Proof of Theorem 5.1.2

Let (Z⁽ⁿ⁾)n≥1 be a sequence of random measure-valued distributed as Z. In this section, we consider the following scaling: X⁽ⁿ⁾ = ¹_nZ⁽ⁿ⁾, and we describe the behaviour of this scaled process whenngoes to infinity.

To understand the behaviour of our model in a large population, we can consider that it starts from a deterministic probability measureX₀, and approach it by the interesting sequence defined by

X⁽ⁿ⁾₀ = 1 n

n

X

k=0

δ_Yk,

where(Y_k)k≥1is a sequence of i.i.d. random variable distributed according toX₀. In other words, we

Limit theorems for some branching measure-valued processes

set

Z⁽ⁿ⁾₀ =

n

X

k=0

δ_Yk.

The sequenceX⁽ⁿ⁾ converges. Indeed, by the branching property, we haveZ⁽ⁿ⁾ =^{d Pn}_k=0Z^Yk, where Z^Y_t^k are i.i.d., distributed as Z and starting from Z^Y₀^k = δ_Yk. Henceforth, if f is a continuous and bounded function then the classical law of large number gives

∀t≥0, lim

n→∞X⁽ⁿ⁾_t (f) =E

hZ^Y_t¹(f)ⁱa.s.

So by corollary5.2.4, it implies thatX⁽ⁿ⁾(pointwise) converges to the solution(µ_t)t≥0 of the follow- ing integro-differential equation:

µ_t(f) = µ₀(f) +

Z t 0

µ_s(Gf) (5.7)

+

Z

E

r(x)^X

k≥0

p_k(x)

Z 1 0

k

X

j=1

f(F_j^(k)(x, θ))dθ−f(x)µ_s(dx)ds.

Theorem5.1.2gives a stronger convergence.

Lemma 5.4.1(Semi-martingale decomposition). If Assumption5.2.1, then for all f ∈ C_c²(E) and t≥0,

X⁽ⁿ⁾_t (f) =X⁽ⁿ⁾₀ (f) +M⁽ⁿ⁾_t (f) +V⁽ⁿ⁾_t (f), where

V⁽ⁿ⁾_t (f) =

Z t 0

Z

E

Gf(x) +r(x)

Z 1 0

X

k∈N k

X

j=1

f(F_j^(k)(x, θ))−f(x)p_kdθX⁽ⁿ⁾_s (dx)ds,

andM⁽ⁿ⁾_t (f)is a square-integrable and càdlàg martingale. Its bracket is defined by hM⁽ⁿ⁾(f)i_t= 1

n

Z t 0

2X⁽ⁿ⁾_s (Gf²)−2X⁽ⁿ⁾_s (f×Gf) +

Z

E

r(x)

Z 1 0

X

k∈N^∗





k

X

j=1

f(F_j^(k)(x, θ))−f(x)





2

p_k(x)dθ X⁽ⁿ⁾_s (dx)ds.

Proof. It is a direct consequence of Lemma 5.2.3. Indeed, if L⁽ⁿ⁾ is the generator of X⁽ⁿ⁾ then it verifies

L⁽ⁿ⁾F_f(µ) =∂_tE[F_f(X⁽ⁿ⁾)|X⁽ⁿ⁾₀ =µ] _t=0 =∂_tE[F_{f /n}(Z⁽ⁿ⁾)|Z⁽ⁿ⁾₀ =nµ]_t=0 =LF_{f /n}(nµ),

Limit theorems for some branching measure-valued processes

whereF_f(µ) =F(µ(f)),F, f are two test functions andLis the generator ofZ.

Remark 5.4.2 (Non explosion). Let us recall that, by Lemma5.2.2, if the assumptions of Theorem 5.1.2hold then Assumption5.2.1holds; that is there is no explosion.

Let us denote byL(U)the law ofU, for any random variableU.

Lemma 5.4.3. Under the assumptions of Theorem5.1.2the sequence(L(X⁽ⁿ⁾))n≥1is uniformly tight in the space of probability measures onD([0, T],(M(E), d_v)).

Proof. We follow the approach of [FM04]. According to [RC86], it is enough to show that, for any continuous bounded functionf, the sequence of laws ofX⁽ⁿ⁾(f)is tight inD([0, T],R). To prove it, we will use the Aldous-Rebolledo criterion. LetC_c^∞ be the sef of functions of classC^∞ with finite support, we set S = C_c^∞∪ { 1}, where1 is the mapping x 7→ 1. We have to prove that, for any functionf ∈S, we have

1. ∀t≥0,X⁽ⁿ⁾_t (f)

n≥0is tight;

2. for alln ∈N, andε, η > 0, there existsδ >0such that for each stopping timeS_nbounded by T,

lim sup

n→+∞

sup

0≤u≤δP(|V⁽ⁿ⁾_Sn+u(f)−V⁽ⁿ⁾_S

n(f)| ≥η)≤ε.

lim sup

n→+∞

sup

0≤u≤δP(|hM⁽ⁿ⁾(f)i_Sn+u− hM⁽ⁿ⁾(f)i_Sn| ≥η)≤ε.

The first point is the tightness of the family of time-marginals(X⁽ⁿ⁾_t (f))_n≥1 and the second point, called the Aldous condition, gives a "stochastic continuity". It looks like the Arzelà-Ascoli Theorem.

Using Lemma5.2.2, there existsC >0such that

P(|X⁽ⁿ⁾_t (f)|> k)≤ kfk∞E[X⁽ⁿ⁾_t (1)]

k

≤ kfk∞CE[X⁽ⁿ⁾₀ (1)]

k ,

which tends to0asktends to infinity. This proves the first point. Let δ >0, we get for all stopping

Limit theorems for some branching measure-valued processes

timesS_n≤T_n≤(S_n+δ)≤T, that there existC⁰, C_f >0such that E[|V⁽ⁿ⁾_Tn(f)−V⁽ⁿ⁾_Sn(f)|] =E

"

Z Tn

Sn

X⁽ⁿ⁾_s (Gf) +

Z

E

r(x)

Z 1 0

X

k∈N k

X

j=1

fF_j^(k)(x, θ)−f(x)p_k(x)dθX⁽ⁿ⁾_s (dx)ds





≤C⁰[kGfk∞+kfk∞]×E[|Tn−Sn|]

≤C_fδ.

In the other hand, there existsC_f⁰ >0such that E[|hM⁽ⁿ⁾(f)i_Tn− hM⁽ⁿ⁾(f)i_Sn|]

=1 nE

"

Z Tn

Sn

2X⁽ⁿ⁾_s (Gf²)−2X⁽ⁿ⁾_s (f Gf) +

Z

E

r(x)

Z 1 0

X

k∈N k

X

j=1

fF_j^(k)(x, θ)−f(x)²p_kdθX⁽ⁿ⁾_s (dx)ds





≤C_f⁰δ n.

Then, for a sufficiently smallδ, the second point is verified and we conclude that (X⁽ⁿ⁾)n≥1 is uni- formly tight inD([0, T],(M(E), d_v)).

Proof of Theorem5.1.2. Let us denote by Xa limit process of(X⁽ⁿ⁾)_n≥1; namely there exists an in- creasing sequence(u_n)n≥1, onN^∗, such that(X^(un))n≥1converges toX. It is almost surely continuous in(M(E), v)since

sup

t≥0

sup

kfk∞≤1

|X⁽ⁿ⁾_t−(f)−X⁽ⁿ⁾_t (f)| ≤ ¯k

n. (5.8)

In the case whereEis compact, the vague and weak topologies coincide. By Doob’s inequality, there existsC >0such that

sup

f E

"

sup

t≤T

M⁽ⁿ⁾_t (f)

#

≤2 sup

f E

hhM⁽ⁿ⁾(f)iT

i≤ C n

where the supremum is taken over all the functionf ∈C_c²(E)such thatkfk∞≤1. Hence,

n→+∞lim sup

f E

"

sup

t≤T

M⁽ⁿ⁾_t (f)

#

= 0. (5.9)

Limit theorems for some branching measure-valued processes

But as

M⁽ⁿ⁾_t (f) = X⁽ⁿ⁾_t (f)−X⁽ⁿ⁾₀ (f)

−

Z t 0

Z

E

Gf(x) +r(x)

Z 1 0

X

k∈N k

X

j=1

fF_j^(k)(x, θ)−f(x)p_k(x)dθX⁽ⁿ⁾_s (dx)ds,

we have

0 =X_t(f)−X₀(f)−

Z t 0

X_s(Gf) +

Z

E

r(x)





k

X

j=1

f(F_j^(K)(x, θ))p_k(x)dθ−f(x)



X_s(dx)ds.

Since this equation has a unique solution, it ends the proof whenE is compact. This approach fails in the non-compact case. Nevertheless, we can use the Méléard-Roelly criterion [MR93]. We have to prove thatXis inC([0, T],(M(E), w))andX⁽ⁿ⁾(1)converges toX(1). By (5.8), Xis continuous.

To prove thatX⁽ⁿ⁾(1)converges toX(1), we use the following lemmas.

Lemma 5.4.4(Approximation of indicator functions). For eachk ∈N, there existsψ_k∈C²(E)such that:

∀x∈E, 1_[k;+∞[(x)≤ψk(x)≤1_[k−1;+∞[(x)and∃C, Gψk≤Cψk−1. Proof. See [JMW12, lemma 4.2] or [MT12, lemma 3.3].

Lemma 5.4.5(Commutation of limits). Under the assumptions of Theorem5.1.2,

k→+∞lim lim sup

n→+∞ E

"

sup

t≤T

X⁽ⁿ⁾_t (ψ_k)

#

= 0, where(ψ_k)k≥0 are defined as in the previous lemma.

The proof is postponed after. Hence, a same computation to [MT12] gives us the convergence in D([0, T],(M(E), w)). Thus, each subsequence converges to the equation (5.7). The end of the proof follow with the same argument of the compact case.

We can give another argument, which does not use the Méléard-Roelly criterion [MR93]. As sup

t≥0

sup

kfk∞≤1

|X⁽ⁿ⁾_t−(f)−X⁽ⁿ⁾_t (f)| ≤ ¯k n,

Xis continuous from[0, T]to(M(E), d_w). LetGbe a Lipschitz function onC([0, T],(M(E), d_w)),

Limit theorems for some branching measure-valued processes

we get,

|E[G(X^(un))]−G(X)| ≤E

"

sup

t∈[0,T]

d_wX^(u_t ⁿ⁾,X_t

#

≤E

"

sup

t∈[0,T]

d_wX^(u_t ⁿ⁾,X^(u_t ⁿ⁾(· ×(1−ψ_k))

#

+E

"

sup

t∈[0,T]

d_wX^(u_t ⁿ⁾(· ×(1−ψ_k)),X_t(· ×(1−ψ_k))

#

+ sup

t∈[0,T]

d_w(X_t(· ×(1−ψ_k)),X_t).

According to Lemma5.4.5, we obtain that

k→+∞lim lim sup

n→+∞ E

"

sup

t∈[0,T]

d_wX^(u_t ⁿ⁾,X^(u_t ⁿ⁾(· ×(1−ψ_k))

#

= 0 and

k→+∞lim sup

t∈[0,T]

d_w(X_t(· ×(1−ψ_k)),X_t) = 0.

Then, we have

d_wX^(u_t ⁿ⁾(· ×(1−ψ_k)),X_t(· ×(1−ψ_k))

=d_vX^(u_t ⁿ⁾(· ×(1−ψ_k)),X_t(· ×(1−ψ_k)). Thus,

k→+∞lim lim sup

n→+∞ E

"

sup

t∈[0,T]

d_wX^(u_t ⁿ⁾(· ×(1−ψ_k)),X_t(· ×(1−ψ_k))

#

= 0, by continuity ofν 7→ν(1−ψ_k)inD(M(E), d_v). And finally,

n→+∞lim GX^(un)=G(X), which completes the proof.

Limit theorems for some branching measure-valued processes

proof of Lemma5.4.5. Ifµ^n,k_t =E(X⁽ⁿ⁾_t (ψ_k))then we have µ^n,k_t =E[X⁽ⁿ⁾₀ (ψ_k)] +

Z t 0 E

Z

E

Gψ_k(x) +r(x)



 X

k≥1 k

X

j=1

p_k(x)

Z 1 0

ψ_k(F_j^(k)(x, θ))−ψ_k(x)



X⁽ⁿ⁾_s (dx)



ds

≤µ^n,k₀ +C

Z t 0

µ^n,k−1_s +µ^n,k_s ds.

Now, by Gronwall’s Lemma, iteration and monotonicity, we deduce that µ^n,k_t ≤C1(µ^n,k₀ +

Z t 0

µ^n,k−1_s ds)

≤C₁µ^n,k₀ +C₁²T µ^n,k−1₀ +

Z t 0

Z s 0

µ^n,k−2_u duds

≤

k−1

X

l=0

µ^n,k−l₀ C₁(C₁T)^l

l! +C₂× (C₁T)^k k!

≤µ^n,bk/2c₀ C₁e^C1T +C₃ ^X

l>bk/2c

(C₁T)^l

l! +C₂× (C₁T)^k k! , whereC₁, C₂ andC₃are three constants. Thus,

k→+∞lim lim sup

n→+∞

µ^n,k_t = 0.

And finally the following expression completes the proof, E

"

sup

t≤T

|Xⁿ_t(ψ_k)|

#

≤µ^n,k₀ +C

Z t 0

µ^n,k−1_s +µ^n,k_s ds+E

"

sup

t≤T

|M⁽ⁿ⁾_t (ψ_k)|

#

.

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