Front propagation and quasi-stationary distributions for one-dimensional Lévy processes

We jointly investigate the existence of quasi-stationary distributions for one-dimensional

Lévy processes and the existence of traveling waves for the Fisher-Kolmogorov-Petrovskii-

Piskunov (F-KPP) equation associated with the same motion. Using probabilistic ideas de-

veloped by S. Harris for the F-KPP equation, we show that the existence of a traveling

wave for the F-KPP equation associated with a centered Lévy processes that branches at rate

r and travels at velocity c is equivalent to the existence of a quasi-stationary distribution for a

Lévy process with the same movement but drifted by -c and killed at zero, with mean absorption time

1/r. This allows to generalize the known existence conditions in both contexts.

Joint work with Pablo Groisman.

sep / 2016
05

Extinction time for the weaker of two competing SIS epidemics

Abstract: We consider a simple stochastic model for the spread of a disease caused by two virus strains in a closed homogeneously mixing population of size *N*. In our model, the spread of each strain is described by the stochastic logistic SIS epidemic process in the absence of the other strain, and we assume that there is perfect cross-immunity between the two virus strains, that is, individuals infected by one strain are temporarily immune to re-infections and infections by the other strain. For the case where one strain has a strictly larger basic reproductive ratio than the other, and the stronger strain on its own is supercritical (that is, its basic reproductive ratio is larger than 1), we derive precise asymptotic results for the distribution of the time when the weaker strain disappears from the population, that is, its extinction time. We further extend our results to the case where the difference between the two reproductive ratios may tend to 0.

In our proof, we set out a simple approach for establishing a fluid limit approximation for a sequence of Markov chains in the vicinity of a stable fixed point of the limit drift equations, valid for a time exponential in the system size. This is a joint work with Malwina Luczak.

ago / 2016
22

Limiting laws for some integrated processes

The study of limiting laws, or penalizations, of a given process may

be seen (in some sense) as a way to condition a probability law by an

a.s. infinite random variable. The systematic study of such problems

started in 2006 with a series of papers by Roynette, Vallois and Yor

who looked at Brownian motion perturbed by several examples of

functionals. These works were then generalized to many families of

processes: random walks, Lévy processes, linear diffusions…

We shall present here some examples of penalization of a non-Markov

process, i.e. the integrated Brownian motion, by its first passage

time, nth passage time, and last passage time up to a finite horizon. We shall show that the penalization principle holds in all these cases, but that the

conditioned process does not always behave as expected. Recent results

around persistence of integrated symmetric stable processes will also

be discussed.

ago / 2016
11

Martingale problem and trace processes applied to metastability.

The Martingale problem is a concept introduced by Stroock and Varadhan

which can be understood as a sort of ordinary differential equation in

which the vector field is replaced by a field of second order

differential operators. A Markov process can be characterized as a

unique solution of a Martingale problem. This fact turns the

martingale problem in a very useful tool to prove convergence of

stochastic processes derived from Markov processes.

In this talk we shall use the martingale problem to prove the

convergence of processes arising in the study of metastable systems.

We shall explain how this tool is used in combination with other ones

like trace processes and potential theory. Finally, we shall show some

examples of systems in which this approach has been applied.

This is a joint work with C. Landim.

ago / 2016
01

Phase Transitions on the Long Range Ising Models in presence of an random external field

We study the ferromagnetic one-dimensiosnal Random Field Ising Model with (RFIM) in presence of an external random field. The interaction between two spins decays as $d^{\alpha-2}$ where $d$ is the distance between two sites and $\alpha \in [0,1/2)$ is a parameter of the model. We consider an external random field on $\mathbb{Z}$ with independent but not identically distributed random variables. Specifically for each $i \in \mathbb{Z}$, the distribution of $h_i$ is

$P[h_i=\pm \theta(1+|i|)^{-\nu/2}]$,

This work, whose main goal is the study of the existence of a phase transition at a strictly positive temperature for different values of $\nu$ is inspired on the very recent article [2] where the 2D Ising Model with spatially dependent but not random external field is studied. In the random case, we combine some of the martingale difference techniques used in the previous articles of Cassandro, Picco and Orlandi [3], and the Aizemann & Wehr method [3]. Some of the classical results, the key parts of this work and some of the technical difficulties will be discussed in this talk.

Joint work with Pierre Picco

References:

[1] M. Aizenman and C. M. Newman. Discontinuity of the percolation density in one-dimensional 1/|x − y|

107(4):611–647, 1986.

[2] Rodrigo Bissacot, Marzio Cassandro, Leandro Cioletti, and Errico Presutti.

Phase transitions in ferromagnetic ising models with spatially dependent

magnetic fields. Communications in Mathematical Physics, 337(1):41–53,

2015.

[3] Marzio Cassandro, Enza Orlandi, and Pierre Picco. Phase transition in the

1d random field Ising model with long range interaction. Communications

in Mathematical Physics, 288(2):731–744, 2009.

jul / 2016
25

Pontificia Universidad Católica de Chile (PUC-Chile)

Av. Vicuña Mackenna 4860, Macul,

Santiago – Chile

(+56 2) 2354 5779

Facultad de Ciencias Físicas y Matemáticas (FCFM)

Universidad de Chile

Beauchef 851, Edificio Norte, Piso 7,

Santiago – Chile