Section: New Results

Probabilistic numerical methods, stochastic modelling and applications

Participants : Mireille Bossy, Nicolas Champagnat, Julien Claisse, Madalina Deaconu, Samuel Herrmann, James Inglis, Antoine Lejay, Sylvain Maire, Sebastian Niklitschek Soto, Denis Talay, Etienne Tanré, Denis Villemonais, Laurent Violeau.

• M. Bossy and J-F. Jabir (University of Valparaíso) [29] , have proved the well-posedness of a conditional McKean Lagrangian stochastic model, endowed with the specular boundary condition, and further the mean no-permeability condition, in a smooth bounded confinement domain $𝒟$. This result extends their previous work [48] , where the confinement domain was the upper-half plane. The extension of the construction to more general confinement domain exhibits difficulties that we handle by combining stochastic calculus and the analysis of kinetic equations. As a prerequisite for the study of the nonlinear case, we construct a Langevin process confined in $\overline{𝒟}$ and satisfying the specular boundary condition. We then use PDE techniques to construct the time-marginal densities of the nonlinear process from which we are able to exhibit the conditional McKean Lagrangian stochastic model.

• N. Champagnat studied in collaboration with S. Méléard (Ecole Polytechnique, Palaiseau) and P.-E. Jabin (Univ. of Maryland) adaptive dynamics and evolutionary branching in individual-based models of populations competing for resources, where resources consumption is modelled similarly as for chemostat systems of ODEs [13] .

• M.  Deaconu and S. Herrmann constructed a new procedure for the simulation of the hitting times of nonlinear boundaries for Bessel processes. This method, called the random walk on moving spheres algorithm, is based on two key properties: first, the explicit distribution of the first hitting time of a particular boundary for the Bessel process; second, the connexion between the Bessel process and the Euclidean norm of a Brownian motion having the same dimension. This result can be applied for the hitting time of a given level for the Cox-Ingersoll-Ross process and thus be used in models arising from finance and neurosciences [15] .

• J. Inglis and E. Tanré completed their study with F. Delarue and S. Rubenthaler (Univ. Nice – Sophia Antipolis) on the global solvability of a networked system of integrate-and-fire neurons proposed in the neuroscience literature. To do this it was necessary to obtain some general estimates of the first hitting times of barriers by non-homogeneous processes, which have been collected together separately in [40] , http://hal.inria.fr/hal-00870991 .

• J. Inglis, in collaboration with O. Faugeras (EPI NeuroMathComp ), studied the well-posedness of stochastic neural field equations within a rigorous framework. The deterministic versions of these equations have been used to great success for the macroscopic modeling of brain activity. Their stochastic counterparts are non-trivial SPDEs, due to the presence of a nonlocal operator [26] , http://hal.inria.fr/hal-00907555 .

• A. Lejay and L. Coutin (Université de Toulouse) have continued their work on the sensitivity of the Itô's map in the context of rough paths [37] .

• With L. Coutin (Université de Toulouse), A. Lejay has provided a framework for considering linear rough differential equations [49] .

• With A. Kohatsu-Higa (Ritsumeikan University) and K. Yasuda (Hosei University), A. Lejay provided bounds on the weak rate of convergence of the Euler scheme when the drift term is discontinuous [41] .

• S. Maire and G. Nguyen have developed a Monte Carlo method to deal with Robin and transmission conditions for elliptic diffusion equations in stratified media. It combines walk on spheres techniques and finite differences [44] .

• D. Villemonais worked on the empirical distribution of Fleming-Viot type particle systems. Using couplings with reflected diffusion processes, he proved the uniform tightness of such empirical distributions and deduced the non-degeneracy of the law of diffusion processes conditioned not to hit a boundary [19] .

• D. Villemonais proved in [18] a general approximation method for Markov processes conditioned not be killed. The method is based on a mean field interacting particles system which is easy to simulate. The study also details the particular case of time/environment dependent diffusion processes.

• N. Champagnat and B. Henry work on the long-time behaviour of the frequency spectrum for the Splitting Tree models under the infinitly-many alleles model. Specificaly, they want to study the asymptotic behavior of the largests families in the “supercritical clonal” case. Such results could be applied to design statistical methods to detect positive selection of a gene in a growing population.

• N. Champagnat, D. Ritchie (Orpailleur team, Inria Nancy) and B. Henry work on the design of a stochastic model for the evolution of 3D structures of proteins. Using Kpax algorithm [52] , which allow to quantify the evolutionary distance between proteins, their goal is to design a statistical method to infer phylogenetic trees with particle systems methods.

• N. Champagnat and D. Villemonais obtained criterions for existence and uniqueness of quasi-stationary distributions and $Q$-processes for general absorbed Markov processes. A quasi-stationary distribution is a stationary distribution conditionnally on non-absorbtion, and the $Q$-process is defined as the original Markov process conditionned to never be absorbed. The criterion that they obtain ensures exponential convergence of the conditionned $t$-marginal of the process conditionned not to be absorbed at time $t$, to the quasi-stationary distribution and also the exponential ergodicity of the $Q$-process. This work is currently being written.

• J. Claisse continued his PhD. under the supervision of N. Champagnat and D. Talay on stochastic control of population dynamics. He completed a finite-horizon optimal control problem on branching–diffusion processes. He also created and studied a hybrid model of tumor growth emphasizing the role of acidity. Key therapeutic targets appear in the model to allow investigation of optimal treatment problems.

• M. Deaconu and S. Herrmann are developing a new algorithm for the simulation of Bessel processes hitting times for non-integer dimensions. The idea is to decompose the dimension into its integer part and its fractional part and use the additivity property for squared Bessel processes. Each simulation step is splitted in two parts: one uses the integer dimension case and the other one considers hitting times of a Bessel process starting from zero.

• M. Deaconu in collaboration with L. Beznea (IMAR Bucarest) and O. Lupaşcu (Université Paris 13 and IMAR Bucarest) studies the connexion between the coagulation/fragmentation phenomena and branching processes.

• J. Inglis and D. Talay are developing a mean-field model of a network of neurons, that contains both a spatial element describing the transmission of a signal along dendrites, as well as non-homogenous weights that represent the strength of the synaptic connections. More generally, this leads to the study of the limiting behavior of non-exchangeable mean-field particle systems.

• J. Inglis and E. Tanré are continuing their collaboration with F. Delarue (Univ. Nice – Sophia Antipolis) by developing approximations to a limiting equation describing the behavior of a large network of neurons all behaving according to the integrate-and-fire model. Both a particle system approximation and an approximation involving delays are considered.

• S. Larnier and A. Lejay have worked on nearshore wave analysis and bathymetry identification through the use of a video installed on the shore [42] , [43] .

• A. Lejay has continued his work with R. Rebolledo (Pontificia Universidad Católica), S. Torres (Universidad de Valparaíso) and E. Mordecki (Universidad de la República) on the parametric estimation of coefficients of diffusion with discontinuous coefficients.

• S. Maire and I. Dimov (Bulgarian academy of sciences) have introduced a new Monte Carlo method to solve real or complex linear systems of equations. Coupled with sequential Monte Carlo this walk on equations method shows a very fast convergence. A similar method is in progress to solve linear integral equations.

• S. Niklitschek Soto and D. Talay have set up and solved a new martingale problem which has allowed them to get a new stochastic representation for solutions of multi-dimensional diffraction parabolic PDEs with general discontinuous coefficients. One of the main difficulties to overcome has been to identify the proper weighted local time process involved in the stochastic dynamics. This work opens the way to innovating Monte Carlo methods for this class of PDEs.

• P. Guiraud (University of Valparaíso) and E. Tanré study the effect of noise in the phenomenon of spontaneous synchronisation in a network of full connected integrate- and-fire neurons. They detail cases in which the phenomenon of synchronization persists in a noisy environment, cases in which noise permits to accelerate synchronization, and cases in which noise permits to observe synchronization while the noiseless model does not show synchronization. (Math Amsud program SIN)

• L. Capietto worked during his internship under the supervision of O. Faugeras (EPI NeuroMathComp ) and E. Tanré on extension of [51] , in a context with several populations of homogeneous neurons. They study the limit mean field equation of the membrane potential as the number of neurons increase in a network with correlated synaptic weights.

• E. Tanré, in collaboration with O. Faugeras (EPI NeuroMathComp ) and the team Inference and Visual Behavior (IViBe) of Institut de Neurosciences de la Timone (INT), studied the motion of eyes, the phenomena of sacades and micro-saccades when monkeys or humans have to fix the center of a picture during a few minutes. They introduce a stochastic model to describe the typical path of the eyes on the picture and evaluate the link between the characteristics of the artificial pictures and the coefficients of the stochastic model.

• L. Violeau continued his PhD. on Stochastic Lagrangian Models and Applications to Downscaling in Fluid Dynamics under the supervision of M. Bossy and A. Rousseau (Lemon team, Inria Sophia Antipolis - Méditerranée). Laurent Violeau has obtained a theoretical rate of convergence of the particle approximation of kinetic conditional McKean-Vlasov stochastic models. This result is the first that explicits the complex relationship between the two sources of spacial errors in such kind of algorithm: the smoothing parameter for the conditional expectation estimator and the number of interacting particles. This theoretical convergence rate was confronted with numerical tests in the case of simplified Lagrangian models that confirm the pertinence of the theoretical bound for the error.

• C. Graham and D. Talay are writing the second volume of their series published by Springer on the Mathematical Foundations of Stochastic Simulations.

• In collaboration with N. Touzi (Ecole Polytechnique), D. Talay is studying stochastic differential equations involving local times with stochastic weights, and extensions of classical notions of viscosity solutions to PDEs whose differential operator has discontinuous coefficients and transmission boundary conditions.