Macroscopic quasi-stationary distribution
and 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
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Introduction to quasi-stationary distributions: Macroscopic model
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Particle systems : Microscopic model
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Selection principle and traveling waves
Outline
I
Introduction to quasi-stationary distributions: Macroscopic model
I
Particle systems : Microscopic model
I
Selection principle and traveling waves
Outline
I
Introduction to quasi-stationary distributions: Macroscopic model
I
Particle systems : Microscopic model
I
Selection principle and traveling waves
Denying eternity
Most phenomena do not last for ever.
However most of them might reach some ”kind of equilibrium” before
vanishing.
What are we observing when considering a macroscopic stochastic
evolution (in biology, physics, populations models,
telecommunications) that has not vanished for (very) large times?
Denying eternity
Most phenomena do not last for ever.
However most of them might reach some ”kind of equilibrium” before
vanishing.
What are we observing when considering a macroscopic stochastic
evolution (in biology, physics, populations models,
telecommunications) that has not vanished for (very) large times?
Denying eternity
Most phenomena do not last for ever.
However most of them might reach some ”kind of equilibrium” before
vanishing.
What are we observing when considering a macroscopic stochastic
evolution (in biology, physics, populations models,
telecommunications) that has not vanished for (very) large times?
Denying eternity
Most phenomena do not last for ever.
However most of them might reach some ”kind of equilibrium” before
vanishing.
What are we observing when considering a macroscopic stochastic
evolution (in biology, physics, populations models,
telecommunications) that has not vanished for (very) large times?
TRANSIENT
Denying eternity
Most phenomena do not last for ever.
However most of them might reach some ”kind of equilibrium” before
vanishing.
What are we observing when considering a macroscopic stochastic
evolution (in biology, physics, populations models,
telecommunications) that has not vanished for (very) large times?
TRANSIENT
STATIONARY
Denying eternity
Most phenomena do not last for ever.
However most of them might reach some ”kind of equilibrium” before
vanishing.
What are we observing when considering a macroscopic stochastic
evolution (in biology, physics, populations models,
telecommunications) that has not vanished for (very) large times?
TRANSIENT
QUASI-STATIONARY
STATIONARY
2 types of quasi-stationarity
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If the stochastic evolution has a strong drift towards extinction,
this quasi-equilibrium might correspond to a large deviation
event.
E.g. observing a player winning at a casino for hours, traffic jam
more than 10 hours,...
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If it tends to vanish more slowly, (spend large time in a subset of
the state space before vanishing) the quasi-equilibrium
corresponds to a metastable state.
E.g. a bottle of cold beer just before freezing.
In the study of population dynamics, this phenomenon is coined
as ”mortality plateau”.
2 types of quasi-stationarity
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If the stochastic evolution has a strong drift towards extinction,
this quasi-equilibrium might correspond to a large deviation
event.
E.g. observing a player winning at a casino for hours, traffic jam
more than 10 hours,...
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If it tends to vanish more slowly, (spend large time in a subset of
the state space before vanishing) the quasi-equilibrium
corresponds to a metastable state.
E.g. a bottle of cold beer just before freezing.
In the study of population dynamics, this phenomenon is coined
as ”mortality plateau”.
2 types of quasi-stationarity
Challenges
Both cases (large deviation and metastability) are interesting to
study theoretically.
These quasi-equilibrium are generally difficult to simulate.
When there are several quasi-equilibrium (an infinity), which one
has a physical meaning?
Markov process conditioned on non-absorption
Let Xt ∈ N , (N = N or R) an irreducible Markov process absorbed in
0. Let T the absorption time of X . Given an initial law µ and a
measurable set A:
φµt (A) = P µ (Xt ∈ A|T > t).
Kolmogorov (1938) proposed to study the long time behavior of
processes conditioned not to being absorbed, i.e. the limits (if it
exists) of φµt (·).
Markov process conditioned on non-absorption
Let Xt ∈ N , (N = N or R) an irreducible Markov process absorbed in
0. Let T the absorption time of X . Given an initial law µ and a
measurable set A:
φµt (A) = P µ (Xt ∈ A|T > t).
Kolmogorov (1938) proposed to study the long time behavior of
processes conditioned not to being absorbed, i.e. the limits (if it
exists) of φµt (·).
Quasi-stationary distribution
We say that ν is a quasi-stationary distribution (QSD ) if there
exists a probability measure µ such that:
lim φµt (·) = ν µ (·).
t→∞
QSD
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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)
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For countable state space, there are different possible
scenarios:
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 (eigenvalue
of the infinitesimal generator): if ν QSD then
P ν (T > t) = exp(−θν t).
There might exist a maximal θ corresponding to the so-called
minimal QSD .
QSD
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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)
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For countable state space, there are different possible
scenarios:
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 (eigenvalue
of the infinitesimal generator): if ν QSD then
P ν (T > t) = exp(−θν t).
There might exist a maximal θ corresponding to the so-called
minimal QSD .
Challenges
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Existence,
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Simulation,
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Properties, extremality
Challenges
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Existence,
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Simulation,
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Properties, extremality
Challenges
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Existence,
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Simulation,
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Properties, extremality
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
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Particle systems: microscopic models
1. Branching processes
2. Fleming-Viot
3. N- Branching Brownian motion
4. Choose the fittest
Outline
I
Particle systems: microscopic models
1. Branching processes
2. Fleming-Viot
3. N- Branching Brownian motion
4. Choose the fittest
Outline
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Particle systems: microscopic models
1. Branching processes
2. Fleming-Viot
3. N- Branching Brownian motion
4. Choose the fittest
Traveling waves for PDE
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An important question in mathematics and physics is the
existence of traveling waves solutions to (in particular parabolic,
reaction-diffusion) PDEs, i.e., solutions of the form
u(x, t) = w(x − ct) where c is the speed of the traveling wave.
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Example: KPP (Kolmogorov-Petrovsky-Piskounov) equation:
ut = 1/2uxx + f (u).
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Links with the maximum of the Branching Brownian motion
(McKean, Bramson,...)
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In general, there may exist an infinity of solutions parametrized
by their speed s.
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Which speed to select?
Traveling waves and QSDs
For the KPP equation with f (u) = u 2 − u, one can prove that:
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the equation has the same traveling wave as the free boundary
equation obtained for the N-BBM,
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There exists a traveling wave (for a given eigenvalue λ) if and
only if there exists a QSD for an associated Brownian motion
with drift.
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This can be extended to Lévy processes dynamics, i.e. more
general 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, though
the velocity of advance might be temporarily enhanced by this
method, yet ultimately, the velocity of advance would adjust itself so as to be the same irrespective of the initial conditions.
If this is so, this equation must omit some essential element of the problem, and it is indeed clear that while a coefficient of diffusion may represent the biological conditions
adequately in places where large numbers of individuals of
both types are available, it cannot do so at the extreme front
and back of the advancing wave, where the numbers of the
mutant and the parent gene respectively are small, and where
their distribution must be largely sporadic.
The missing links: particle systems
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Macroscopic models (QSDs and traveling waves) forget that a
population can be very large but finite.
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Microscopic models are intrinsically corresponding to finite
population. They do select the minimal QSD/traveling wave.
references
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Simulation of quasi-stationary distributions on countable spaces,
P. Groisman, M. J., Markov processes and related fields 2013
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Fleming-Viot selects the minimal quasi-stationary distribution:
The Galton-Watson case
A. Asselah, P. Ferrari, P. Groisman, M. J. Ann Inst H. Poincaré,
2015
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Front propagation and quasi-stationary distributions: the same
selection principle.
P. Groisman, M. J.
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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
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limx→∞ P x (T > t) = ∞,
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There exists γ > 0 and z ∈ N such that E z (exp(γT0 )) < ∞,
then there exists a QSD .
QSD and QLD : examples
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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)
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p x/2
.
q
Population process with linear drift:
q(x, x + 1) = px1x>0 , q(x, x − 1) = qx1x>0 .
Infinite family of QSD . Minimal QSD :
ν ∗ (x) = c
p x
.
q
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)
I
p x/2
.
q
Population process with linear drift:
q(x, x + 1) = px1x>0 , q(x, x − 1) = qx1x>0 .
Infinite family of QSD . Minimal QSD :
ν ∗ (x) = c
p x
.
q