Section: New Results

Probabilistic numerical methods, stochastic modelling and applications

Participants : Mireille Bossy, Nicolas Champagnat, Julien Claisse, Madalina Deaconu, Benoît Henry, James Inglis, Antoine Lejay, Oana Valeria Lupascu, Sylvain Maire, Sebastian Niklitschek Soto, Denis Talay, Etienne Tanré, Denis Villemonais.

Published works and preprints

• M. Bossy and J.-F. Jabir (University of Valparaíso) [13] 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 $𝒟$.

• M. Bossy, N. Champagnat, S. Maire and L. Violeau worked with H. Leman (CMAP, Ecole Polytechnique) and M. Yvinec (Inria Sophia, EPI Geometrica ) on Monte Carlo methods for the linear and non-linear Poisson-Boltzmann equations [12] . These methods are based on walk on spheres algorithm, simulation of diffusion processes driven by their local time, and branching Brownian motion. Their code for the linear equation can deal with bio-molecules of arbitrary sizes, based on computational geometry tools from the CGAL C++ Library developed by the Geometrica team. The non-linear equation is solved using branching Brownian motion.

• M. Bossy, O. Faugeras (Inria Sophia, EPI NeuroMathComp ), and D. Talay have clarified the well-posedness of the limit equations to the mean-field $N$-neuron models proposed in [42] and proven the associated propagation of chaos property. They also have completed the modeling issue in [42] by discussing the well-posedness of the stochastic differential equations which govern the behavior of the ion channels and the amount of available neurotransmitters. See [29] .

• N. Champagnat and D. Villemonais obtained criterions for existence and uniqueness of quasi-stationary distributions and $Q$-processes for general absorbed Markov processes [31] . A quasi-stationary distribution is a stationary distribution conditionally on non-absorbtion, and the $Q$-process is defined as the original Markov process conditioned to never be absorbed. The criterion that they obtain ensures exponential convergence of the conditioned $t$-marginal of the process conditioned not to be absorbed at time $t$, to the quasi-stationary distribution and also the exponential ergodicity of the $Q$-process.

• M. Deaconu and S. Herrmann continued and completed the study of the simulation of the hitting time of some given boundary for Bessel processes. They constructed an original approximation method for hitting times of a given threshold by Bessel processes with non-integer dimension. In this work, they combine the additivity property of the laws of squared Bessel processes with their previous results on the simulation of hitting times of Bessel processes with integer dimension, based on the method of images and on the connexion with the Euclidiean norm of the Brownian motion [33] .

• M. Deaconu, S. Herrmann and S. Maire introduced a new method for the simulation of the exit time and position of a $\delta$-dimensional Brownian motion from a domain. The main interest of this method is that it avoids splitting time schemes as well as inversion of complicated series. The idea is to use the connexion between the $\delta$-dimensional Bessel process and the $\delta$-dimensional Brownian motion thanks to an explicit Bessel hitting time distribution associated with a particular curved boundary. This allows to build a fast and accurate numerical scheme for approximating the hitting time [34] .

• M. Deaconu and O. Lupaşcu worked with L. Beznea (Bucharest, Romania) on the construction and the branching properties of the solution of the fragmentation equation and properly associate a continuous time càdlàg Markov process. The construction and the proof of the path regularity of the Markov processes are based on several newly developed potential theoretic tools.

• J. Inglis, together with O. Faugeras (Inria NeuroMathComp ) finalized their article [18] on the well-posedness of stochastic neural field equations within a rigorous framework.

• J. Inglis and E. Tanré together with F. Delarue and S. Rubenthaler (Univ. Nice – Sophia Antipolis) finalized their article [16] on the global solvability of a networked system of integrate-and-fire neurons proposed in the neuroscience literature.

• J. Inglis and E. Tanré together with F. Delarue and S. Rubenthaler (Univ. Nice – Sophia Antipolis) completed their study of the mean-field convergence of a highly discontinuous particle system modeling the behavior of a spiking network of neurons, based on the integrate-and-fire model [17] . Due to the highly singular nature of the system, it was convenient to work with a relatively unknown Skorohod topology.

• J. Inglis and D. Talay introduced in [38] a new model for a network of spiking neurons that attempted to address several criticisms of previously considered models. In particular the new model takes into account the role of the dendrites, and moreover includes non-homogeneous synaptic weights to describe the fact that not all neurons have the same effect on the others in the network. They were able to obtain mean-field convergence results, using new probabilistic arguments.

• A. Lejay have worked with G. Pichot (EPI Sage ) on benchmarks for testing Monte Carlo methods to simulate particles in one-dimensional media, and applied this statistical methodology to four methods, including the exact method developed previously [45] . This work led also to empirical observations that should guide the design of new methods [24] .

• S. Maire is working with the Bulgarian Academy of sciences on Monte Carlo algorithms for linear equations based on killed random walks. In a first work, with I. Dimov and J-M. Sellier [37] , a new Monte Carlo method to solve linear systems of equations has been introduced. This method can either compute one component of the solution or all components simultaneously. In a second work, with Ivan Dimov and Rayna Georgieva, a new Monte Carlo method to solve Fredhom integral equations of the second kind is developed [36] .

• D. Villemonais worked with P. Del Moral (Univ. Sydney) on the conditional ergodicity of time inhomogeneous diffusion processes [35] . They proved that, conditionally on non extinction, an elliptic time-inhomogeneous diffusion process forgets its initial distribution exponentially fast. An interacting particle scheme to numerically approximate the conditional distribution is also provided.

• D. Villemonais proved a Foster-Lyapunov type criterion which ensures the exponential ergodicity of a Fleming-Viot type particle system whose particles evolve as birth and death processes. The criterion also ensures the tightness of the sequence of empirical stationary distributions considered as a family of random measures. A numerical study of the speed of convergence of the particle system is also obtained under various settings [41] .

Other works in progress

• M. Bossy and J-F. Jabir (University of Valparaíso) proved the validity of a particle approximation of a (simplified) Lagrangian Stochastic Model submitted to specular reflections at the boundary and satisfying the mean no-permeability condition. This work achieves to extend our previous study [43] to the multidimensional case.

• N. Champagnat and D. Villemonais obtained criterions for existence, uniqueness and exponential convergence in total variation of quasi-stationary distributions and $Q$-processes for general absorbed and killed diffusion processes. The criterion obtained is equivalent to the property that a diffusion on natural scale coming down from infinity has uniformly (w.r.t. the initial condition) bounded expectation at a fixed time $t$. A study of nearly critical cases allow to conjecture that this property is true for all diffusion processes on natural scale coming down from infinity. This work is currently being written.

• N. Champagnat and B. Henry worked on the long-time behavior of the frequency spectrum for the Splitting Tree models under the infinitly-many alleles model. They obtained, using a new method for computing the expectation of an integral with respect to a random measure, the asymptotic behavior of the moments of the frequency spectrum. As an application, they derived the law of large number and a new central limit theorem for the frequency spectrum. This work is currently being written.

• J. Claisse defended 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.

• J. Claisse and D. Talay in collaboration with X. Tan (Univ. of Paris Dauphine) extended their previous work on a pseudo-Markov property enjoyed by the solutions of controlled stochastic differential equations and its application to the proof of the dynamic programming principle. A paper is being finished.

• M. Deaconu and O. Lupascu are working with L. Beznea (Bucharest, Romania) on a stochastic model for avalanche phenomena involving rupture properties that occur in the physical and deterministic models for snow avalanches. This approach is based on their recent results on fragmentation processes by stochastic differential equation and branching processes.

• M. Deaconu and O. Lupascu are working on a numerical probabilistic algorithm for an avalanche-type process. The originality of this approach is to use a coagulation/fragmentation model to describe the avalanche phenomenon. More precisely, they consider a particular fragmentation kernel which introduces “rupture-type” properties of deterministic models for snow avalanches.

• An important issue in neuroscience is the modelling of spike trains of a single neuron. In this context, the membrane potential of a neuron can be described by using a simple stochastic differential equation with periodic input, that is reset to a rest potential each time it hits a certain threshold. J. Inglis, A. Richard, D. Talay, and E. Tanré study how the law of these hitting times is affected when one changes the white noise (in the SDE) into a correlated noise. Practically, they use a fractional Brownian motion, and since the computation of the hitting times of such a non-Markovian, non-semimartingale process is still an open question, they rather try to compute the deviations from the white noise model. This is expected to give insights on the relevance of models with memory and long-range dependence.

• J. Inglis started a collaboration with B. Hambly and S. Ledger at the University of Oxford, in which interacting mean-field particle systems with common noise are being studied. Such systems are representative of systems of spiking neurons or portfolio defaults. In previous studies each particle was driven by a noise that was assumed independent from particle to particle (i.e. intrinsic noise). By considering a common driving noise in addition to the intrinsic noise, it is possible to model the fact that the environment in which the particles live is also noisy. This leads to the study of a new type of conditional McKean-Vlasov equation.

• J. Inglis, in collaboration with J. Maclaurin (EPI NeuroMathComp ) and W. Stannat (Berlin), has begun working on a new framework to understand the effect of noise on neural field equations. Deterministic neural field equations exhibit traveling wave solutions, and so the effect of noise on these solutions is of great interest. The idea is to decompose the solution into various components, which allow one to see directly how the noise affects the solution in the direction of the moving wave front. In particular, the goal is to reconcile mathematically the previous works of P. Bressloff and W. Stannat on the same subject, and to obtain a large deviation principle.

• J. Inglis and D. Talay are in the process of studying the emergence of spatio-temporal noise starting from microscopic models of neuron conductance.

• A. Lejay continued his collaboration with S. Torres (Universidad de Valapraíso, Chile) and E. Mordecki (Universidad de la República, Uruguay) on the estimation of the parameter of the Skew Brownian motion. This work is related to the modelling of diffusion processes in media with interfaces and has potential applications in many domains, such as population ecology.

• Together with R. Rebolledo (Pontificia Universidad Católica, Santiago, Chile), A. Lejay continued his review work on the mathematical modelling of the Wave Energy Converter Called the Oscillating water column, within the framework of the CIRIC project.

• A. Lejay continued his work on the Snapping out Brownian motion to perform numerical tests for the computation of the mean residence time in a diffusive medium with semi-permeable membranes, such as the one encountered in the mathematical modelling of diffusion Magnetic Resonance Imaging.

• A. Lejay continued his collaboration with L. Coutin (Universté Paul Sabatier, Toulouse) on the sensitivity of rough linear differential equations, by providing general results on the derivatives of the solution of rough differential equations with respect to parameters or the starting point.

• S. Niklitschek Soto and D. Talay completed their stochastic analysis of diffraction parabolic PDEs with general discontinuous coefficients in the multidimensional case.

• P. Guiraud (University of Valparaíso) and E. Tanré study the effect of noise in the phenomenon of spontaneous synchronisation in a network of connected leaky 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)

• O. Faugeras (EPI NeuroMathComp ) and E. Tanré worked on an extension of [44] to a context of several populations of homogeneous neurons. They study the limit mean field equation of the membrane potential as the number of neurons increases in a network with correlated synaptic weights. A paper is in preparation.

• C. Graham (CMAP, Ecole polytechnique) 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 (CMAP, 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.