Université Paris Diderot
UFR de Mathématiques and LPMA
Bâtiment Sophie Germain
8 place Aurélie Nemours 75013 Paris
Office 545 (5th floor): directions
Trimester IHP (April 3rd-July 7th 2017): STOCHASTIC DYNAMICS OUT OF EQUILIBRIUM
Research interests: probability and applications
- Statistical Mechanics, Disordered Systems
- Nonequilibrium Statistical Mechanics
- Applications to Biology and Life Science
Disorder and critical phenomena through basic probability models Lecture Notes in Mathematics 2025, Springer, 2011.
Random Polymer Models IC Press, World Scientific, 2007.
Coauthors (chronological order)
On the long time effect of small noise on stochastic ODEs (in presence of attracting periodic trajectories)
Stable limit cycles are omnipresent when modeling real life systems and this just reflects the central role that rhythms play in our life and in just about everything that surrounds us. In the image, in black, the limit cycle of an Ordinary Differential Equation (ODE) system dx(t)/dt=F(x(t)): the particular instance is the Fitzhugh-Nagumo system, a basic two dimensional model of one neuron. Perturbing stochastically the system is a way of going farther from a modeling viewpoint. The figure shows two (blue and red) one period realizations of the solution to the Itô stochastic system dX(t)=F(X(t))dt+εG(X(t))dB(t), with G(·) a matrix valued function and B a (multi-dimensional) Brownian motion and ε is a small positive parameter. Initial condition is kept the same in all cases. The first remark is that for ε tending to zero stochastic and deterministic and stochastic trajectories get closer on every finite time horizon. The question is: what happens if we let the time horizon grow with ε? When does this proximity breaks down and what do we observe? In Small noise and long time phase diffusion in stochastic limit cycle oscillators (joint work with C. Poquet and A. Shapira) we attempt a review the very wide applied science literature on this important issue and we establish rigorously (and quantitatively) that a phase diffusion phenomenon takes place on the time scale ε-2. One can have a glimpse on this phenomenon by clicking on the image (The blue dot is the deterministic system, the red dot is the stochastic one). It is only going on times of about 100 (ε=1/10 in the simulation) periods that one starts appreciating the noise induced frequency shift induced by the noise (in mathematical terms: the phase diffusion has a net drift that happens to be in the sense of rotation of the limit cycle for the specific choice of F and G). Our work provides estimates on the effect of the noise up to times of the order of exp(cε-2), c>0, where Large Deviations effects enter the game.
The generalized Poland-Scheraga DNA denaturation model and bivariate renewals
The Poland-Scheraga (PS) is the standard basic model for DNA denaturation, that is the transition that happens at high temperature from two complementary DNA strands that are bind together (localized state) to two free strands (delocalized state). The original PS model is limited to exact complementarity of the two strands - equal length strands and no mismatches are allowed - and it boils down to a sequence of bind pairs and symmetric loops (i.e., the number of bases contributed by each strand is the same). This model enjoyed and still enjoys a large popularity also because, in its homogeneous version, it is exactly solvable (in the sense of statistical mechanics: the PS model is a Gibbs measure) and the denaturation transition can be understood in detail. This solvable character is ultimately related to the fact that the homogeneous PS model is very closely related to a class of discrete renewal processes (it can even be mapped to a renewal). Very remarkably, a natural generalization of the PS model - the generalized PS (gPS) model - allowing unequal length strands and asymmetric loops turns out to retain the solvable character of the PS model. A direct representation of a trajectory of the model with two strands of respective lengths twelve and nineteen bases is shown in the first figure. In Generalized Poland-Scheraga denaturation model and two-dimensional renewal processes (joint work with M. Khatib) we considered the homogeneous gPS model and we have exploited the fact that this model can be mapped to a two dimensional (or bivariate) renewal. The mapping at the level of trajectory transforms base pairs into points in the plane: the base pairs (1,1), (2,2), (3,6),… become an increasing sequence in the plane (see the second figure: we made the conventional choice to shift all coordinates down by one). This allows an analysis which is parallel to the original PS model, with the novelties introduced by the higher dimensional character. These novelties are not only technical, because the gPS has a phenomenology that is substantially richer than the PS model. The most evident novelty is the appearance of transitions inside the localized regime. From a mathematical viewpoint these transitions between different localized states can be interpreted as the switching of the underlying renewal (that is under the effect of pinning potentials) from Large Deviations regimes that are of Cramér type, i.e. that correspond to tilting the measure, to regimes that are not.