Macroscopic quasi-stationary distribution and microscopic particle systems

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 Introduction to quasi-stationary distributions: Macroscopic model Particle systems : Microscopic model 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?

2 types of quasi-stationarity 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,... 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 X t 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: φ µ t (A) = Pµ (X t 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 φ µ t ( ).

Quasi-stationary distribution We say that ν is a quasi-stationary distribution (QSD ) if there exists a probability measure µ such that: lim t φµ t ( ) = νµ ( ).

QSD 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) 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( θ νt). There might exist a maximal θ corresponding to the so-called minimal QSD.

Challenges Existence, Simulation, 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 Particle systems: microscopic models 1. Branching processes 2. Fleming-Viot 3. N- Branching Brownian motion 4. Choose the fittest

Traveling waves for PDE 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. Example: KPP (Kolmogorov-Petrovsky-Piskounov) equation: u t = 1/2u xx + f (u). Links with the maximum of the Branching Brownian motion (McKean, Bramson,...) In general, there may exist an infinity of solutions parametrized by their speed s. Which speed to select?

Traveling waves and QSDs For the KPP equation with f (u) = u 2 u, one can prove that: the equation has the same traveling wave as the free boundary equation obtained for the N-BBM, 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. 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 Macroscopic models (QSDs and traveling waves) forget that a population can be very large but finite. Microscopic models are intrinsically corresponding to finite population. They do select the minimal QSD/traveling wave.

references Simulation of quasi-stationary distributions on countable spaces, P. Groisman, M. J., Markov processes and related fields 2013 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 Front propagation and quasi-stationary distributions: the same selection principle. P. Groisman, M. J. Kesten-Stigum theorems in L 2 beyond R-positivity. S. Saglietti, M.J.

Thanks

Existence of QSD Very few general result: Proposition (Ferrari et. al. 1995) X t N. If lim x P x (T > t) =, There exists γ > 0 and z N such that E z (exp(γt 0 )) <, then there exists a QSD.

QSD and QLD : examples Birth and death process (non-empty M/M/1 queue): q(x, x + 1) = p1 x>0, q(x, x 1) = q1 x>0. Infinite family of QSD. Minimal QSD : ν (x) = c(x + 1) ( p ) x/2. q Population process with linear drift: q(x, x + 1) = px1 x>0, q(x, x 1) = qx1 x>0. Infinite family of QSD. Minimal QSD : ν (x) = c ( p ) x. q