Florent Malrieu

Qualitative properties of certain piecewise deterministic Markov processes

We study a class of Piecewise Deterministic Markov Processes with state space Rd x E where E is a finite set. The continuous component evolves according to a smooth vector field that is switched at the jump times of the discrete coordinate. The jump rates may depend on the whole position of the process. Working under the general assumption that the process stays in a compact set, we detail a possible construction of the process and characterize its support, in terms of the solutions set of a differential inclusion. We establish results on the long time behaviour of the process, in relation to a certain set of accessible points, which is shown to be strongly linked to the support of invariant measures. Under Hormander-type bracket conditions, we prove that there exists a unique invariant measure and that the processes converges to equilibrium in total variation. Finally we give examples where the bracket condition does not hold, and where there may be one or many invariant measures, depending on the jump rates between the flows.

On the stability of planar randomly switched systems

Consider the random process (Xt) t>0 solution of u Xt = AItXt where (It) t>0 is a Markov process on f 0;1g and A0 and A1 are real Hurwitz matrices on R 2 . Assuming that there exists � 2 (0;1) such that (1 − � )A0 + �A 1 has a positive eigenvalue, we establish that k Xtk may converge to 0 or +1 depending on the the jump rate of the process I. An application to product of random matrices is studied. This paper can be viewed as a probabilistic counterpart of the paper [2] by Balde, Boscain and Mason.

Quantitative estimates for the long-time behavior of an ergodic variant of the telegraph process

Motivated by stability questions on piecewise-deterministic Markov models of bacterial chemotaxis, we study the long-time behavior of a variant of the classic telegraph process having a nonconstant jump rate that induces a drift towards the origin. We compute its invariant law and show exponential ergodicity, obtaining a quantitative control of the total variation distance to equilibrium at each instant of time. These results rely on an exact description of the excursions of the process away from the origin and on the explicit construction of an original coalescent coupling for both the velocity and position. Sharpness of the obtained convergence rate is discussed.

Rare sex or out of reach equilibrium? The dynamics of F IS in partially clonal organisms

BackgroundPartially clonal organisms are very common in nature, yet the influence of partial asexuality on the temporal dynamics of genetic diversity remains poorly understood. Mathematical models accounting for clonality predict deviations only for extremely rare sex and only towards mean inbreeding coefficient FIS¯<0

Probabilistic and Piecewise Deterministic models in Biology

\overline{F_{IS}}<0

Algorithme EM et mélanges

. Yet in partially clonal species, both FIS < 0 and FIS > 0 are frequently observed also in populations where there is evidence for a significant amount of sexual reproduction. Here, we studied the joint effects of partial clonality, mutation and genetic drift with a state-and-time discrete Markov chain model to describe the dynamics of FIS over time under increasing rates of clonality.ResultsResults of the mathematical model and simulations show that partial clonality slows down the asymptotic convergence to FIS = 0. Thus, although clonality alone does not lead to departures from Hardy-Weinberg expectations once reached the final equilibrium state, both negative and positive FIS values can arise transiently even at intermediate rates of clonality. More importantly, such “transient” departures from Hardy Weinberg proportions may last long as clonality tunes up the temporal variation of FIS and reduces its rate of change over time, leading to a hyperbolic increase of the maximal time needed to reach the final mean FIS,∞¯

Urnes d’Ehrenfest

\overline{F_{IS,\infty }}