Moderate deviations and extinction of an epidemic

Consider an epidemic model with a constant flux of susceptibles, in a situation where the corresponding deterministic epidemic model has a unique stable endemic equilibrium. For the associated stochastic model, whose law of large numbers limit is the deterministic model, the disease free equilibrium is an absorbing state, which is reached soon or later by the process. However, for a large population size, i.e. when the stochastic model is close to its deterministic limit, the time needed for the stochastic perturbations to stop the epidemic may be enormous. In this paper, we discuss how the Central Limit Theorem, Moderate and Large Deviations allow us to give estimates of the extinction time of the epidemic, depending upon the size of the population

A path-valued Markov process indexed by the ancestral mass

A family of Feller branching diffusions $Z^x$, $x \ge 0$, with nonlinear drift and initial value $x$ can, with a suitable coupling over the {\em ancestral masses} $x$, be viewed as a path-valued process indexed by $x$. For a coupling due to Dawson and Li, which in case of a linear drift describes the corresponding Feller branching diffusion, and in our case makes the path-valued process Markovian, we find an SDE solved by $Z$, which is driven by a random point measure on excursion space. In this way we are able to identify the infinitesimal generator of the path-valued process. We also establish path properties of $x\mapsto Z^x$ using various couplings of $Z$ with classical Feller branching diffusions.Comment: 23 pages, 1 figure. This version will appear in ALEA. Compared to v1, it contains amendmends mainly in Sec. 2 and in the proof of Proposition 4.

Continuity of the Feynman-Kac formula for a generalized parabolic equation

It is well-known since the work of Pardoux and Peng [12] that Backward Stochastic Differential Equations provide probabilistic formulae for the solution of (systems of) second order elliptic and parabolic equations, thus providing an extension of the Feynman-Kac formula to semilinear PDEs, see also Pardoux and Rascanu [14]. This method was applied to the class of PDEs with a nonlinear Neumann boundary condition first by Pardoux and Zhang [15]. However, the proof of continuity of the extended Feynman-Kac formula with respect to x (resp. to (t,x)) is not correct in that paper. Here we consider a more general situation, where both the equation and the boundary condition involve the (possibly multivalued) gradient of a convex function. We prove the required continuity. The result for the class of equations studied in [15] is a Corollary of our main results

Continuous branching processes : the discrete hidden in the continuous : Dedicated to Claude Lobry

International audienceFeller diffusion is a continuous branching process. The branching property tells us that for t > 0 fixed, when indexed by the initial condition, it is a subordinator (i. e. a positive–valued Lévy process), which is fact is a compound Poisson process. The number of points of this Poisson process can be interpreted as the number of individuals whose progeny survives during a number of generations of the order of t × N, where N denotes the size of the population, in the limit N ―>µ. This fact follows from recent results of Bertoin, Fontbona, Martinez [1]. We compare them with older results of de O’Connell [7] and [8]. We believe that this comparison is useful for better understanding these results. There is no new result in this presentation.La diffusion de Feller est un processus de branchement continu. La propriété de branchement nous dit que à t > 0 fixé, indexé par la condition initiale, ce processus est un subordinateur (processus de Lévy à valeurs positives), qui est en fait un processus de Poisson composé. Le nombre de points de ce processus de Poisson s’interprète comme le nombre d’individus dont la descendance survit au cours d’un nombre de générations de l’ordre de t × N, où N désigne la taille de la population, dans la limite N --> µ. Ce fait découle de résultats récents de Bertoin, Fontbona, Martinez [1]. Nous le rapprochons de résultats plus anciens de O’Connell [7] et [8]. Ce rapprochement nous semble aider à mieux comprendre ces résultats. Cet article ne contient pas de résultat nouveau