General criteria for the study of quasi-stationarity

Denis VillemonaisCollaborations with Nicolas Champagnat and Cecile Mailler

Universite de Lorraine, Nancy, France

UZH, Stochastische Prozesse, March 27, 2019

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1. Absorbed processes

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A generic example

Process evolving stochastically in a domain E ⊂R2, absorbed at theboundary

→ ∂ ∉ E unique absorbing point

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A generic example

Process evolving stochastically in a domain E ⊂R2, absorbed at theboundary

→ ∂ ∉ E unique absorbing point

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A generic example

Process evolving stochastically in a domain E ⊂R2, absorbed at theboundary

→ ∂ ∉ E unique absorbing point

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A generic example

Process evolving stochastically in a domain E ⊂R2, absorbed at theboundary

→ ∂ ∉ E unique absorbing point

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A generic example

Process evolving stochastically in a domain E ⊂R2, absorbed at theboundary

→ ∂ ∉ E unique absorbing point

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A generic example

Process evolving stochastically in a domain E ⊂R2, absorbed at theboundary

→ ∂ ∉ E unique absorbing point

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A generic example

Process evolving stochastically in a domain E ⊂R2, absorbed at theboundary

→ ∂ ∉ E unique absorbing point

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A generic example

Process evolving stochastically in a domain E ⊂R2, absorbed at theboundary

→ ∂ ∉ E unique absorbing point

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A generic example

Process evolving stochastically in a domain E ⊂R2, absorbed at theboundary

→ ∂ ∉ E unique absorbing point

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Formal defintion

Let (Xt)t∈[0,+∞[ evolving E ∪ {∂}, where ∂ ∉ E is absorbing.

Denoting by τ∂ = inf{t ≥ 0, Xt = ∂} the hitting time of ∂,

Xt = ∂, ∀t ≥ τ∂ almost surely.

Example : Stochastic Lotka-Volterra system in R2

The process (Xt)t∈[0,+∞) describes the population dynamic of twocompeting populations. It evolves in R2+ and satisfies the followingSDE

dX 1t =

√X 1

t dB1t +X 1

t (r− c11X 1t − c12X 2

t )dt,

dX 2t =

√X 2

t dB2t +X 2

t (r− c21X 1t − c22X 2

t )dt.

∂= (0,0) or {0}×R+∪R+× {0} : extinction of the whole populationor extinction of exactly one type.

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Formal defintion

Let (Xt)t∈[0,+∞[ evolving E ∪ {∂}, where ∂ ∉ E is absorbing.

Denoting by τ∂ = inf{t ≥ 0, Xt = ∂} the hitting time of ∂,

Xt = ∂, ∀t ≥ τ∂ almost surely.

Example : Stochastic Lotka-Volterra system in R2

The process (Xt)t∈[0,+∞) describes the population dynamic of twocompeting populations. It evolves in R2+ and satisfies the followingSDE

dX 1t =

√X 1

t dB1t +X 1

t (r− c11X 1t − c12X 2

t )dt,

dX 2t =

√X 2

t dB2t +X 2

t (r− c21X 1t − c22X 2

t )dt.

∂= (0,0) or {0}×R+∪R+× {0} : extinction of the whole populationor extinction of exactly one type.

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Limiting behaviour

In many interesting cases,

Px0 (Xt ∈ ·) −−−→t→∞ δ∂, ∀x0 ∈ E ∪ {∂}.

Question : What about the distribution of the process conditionednot to be absorbed, namely Px0 (Xt ∈ · | t < τ∂) ?

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Limiting behaviour

In many interesting cases,

Px0 (Xt ∈ ·) −−−→t→∞ δ∂, ∀x0 ∈ E ∪ {∂}.

Question : What about the distribution of the process conditionednot to be absorbed, namely Px0 (Xt ∈ · | t < τ∂) ?

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Limiting behaviour

In many interesting cases,

Px0 (Xt ∈ ·) −−−→t→∞ δ∂, ∀x0 ∈ E ∪ {∂}.

Question : What about the distribution of the process conditionednot to be absorbed, namely Px0 (Xt ∈ · | t < τ∂) ?

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Limiting behaviour

In many interesting cases,

Px0 (Xt ∈ ·) −−−→t→∞ δ∂, ∀x0 ∈ E ∪ {∂}.

Question : What about the distribution of the process conditionednot to be absorbed, namely Px0 (Xt ∈ · | t < τ∂) ?

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Limiting behaviour

In many interesting cases,

Px0 (Xt ∈ ·) −−−→t→∞ δ∂, ∀x0 ∈ E ∪ {∂}.

Question : What about the distribution of the process conditionednot to be absorbed, namely Px0 (Xt ∈ · | t < τ∂) ?

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Limiting behaviour

In many interesting cases,

Px0 (Xt ∈ ·) −−−→t→∞ δ∂, ∀x0 ∈ E ∪ {∂}.

Question : What about the distribution of the process conditionednot to be absorbed, namely Px0 (Xt ∈ · | t < τ∂) ?

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2. Quasi-stationary distributions

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Definition

A quasi-stationary distribution (QSD) is a probability measure αon E such that

α= limt→∞Pµ (Xt ∈ ·|t < τ∂)

for some initial probability measure µ on E.

Proposition

A probability measure α is a QSD if and only if, for any t ≥ 0,

α=Pα(Xt ∈ .|t < τ∂).

→ Surveys and book

Meleard, V. 2012, Van Doorn, Pollett 2013Collet, Martınez, San Martın 2013 (prior work)

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Definition

Let α be a QSD. The domain of attraction of α is the set of initialdistributions µ such that

limt→∞Pµ(Xt ∈ .|t < τ∂) =α.

In the general case :

Existence of a QSD is not true

Existence does not imply uniqueness of a QSD

Uniqueness does not imply attraction of all initialdistributions

Attraction of all initial distributions does not imply uniformconvergence

Question : how to guarantee some or all of the above properties ?

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Definition

Let α be a QSD. The domain of attraction of α is the set of initialdistributions µ such that

limt→∞Pµ(Xt ∈ .|t < τ∂) =α.

In the general case :

Existence of a QSD is not true

Existence does not imply uniqueness of a QSD

Uniqueness does not imply attraction of all initialdistributions

Attraction of all initial distributions does not imply uniformconvergence

Question : how to guarantee some or all of the above properties ?

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Definition

Let α be a QSD. The domain of attraction of α is the set of initialdistributions µ such that

limt→∞Pµ(Xt ∈ .|t < τ∂) =α.

In the general case :

Existence of a QSD is not true

Existence does not imply uniqueness of a QSD

Uniqueness does not imply attraction of all initialdistributions

Attraction of all initial distributions does not imply uniformconvergence

Question : how to guarantee some or all of the above properties ?

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3. Uniqueness of QSDs and exponential convergence

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Probabilistic approach

Let X evolving in E ∪ {∂} absorbed at ∂.

→ Assumption A1 (Doeblin condition)There exists a probability measure ν and c1 > 0 such that

Px(X1 ∈ ·|1 < τ∂) ≥ c1ν(·), ∀x ∈ E.

→ Assumption A2 (Harnack inequality)

Pν(t < τ∂)

Px(t < τ∂)> c2 > 0, ∀x ∈ E,t ≥ 0.

Theorem (Champagnat, V. 2016)

A1 and A2 ⇔ there exists C > 0, γ> 0 and α ∈M 1(E) such that,for all µ ∈M1(E),∥∥Pµ(Xt ∈ ·|t < τ∂)−α∥∥

TV ≤ Ce−γt .

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Probabilistic approach

Let X evolving in E ∪ {∂} absorbed at ∂.

→ Assumption A1 (Doeblin condition)There exists a probability measure ν and c1 > 0 such that

Px(X1 ∈ ·|1 < τ∂) ≥ c1ν(·), ∀x ∈ E.

→ Assumption A2 (Harnack inequality)

Pν(t < τ∂)

Px(t < τ∂)> c2 > 0, ∀x ∈ E,t ≥ 0.

Theorem (Champagnat, V. 2016)

A1 and A2 ⇔ there exists C > 0, γ> 0 and α ∈M1(E) such that,for all µ ∈M1(E),∥∥Pµ(Xt ∈ ·|t < τ∂)−α∥∥

TV ≤ Ce−γt .

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Probabilistic approach

Let X evolving in E ∪ {∂} absorbed at ∂.

→ Assumption A1 (Doeblin condition)There exists a probability measure ν and c1 > 0 such that

Px(X1 ∈ ·|1 < τ∂) ≥ c1ν(·), ∀x ∈ E.

→ Assumption A2 (Harnack inequality)

Pν(t < τ∂)

Px(t < τ∂)> c2 > 0, ∀x ∈ E,t ≥ 0.

Theorem (Champagnat, V. 2016)

A1 and A2 ⇔ there exists C > 0, γ> 0 and α ∈M1(E) such that,for all µ ∈M1(E),∥∥Pµ(Xt ∈ ·|t < τ∂)−α∥∥

TV ≤ Ce−γt .

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Comments and examples

A1 and A2 have been used in several situations· general one dimensional diffusion processes· multi-dimensional diffusion processes (with K. Coulibaly-P)· birth and death processes with catastrophes· multi-dimensional birth and death processes· branching/dying Brownian motions· time-inhomogeneous processes· Benaım, Cloez, Panloup 2016, Chazotte, Collet, Meleard 2017

A1 and A2 also imply several interesting properties· Spectral properties of the infinitesimal generator· Uniform convergence of eλ0tPx(t < τ∂) toward an eigenfunction· Existence and exponential ergodicity of the Q-process

Intrinsec limitations· Uniform convergence and uniqueness of QSD=⇒ compact state spaces or entrance boundary at infinity

or regularity of the boundaries· Not suited for the study of classical models (linear BD,

Orstein-Uhlenbeck, AR-1, Galton-Watson, etc...)

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Comments and examples

A1 and A2 have been used in several situations· general one dimensional diffusion processes· multi-dimensional diffusion processes (with K. Coulibaly-P)· birth and death processes with catastrophes· multi-dimensional birth and death processes· branching/dying Brownian motions· time-inhomogeneous processes· Benaım, Cloez, Panloup 2016, Chazotte, Collet, Meleard 2017

A1 and A2 also imply several interesting properties· Spectral properties of the infinitesimal generator· Uniform convergence of eλ0tPx(t < τ∂) toward an eigenfunction· Existence and exponential ergodicity of the Q-process

Intrinsec limitations· Uniform convergence and uniqueness of QSD=⇒ compact state spaces or entrance boundary at infinity

or regularity of the boundaries· Not suited for the study of classical models (linear BD,

Orstein-Uhlenbeck, AR-1, Galton-Watson, etc...)

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Comments and examples

A1 and A2 have been used in several situations· general one dimensional diffusion processes· multi-dimensional diffusion processes (with K. Coulibaly-P)· birth and death processes with catastrophes· multi-dimensional birth and death processes· branching/dying Brownian motions· time-inhomogeneous processes· Benaım, Cloez, Panloup 2016, Chazotte, Collet, Meleard 2017

A1 and A2 also imply several interesting properties· Spectral properties of the infinitesimal generator· Uniform convergence of eλ0tPx(t < τ∂) toward an eigenfunction· Existence and exponential ergodicity of the Q-process

Intrinsec limitations· Uniform convergence and uniqueness of QSD=⇒ compact state spaces or entrance boundary at infinity

or regularity of the boundaries· Not suited for the study of classical models (linear BD,

Orstein-Uhlenbeck, AR-1, Galton-Watson, etc...)

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4. A general criterion for the study of quasi-stationarity

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A direct application : perturbed dynamical systems

Let E be a measurable set of Rd with positive Lebesgue measureand let ∂ 6∈ E. Assume that

Xn+1 ={

f (Xn)+ξn if Xn 6= ∂ and f (Xn)+ξn ∈ E,

∂ otherwise,

where f :Rd →Rd is measurable and (ξn)n∈N is an i.i.d.non-degenerate Gaussian sequence in Rd.

Theorem (Champagnat, V. 2017)

If f is locally bounded such that |x|− |f (x)| −−−−−→|x|→+∞

+∞, then there

exists a quasi-stationary distribution νf attracting all initialdistributions on E admitting an exponential moment.

We will denote (later) by θf < 1 the associated eigenvalue :PνQSD (n < τ∂) = θn

f .

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The abstract criterion

Assumption (E). ∃ n1 ∈N, θ1,θ2,c1,c2,c3 > 0, ϕ1,ϕ2 : E →R+ and aprobability measure ν on K ⊂ E such that

(Local A1) ∀x ∈ K , Px(Xn1 ∈ ·) ≥ c1ν(·∩K ).

(Local A2) ∀x ∈ K ,

Pν(n < τ∂)

Px(n < τ∂)≥ c2, ∀n ∈N.

(Aperiodicity) ∀x ∈ K , ∃n4(x) such that

∀n ≥ n4(x), Px(Xn ∈ K ) > 0.

(Lyapunov) θ1 < θ2, ϕ1 ≥ 1, supK ϕ1 <∞, infK ϕ2 > 0, ϕ2 ≤ 1,

E(ϕ1(X1)) ≤ θ1ϕ1(x)+ c21K (x), ∀x ∈ E

E(ϕ2(X1)) ≥ θ2ϕ2(x), ∀x ∈ E.

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The abstract criterion

Assumption (E). ∃ n1 ∈N, θ1,θ2,c1,c2,c3 > 0, ϕ1,ϕ2 : E →R+ and aprobability measure ν on K ⊂ E such that

(Local A1) ∀x ∈ K , Px(Xn1 ∈ ·) ≥ c1ν(·∩K ).

(Local A2) ∀x ∈ K ,

Pν(n < τ∂)

Px(n < τ∂)≥ c2, ∀n ∈N.

(Aperiodicity) ∀x ∈ K , ∃n4(x) such that

∀n ≥ n4(x), Px(Xn ∈ K ) > 0.

(Lyapunov) θ1 < θ2, ϕ1 ≥ 1, supK ϕ1 <∞, infK ϕ2 > 0, ϕ2 ≤ 1,

E(ϕ1(X1)) ≤ θ1ϕ1(x)+ c21K (x), ∀x ∈ E

E(ϕ2(X1)) ≥ θ2ϕ2(x), ∀x ∈ E.

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Main result

Theorem (Champagnat, V. 2017)

If Assumption E is satisfied, then ∃α ∈ (0,1), C > 0 such that

∣∣Eµ [f (Xn) | n < τ∂

]−νQSD(h)∣∣≤ Cαnµ(ϕ1)

µ(ϕ2)

for all µ ∈M1(E) such that µ(ϕ1)/µ(ϕ2) <+∞ and all measurablefunction f : E →R such that |f | ≤ϕ1.

Moreover, ∃η : E →R+ and θ0 ≥ θ2 such that, for all |f | ≤ϕ1,∣∣θ−n0 Ex[f (Xn)]−η(x)α(f )

∣∣≤ Cαnϕ1(x), ∀n ≥ 0,

Note : the constants C and α only depend on quantities appearingin Assumption (E).

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Main result

Theorem (Champagnat, V. 2017)

If Assumption E is satisfied, then ∃α ∈ (0,1), C > 0 such that

∣∣Eµ [f (Xn) | n < τ∂

]−νQSD(h)∣∣≤ Cαnµ(ϕ1)

µ(ϕ2)

for all µ ∈M1(E) such that µ(ϕ1)/µ(ϕ2) <+∞ and all measurablefunction f : E →R such that |f | ≤ϕ1.

Moreover, ∃η : E →R+ and θ0 ≥ θ2 such that, for all |f | ≤ϕ1,∣∣θ−n0 Ex[f (Xn)]−η(x)α(f )

∣∣≤ Cαnϕ1(x), ∀n ≥ 0,

Note : the constants C and α only depend on quantities appearingin Assumption (E).

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Main ingredients of the proof

1 The main point of the proof is to use Hairer and Mattingly[2011] to prove that∥∥∥δxSn0n

0,n0n −δySn0n0,n0n

∥∥∥TV

≤ Cαn(2+ψn0n(x)+ψn0n(y))

for some time dependent Lyapunov function ψn and thesemi-group

Sn+mn,n+1f (x) = EXn=x

(f (Xn+1) | n+m < τ∂

)

2 In order to do so, we prove various estimates onEx(ϕ1(Xn))/Ex(ϕ2(Xn)), on Px(Xn ∈ K | n+m < τ∂) and so on.For instance, one central estimate is, ∀θ ∈ (θ1/θ2,1),

Ex(ϕ1(Xn) | n < τ∂)) ≤ Ex(ϕ1(Xn)1n<τ∂)

Ex(ϕ2(Xn)1n<τ∂)≤

(θnϕ1(x)

ϕ2(x)

)∨Cθ.

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Main ingredients of the proof

1 The main point of the proof is to use Hairer and Mattingly[2011] to prove that∥∥∥δxSn0n

0,n0n −δySn0n0,n0n

∥∥∥TV

≤ Cαn(2+ψn0n(x)+ψn0n(y))

for some time dependent Lyapunov function ψn and thesemi-group

Sn+mn,n+1f (x) = EXn=x

(f (Xn+1) | n+m < τ∂

)

2 In order to do so, we prove various estimates onEx(ϕ1(Xn))/Ex(ϕ2(Xn)), on Px(Xn ∈ K | n+m < τ∂) and so on.For instance, one central estimate is, ∀θ ∈ (θ1/θ2,1),

Ex(ϕ1(Xn) | n < τ∂)) ≤ Ex(ϕ1(Xn)1n<τ∂)

Ex(ϕ2(Xn)1n<τ∂)≤

(θnϕ1(x)

ϕ2(x)

)∨Cθ.

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Application 1. Convergence of a reinforced algorithm

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Consider the process (Yt)t≥0 in Rd evolving as follows :

→ Y evolves following the SDE

dYt = dBt +b(Yt)dt, Y0 ∈Rd (1)

→ and, with rate κ(Yt) ≥ 1, the process jumps with respect to itsempirical occupation measure 1

t

∫ t0 δYs ds.

Theorem (Champagnat, V. 2017 ; Mailler, V. 2018)

Assume that limsupx→+∞⟨b(x),x⟩

|x| <−32‖κ‖

1/2∞. Then

the solution to (1) admits a unique QSD νQSD with anexponential moment of order ‖κ‖1/2∞almost surely,

1

t

∫ t

0δYs ds

weak−−−−→t→+∞ νQSD.

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Consider the process (Yt)t≥0 in Rd evolving as follows :

→ Y evolves following the SDE

dYt = dBt +b(Yt)dt, Y0 ∈Rd (1)

→ and, with rate κ(Yt) ≥ 1, the process jumps with respect to itsempirical occupation measure 1

t

∫ t0 δYs ds.

Theorem (Champagnat, V. 2017 ; Mailler, V. 2018)

Assume that limsupx→+∞⟨b(x),x⟩

|x| <−32‖κ‖

1/2∞. Then

the solution to (1) admits a unique QSD νQSD with anexponential moment of order ‖κ‖1/2∞

almost surely,1

t

∫ t

0δYs ds

weak−−−−→t→+∞ νQSD.

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Consider the process (Yt)t≥0 in Rd evolving as follows :

→ Y evolves following the SDE

dYt = dBt +b(Yt)dt, Y0 ∈Rd (1)

→ and, with rate κ(Yt) ≥ 1, the process jumps with respect to itsempirical occupation measure 1

t

∫ t0 δYs ds.

Theorem (Champagnat, V. 2017 ; Mailler, V. 2018)

Assume that limsupx→+∞⟨b(x),x⟩

|x| <−32‖κ‖

1/2∞. Then

the solution to (1) admits a unique QSD νQSD with anexponential moment of order ‖κ‖1/2∞almost surely,

1

t

∫ t

0δYs ds

weak−−−−→t→+∞ νQSD.

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Application 2. Law of large numbers for branching processes

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Consider a branching version of the perturbed dynamical systemXn+1 = f (Xn)+ξn with the following branching mecanism :

→ starting from any particle at position x0 ∈ E, N0 ∼ Geom(p) isdrawn and the particle gives birth to N0 iid particles with thesame law as f (x0)+ξ0.

→ each new particle evolve and give birth as above,independently from the others. Denote by Popn the set ofparticles in the population at time n.

Theorem (V. 2019*)

If f is locally bounded such that

|x|− |f (x)| −−−−−→|x|→+∞

+∞,

and if θf /(1−p) > 1, then there exists a random variable W∞ ∈ L2

and three constants θ0 > 0,C > 0,α ∈ (0,1) such that

E

(∣∣∣∣∣θ−n0

∑xk∈Popn

g(xk)−W∞νQSD(g)

∣∣∣∣∣2)

≤ Cαn exp(x0)

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Consider a branching version of the perturbed dynamical systemXn+1 = f (Xn)+ξn with the following branching mecanism :

→ starting from any particle at position x0 ∈ E, N0 ∼ Geom(p) isdrawn and the particle gives birth to N0 iid particles with thesame law as f (x0)+ξ0.

→ each new particle evolve and give birth as above,independently from the others. Denote by Popn the set ofparticles in the population at time n.

Theorem (V. 2019*)

If f is locally bounded such that

|x|− |f (x)| −−−−−→|x|→+∞

+∞,

and if θf /(1−p) > 1, then there exists a random variable W∞ ∈ L2

and three constants θ0 > 0,C > 0,α ∈ (0,1) such that

E

(∣∣∣∣∣θ−n0

∑xk∈Popn

g(xk)−W∞νQSD(g)

∣∣∣∣∣2)

≤ Cαn exp(x0)

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Thank you !

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