

a)W is the initial kernel of a proper sub-Markovian resolvent of kernels W = (Wα)α>0 on (E,B) exactly subordinate to U. In addition W is diagonal and P f ∈ E(W) for all f ∈pB.

b) There exists a right Markov process with state space E having W as associated resolvent.

Proof. For every f ∈pBwith u=U f <∞ we have (P u)i = (R^E\Eiu)i≤ui for all i∈I.

Consequently, P U f ≤ U f for all f ∈ pB and therefore the map f 7→ U f − P U f =:W f extends to a kernel on E.

We prove that P is a subordination operator with respect to U. Since the sets E_i and E\E_i are finely open, the kernel P_i = R^E\Ei on (E,B) is a subordination operator for alliandEPi =Ei. By Proposition 3.3 we get that P is a subordination operator such thatE_P =E. Hence by (3.1) i) there exists a sub-Markovian resolvent W = (Wα)α>0 on (E,B) such that W is exactly subordinate toU and havingW as initial kernel. By the last assertion of (3.1) iii) we deduce that P f ∈ E(W) for all f ∈pB. Assertion b) follows now by (3.1) iii) sinceE_P =E.

We show thatW is diagonal. Considerf ∈pBandF = [f >0]. Because U satisfies the complete maximum principle (see, e.g., [2, Section 1.1]) we have

(4.1) R^FU f =U f on E.

SinceE\Ej ⊃Ei we getR^E\EjUfei ≥R^E\EiUfeiand by (4.1) we getR^E\EjUfei

= Ufe_i, henceW_ijf_i = 0 on E_j fori6= j. Because W_αf ≤W f for all α >0, f ∈pB and using the fact that W is diagonal, we deduce thatWα is diagonal for all α >0, i.e.,W is diagonal.

Remark 4.2. i) Theorem 4.1 was inspired by an analytic result of K. Jans- sen ([8, Theorem 1.1]), presented in terms of perturbations of balayage spaces.

In particular, E is there a locally compact space with countable base and the setsEi,i∈I, are open.

ii) By Theorem 4.1 it follows that the kernelP, P f =X

i∈I

1Ei R^E\Eif,

is a subordination operator with respect to U. It is a concrete example which belongs to the class of subordination operators treated in Proposition 3.3.

iii) In a forthcoming work we shall apply the above results to a class of discrete branching processes, in the sense of N. Ikeda, M. Nagasawa and S. Watanabe (cf. [7] and [9]). We close this section by summarizing the ap- propriate frame for this application; see the survey article [1] for more details:

U = (Uα)α>0 will be the resolvent of a Borel right Markov process with state space D; E_i will be the symmetric ith power of D. Notice that the direct sum E is identified with the space of all positive measures µ on D which are finite sums of Dirac measures: µ= Pn

k=1δx_k, where x1, . . . , xn ∈ D. By Theorem 4.1 the resolvent W on E will be a diagonal one.

Acknowledgement. Support from the Romanian Ministry of Education, Research and Innovation (PN II Program of CNCSIS) is gratefully acknowledged.

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Received 15 October 2009 Romanian Academy

“Simion Stoilow” Institute of Mathematics P.O. Box 1-167, 014700 Bucharest, Romania

lucian.beznea@imar.ro and

oandrei22@yahoo.com