LUCRARE DE DIZERTAŢIE

Transcription

1 ŞCOALA NORMALĂ SUPERIOARĂ BUCUREŞTI Departamentul de Matematică LUCRARE DE DIZERTAŢIE Puncte de neramificare pentru rezolvante de nuclee Oana Valeria Lupaşcu Conducator ştiinţific: Prof. Dr. Lucian Beznea Bucureşti, septembrie 211

2 CONTENTS Introduction Preliminaries Sub-Markovian resolvent of kernels Subordination by convolution semigroups References

3 Introduction The excessive functions with respect to a right continuous Markov process are related to the generator of the process (or to its inverse, the potential kernel) in the same way as the classical superharmonic functions on an Euclidean open set are related to the Laplace operator on that set (or to the Newtonian kernel). The aim of this work is to present systematically several properties of the excessive functions with respect to a transition function (a semigroup of sub-markovian kernels), possible associated to a Markov process. We follow essentially the first Chapter from the monograph [BeBo 4], the monograph [Sh 88], and the article [St 89] of J. Steffens; cf. also [Be 11b] and [BeBoRö 6]. The main property we characterize is the stability to the pointwise infimum of the convex cone of all excessive functions. From the probabilistic point of view, this property is precisely the fact that all the points of the state space of the process are nonbranch point. The final aim is to present an application to the construction of Right Markov processes in infinite dimensional situations. The plan of the work is the following. In the first section (Preliminaries) we present some useful basic results on the Gaussian semigroup in R d, transition functions, the associated resolvent of kernels, the infinitesimal generator, the definition of the Markov processes, and the Brownian motion as an example. In Ssection 2 we give results on the excessive functions with respect to a sub-markovian resolvent of kernels: Hunt s Approximation Theorem (Theorem 2.1), the C -resolvent of contractions on L p -spaces induced by such a resolvent (Theorem 2.2). The main result is Theorem 2.5, stating the above mentioned characterization of the min-stability of the excessive functions. It turns out that this characterization of the property that all the points are nonbranch points is an esential step in the construction of the measure-valued branching Markov processes; cf. [Be 11a] for the case of continuous branching, using several potential theoretical tools. We expect that a similar procedure will be efficient in the case of the discrete branching processes, as treated in the Addendum of [BeOp11]; see also [INW 68], [Si68] and [Na 76]. 2

4 The announced application is developed in Section 3. More precisely, we prove in Theorem 3.2 that the min-stability property of the excessive functions is preserved by subordination with a convolution semigroup, the so called Bochner subordination; we follow the notations and the approach from [BlHa 86]. We complete in this way results from the recent article [BeRö 11], where this property was supposed to be satisfied by the subordinate resolvent, in order to associate to it a cadlag Markov process; see the Example following Corollary 5.4 from [BeRö 11]. 3

5 1 Preliminaries The n-dimensional Brownian motion: the Gaussian semigroup and the Newtonian potential kernels For each f pb(r n ) and t >, the Gaussian kernel P t on R n is defined by: 2 1 P t f(x) = (2πt) n/2 2t f(y)dy, R n e x y where pb = pb(r n ) denotes the set of all positive, real-valued Borelian functions on R n. The family (P t ) t, P = Id is called the Gaussian semigroup on R n. Let g t, g t : R n R, be the density of the Gaussian kernel P t, g t (x) = 1 (2πt) x 2 e 2t. n/2 Clearly, the kernel P t is defined by a convolution, P t f(x) = g t f(x) = g t (x y)f(y)dy R n for all f pb. Proposition 1.1. (i) For each t > the kernel P t is Markovian, i.e., P t 1 = 1. In particular, for every x R n the measure f P t f(x) is a probability on R n. (ii) P t is a linear operator on the space bb of all bounded Borel measurable functions on R n and if f then P t f. (iii) The family (P t ) t is a semigroup of kernel on R n, i.e., P s P t = P t+s for all s, t. Proof. (i) If n = 1, then we have: P t 1(x) = 1 2πt e (y x)2 2t dy = 2t 2πt The case n > 1 follows by Fubini s Theorem: 1 x y n P t 1(x) = 2 (2πt) n/2 2t dy = R n e 4 i=1 [ 1 2πt e z2 dz = 1. e (x i y i )2 2t dy i ] = 1.

6 (ii) Let f bb, f M. Then P t f P t ( f ) P t M = M P t 1 = M. Consequently P t f bb. (iii) Since P t f = g t f and P t P s (f) = g t g s f for all f bb, it follows that in order to prove the semigroup property of (P t ) t we have to show that g t g s = g t+s. We check the above equality in the case n = 1: = g s g t (x) = 1 2πt 1 2πs 2π x 2 ts e 2s = 1 e s+t 2st (y t s+t x)2 dy = 1 2πt e (x y)2 2s e y2 2t e s+t 2st y2 + x s y dy dy 1 2π(s + t) e x2 /2(s+t) = g s+t (x). Let E be a metrizable Lusin topological space and B the Borel σ-algebra on E Transition function. A family of kernels (P t ) t on (E, B) which are sub-markovian (i.e., P t 1 1 for all t ), such that P f = f and P s (P t f) = P s+t f for all s, t and f pb is called transition function on E. We assume further that for all f bpb the real-valued function (t, x) P t f(x) is B ( [, ) ) B-measurable. Resolvent of kernels. The resolvent of kernels associated with the transition function (P t ) t is the family U = (U α ) α> on (E, B) defined by U α f := e αt P t f dt. The following two properties hold for the resolvent of kernels associated with a transition function. The family U = (U α ) α> satisfies the resolvent equation, i.e., for all f bpb we have U α = U β + (β α)u α U β for all α, β >. 5

7 Note that in particular we have: U α U β = U β U α for all α, β >. The resolvent family U = (U α ) α> is sub-markovian, i.e., αu α 1 1 for all α >. Indeed, we have: U α 1 = e αt P t 1dt e αt dt = 1. α Right continuous Markov process. A system X = (Ω, G, G t, X t, θ t, P x ) is called right continuous Markov process with state space E, with transition function (P t ) t provided that the following conditions are satisfied: a) (Ω, G) is a measurable space, (G t ) t is a family of sub σ- algebras of G such that G s G t if s < t; for all t b) X t : Ω E is a G t /B -measurable map such that X t (ω) = for all t > t if X t (ω) =, where is a cemetery state adjoined to E as an isolated point of E := E { } and B is the Borel σ-algebra on E. c) ζ(ω) := inf { t Xt (ω) = } (the lifetime of X) d) For each t, the map θ t : Ω Ω is such that X s θ t = X s+t for all s > (ii) (Markov property). For all x E, P x is a probability measure on (Ω, G) such that x P x (F ) is universally B-measurable for all F G, E x (f X ) = f(x), and E x (f X s+t G) = E x (P t f X s G) for all x E, s, t, f pb and G pg s, where Pt is the Markovian kernel on (E, B ) such that Pt 1 = 1 and Pt E = P t. Brownian motion. A (right) continuous Markov process (B t ) t with state space R n is called n-dimensional Brownian motion provided that its transition function is the Gaussian semigroup: P x (B t A) = 1 e x y 2 (2πt) n/2 2t dy, for all A B(R n ). A 6

8 The generator. Let F be a Banach space and (P t ) t be a semigroup of contractions on F. We define { } P t u u D(L) := u F : there exists lim F. t t For u D(L) we define: P t u u Lu := lim, t t The linear operator (L, D(L)) is called the infinitesimal operator (or generator) of the semigroup (P t ) t. Example. The infinitesimal operator of the Gaussian semigroup (P t ) t (regarded, e.g., as a C -semigroup of contractions on F = L 2 (R n, λ)) is the Laplace operator, we write P t = e t. More precisely, if u C(R), 2 then we have in L 2 (R n P, λ): lim tu u t = u. t 7

9 2 Sub-Markovian resolvent of kernels Let U = (U α ) α> be a sub-markovian resolvent of kernels on the Lusin measurable space (E, B). We shall denote by U the initial kernel of U : U = sup α> U α. If q >, then the family U q = (U q+α ) α> is also a sub-markovian resolvent of kernels on (E, B), having U q as (bounded) initial kernel. Excessive function. A function v pb is called U-supermedian if αu α v v for all α >. A U-supermedian function v is named U-excessive if in addition sup α> αu α v = v. We denote by E(U) (resp. S(U)) the set of all B- measurable U-supermedian functions. It is easy to check that S(U) and E(U) are convex cones. If v S(U) then the function v := sup α> αu α v is U-excessive and the set M = [v v] is U-neglijable, i.e., U β (1 M ) = for some (and hence all) β >. In addition the following assertions hold: (2.1) If u S(U) then: u E(U) if and only if u = û. (2.2) If (u n ) n is a sequence of U-supermedian functions which is pointwise increasing to u, then the function u is also U-supermedian and the sequence (û n ) n increases to û. In particular, (2.3) if (u n ) n is a sequence of U-excessive functions which is pointwise increasing to u, then the function u is also a U-excessive. A first main results on the U-excessive function is the following approximation result of G.A. Hunt. Theorem 2.1. Hunt s Approximation Theorem. Let U = (U α ) α> be a sub-markovian resolvent of kernels on the measurable space (E, B) and let us fix q >. Then for each v E(U q ) there exists a sequence (f n ) n in bpb such that U q f n is bounded for all n and the sequence (U q f n ) n is pointwise increasing to v. Proof. Let v n := inf(v, nuq 1). Note first that if x E is such that 8

10 U q 1(x) = and v E(U q ), then v(x) =. Indeed, we have v = sup n inf(v, n) and so U q+α (inf(v, n))(x) U q (n)(x) =, U q+α v(x) = sup n U q+α (inf(v, n))(x) =. Therefore v(x) = sup α αu q+α v(x) =. By (2.3) we deduce that the sequence (v n ) n is increasing and sup n v n = sup n inf(v, nw q 1) = v = v. Since v E(U q ) it follows that sup n nu q+n v n = v. If we set f n := n(v n nu q+n v n ), then U q f n = nu q+n v n. We conclude that the sequence (U q f n ) n is increasing and sup n U q f n = sup n nu q+n v n = v. Excessive measure. Recall that a σ-finite measure ξ on (E, (B)) is called U-excessive provided that ξ αu α ξ for all α >. We denote by Exc(U) the set of all U-excessive measures. Let further m be a fixed U-excessive measure. Notation: We denote by pb L p (E, m) the set of all B-measurable functions which belong to L p (E, m). If f L p (E, m) and f pb L p (E, m) is a m-version of f then clearly U α f is the element of L p (E, m) having the function U α f as m-version. Usually we shall identify a function from pb L p (E, m) with its class in L p (µ). For instance if f pb L p (E, m) then U α f denotes in the same time a function from pb L p (E, m) and the element from L p (E, m) having U α f as m-version. Theorem 2.2. Assume that the set E(U q ) of all B-measurable U q - excessive functions generates the σ-algebra B. If 1 < p < is fixed then the following assertions hold. i) If f pb and f = m-a.e. then U α f = m-a.e. for all α >. ii) If α >, 1 < p < and f L p (E, m) then U α f L p (E, m) and αu α f p f p. iii) For every f L p (E, m) we have lim α αu α f f p = Consequently, the family (U α ) α> becomes a C -resolvent of sub- Markovian contractions on L p (E, m). 9

11 Proof. By hypothesis we get αu α fdm fdm for all f pb and consequently if f = m-a.e. then U α f = m-a.e. for all α >. Also if f 1 m-a.e. then αu α f αu α 1 1 m-a.e. and thus U α becomes a continuous linear operator on L (E, m) and L 1 (E, m) respectively, such that αu α L (E,m) 1 and αu α L 1 (E,m) 1. ii) If α > and x E then αu α f(x) (αu α (f p )(x)) 1 p (αuα 1(x)) 1 p (αu α (f p )(x)) 1 p, where p is such that = 1. So, if f p p L p (E, m) then αu α f p dm αu α ( f p )dm f p dm. We conclude that U α f L p (E, m) and αu α p 1. iii) Because αu α f p 1 for all α >, it follows that the set A := {f L p (E, m)/ lim α αu α f f p = } is a closed subspace of L p (E, m). If q > and v be(u q ) L p (E, m) then v A. Indeed, from αu q+α v v we get lim α v αu α v p = lim α v αu q+α v p =. Let now g L p (E, m) be such that g, f = for all f A. Particularly, we have g U q fdm = g + U q fdm for all f L p +(E, m). Thus the last equality holds for all f pb. By the mass uniqueness we get g + dm = g dm, i.e., g =, hence A = L p (E, m) (Hahn-Banach Theorem). Nonbranch point. A point x E is called nonbranch point with respect to U provided that (N1) inf(u, v)(x) = inf(u, v)(x) for all u, v E(U) and (N2) 1(x) = 1 We denote by D U the set of all nonbranch points with respect to U. (2.4) By Hunt s approximation theorem and using (2.2) and (2.3), one can easyly see that: a point x E is a nonbranch point with respect to U if and only if (N2) holds and (N1) is verified for all bounded functions u, v E(U) of the form u = Uf and v = Ug with f, g bb. A U-excessive measure of the form µ U (where µ is a σ-finite measure) is called potential. We denote by Pot(U) the convex cone af all potential U-excessive measuares. 1

12 Further let L : Exc(U) E(U) R + be the energy functional (associated with U) defined by L(ξ, v) := sup{µ(v), Pot(U) µ U ξ} for all ξ Exc(U) and v E(U). The energy functional associated with U q will be denoted by L q. For the rest of the section we assume that q =. Let P denotes the transition function (P t ) t and define E(P) := {v : E R +, P t v v for all t > and lim t P t v = v} Proposition 2.3. We have E(U) = E(P). Proof. Let u E(P). From P t u u for all t > we obtain U α u = and so, u S(U). e αt P t u dt e αt u dt = u e αt dt = u α Because u E(P), the map t P t u is descreasing and there exists the pointwise limit lim P t u = sup P t u = lim P tn u = u, t n t> where (t n ) n is a sequence of positive numbers decreasing to zero. We have αu α u = α e αt P t u dt = e s P s/α u ds. For each fixed s > we have P s/α u u as α, therefore by dominated convergence we get lim α It follows that e s P s/α u ds = û = lim α αu α u = lim α 11 e s u ds = u e s P s/α u ds = u. e s ds = u.

13 Hence u E(U) and we conclude that E(P) E(U). Let now u E(U). By Hunt s approximation Theorem there exists a sequence (f n ) n bpb such that Uf n u. Consequently, to prove that the function v belongs to E(P), it is enough to show that P t Uf Uf for all t > and that lim t P t Uf = Uf. We have and P t Uf = P t+s f ds = lim P t Uf = lim t t t t P s f ds = P s f ds P s f ds = Uf P s f ds = Uf. Proposition 2.4. The following assertions hold for a resolvent of kernels U = (U α ) α> on (E, B). (i) The following two conditions are equivalent. (i.a) All the points of E are nonbranch points with respect to U. (i.b) The convex cone E(U) is min-stable and contains the positive constant functions, i.e., for all u, v E(U) we have inf(u, v) E(U) and 1 E(U). (ii) If U is the resolvent of right Markov process with state space E, then all the points of E are nonbranch points with respect to U. Proof. Since U is sub-markovian, we clearly have that 1 S(U). Let u, v be(u) and set w := inf(u, v). Then clearly w S(U) (i) The equivalence between (i.a) and (i.b) is a direct consequence of (2.1). (ii) Let X be the right Markov process having U as associated resolvent. Then every U-excessive function is a.s. right continuous along the paths of X, i.e., (2.5) If u E(U) then the function t u X t is a.s. right continuous on [, ). We already noted that the constant function 1 is U-supermedian. If x E then P t 1(x) = E x ([t < ζ]) and since E x (X = x) = 1 we deduce that a.s. ζ > and therefore lim t P t 1(x) = E x ([ < ζ]) = 1, hence 1 E(U) since by Proposition 2.2 we have E(U) = P(U). We also noted that if u, v be(u) then the function w = inf(u, v) is U-supermedian and by (2.4) w is also right continous along the 12

14 paths of X. By dominated convergence and again since E x (X = x) = 1 we get lim P t w(x) = lim P x (w X t ) = P x (w X ) = w(x). t t It follows that w P(U), so, again by Proposition 2.2, it is U- excessive, hence (i.b) holds and therefore also (i.a) is verified. We conclude that all the points of E are nonbranch points with respect to U and the proof is complete. We can state now the central result of this section. Theorem 2.5. Let (U α ) α> be a sub-markovian resolvent of kernels on (E, B), q > be fixed, and assume that the σ-algebra generated generated by E(U q ) is precisely B. Then the following three assertions are equivalent. (i) All the points in E are nonbranch points with respect to U. (ii) The following two properties hold. (UC) Uniqueness of charges: If µ, ν are two finite measures such that µ U q = ν U q then µ = ν. (SSP ) Specific solidity of potentials: If ξ, eta Exc(U q ) such that ξ + η = µ U q, then there exists a measure ν on E such that ξ = ν U q. (iii) The linear space [be(u q )] spanned by be(u q ) is an unitary algebra. Proof. We show first that (2.6) If v : E R + and ϕ : I R + is an increasing concave function, where I is an interval, I, such that Im(v) I, and if v S(U q ), then ϕ v S(U q ). Particularly, the vector space [bs(u q )] spanned by S(U q ) is an algebra. The first assertion follows by Jensen inequality, applied to the sub-probability µ x := αu q+α (x, dy) for all x E. Indeed, for all x E we have αu q+α (ϕ v)(x) = ϕ v dµ x ϕ(µ x (v)) = ϕ(αu q+α v(x)) ϕ(v(x)), 13

15 where the last inequality holds because ϕ is incresing and note that µ x (v) I because µ x (v) v(x). To prove that [bs(u q )] is an algebra, it is sufficient to show that v 2 [bs(u q )], for every v bs(u q ). We may assume that v 1 and let ϕ : [, 1] R + defined by ϕ(x) = 2x x 2. Then ϕ is concave and increasing, hence ϕ v bs(u a ) and therefore v 2 [bs(u q )]. (i) = (iii). As before, to prove that [be(u q )] is an algebra, it is sufficient to show that v 2 [be(u q )] for every v be(u q ). We may assume that v 1. By (2.6) it follows that v 2 belongs to [bs(u q )], v 2 = 2v w with w := 2v v 2 bs(u q ). It remains to show that w E(U q ). But w is a finely continuous U q -supermedian function, hence it is U q -excessive. (iii) = (i). Let A be the closure of [bs(u q )] in the supremum norm, it is a Banach algebra and therefore a lattice with respect to the pointwise order relation. Since lim α αu q+α v = v, pointwise for all v E(U q ) it follows that the same property holds for all v A. Consequently, since 1 A, we have 1 = 1 and if u 1, u 2 E(U q ) then the U q -supermedian function v = inf(u 1, u 2 ) belongs to A and therefore v = v, D Uq = E. (i) = (ii). We show that if µ, ν are two measures on (E, B) such that their potentials µ U q and ν U q are σ-finite and µ U q = ν U q, then µ = ν. Indeed, the resolvent equation implies that if β > then the measures µ U q+β and µ U q U q+β are σ-finite, hence (2.7) µ U q+β = ν U q+β for all β >. Let further g bpb, g >, be such that µ U q (g) = ν U q (g) < and set h := U q g, so < h L 1 (E, µ) L 1 (E, ν). If f [be(u q )], f 1, then fh [be(u q )] (because by the already proved implication (i) = (iii) it is an algebra) and therefore lim n nu q+n (fh) = fh. Since nu q+n (fh) nu q+n h h L 1 (E, µ) L 1 (E, ν), by (2.7) and the dominated convergence, we obtain that µ(fh) = ν(fh) for all f [be(u q )] (which is an algebra of bounded functions generating the σ-algebra B). By the monotone class theorem we conclude that µ = ν. Hence the unqueness of charges property (UC) holds. 14

16 We prove now that the specific solidity of potentials property (SSP) also holds. Let ξ, ηmu U q Exc(U q )such ξ + η = µ U q. We may assume that the measure µ is finite. Consider the functional ϕ ξ : be(u q ) R + defined as ϕ ξ (v) := L q (ξ, v) for all v be(u q ). Note that L q (ξ, v) L q (µ U q, v) = µ(v) <. We may extand ϕ ξ to a real valued linear functional on [be(u q )] and we get (2.8) ϕ ξ (f) + ϕ η (f) = µ(f) for all f [be(u q )]. Note that ϕ ξ is positive, i.e. (2.9) ϕ ξ (f) provided that f [be(u q )] is positive. This follows because (by the properties of the energy functional L q ) ϕ ξ is increasing as a functional on E(U q ): if u, v E(U q ) and u v, then ϕ ξ (u) ϕ η (v). We claim taht if (f n ) n [be(u q )] is decreasing poinwise to zero then the sequence (ϕ ξ (f n )) n also decreases to zero.note first that by monotene convergence we have lim n µ(f n ) =. From (2.8) and (2.9) it follows that ϕ ξ (f n ) µ(f n ) for all n and thus lim n ϕ ξ (f n ) lim n µ(f n ) =. We can apply now Daniell s theorem on the vector lattice [be(u q )], for the functional ϕ ξ. Hence there exists a positive measure ν on B such that ϕ ξ (f) = ν(f) for all f bpb. (Recall again that the σ-algebra generated by [be(u q )] is precisely B.) Taking f = U q g with g bpb, we get ξ(g) = L q (ξ, U q g) = ϕ ξ (f) = ν(u q g) for all g bpb, so ξ = ν U q. For the proof of (ii) = (i) see [St 89]. 15

17 3 Subordination by convolution semigroups A family (µ t ) t> of measures on R + is called a (vaguely continuous) convolution semigroup on R + if the following conditions are satisfied: (i) µ t (R +) 1 for all t >, (ii) µ s µ t = µ s+t for all s, t >, (iii) lim t µ t = ɛ (vaguely). Note that (i) and (iii) imply that lim t µ t (f) = f() for every f C b (R + ). In the sequel we fix a transition function P = (P t ) t> on (E, B) and a convolution semigroup (µ t ) t> on R +. For each t > we define the kernel kernel P µ t P µ t f := P s fµ t (ds) for all f bpb. on (E, B) by Proposition 3.1. The family P µ = (P µ t ) t is a sub-markovian semigroup of kernels on (E, B) and E(P) E(P µ ). The semigroup P µ = (P µ t ) t> is called the sub-markovian semigroup subordinated to P by means of (µ t ) t>. Proof. Since for all t 1, t 2 >, and f bpb we have P µ t 1 P µ t 1 f = = P s1 (P µ t 2 f)µ t1 (ds 1 ) = P s1 +s 2 fµ t2 (ds 2 )µ t1 (ds 1 ) = ( P s f(µ t1 +t 2 (ds) = P µ t 1 +t 2, ) P s1 P s2 fµ t2 (ds 2 ) µ t1 (ds 1 ) P s f(µ t1 µ t2 (ds) it follows that the family of kernels P µ is indeed a semigroup which is certainly sub-markovian. We prove now that E(P) E(P µ ) : 16

18 Fix u E(P). Then obviously P µ t u u for every t >. Let x X, a < u(x) and < b < 1. Then there exist s > and t > such that P s u(x) > a for every < s < s and µ t (], s [) > b, hence P µ t u(x) This implies that u E(P µ ). s P s u(x)µ t (ds) ab. Theorem 3.2. Assume that the resolvent U is proper and that all the points of E are nonbranch points with respect to U. Then the same property holds for the resolvent U µ associated with P µ. Proof. By Theorem 2.5 we have to show that conditions (UC) and (SSP ) are verified by the resolvent U µ. Let ν 1 and ν 2 be two positive finite measures on E such that ν 1 U q µ = ν 2 U q µ. Using Theorem 2.1 (Hunt s Approximation Theorem) it follows that ν 1 (v) = ν 2 (v) for all v E(U q µ ). Because E(P) E(P µ ) = E(U µ ) E(U q µ ), we get that ν 1 (v) = ν 2 (v) for all v E(U). In particular, we have ν 1 (Uf) = ν 2 (Uf) for all f bpb, hence ν 1 U = ν 2 U. Because by hypothesis all the points of E are nonbranch points with respect to U, by Theorem 2.5 we deduce that the uniqueness of charges property holds for U. It follows that ν 1 = ν 2 and we conclude that (UC) also holds for U µ. We check now that the specific solidity of potentials property (SSP ) holds with respect to U µ. Let ξ, η, and ν U q µ be U q µ -excessive measures such that (3.1) ξ + η = ν U µ q. We may assume that ν is a finite measure, consequently the measures ξ and η are also finite. We define the positive measures ξ and η on E by ξ (f) := L µ q (ξ, Uf), η (f) := L µ q (η, Uf) for all f bpb. We claim that ξ and η are U-excessive measures. Indeed, if α > then ξ αu α (f) = L µ q (ξ, αu α Uf) L µ q (ξ, Uf) = ξ (f). We show now that ξ is a σ-finite measure. If f o bpb, f o >, is such that Uf o 1, then ξ (f o ) = L µ q (ξ, Uf o ) L µ q (ν U µ q, Uf o ) = ν(uf o ) ν(1) <. 17

19 Hence the measure ξ is σ-finite. We conclude that ξ is U-excessive and analogously one gets that η is also a U-excessive measure. Using (3.1), we have for every f bpb ξ (f) + η (f) = L µ q (ξ, Uf) + L µ q (η, Uf) = L µ q (ξ + η, Uf) = L µ q (ν U µ q, Uf) = ν(u q f). We obtained that the following equality of U-excessive measures holds: ξ + η = ν U. Since by hypothesis the property (SSP ) holds for U, we deduce from the last equality that there exists a measure λ on E such that ξ = λ U. It follows that hence L µ q (ξ, Uf) = ξ (f) = λ(uf) = L µ q (λ U µ q, Uf), L µ q (ξ, Uf) = L µ q (λ U µ q, Uf) for all f bpb. In particular, taking f = U µ q g, with g bpb, and since UU µ q = U µ q U, it follows that for all g bpb we have ξ(ug) = L µ q (ξ, UU µ q g) = L µ q (λ U µ q, U µ q Ug) = λ U µ q (Ug) Note that in addition we have ξ(ug) ν(u µ q Ug) 1 q ν(ug). In particular, the measures ξ U and (λ U µ q ) U are σ-finite and equal. Because by the resolvent equation we have U q g = U(g qu q g) for all g bpb, g f o, it follows that ξ U q g = (λ U µ q ) U q g, and therefore ξ U q = (λ U µ q ) U q. By the uniqueness of charges propert (UC) for the resolvent U (see the proof of the implication (i) = (ii) in the proof of Theorem 2.5) we conclude that ξ = λ U µ q, so, the property (SSP ) holds with respect to U µ. 18

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