Exponential convergence to quasi-stationary distribution and Q-process

TLDR: In this article, the authors obtain necessary and sufficient conditions for exponential convergence to a unique quasi-stationary distribution in the total variation norm of the $$Q$$ -process (the process conditioned to never be absorbed), and apply these results to one-dimensional birth and death processes with catastrophes, multi-dimensional and infinite-dimensional population models with Brownian mutations and neutron transport dynamics absorbed at the boundary of a bounded domain.

Abstract: For general, almost surely absorbed Markov processes, we obtain necessary and sufficient conditions for exponential convergence to a unique quasi-stationary distribution in the total variation norm. These conditions also ensure the existence and exponential ergodicity of the $$Q$$ -process (the process conditioned to never be absorbed). We apply these results to one-dimensional birth and death processes with catastrophes, multi-dimensional birth and death processes, infinite-dimensional population models with Brownian mutations and neutron transport dynamics absorbed at the boundary of a bounded domain.

Figures

Figure 2: The sets Kx and Lx of Assumption (B2).

Figure 3: The triangle [u, v, w] and the point v′.

Figure 1: A sample path of the neutron transport process (X,V ). The times J1 < J2 < . . . are the successive jump times of V .

Figure 4: Definition of uz, wz, ŭzwz, z0, u0 and w0. The angle marked M is larger than the one marked ).

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Exponential convergence to quasi-stationary distribution
and Q-process
Nicolas Champagnat, Denis Villemonais
To cite this version:
Nicolas Champagnat, Denis Villemonais. Exponential convergence to quasi-stationary distribution
and Q-process. Probability Theory and Related Fields, Springer Verlag, 2016, 164 (1), pp.243-283.
�10.1007/s00440-014-0611-7�. �hal-00973509v2�

Exponential convergence to quasi-stationary
distribution and Q-process
Nicolas Champagnat
1,2
, Denis Villemonais
1,2
December 23, 2014
Abstract
For general, almost surely absorbed Markov processes, we obtain
necessary and sufficient conditions for exponential convergence to a
unique quasi-stationary distribution in the total variation norm. These
conditions also ensure the existence and exponential ergodicity of the
Q-process (the process conditioned to never be absorbed). We ap-
ply these results to one-dimensional b ir t h and deat h processes with
catastrophes, multi-dimensional birth and death p rocesses, infinite-
dimensional population models with Brownian mutations and neutron
transport dynamics absorbed at the boundary of a bounded domain.
Keywords: process with absorption; quasi-stationary distribution; Q-process;
Dobrushin’s ergodicity coefficient; uniform mixing property; birth and deat h
process; neutron transport process.
2010 Mathematics Subject Classification. Primary: 60J25; 37A25; 60B10;
60F99. S econ d ary : 60J80; 60G10; 92D25.
1 Introduction
Let (Ω, (F
t
)
t≥0
, (X
t
)
t≥0
, (P
t
)
t≥0
, (P
x
)
x∈E∪{∂}
) be a time homogeneous Markov
process with state space E ∪ {∂} [31, Definition III.1.1], where (E, E) is a
measurable space and ∂ 6∈ E. We recall that P
x
(X
0
= x) = 1, P
t
is the tran-
sition function of the process satisfying the usual m easu r abi l i ty assumptions
and Chapman-K ol mogor ov equation. The famil y (P
t
)
t≥0
defines a semi-
group of operators on the set B(E ∪ {∂}) of bounded Borel f u nc ti on s on
1
Universit´e de Lorraine, IECN, Campus Scientifique, B.P. 70239, Vandœuvre-l`es-Nancy
Cedex, F-54506, France
2
Inria, TOSCA team, Villers-l`es-Nancy, F-54600, France.
E-mail: Nicolas.Champagnat@inria.fr, Denis.Villemo n ai s@ u n iv -l o rra in e. fr
1

E ∪ ∂ en dowed with the uniform norm. We will also denote by p(x; t, dy) its
transition kernel, i.e. P
t
f(x) =
R
E∪{∂}
f(y)p(x; t, dy) for all f ∈ B(E ∪ {∂}).
For all probability measure µ on E ∪ {∂}, we will use t h e n ot at ion
P
µ
(·) :=
Z
E∪{∂}
P
x
(·)µ(dx).
We shall denote by E
x
(resp. E
µ
) the expectation corresp on di n g to P
x
(resp.
P
µ
).
We consider a Markov processes absorbed at ∂. More precisely, we as-
sume that X
s
= ∂ implies X
t
= ∂ for all t ≥ s. This implies that
τ
∂
:= inf{t ≥ 0, X
t
= ∂}
is a stopping time. We also assume that τ
∂
< ∞ P
x
-a.s. for all x ∈ E and
for all t ≥ 0 and ∀x ∈ E, P
x
(t < τ
∂
) > 0.
Our first goal is to prove that Assumption (A) below is a necessary and
sufficient criteri on for the existence of a unique quasi-limiting distribution
α on E for the process (X
t
, t ≥ 0), i.e. a probability measure α such that
for all probability measure µ on E and all A ∈ E,
lim
t→+∞
P
µ
(X
t
∈ A | t < τ
∂
) = α(A), (1.1)
where, in addition, the convergence is exponential and uniform with respect
to µ and A. In particular, α is also the unique quasi-stati onar y distribu-
tion [28], i.e. t h e unique prob abi l i ty measure α such that P
α
(X
t
∈ · | t <
τ
∂
) = α(·) for all t ≥ 0.
Assumption (A) There exists a probability measure ν on E such that
(A1) there exists t
0
, c
1
> 0 su ch t h at for all x ∈ E,
P
x
(X
t
0
∈ · | t
0
< τ
∂
) ≥ c
1
ν(·);
(A2) there exists c
2
> 0 su ch t h at for all x ∈ E and t ≥ 0,
P
ν
(t < τ
∂
) ≥ c
2
P
x
(t < τ
∂
).
Theorem 1.1. Assumption (A) implies the existence of a probability mea-
sure α on E such that, for any initial distribution µ,
kP
µ
(X
t
∈ · | t < τ
∂
) − α(·)k
T V
≤ 2(1 − c
1
c
2
)
⌊t/t
0
⌋
, (1.2)
where ⌊·⌋ is the integer part function and k · k
T V
is the total variation norm.
Conversely, if there is uniform exponential convergence for the total vari-
ation norm in (1.1), then Assumption (A) hold s true.
2

Stronger versions of this theorem and of t he other results presented in
the introduction will be given in the next sections.
The quasi-stationary distribution describes the dis t ri b ut i on of the pro-
cess on the event of non-absorption. It is well known (see [28]) that when
α is a quasi-stationary distribution, ther e exists λ
0
> 0 such that, for all
t ≥ 0,
P
α
(t < τ
∂
) = e
−λ
0
t
. (1.3)
The following proposition characterizes the limiting behaviour of the ab-
sorption probability for other initial distrib u t ion s.
Proposition 1.2. There exists a non-negative function η on E ∪ {∂}, pos-
itive on E and vanishing on ∂, such that
µ(η) = lim
t→∞
e
λ
0
t
P
µ
(t < τ
∂
),
where the convergence is u ni f or m on the set of probability measu res µ on E.
Our second goal is to study consequences of Assumption (A) on the
behavi or of the process X conditioned to never be absorbed, usually referred
to as the Q-process (see [1] in discrete time and for example [5] in continuous
time).
Theorem 1.3. Assumption (A) implies that the family (Q
x
)
x∈E
of proba-
bility measures on Ω defined by
Q
x
(A) = lim
t→+∞
P
x
(A | t < τ
∂
), ∀A ∈ F
s
, ∀s ≥ 0,
is well defined and the process (Ω, (F
t
)
t≥0
, (X
t
)
t≥0
, (Q
x
)
x∈E
) is an E-valued
homogeneous Markov process. In addition, this process admits the unique
invariant distribution
β(dx) =
η(x)α(dx)
R
E
η(y)α(dy)
and, for any x ∈ E,
kQ
x
(X
t
∈ ·) − βk
T V
≤ 2(1 − c
1
c
2
)
⌊t/t
0
⌋
.
The st ud y of quasi-stationar y distr i but ion s goes back to [38] for branch-
ing processes and [12, 33, 13] for Markov chains in finite or denumerable
state spaces, satisfying irreducibility assumptions. In these works, the exis-
tence and the convergence to a quasi-st at ion ar y distr ib u t ion are proved using
3

spectral propertie s of the generator of t h e absorbed Markov process. This is
also the case for most further wor ks . For examp l e, a extensively developed
tool to study birth and death processes is based on orthogonal polynomials
techniques of [23], applied to quasi -st at i onar y distributions in [22, 6, 34]. For
diffusion processes, we can refer to [30] and more recently [4, 5, 26], all b ase d
on the spectral decomposition of the generator. Most of these works only
study one-dimensional processes, whose reversibility help s for the spectral
decomposition. Processes in higher dimensions were studied either assuming
self-adjoint generator in [5], or using abstract cri t er i a from spectral theory
like in [30, 10] (the second one in infinite dimension). Other formulations in
terms of abstract spectral theoretical criteria were also stud i ed in [24]. The
reader can refer to [28, 11, 36] for introductory pr es entations of the topic.
Most of the previously cited works do not provide convergence results
nor estimates on the speed of convergence. The articles studying these
questions either assu me abstract conditions which are very difficult to check
in practice [33, 24], or prove expon ential convergence for very weak norms [4,
5, 26].
More probabi li s t ic methods were also developed . The older reference is
based on a rene wal technique [21] and proves the existence and convergence
to a q u asi -s tat i on ary distribution for discrete processes for which Assump-
tion (A1) is not satisfied. More recently, one-dimensional birth and death
processes with a unique quasi-stationary distribution have been shown to
satisfy (1.1) with uniform convergence in total variation [27]. Convergence
in total variation for processes in discrete state space satisfying strong mix-
ing conditions was obtained in [9] using Fleming-Viot particle systems whose
empirical distribution approximates conditional distributions [37]. Sufficient
conditions for exponent i al convergence of condition ed system s in discrete
time can be found in [15] wit h applications of discrete generation particle
techniques in signal processing, statistical machine learning, an d quantum
physics. We al s o refer the reader t o [16, 17, 18] for approximations tech-
niques of non absorbed trajectories in terms of genealogical trees.
In this work, we obtain in Section 2 necessary and sufficient conditions
for exponential convergence to a unique quasi-stationary distribution for
general (virtually any) Markov processes (we state a stronger form of Th e-
orem 1.1). We also obtain spectral properties of the infinitesimal generator
as a corollary of our main result. Our non-spectral approach and results
fundamentally differ from all the previously cited references, except [27, 9]
which only focus on very specific cases. In Section 3, we show, using penal-
isation techn iq u es [32], that the same conditions are sufficient to prove the
existence of the Q-process and its exponential ergodicity, uniformly in total
4

Frequently asked questions

Q1. What is the simplest way to determine if a distribution is invariant?

Since η is bounded, β is well defined and is an invariant distribution for X under Q.By (5.1) and the semi-group property of (RTs,t)s,t, the authors have for all µ1, µ2 ∈ M1(E) and all t ≤ T∥ ∥ ∥µ1R T 0,t − µ2R T 0,t ∥ ∥ ∥TV ≤ (1− c1c2)⌊t/t0⌋‖µ1 − µ2‖TV .By definition of RT0,t and by dominated convergence when T → ∞, the authors obtain‖Qµ1(Xt ∈ ·)−Qµ2(Xt ∈ ·)) ‖TV ≤ (1− c1c2) ⌊t/t0⌋‖µ1 − µ2‖TV .Taking µ2 = β, this implies that X is exponentially ergodic under Q with unique invariant distribution β.Step 1: computation of L̃w and a first inclusion in (3.2). 

Q2. What were the methods used to study the processes in higher dimensions?

Processes in higher dimensions were studied either assuming self-adjoint generator in [5], or using abstract criteria from spectral theory like in [30, 10] (the second one in infinite dimension). 

Q3. What is the proof of Lemma 5.2?

Then µ is the largest non-negative measure on E such that µ ≤ µx for all x ∈ F and is called the infimum measure of (µx)x∈F .Proof of Lemma 5.2. 

Q4. What is the weak infinitesimal generator of the semi-group (Pt)?

If in addition E is a topological space and E is the Borel σ-field, and if for all open set U ⊂ E and x ∈ U ,lim h→0 p(x;h, U) = lim h→0 Ph1U (x) = 1, (3.3)then the semi-group (P̃t) is uniquely determined by its weak infinitesimal generator L̃w. 

Q5. What is the first general result showing the link between quasistationary distributions and Q-process?

This is the first general result showing the link between quasistationary distributions and Q-processes, since the authors actually prove that, for general Markov processes, the uniform exponential convergence to a quasistationary distribution implies the existence and ergodicity of the Q-process. 

Q6. What is the proof of the proof for Theorem 4.1?

Since absorption occurs only from states with one individual, this is equivalent to: there exists a constant C > 0 such that, for all t ≥ 0, x0 ∈ T,Pδx0 (t < τ∂) ≥ C sup ξ∈Kn0 Pξ(t < τ∂). (4.12)If this holds, the authors conclude the proof as for Theorem 4.1. 

Q7. What is the transition function of the process?

Let (Ω, (Ft)t≥0, (Xt)t≥0, (Pt)t≥0, (Px)x∈E∪{∂}) be a time homogeneous Markov process with state space E ∪ {∂} [31, Definition III.1.1], where (E, E) is a measurable space and ∂ 6∈ E. The authors recall that Px(X0 = x) = 1, Pt is the transition function of the process satisfying the usual measurability assumptions and Chapman-Kolmogorov equation. 

Q8. What is the unique invariant distribution of X under Q?

is the unique invariant distribution of X under Q. Moreover, for any initial distributions µ1, µ2 on E,‖Qµ1(Xt ∈ ·)−Qµ2(Xt ∈ ·)‖TV ≤ (1− c1c2) ⌊t/t0⌋‖µ1 − µ2‖TV ,where Qµ = ∫E Qx µ(dx). 

Q9. What is the limiting behaviour of the absorption probability for other initial distributions?

It is well known (see [28]) that when α is a quasi-stationary distribution, there exists λ0 > 0 such that, for all t ≥ 0,Pα(t < τ∂) = e −λ0t. (1.3)The following proposition characterizes the limiting behaviour of the absorption probability for other initial distributions. 

Q10. What is the weak infinitesimal generator of Pt?

The weak infinitesimal generator Lw of (Pt) is defined asLwf = b.p.- lim h→0Phf − fh ,for all f ∈ B(E ∪ {∂}) such that the above b.p.–limit exists andb.p.- lim h→0PhL wf = Lwf. 

Q11. What is the spectral property of the generator of the absorbed Markov process?

In these works, the existence and the convergence to a quasi-stationary distribution are proved usingspectral properties of the generator of the absorbed Markov process. 

Q12. How do the authors denote the boundary of the domain D?

The authors denote by ∂D the boundary of the domain D, diam(D) its diameter, and for all A ⊂ R2 and x ∈ R2, by d(x,A) the distance of x to the set A: d(x,A) = infy∈A |x − y|. 

Q13. what is the ergodicity of the Q-process?

In other words, for all ϕ ∈ B(E) and t ≥ 0,P̃tϕ(x) = eλ0tη(x) Pt(ηϕ)(x) (3.1)where (P̃t)t≥0 is the semi-group of X under Q.(iii) Exponential ergodicity. 

Q14. What can be done to illustrate the simplest case?

The assumptions of contant velocity, uniform jump distribution, uniform jump rates and on the dimension of the process can be relaxed but the authors restrict here to the simplest case to illustrate how conditions (A1) and (A2) can be checked.