Limit theorems

for anyφ∈L²(X;h²m). This together with (4.1) yields that

tlim→∞e^λ1tEx[exp (A^µ_t)f(Xt)] =h(x) Z

X

f h dm, x∈X (4.6)

for anyf ∈L²(X;m).

By (1.20),

Ex

·³

e^λ1tZt(g)

´₂¸

= I + II, (4.12)

where

I :=e^2λ1tEx

h exp

³

A^(Q_t ^−1)µ

´ g(Xt)²

i

and

II :=E_x

·Z _t

0

exp

³

2λ₁s+A^(Q_s ^−1)µ

´ ³

e^λ1(t−s)E_Xs h

exp

³

A^(Q_t−s^−1)µ

´

g(X_t−s) i´2

dA^Rµ_s

¸ .

Recall that λ^h₂ := λ^h₂((Q−1)µ) > 0 as we saw in (4.2). Then, since for any positive ε <

(−λ₁)∧(2λ^h₂),

sup

(x,t)∈X×(0,∞)

E_x h

exp

³

(2λ₁+ε)t+A^(Q_t ^−1)µ

´i

<∞ by Theorem 5.1 of [14] or Theorem 2.4 of [52], it follows that

I≤e^−εtE_x h

exp

³

(2λ₁+ε)t+A^(Q_t ^−1)µ

´i∥g∥²_L∞(X;m) ≤c₁e^−εt∥g∥²_L∞(X;m).

By (4.4),

II ≤ c₂E_x

·Z _t

0

exp

³

2λ₁s+A^(Q_s ^−1)µ

´

e^{−2λh2(t−s)}dA^Rµ_s

¸

∥g∥²_L²_(X;m)

≤ c2e^−εtEx

·Z _ζ

0

exp

³

(2λ1+ε)s+A^(Q_s ^−1)µ

´ dA^Rµ_s

¸

∥g∥²_L²_(X;m). Since

inf

½

E(u, u)− Z

X

u²(Q−1)dµ−(2λ₁+ε) Z

X

u²dm:u∈ F, Z

X

u²dm= 1

¾

=−λ₁−ε

>0 by the definition ofλ₁, we deduce from Theorem 1.2 and [52, Lemma 3.5] that

sup

x∈X

E_x

·Z _ζ

0

exp

³

(2λ₁+ε)s+A^(Q_s ^−1)µ

´ dA^Rµ_s

¸

<∞, and thus II≤c₃e^−εt∥g∥²_L∞(X;m). Combining these estimates implies that

Ex

·³

e^λ1tZt(g)

´₂¸

≤(c1+c3)e^−εt∥g∥²_L∞(X;m). (4.13) Furthermore, by Chebyshev’s inequality,

P_x³¯¯¯e^λ1tZ_t(g)¯¯¯> δ

´≤ 1 δ²E_x

·³

e^λ1tZ_t(g)

´2¸

≤ C

δ²e^−εt∥g∥²_L∞(X;m)

for anyδ >0, and the last term above goes to 0 ast→ ∞. Therefore (4.10) follows.

Let f and g be the same as above, respectively. By (4.13), X∞

n=1

E_x

·³

e^λ1tnZ_t(g)

´2¸

≤C X∞ n=1

e^−εtn <∞.

By Borel-Cantelli’s lemma, we have limn→∞e^λ1tnZtn(g) = 0 Px-a.s., and so (4.11) as limt→∞Mt= M_∞ P_x-a.s.

We will assume the following on X and the branching rate:

Assumption 4.4. Either (i) or (ii) holds:

(i) It holds that Px(ζ < ∞) = 1 for every x ∈ X and the branching rate µ ∈ K∞ satisfies RR

X×XG^µ(x, y)µ(dx)µ(dy)<∞.

(ii) Mis Harris recurrent and the branching rate µ∈ K_∞ satisfiesµ(X)<∞.

Recall that we proved Lemma 3.1 and Proposition 3.3 thatMtin (4.9) is a square integrable martingale and that

{e0 =∞}={M_∞∈(0,∞)} Px-a.s., (4.14) wheree0 is the extinction timeof Mdefined by

e₀ = inf{t >0 :Z_t= 0}.

We then get the following immediately from the above, Lemma 4.2 and Proposition 4.3.

Corollary 4.5. (i) It holds that

tlim→∞

Zt(A)

Ex[Zt(A)] = M_∞

h(x) in Px-probability for every Borel subsetA in X such that 0< m(A)<∞.

(ii) Suppose that Assumption 4.4 holds. If m(X)<∞, then

tlim→∞

Zt(A) Z_t =

R

Ah dm R

Xh dm in P^e_x-probability for every Borel subsetA in X, where P^e_x(·) =Px(· |e0=∞).

(iii) Suppose that Assumption 4.4 holds. Let {t_n} be any sequence as in Proposition 4.3. If m(X)<∞, then

nlim→∞

Z_tn(A) Ztn

= R

Ah dm R

Xh dm P^e_x-a.s.

for every Borel subsetA in X.

Proof. LetA be a Borel subset inX such that 0< m(A)<∞. Then combining Proposition 4.3 with Lemma 4.2 implies that

tlim→∞

Z_t(A)

E_x[Z_t(A)] = lim

t→∞

e^λ1tZ_t(A) e^λ1tE_x[Z_t(A)]

= M_∞R

Ah dm h(x)R

Ah dm = M_∞

h(x) inPx-probability,

whence (i) holds. We now that assume that Assumption 4.4 holds and that 0 < m(X) <∞. Then, since the constant function belongs toL²(X;m), we obtain by Proposition 4.3 and (4.14),

tlim→∞

Z_t(A) Zt

= lim

t→∞

e^λ1tZ_t(A) e^λ1tZt

= M_∞R

Ah dm M_∞R

Xh dm = R

Ah dm R

Xh dm inP^e_x-probability, which yields (ii). By the same way, (iii) follows.

Corollary 4.5 is an extension of the result for branching Brownian motions by S. Watanabe [61, Corollary on p.397] to branching symmetric Hunt processes with state dependent branching rates and branching mechanisms.

Remark 4.6. Let M^α = (Xt, Px) be the symmetric α-stable process on R^d. Since (4.6) is true for any f ∈ Bb(R^d) by [55, Corollary 4.7], Lemma 4.2 holds for any f ∈ Bb(R^d). We now consider the branching symmetricα-stable process with motion component M^α and branching rate µ∈ K^R_∞^d. Then for any f ∈ Bb(R^d),

sup

(x,t)∈R^d×(1,∞)

¯¯¯e^λ1tE_x h

exp

³

A^(Q_t ^−1)µ

´

f(X_t)i¯¯¯

≤C_p∥f∥L^∞(R^d;dx) sup

x∈R^d∥h(x)p^h₁(x,·)∥L^p(R^d;h²dx)

°°°°1 h

°°°°

L^q(R^d;h²dx)

<∞

(4.15)

for any p >2 +n/α and q =p/(p−1) by Lemmas 4.4 and 4.6 of [55], where Cp is a positive constant depending on p. Thus II in the proof of Proposition 4.3 converges to 0 as t → ∞ for any f ∈ Bb(R^d) by combining (4.15) with the dominated convergence theorem, instead of the inequality (4.4). As a result, (4.10) holds for anyf ∈ Bb(R^d), which leads us to that Corollary 4.5 (i) and (ii) hold for every Borel subsetA inR^d.

We are now in a position to establish the following almost sure convergence ofe^λ1tZ_t(f).

Theorem 4.7. There exists a subspace Ω₀ ⊂Ω of full probability such that, for every ω ∈Ω₀ and for every bounded Borel measurable function f on X with compact support whose set of discontinuous points has zero m-measure,

tlim→∞e^λ1tZt(f)(ω) =M_∞(ω) Z

X

f h dm. (4.16)

Observe thathis strictly positive and continuous onX. So every bounded Borel measurable functionf with compact support is bounded by chfor somec >0.

Our approach to Theorem 4.7 is similar to that to [4, Theorem 1’]. We now prove two lemmas. Letδ be a positive constant 0< δ <(−λ1)∧2λ^h₂ and denote byX^nδ,i_t the particles at timet≥nδsuch that whose parent isXⁱ_nδ. LetU be a nearly Borel subset ofX, and forx∈X and ε >0,

U^ε(x) :=

½

y∈U : h(y)≥ 1 1 +εh(x)

¾ . Define

Y_n,i^δ,ε= 1

1 +εh(Xⁱ_nδ)1ⁿ

X^nδ,i_t ∈U^ε(Xⁱ_nδ) for everyt∈[nδ,(n+1)δ]o

and Sn^δ,ε=e^λ1nδP_Znδ

i=1Y_n,i^δ,ε.

Lemma 4.8. It holds that

nlim→∞

µ

S_n^δ,ε−E_x

· S_n^δ,ε¯¯

¯¯Gnδ

¸¶

= 0 P_x-a.s.

Proof. A direct calculation implies that Ex

"µ

S_n^δ,ε−Ex

· S_n^δ,ε¯¯

¯¯Gnδ

¸¶2#

=Ex

"³ S_n^δ,ε

´2

−2S_n^δ,εEx

· S_n^δ,ε¯¯

¯¯Gnδ

¸ +Ex

· S_n^δ,ε¯¯

¯¯Gnδ

¸2#

=E_x

"

E_x

·³ S_n^δ,ε

´2 ¯¯

¯¯Gnδ

¸

−E_x

· S_n^δ,ε¯¯

¯¯Gnδ

¸2# .

(4.17) Since

Ex

·³ S_n^δ,ε

´₂ ¯¯

¯¯Gnδ

¸

=e^2λ1nδEx



X^Znδ

i=1

³ Y_n,i^δ,ε

´₂

+ X

1≤i,j≤Znδ,i̸=j

Y_n,i^δ,εY_n,j^δ,ε¯¯

¯¯Gnδ





=e^2λ1nδ

Znδ

X

i=1

E_x

·³ Y_n,i^δ,ε

´2 ¯¯

¯¯Gnδ

¸

+e^2λ1nδ X

1≤i,j≤Znδ,i̸=j

E_x

· Y_n,i^δ,ε¯¯

¯¯Gnδ

¸ E_x

· Y_n,j^δ,ε¯¯

¯¯Gnδ

¸

and E_x

· S_n^δ,ε¯¯

¯¯Gnδ

¸2

= Ã

e^λ1nδ

Znδ

X

i=1

E_x

· Y_n,i^δ,ε¯¯

¯¯Gnδ

¸!²

=e^2λ1nδ

Znδ

X

i=1

E_x

· Y_n,i^δ,ε¯¯

¯¯Gnδ

¸2

+e^2λ1nδ X

1≤i,j≤Znδ,i̸=j

E_x

· Y_n,i^δ,ε¯¯

¯¯Gnδ

¸ E_x

· Y_n,j^δ,ε¯¯

¯¯Gnδ

¸ ,

the last term of (4.17) is equal to e^2λ1nδE_x

"_Z Xnδ

i=1

à E_x

·³ Y_n,i^δ,ε

´2 ¯¯

¯¯Gnδ

¸

−E_x

· Y_n,i^δ,ε¯¯

¯¯Gnδ

¸2!#

≤e^2λ1nδE_x

"_Z Xnδ

i=1

E_x

·³ Y_n,i^δ,ε

´2 ¯¯

¯¯Gnδ

¸#

.

By the Markov property and (1.19), the last term above is equal to e^2λ1nδE_x

"_Z Xnδ

i=1

E_Xi nδ

·³ Y_0,1^δ,ε

´2¸#

=e^2λ1nδE_x

· exp

³

A^(Q_nδ^−1)µ

´ E_Xnδ

·³ Y_0,1^δ,ε

´2¸¸

≤e^2λ1nδEx

h exp

³

A^(Q_nδ^−1)µ

´

h(Xnδ)² i

≤e^2λ1nδEx

h exp

³

A^(Q_nδ^−1)µ

´

h(X_nδ)

i∥h∥L^∞(X;m)

=e^λ1nδh(x)∥h∥L^∞(X;m). Therefore

X∞ n=0

Ex

"µ

S_n^δ,ε−Ex

· S_n^δ,ε¯¯

¯¯Gnδ

¸¶2#

<∞,

which yields the desired result by an application of Borel-Cantelli’s lemma.

Lemma 4.9. It holds that lim inf

t→∞ e^λ1tZt(1Uh)≥M_∞ Z

U

h²dm Px-a.s. (4.18)

for every open subsetU in X.

Proof. Since e^λ1tZ_t(1_Uh) ≥ e^λ1δS_n^δ,ε for any t ∈ [nδ,(n+ 1)δ], the Markov property and Lemma 4.8 yield that

lim inf

t→∞ e^λ1tZt(1_Uh)≥e^λ1δlim inf

n→∞ S_n^δ,ε

=e^λ1δlim inf

n→∞ e^λ1nδ

Znδ

X

i=1

E_Xⁱ

nδ

h S₀^δ,ε

i

= e^λ1δ

1 +εlim inf

n→∞ e^λ1nδ

Znδ

X

i=1

h(Xⁱ_nδ)P_Xⁱ

nδ

³

Xt∈U^ε(X0) for everyt∈[0, δ]

´ .

By (4.11), the right hand side above is equal to e^λ1δ

1 +εM_∞ Z

X

Px

¡Xt∈U^ε(X0), for everyt∈[0, δ]¢

h(x)²m(dx)

= e^λ1δ 1 +εM_∞

Z

X

Ex

h

e^−Aµδ;δ < τε

i

h(x)²m(dx)

≥ e^λ1δ 1 +εM_∞

Z

U

Ex

h

e^−Aµδ;δ < τε

i

h(x)²m(dx),

whereτε= inf{t >0 :Xt∈/ U^ε(X0)}.SinceXtis right continuous, the last term above converges toM_∞R

Uh²dm by letting firstδ→0 and then ε→0, whence (4.18) holds.

Proof of Theorem 4.7. Since X is a locally compact separable metric space, there exists a countable baseU of open set{U_k, k≥1}that is closed under finite union. By Lemma 4.9, there existsΩ₀⊂Ωof full probability so that for every ω∈Ω₀,

lim inf

t→∞ e^λ1tZt(1U_kh)(ω)≥M_∞(ω) Z

Uk

h²dm for everyU_k∈ U.

For any open set U, there exists a sequence of increasing open sets {U_nk, k ≥ 1} inU so that

∪^∞_k=1U_nk =U. We have for everyω∈Ω₀, lim inf

t→∞ e^λ1tZ_t(1_Uh)(ω)≥lim inf

t→∞ e^λ1tZ_t(1_Unkh)(ω)

≥M_∞(ω) Z

U_nk

h²dm for everyk≥1.

Passingk→ ∞yields that lim inf

t→∞ e^λ1tZ_t(1_Uh)(ω)≥M_∞(ω) Z

U

h²dm. (4.19)

We now consider (4.16) on {M_∞ > 0}. For each fixed ω ∈ Ω₀ ∩ {M_∞ > 0}, define the probability measures µt andµ onX respectively, by

µ_t(A)(ω) = e^λ1tZ_t(1_Ah)(ω)

Mt(ω) and µ(A) =

Z

A

h²dm, A∈ B(X)

for every t≥0. Note that the measure µ_t is well-defined for every t≥0. The inequality (4.19) tells us that µt converges weakly to µ (for example, see [25, Theorem 9.1 on p.164]). Since h is strictly positive and continuous on X, for every function f on X with compact support on X whose discontinuity set has zero m-measure (equivalently zero µ-measure), ϕ := f /h is a bounded function having compact support with the same set of discontinuity with f. We thus have

tlim→∞

Z

X

ϕ dµt= Z

X

ϕ dµ, (4.20)

which is equivalent to say that

tlim→∞e^λ1tZt(f)(ω) =M_∞(ω) Z

X

f h dm for everyω∈Ω0∩ {M_∞>0}. (4.21) Since, for every function f on X such that|f|is bounded bychfor somec >0,

e^λ1t|Z_t(f)| ≤e^λ1tZ_t(|f|)≤cM_t,

(4.21) holds automatically on{M_∞= 0}. This completes the proof of the theorem.

In a similar way to that yielding Corollary 4.5, we obtain from Theorem 4.7 and Lemma 4.2 the following:

Corollary 4.10. Let Ω₀ be the same as in Theorem 4.7.

(i) (A law of large numbers) It holds that

tlim→∞

Z_t(A)(ω)

Ex[Zt(A)] = M_∞(ω) h(x)

for every ω ∈ Ω₀ and for every relatively compact Borel subset A in X having m(A) > 0 and m(∂A) = 0.

(ii) Suppose also that Assumption 4.4 holds. Then

tlim→∞

Zt(A)(ω) Zt(B)(ω) =

R

Ah dm R

Bh dm

for every ω∈Ω₀∩ {e₀ =∞}and for every pair of relatively compact Borel subsets A andB in X having m(A), m(B)>0 and m(∂A) =m(∂B) = 0 respectively.

ドキュメント内 PDF Asymptotic Properties of Branching Symmetric Markov Processes - Osaka U (ページ 53-59)