Convergence exponentielle uniforme vers la distribution quasi-stationnaire de processus de Markov absorbés

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1 Convergence exponentielle uniforme vers la distribution quasi-stationnaire de processus de Markov absorbés Nicolas Champagnat, Denis Villemonais Séminaire commun LJK Institut Fourier, Méthodes probabilistes et statistiques en dynamique des populations, Grenoble, 15/12/2016

2 Introduction Population dynamics with extinction Stationary behavior of population dynamics: usually extinction However, we only observe populations which are not extinct, and most often, they exhibit a stationary behavior Quasi-stationary distributions (QSD) are stationary distributions conditionally on non-extinction (goes back to Seneta and Vere-Jones, 1966) Introduction aux processus de Markov avec extinction Existence et unicité des distributions quasi-stationnaires Calcul des distributions quasi-stationnaires Quasi-stationarity can explain e.g. mortality plateau Quelques simulations Les échelles de temps References Questions: Speed of convergence of conditional distributions? Dependence w.r.t. initial distribution? Un objet spécifique Plateaux de mortal Observations et défi Retour aux applica Slowing of Mortality Rates at Older Ages In La (1992) Carey et al. Sur une éch courte (que observe une parmis les i

3 Introduction Another example of application: molecular dynamics Parallel replica algorithm: For simulation of metastable dynamics (e.g. bio-molecular dynamics), to speed up the exit from an attracting domain of the dynamics: when the dynamics is blocked in some attracting part of the space, run N copies of the process in parallel, wait for the first one which exits the attracting set, at a time T continue the dynamics from this exit point and approximate the true exit time as NT. The theoretical justification of this algorithm (cf. Le Bris, Lelièvre, Luskin, Perez, 2012) relies on quasi-stationary distributions, and more precisely on the speed of convergence of conditional distributions to the QSD.

4 Introduction Notations We consider a general stochastic population dynamics (X t, t 0) with values in E { } s.t. E is arbitrary (measurable space), e.g. E = N, (N ) d (size of populations), (0, ) or (0, ) d (population densities) E is an absorbing (cemetary) point, e.g. =. 0, N d \ (N ) d or {x R d + : x i = 0 for some i}. X is a Markov process P x denotes its law given X 0 = x E { }, and for µ probability measure on E, P µ = E P x dµ(x) τ := inf{t 0 : X t = } Assumptions: t τ, X t = (absorbing point) x E, P x (τ < ) = 1 (almost sure absorption) x E, t 0, P x (t < τ ) > 0 (possible survival up to any time)

5 Introduction Quasi-stationary distributions (cf. survey [Méléard-Villemonais, 2012]) Définition Quasi-stationary distribution (QSD): α P(E) s.t. t 0, P α (X t t < τ ) = α Quasi-limiting distribution (QLD): α P(E) s.t. µ P(E), P µ (X t t < τ ) α (for some topology) Yaglom limit: α P(E) s.t. x E, P x (X t t < τ ) α Universal QLD: α P(E) s.t. µ E, P µ (X t t < τ ) α Proposition α universal QLD α Yaglom limit α QLD α QSD If α is a QSD, then λ 0 > 0 s.t. P α (t < τ ) = e λ0t L α = λ 0 α, where L is the infinitesimal generator of X

6 Introduction Observation of a QSD and domain of attraction Even in cases where all states of the population communicate (except ), there is not necessarily uniqueness of a QSD for example, subcritical branching processes. Then, which QSD is observed in a population? After what time? When is it independent of the initial distribution? To each QSD α is associated a basin of attraction, which is the set of initial distributions for which α is the QLD. Usually, the basin of attraction of a QSD (and the speed of convergence to the QSD) is determined by the tail of the initial distribution (e.g. for branching processes, finiteness of moments) impossible to observe/estimate in practice

7 Introduction Aim of the talk First goal: necessary and sufficient conditions for the existence (and uniqueness) of a universal QSD with uniform exponential rate of convergence. Second goal: apply these criteria to various classes of population dynamics and types of extinction. The mathematical study of QSD makes usually use of spectral theoretic tools (Perron-Frobenius, Krein-Rutman, Sturm-Liouville, L 2 theory for self-adjoint operators...) This talk: probabilistic criteria and tools (coupling) Our criteria are inspired from classical criteria in the irreducible case (ergodicity coefficient of Dobrushin, Doeblin condition, cf. Meyn and Tweedie, 1993)

8 The general criterion The criterion Assumption (A) There exists a probability measure ν on E s.t. (A1) t 0, c 1 > 0 s.t. x E, P x (X t0 t 0 < τ ) c 1 ν( ) (A2) c 2 > 0 s.t. x E and t 0, P ν (t < τ ) c 2 P x (t < τ )

9 The general criterion Interpretation of Conditions (A) (A1) Coupling of conditional laws independently of the initial condition. Typically, ν = c1 K where K is a compact set. Example: diffusion on E = R + and = 0, K = {a x b}. if x > b, we need P x (τ K < t 0) c (here conditioning plays no role) if x < a, we need P x (τ K < t 0 t 0 < τ ) c. = the diffusion must go down from infinity (+ is an entrance boundary) go away from 0 conditionally on non-absorption when it starts close to 0 couple inside K (A2) Sort of Harnack inequality: the populations survives nowhere much better than starting from ν. True for example if inf y K P y (t < τ ) c 2 sup x E P x (t < τ )

10 The general criterion Ergodicity coefficient of Dobrushin In the case where (X t, t 0) is irreducible, if there exist t 0, c 1 > 0 and ν P(E) s.t. x E, P x (X t0 ) c 1 ν, then x, y E, P x (X t0 ) P y (X t0 ) TV 2(1 c 1 ) = (1 c 1 ) δ x δ y TV < 2 µ 1, µ 2 P(E), P µ1 (X t0 ) P µ2 (X t0 ) TV (1 c 1 ) µ 1 µ 2 TV, and by the Markov property, x, y E, P x (X kt0 ) P y (X kt0 ) TV (1 c 1 ) P x (X (k 1)t0 ) P y (X (k 1)t0 ) VT... 2(1 c 1 ) k exponential convergence in total variation to a unique invariant measure.

11 Main results Necessary and sufficient condition for the existence of a universal QLD [C.-Villemonais, 2016] Theorem Condition (A) is equivalent to the existence of a universal QLD α s.t. µ P(E) P µ (X t t < τ ) α VT Ce γt for two constants C, γ > 0 independent of µ. In addition, under Condition (A), we can take C = 2/(1 c 1 c 2 ) and γ = log(1 c 1 c 2 )/t 0.

12 Main results Some comments We obtain a necessary and sufficient condition The convergence is uniform w.r.t. the initial distribution A bound on the exponential rate of convergence can be computed from the constants c 1, c 2, but this is non-optimal Difficulty: how to check (A1) and (A2) in practice?

13 Main results Idea of the proof of sufficiency (t 0 = 1) Step 1: P x (X 1 t < τ ) c 1 c 2 ν t where ν t P(E). P x (X 1 A t < τ ) = P x (1 < τ ) P x (t < τ ) E x [1 A (X 1 )P X1 (t 1 < τ ) 1 < τ ] where ν t (A) = ν[1 A( )P (t 1 < τ )]. P ν (t 1 < τ ) c 1 P x (1 < τ ) P x (t < τ ) ν[1 A( )P (t 1 < τ )] c 1 ν[1 A ( )P (t 1 < τ )] sup y E P y (t 1 < τ ) c 1c 2 ν t (A)

14 Main results Idea of the proof (continued) Step 2: define the time-inhomogeneous semigroup of the process (X t, t [0, T ]) conditionned on {T < τ }: R T s,tf (x) = E x (f (X t s ) T s < τ ) = E(f (X t ) X s = x, T < τ ). Then, for all u s t T, R T u,sr T s,tf = R T u,tf. Step 3: Step 1 says that δ x R T s,s+1 c 1 c 2 ν T s 0, with mass 1 c 1 c 2, and so δ x R T s,s+1 δ y R T s,s+1 VT 2(1 c 1 c 2 ) = (1 c 1 c 2 ) δ x δ y VT. The end of the proof is similar to the classical coupling coefficient argument of Dobrushin.

15 Asymptotic behavior of the survival probability Asymptotic behavior of survival probabilities and eigenfunction We can also deduce from Condition (A) the spectral properties of the QSD. Proposition Under Condition (A), there exists η : E { } R + bounded such that η( ) = 0 and η > 0 on E s.t. η(x) = lim t P x (t < τ ) P α (t < τ ) = lim t eλ0t P x (t < τ ) where the convergence is uniform w.r.t. x. In addition η D(L) where L is the infinitesimal generator of P t, and Lη = λ 0 η.

16 Q-process Exponential ergodicity of the Q-process Theorem Assume (A), then (i) (Q x ) x E proba s.t. for all s > 0 and A F s, lim P x (A t < τ ) = Q x (A), t and X t is a homogeneous Markov process under (Q x ) x E (ii) Under (Q x ) x E, X t has semi-group P t defined by P t ϕ(x) = eλ0t η(x) P t(ηϕ)(x) (iii) β(dx) = η(x)α(dx) is the unique invariant distribution of X η dα under (Q x ) and µ P(E), Q µ (X t ) β VT Ce γt

17 1D birth and death processes Applications to 1D birth and death processes Let X t be a BDP on Z + with birth rate λ n and death rate µ n in state n. = 0 is absorbing if λ 0 = µ 0 = 0; we assume λ n > 0 and µ n > 0 for all n 1. (A1) is true for ν = δ 1 if the BDP comes down from infinity, i.e. if inf n P n (T k < t) > 0 for some k 1 and t > 0, which is equivalent to S = ( λ n λ ) n... λ 1 < µ n+1 µ n µ n... µ 1 n 0 The difficulty is to prove (A2) Proposition (Martinez, San Martin, Villemonais, 2012) Assume S <. Then (A2) is satisfied, and there exists a unique universal QLD, with exponential convergence in total variation.

18 1D birth and death processes Proof of (A2) 1 Large rate of coming back in compact sets: We have sup x E x (τ ) = k 1 E k (T k 1 ) = S <, where T k is the first hitting time of k. So ε > 0, k s.t. sup x k E X T k ε So sup x k P x (T k 1) ε and hence sup x k P X (T k n) ε n Hence λ > 0, k 1 s.t. sup x k E x (e λ T k ) <. 2 Small(er) rate of absorption from compact sets: P 1 (t + s < τ ) P 1 (X t = 1)P 1 (s < τ ) e (λ1+µ1)t P 1 (s < τ ). we can choose λ > λ 1 + µ 1 in Point 1. 3 Uniform Irreducibility in compact sets: for all k 1, P 1 (t τ ) P 1 (t + 1 < τ ) P 1 (X 1 = k)p k (t < τ ) C k P k (t < τ ).

19 1D birth and death processes Proof of (A2) (continued) Therefore, P (t < τ ) P (t < T k ) + P (T k < t < τ ) Hence, Ce λ t + t 0 Ce λ t + C 1 k P k (t s < τ )dp (T k > s) t Ce (λ1+µ1)t + C 1 k 0 P 1 (t s < τ )dp (T k > s) t C P 1 (t < τ ) + C 1 k P 1 (t < τ ) 0 P 1 (t s < τ )dp (T k > s) t P (t < τ ) C P 1 (t < τ ). 0 e (λ1+µ1)s dp (T k > s).

20 1D birth and death processes Comments and extensions Most of the classical biological models with density-dependence satisfy S < (e.g. the logistic BDP, Gompertz BDP...). Subcritical branching processes do not satisfy S <. For 1D BDP, when S =, it is known that there is either no existence or no uniqueness of the QSDs (van Doorn, 1991). The method is easy to extend to a lot of cases where the absorption rate is bounded. For example, multi-dimensional BDP aborbed at (0,..., 0), BDP processes with killing (or catastrophes) at bounded rate... Cases with unbounded absorption rate are harder to study next parts.

21 Multi-dimensional BDP absorbed at the boundary Multi-dimensional BDP absorbed at the boundary [C-V, preprint, 2016] We consider a logistic (competitive Lotka-Volterra) BDP in N r, absorbed at = N r \ (N ) r, with transition rates from n = (n 1,..., n r ) to { n + e j with rate n j b j, n e j with rate n j (d j + r k=1 c jkn k ), where b j, d j, c jk > 0. b j is the individual birth rate of type j d j is the individual death rate of type j c ij is the death rate of an individual of type i due to the competition exerted by individuals of type j Theorem Under the previous conditions, Condition (A) is satisfied, and there exists a unique universal QLD, with exponential convergence in total variation.

22 Multi-dimensional BDP absorbed at the boundary Sketch of proof A first idea: for all nonnegative bounded function V on N r, E x V (X t ) t [ P x (t < τ ) = V (x)+ Ex LV (X s ) P x (s < τ ) E ] x L1 (N (X )r s ) E x V (X s ) P x (s < τ ) P x (s < τ ) 0 Hence one can try to find a non-linear Lyapunov function V for the conditional distributions (µ t ) t 0, satisfying for all µ P((N ) r ) µ(lv ) µ(v )µ(l1 (N ) r ) A Bµ(W ) for a function W going to + at infinity. Proof of (A1): since E x (V (X t ) t < τ ) is nonnegative and V (x) is bounded, there exists a compact set K and a time t 0 such that inf P x (X t0 K t 0 < τ ) > 0. x E Then, by irreducibility in K, (A1) is true with ν = δ (1,...,1). Proof of (A2): similar to the 1D case. ds.

23 Multi-dimensional BDP absorbed at the boundary Sketch of proof (continued) Problem: a Lyapunov function V as above does not seem to exist for all choices of c ij. Second idea: for all nonnegative bounded functions V and ϕ on N r such that ϕ > 0 on (N ) r E x V (X t ) E x ϕ(x t ) = V (x) t [ ϕ(x) + Ex LV (X s ) E x ϕ(x t ) E ] x Lϕ(X s ) E x V (X s ) ds. E x ϕ(x t ) E x ϕ(x t ) 0 Now, since V (x)/ϕ(x) may be unbounded, we need a stronger form of non-linear Lyapunov inequality: there exists ε > 0 such that, for all µ P((N ) r ) µ(lv ) µ(ϕ) µ(v ) µ(lϕ) µ(ϕ) µ(ϕ) µ(v )1+ε A B. (1) µ(ϕ) 1+ε We can check that (1) holds for ϕ(x) = x α and V (x) = 1 x β for appropriate α, β > 0. One can then adapt the previous proof to check (A1 A2).

24 Multi-dimensional BDP absorbed at the boundary Comments The quasi-stationary behavior of multi-dimensional BDP process absorbed at the boundary have been the object of very few works (except for Galton-Watson-type processes). The case of diffusions on R r + was more studied (cf. Cattiaux, Méléard, 2010). The theory of R-positive matrices (see Kesten, Ferrari, Martinez, 1996) can also be used to prove the existence and uniqueness of a QSD, but not the uniform convergence. The non-linear Lyapunov criteria used in the last proof apply to many other cases: BDP in one or several dimension with unbounded killing rates (first Lyapunov criterion) Diffusions in one or several dimensions absorbed at the boundary of a domain (second Lyapunov criterion, work in progress)

25 Diffusions in dimension 1 Application to one-dimensional diffusions [C-V, preprint, 2015] We consider a diffusion (X t, t 0) on [0, ) absorbed at 0 on natural scale, i.e. (typically) dx t = σ(x t )db t, t < τ, (e.g. Fisher diffusion, or Wright-Fisher on [0, 1]). We assume that 0 is accessible (i.e. either exit or regular). Theorem Condition (A) is equivalent to the condition (B) The diffusion (X t, t 0) comes down from infinity (i.e. + is an entrance boundary) and there exist t, A > 0 s.t. P x (t < τ ) Ax, x > 0. We also prove that nearly all diffusions coming down from infinity and a.s. absorbed at 0 satisfy (B), and hence (A).

26 Diffusions in dimension 1 A glimpse to the proof We focus on (A) = (B), and only explain the... New difficulty: prove that conditionally on extinction, the diffusion started close to 0 goes away from 0: ε, c > 0 s.t. P x (X t1 ε t 1 < τ ) c, x > 0. Key argument: Since X is a local martingale, x = E x (X t1 T 1 ) = P x (t 1 < τ )E x (X t1 T 1 t 1 < τ ) + P x (T 1 < τ t 1 ) P x (t 1 < τ )E x (X t1 T 1 t 1 < τ ) + P x (T 1 < τ )P 1 (τ t 1 ) AxE x (X t1 T 1 t 1 < τ ) + xp 1 (τ t 1 ), and so E x (X t1 T 1 t 1 < τ ) c > 0. This implies that, conditionally on non-extinction, X hits some ε > 0 before t 1 with a positive probability.

27 Diffusions in dimension 1 Comments The quasi-stationary behavior of 1D diffusion has been extensively studied in the literature (Cattiaux et al., 2009, Kolb and Steinsaltz, 2012, Littin, 2012, Miura, 2014). We do not improve the best known conditions for existence and uniqueness of the QSD, but we improve the convergence. It is possible to extend our method to models with killing, or to cases with jump.

28 Concluding remarks We obtained explicit conditions ensuring the existence of a universal QLD. We showed how these conditions can be checked in classical models of population dynamics. The method is flexible enough to cover basic extensions of the models, including processes with killing, jumps, and time inhomogeneous processes. Open problems/work in progress: Simple criteria expressed in terms of (standard) Lyapunov functions (like those used in Collet et al., 2011). Criteria applying in cases of non-uniform convergence w.r.t. the initial distribution, and characterization of (part of) the domain of attraction of the minimal QSD. Study of the asymptotic behavior of Fleming-Viot particule systems approximating QSD convergence to the minimal QSD (Thai, Asselah, Chafaï, 2015).