Macroscopic quasi-stationary distribution and microscopic particle … · 2016. 5. 23. · Macroscopic quasi-stationary distribution and microscopic particle systems Matthieu Jonckheere

Macroscopic quasi-stationary distributionand microscopic particle systems

Matthieu Jonckheere, UBA-CONICET, BCAM visiting fellow

Coauthors:A. Asselah, P. Ferrari, P. Groisman, J. Martinez, S. Saglietti.

BCAM, May 2016

Outline

I Introduction to quasi-stationary distributions: Macroscopic model

I Particle systems : Microscopic model

I Selection principle and traveling waves

Outline




Outline




Denying eternity

Most phenomena do not last for ever.

However most of them might reach some ”kind of equilibrium” beforevanishing.

What are we observing when considering a macroscopic stochasticevolution (in biology, physics, populations models,telecommunications) that has not vanished for (very) large times?

Denying eternity




Denying eternity




Denying eternity




TRANSIENT

Denying eternity




TRANSIENT STATIONARY

Denying eternity




TRANSIENT QUASI-STATIONARY STATIONARY

2 types of quasi-stationarity

I If the stochastic evolution has a strong drift towards extinction,this quasi-equilibrium might correspond to a large deviationevent.

E.g. observing a player winning at a casino for hours, traffic jammore than 10 hours,...

I If it tends to vanish more slowly, (spend large time in a subset ofthe state space before vanishing) the quasi-equilibriumcorresponds to a metastable state.

E.g. a bottle of cold beer just before freezing.

In the study of population dynamics, this phenomenon is coinedas ”mortality plateau”.


I If the stochastic evolution has a strong drift towards extinction,this quasi-equilibrium might correspond to a large deviationevent.

E.g. observing a player winning at a casino for hours, traffic jammore than 10 hours,...

I If it tends to vanish more slowly, (spend large time in a subset ofthe state space before vanishing) the quasi-equilibriumcorresponds to a metastable state.

E.g. a bottle of cold beer just before freezing.

In the study of population dynamics, this phenomenon is coinedas ”mortality plateau”.


Challenges

Both cases (large deviation and metastability) are interesting tostudy theoretically.

These quasi-equilibrium are generally difficult to simulate.

When there are several quasi-equilibrium (an infinity), which onehas a physical meaning?

Markov process conditioned on non-absorption

Let Xt ∈ N , (N = N or R) an irreducible Markov process absorbed in0. Let T the absorption time of X . Given an initial law µ and ameasurable set A:

φµt (A) = Pµ(Xt ∈ A|T > t).

Kolmogorov (1938) proposed to study the long time behavior ofprocesses conditioned not to being absorbed, i.e. the limits (if itexists) of φµt (·).

Markov process conditioned on non-absorption

Let Xt ∈ N , (N = N or R) an irreducible Markov process absorbed in0. Let T the absorption time of X . Given an initial law µ and ameasurable set A:

φµt (A) = Pµ(Xt ∈ A|T > t).

Kolmogorov (1938) proposed to study the long time behavior ofprocesses conditioned not to being absorbed, i.e. the limits (if itexists) of φµt (·).

Quasi-stationary distribution

We say that ν is a quasi-stationary distribution (QSD ) if thereexists a probability measure µ such that:

limt→∞

φµt (·) = νµ(·).

QSD

I Finite state space: S ⊂ N : there exists a unique QSD .Spectral point of view: QSD = maximal left eigenvector of Q(infinitesimal generator of the killed process)

I For countable state space, there are different possiblescenarios:

1. No QSD .2. Unique QSD and convergence from any initial distribution towards

this measure.3. Infinity of QSD :

Parametrization of the family of QSD with a parameter θ (eigenvalueof the infinitesimal generator): if ν QSD then

Pν(T > t) = exp(−θν t).

There might exist a maximal θ corresponding to the so-calledminimal QSD .

QSD

I Finite state space: S ⊂ N : there exists a unique QSD .Spectral point of view: QSD = maximal left eigenvector of Q(infinitesimal generator of the killed process)

I For countable state space, there are different possiblescenarios:

1. No QSD .2. Unique QSD and convergence from any initial distribution towards

this measure.3. Infinity of QSD :

Parametrization of the family of QSD with a parameter θ (eigenvalueof the infinitesimal generator): if ν QSD then

Pν(T > t) = exp(−θν t).

There might exist a maximal θ corresponding to the so-calledminimal QSD .

Challenges

I Existence,

I Simulation,

I Properties, extremality

Challenges

I Existence,

I Simulation,


Challenges

I Existence,

I Simulation,


Bibliography on QSDs

- van Doorn, Ferrari, Martinez, Pollet, Seneta, Vere-Jones,...

See P. Pollett bibiliography:

http://www.maths.uq.edu.au/ pkp/papers/qsds/qsds.pdf

Outline

I Particle systems: microscopic models

1. Branching processes2. Fleming-Viot

3. N- Branching Brownian motion4. Choose the fittest

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Outline




Traveling waves for PDE

I An important question in mathematics and physics is theexistence of traveling waves solutions to (in particular parabolic,reaction-diffusion) PDEs, i.e., solutions of the formu(x , t) = w(x − ct) where c is the speed of the traveling wave.

I Example: KPP (Kolmogorov-Petrovsky-Piskounov) equation:

ut = 1/2uxx + f (u).

I Links with the maximum of the Branching Brownian motion(McKean, Bramson,...)

I In general, there may exist an infinity of solutions parametrizedby their speed s.

I Which speed to select?

Traveling waves and QSDs

For the KPP equation with f (u) = u2 − u, one can prove that:

I the equation has the same traveling wave as the free boundaryequation obtained for the N-BBM,

I There exists a traveling wave (for a given eigenvalue λ) if andonly if there exists a QSD for an associated Brownian motionwith drift.

I This can be extended to Levy processes dynamics, i.e. moregeneral equations...

Need for a selection principle: Quoting Fisher

(About the ”velocity of advance” for genetic evolutions):

Common sense would, I think, lead us to believe that, thoughthe velocity of advance might be temporarily enhanced by thismethod, yet ultimately, the velocity of advance would adjust it-self so as to be the same irrespective of the initial conditions.If this is so, this equation must omit some essential ele-ment of the problem, and it is indeed clear that while a co-efficient of diffusion may represent the biological conditionsadequately in places where large numbers of individuals ofboth types are available, it cannot do so at the extreme frontand back of the advancing wave, where the numbers of themutant and the parent gene respectively are small, and wheretheir distribution must be largely sporadic.

The missing links: particle systems

I Macroscopic models (QSDs and traveling waves) forget that apopulation can be very large but finite.

I Microscopic models are intrinsically corresponding to finitepopulation. They do select the minimal QSD/traveling wave.

references

I Simulation of quasi-stationary distributions on countable spaces,P. Groisman, M. J., Markov processes and related fields 2013

I Fleming-Viot selects the minimal quasi-stationary distribution:The Galton-Watson caseA. Asselah, P. Ferrari, P. Groisman, M. J. Ann Inst H. Poincare,2015

I Front propagation and quasi-stationary distributions: the sameselection principle.P. Groisman, M. J.

I Kesten-Stigum theorems in L2 beyond R-positivity.S. Saglietti, M.J.

Thanks

Existence of QSD

Very few general result:

Proposition (Ferrari et. al. 1995)Xt ∈ N. If

I limx→∞ Px(T > t) =∞,I There exists γ > 0 and z ∈ N such that Ez(exp(γT0)) <∞,

then there exists a QSD .

QSD and QLD : examples

I Birth and death process (non-empty M/M/1 queue):

q(x , x + 1) = p1x>0, q(x , x − 1) = q1x>0.

Infinite family of QSD . Minimal QSD :

ν∗(x) = c(x + 1)(p

q)x/2

.

I Population process with linear drift:

q(x , x + 1) = px1x>0, q(x , x − 1) = qx1x>0.


ν∗(x) = c(p

q)x.

QSD and QLD : examples

I Birth and death process (non-empty M/M/1 queue):

q(x , x + 1) = p1x>0, q(x , x − 1) = q1x>0.


ν∗(x) = c(x + 1)(p

q)x/2

.

I Population process with linear drift:

q(x , x + 1) = px1x>0, q(x , x − 1) = qx1x>0.


ν∗(x) = c(p

q)x.

Macroscopic quasi-stationary distribution and microscopic particle … · 2016. 5. 23. · Macroscopic quasi-stationary distribution and microscopic particle systems Matthieu Jonckheere

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