Section: New Results

Probabilistic numerical methods, stochastic modelling and applications

Participants : Mireille Bossy, Nicolas Champagnat, Madalina Deaconu, Coralie Fritsch, Benoît Henry, James Inglis, Antoine Lejay, Oana-Valeria Lupascu, Sylvain Maire, Paolo Pigato, Alexandre Richard, Denis Talay, Etienne Tanré, Denis Villemonais.

Published works and preprints

• M. Bossy with H. Quinteros (UChile) submitted a paper [36] on the strong convergence of the symmetrized Milstein scheme for some CEV-like SDEs.

• M. Bossy and J.-F. Jabir (University of Valparaíso) submitted a paper [35] on the particle approximation for Lagrangian stochastic models with specular boundary condition.

• M. Bossy with N. Maizi (Mines ParisTech) and O. Pourtallier (Inria) published a book chapter [31] on game theory analysis for carbon auction market through electricity market coupling hypothesis.

• M. Bossy, O. Faugeras (Inria Sophia, EPI NeuroMathComp ), and D. Talay published a clarification on the well-posedness of the limit equations to the mean-field $N$-neuron models proposed in [58] and proven the associated propagation of chaos property. They also have completed the modeling issue in [58] 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 [15] .

• M. Bossy, N. Champagnat, S. Maire and L. Violeau worked with H. Leman (CMAP, Ecole Polytechnique) and M. Yvinec (Inria Sophia, Geometrica team) on Monte Carlo methods for the linear and non-linear Poisson-Boltzmann equations [14] . These methods are based on walk on spheres algorithm, simulation of diffusion processes driven by their local time, and branching Brownian motion to deal with the nonlinear case.

• Together with M. Baar and A. Bovier (Univ. Bonn), N. Champagnat studied the adaptive dynamics of populations under the assumptions of large population, rare and small mutations [34] . In this work, the three limits are taken simultaneously, contrary to the classical approach, where the limits of large population and rare mutations are taken first, and next the limit of small mutations  [59] . We therefore obtain the precise range of assumptions under which these limits can be taken, and provide explicit biological conditions for which our approximation is valid.

• N. Champagnat and C. Fritsch worked with F. Campillo (Inria Sophia-Antipolis, Lemon team) on the links between a branching process and an integro-differential equation of a growth-fragmentation-death model [37] . They proved that the two representations of the model lead to the same criteria of invasion of a population in a given environment.

• Using a new method to compute the expectation of an integral with respect to a random measure, N. Champagnat and B. Henry obtained explicit formulas for the moments of the frequency spectrum in the general branching processes known as Splitting Trees, with neutral mutations and under the infinitely-many alleles model [40] . This allows them to obtain a law of large numbers for the frequency spectrum in the limit of large time.

• N. Champagnat and P.-E. Jabin (Univ. Maryland) improved significantly the description of the functional spaces in the preprint [41] , devoted to the study of strong existence and pathwise uniqueness for stochastic differential equations (SDE) with rough coefficients, typically in Sobolev spaces.

• N. Champagnat and D. Villemonais obtained criteria for existence and uniqueness of quasi-stationary distributions (QSD) and $Q$-processes for general absorbed Markov processes [17] . A QSD 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 criteria ensure exponential convergence of the $t$-marginal of the process conditioned not to be absorbed at time $t$, to the QSD and also the exponential ergodicity of the $Q$-process.

• N. Champagnat and D. Villemonais obtained criteria for existence, uniqueness and exponential convergence in total variation to QSD for general absorbed and killed diffusion processes [43] , [42] . For diffusions without killing [43] , 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 $t$. The criteria obtained for diffusion processes with killing on $[0,\infty )$ [42] combine the last criteria and conditions on the killing time only close to 0, provided $\infty$ is an entrance boundary.

• N. Champagnat and D. Villemonais obtained criteria for existence, uniqueness and exponential convergence in total variation to QSD for general multi-dimensional birth and death processes in ${\mathbb{Z}}_{+}^d$ absorbed at the boundary ${\mathbb{Z}}_{+}^d\setminus {\mathbb{N}}^d$ [44] . These birth and death models are motivated by population dynamics and the criteria obtained assume stronger intra-spectific competition than inter-specific competition. These resuls are the first one for such processes, except for the particular case of branching processes, which can be studied using very specific methods.

• 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. This method is based on 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 brownian hitting time [19] .

• M. Deaconu and O. Lupaşcu worked with L. Beznea (Bucharest, Romania) on the probabilistic interpretation of fragmentation phenomena. They constructed a continuous time branching process and characterized its behavior by using new potential theoretical tools [12] .

• M. Deaconu, O. Lupaşcu and L. Beznea (Bucharest, Romania) started a new challenging work on the description of rupture phenomena like avalanches, by using fragmentation models. The physical properties of the model are deeply involved in this study. The first results concern a stochastic equation of fragmentation and branching processes related to avalanches [13] .

• M. Deaconu, B. Dumortier and E. Vincent are working with the Venathec SAS on the acoustic control of wind farms. They constructed a new approach to control wind farms with a control model based on real-time source separation. They first designed a deterministic algorithm in order to maximize the electric production of the wind farms under the legal acoustic constraints. They showed that it is a non linear knapsack optimization problem and they proposed an efficient solution in that context using a branch and bound algorithm based on continuous relaxation. This work was published at the EWEA 2015 [30] .

• In [49] , B. Henry showed a central limit theorem for the population counting process of a supercritical Splitting Tree in the limit of large time. Thanks to the results of [40] , he also obtained a central limit theorem for the frequency spectrum of Splitting Trees with neutral mutations and under the infinitely-many alleles model.

• S. Herrmann and E. Tanré have proposed a new very efficient algorithm to simulate the first-passage-time of a one-dimensional Brownian motion over a continuous curved boundary [23] .

• 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 [21] .

• In collaboration with J. Maclaurin (Inria Sophia, EPI NeuroMathComp ) J. Inglis has presented a general framework to rigorously study the effect of spatio-temporal noise on traveling waves and stationary patterns. In particular the framework can incorporate versions of the stochastic neural field equation that may exhibit traveling fronts, pulses or stationary patterns. They have formulated a local SDE that describes the position of the stochastic wave up until a discontinuity time, at which point the position of the wave may jump and studied the local stability of this stochastic front and the long-time behavior of the stochastic wave  [50] .

• A. Lejay has continued his work on the Snapping Out Brownian motion, especially with regard to the simulation issues, with potential application to brain imaging techniques [33] , [53] .

• A. Lejay has continued his work on the simulation of processes with either discontinuous drift (with Arturo Kohatsu-Higa, Ritsumeikan Universitey and Kazuhiro Yasuda, Hosei University, Japan) [52] or with discontinuous coefficients (with Lionel Lenêtre and Géraldine Pichot, EPI Sage , Irisa) [54] .

• A. Lejay has continued his work on the theory of rough paths, notably with the sensitivity aspects with Laure Coutin (Univ. Toulouse III) [47] .

• In collaboration with Ivan Dimov and Jean-Michel Sellier (BAS), S. Maire developed a new Monte Carlo method, called the walk on equations, to solve linear systems of equations [22] .

• In collaboration with Xuan Vu, Caroline Chaux-Moulin and Nadege Thirion-Moreau, S. Maire developed a stochastic algorithm to decompose large non-negative tensors with applications in spectroscopy [28] .

• In collaboration with Martin Simon, Sylvain Maire developed a variant of the walk on spheres method to deal with diffusion equations appearing in electrical impedance tomography.

• With Giang Nguyen, Sylvain Maire worked on finite differences techniques to deal with many kinds of boundary conditions that are met during the Monte Carlo simulation of diffusions [25] .

• A. Richard submitted a paper [56] on the spectral representation of $L^2$-indexed increment-stationary processes. The main result states that any random field (i.e. process indexed by a multidimensional parameter of a function in $L^2$) with stationary increments can be written as an integral against a random measure satisfying certain properties. Applications to sample path properties of a multiparameter fractional Brownian motion are exhibited.

• D. Villemonais worked with P. Del Moral (Univ. Sydney) on the conditional ergodicity of time inhomogeneous diffusion processes [48] . 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 $\alpha$-positive recurrence of birth and death processes. This criterion also provides a non-trivial subset of the domain of attraction for quasi-stationary distributions. Finally, this study leads to 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 [29] .

• J. Inglis and D. Talay ended their work on mean-field limits of a stochastic particle system smoothly interacting through threshold hitting-times and applications to neural networks with dendritic component [51] .

Other works in progress

• Together with M. Andrade (Univ. Paris 7) and R. Ferrière (ENS Paris and Univ. Arizona), N. Champagnat is working on the phenomenon of clustering in populations structured by space and traits for which local adaptation favors different trait values at different spatial locations. Two methods are used and numerically validated: a Turing instablity method and a Hamilton-Jacobi approximation of the population density. This work is currently being written.

• N. Champagnat and J. Claisse (Ecole Polytechnique) are currently working on the ergodic and infinite horizon controls of discrete population dynamics with almost sure extinction in finite time. This can either correspond to control problems in favor of survival or of extinction, depending on the cost function. They have proved that these two problems are related to the QSD of the processes controled by Markov controls. This work is currently being written.

• N. Champagnat and C. Fritsch worked with F. Campillo (Inria Sophia-Antipolis, Lemon team) on the variations of the principal eigenvalue (resp. the survival probability) of an integro-differential equation (resp. branching process) of growth-fragmentation-death models with respect to an environmental parameter. This work is currently being written.

• N. Champagnat, K. Coulibaly-Pasquier (Univ. Lorraine) and D. Villemonais are currently working on general criteria for existence, uniqueness and exponential convergence in total variation to QSD for multi-dimensional diffusions in a domain absorbed at its boundary. These results both improve and simplify the existing results and methods. This work is currently being written.

• N. Champagnat and D. Villemonais are currently working on extensions of their work [17] to general penalized processes, including time-inhomogeneous Markov processes with absorption. Their method allows to improve significantly the former results of  [60] , [61] . This work is currently being written.

• N. Champagnat and D. Villemonais are also working on extensions of the criteria of [17] in the form of Foster-Lyapunov criteria allowing to deal with cases where the convergence of conditional distribution to the QSD is not uniform with respect to the initial distribution. This work is currently being written.

• M. Deaconu and S. Herrmann are working on the numerical approach of the time-space Dirichlet problem.

• M. Deaconu, O. Lupaşcu and L. Beznea (Bucharest, Romania) worked on the numerical scheme for the simulation of an avalanche by using the fragmentation model. This work is currently being written.

• M. Deaconu, B. Dumortier and E. Vincent are working with the Venathec SAS on the acoustic control of wind farms. They plan to submit another article to IEEE transaction on sustainable energy soon. Currently they work on handling uncertainties in the model in order to design a stochastic algorithm.

• C. Fritsch worked with F. Campillo (Inria Sophia-Antipolis, Lemon team) and O. Ovaskainen (Univ. Helsinki) about the numerical analysis of the invasion of mutant populations in a chemostat, using branching processes and integro-differential models.

• C. Fritsch started a collaboration with B. Cloez (INRA, Montpellier) on a central limit theorem of mass-structured individual-based chemostat model.

• With P. Pigato, A. Lejay has continued his work on the estimation of parameters of skew diffusions.

• Within the ANESTOC Associate Team, R. Rebolledo (Pontificia Universidad Católica de Chile) and A. Richard initiated a work on the long-term behavior of a class of non-Markovian stochastic differential equations. These equations of Volterra type can be used to model the motion of a particle subject to friction forces in a heat bath, which could also be interesting in neuroscience for ion channels.

• A. Richard and E. Tanré are working with P. Orio (CINV, Chile) on the measurement of long-range dependence in series of neuronal spikes, and are providing a leaky integrate-and-fire model with fractional noise to include this effect. So far, we produced numerical experiments that confirm the existence of memory in our model, and A. Richard and E. Tanré now work on the convergence of the statistical estimator that measures this phenomenon.

• A. Richard, E. Tanré and S. Torres (Universidad de Valparaíso, Chile) are working on the definition of a skew fractional Brownian motion. The skew Brownian motion (sBm) is a process which is partly reflected when it reaches the horizontal line, making it a natural model for the motion of a particle crossing media with different diffusion properties. The fractional sBm is a modification of this process to incorporate long-range dependences. So far, we constructed a reflected fractional Brownian motion, and we are now investigating its approximation by a discrete-time process.

• During her internship supervised by E. Tanré and Romain Veltz (Neuromathcomp team), Roberta Evangelista worked on “A stochastic model of gamma phase modulated orientation selectivity”. Neurons in primary visual cortex (V1) are known to be highly selective for stimulus orientation. Recent experimental evidence has shown that, in awake monkeys, the orientation selectivity of V1 neurons is modulated by gamma oscillations. In particular, neurons’ firing rate in response to the preferred orientation changes as a function of the gamma phase of spiking. The effect is drastically reduced for non-preferred orientations. We have introduced a stochastic model of a network of orientation-dependent excitatory and inhibitory spiking neurons. We have found conditions on the parameters such that the solutions of the mathematical model reproduce the experimental behavior.

• During his internship supervised by E. Tanré and Romain Veltz (Neuromathcomp team), Quentin Cormier studies numerically and theoretically a model of spiking neuron in interaction with plasticity. The synaptic weights evolve according to biological law of plasticity. We study the existence of separable time scales. During his internship, Quentin Cormier also develop a numerical code to simulate large networks of neurons evolving according to this dynamics.

• C. Graham (Ecole Polytechnique) and D. Talay have written a large part of the second volume of their series on Mathematical Foundation of Stochastic Simulation.