## Section: New Results

### Probabilistic numerical methods, stochastic modelling and applications

Participants : Mireille Bossy, Nicolas Champagnat, Julien Claisse, Olivier Davidau, Madalina Deaconu, Samuel Herrmann, Pierre-Emmanuel Jabin, Antoine Lejay, Pierre-Louis Lions, Sylvain Maire, Nicolas Perrin, Denis Talay, Etienne Tanré, Julian Tugaut.

#### Monte Carlo methods for diffusion processes and PDE related problems

A. Lejay and A. Kohatsu-Higa (Ōsaka University) are studying the rate of convergence of solutions of Stochastic Differential Equations with an irregular drift. The main tool is the Itô-Krylov estimate and the Malliavin calculus.

#### Kinetic approximation of divergence form operators

A. Lejay and S. Maire have developed in [53] a numerical method to simulate diffusion processes associated to divergence form operators with piecewise constant coefficients in any dimension. It relies on the combination of efficient random walks in each subdomain and a kinetic approximation at the interfaces between the subdomains.

#### Poisson-Boltzmann equation in molecular dynamics

M. Bossy, N. Champagnat, S. Maire and D. Talay have studied in [17] the probabilistic interpretation and associated numerical methods for the Poisson-Boltzmann equation of molecular dynamics that allows one to compute the electrostatic potential around a bio-molecular assembly (for example a protein). The first part of the study concerns the probabilistic interpretation of divergence form operators with piecewise constant coefficients in any dimension. The second part of the study concerns several extensions and improvements of previous numerical methods [62] based on a combination of walk on spheres methods and jump procedures at the interfaces. One of these jump procedures relies on the kinetic approximation developed in [53] .

During his internship, J. Schneider has developed C/C++ and FORTRAN programs corresponding to these numerical methods.

#### Stochastic methods in molecular dynamics

N. Perrin has started his Ph.D. thesis under the supervision of M. Bossy, N. Champagnat and D. Talay on stochastics methods in molecular dynamics. In addition to the Poisson-Boltzman equation of the previous paragraph, he is also studying methods due to P. Malliavin (French Academy of Science) and based on the Fourier analysis of covariance matrices with delay in order to identify the fast and slow components of a molecular dynamics and to construct simplified projected dynamics.

#### Optimal stochastic control of population dynamics

After his four months internship in Tosca , J. Claisse has started his Ph.D. under the supervision of N. Champagnat and D. Talay on stochastic control of population dynamics. He already proved that the optimal expected value function of a basic controlled stochastic model satisfies a Hamilton-Jacobi-Bellman equation.

#### Stochastic spectral formulation for elliptic problems

S. Maire and E. Tanré [40] have developed spectral methods for elliptic PDEs by combining standard deterministic linear approximations and the Feynman-Kac formula. These methods require to invert a matrix having asymptotically condition number 1.

#### Adaptive numerical approximations on hypercubes

C. De Luigi and S. Maire are working on a numerical adaptive method to compute piecewise sparse polynomial approximations and the integral of a multivariate function in medium dimensions (up to 10). This method relies mainly on the quadrature formulae previously developed by the same authors [61] .

#### Selection dynamics for competitive interactions

In [26] , P.-E. Jabin and G. Raoul have studied the large time behaviour of population dynamics models with competition between individuals.

#### Mathematical modelling of immune competition in cancer dynamics

In [18] , I. Brazzoli, E. De Angelis and P.-E. Jabin have introduced a simple model of competition between the immune system, cancer cells and endothelial cells. This model takes into account the aggressiveness of cancer cells. When time is large, wave propagation patterns with oscillations are observed.

#### On existence and well-posedness for the dynamics of a particle in a BV force field in any dimension

In [20] , N. Champagnat and P.-E. Jabin have shown well-posedness for the solutions of ordinary differential equations in the phase space when the force field admits less than one derivative. They also obtained a lower bound on the minimal fractional derivative needed to ensure well-posedness in the specific sense they defined.

#### Some regularizing methods for transport equations and the regularity of solutions to scalar conservation laws

In this work [36] , P.-E. Jabin have studied the connection between three regularization phenomena: stationary phase arguments, averaging lemmas and regularizing effects for scalar conservation laws. When the coefficients are smooth enough, all these regularization phenomena can be deduced from a classical stationary phase argument. However, considerable differences appear in more singular cases.

#### Averaging Lemmas and Dispersion Estimates for kinetic equations

In [25] , P.-E. Jabin first reviews some of the main results on averaging lemmas and present their proofs in as self contained a way as possible. The use of kinetic formulations for the well posedness of scalar conservation laws is then explained as an example of application.

#### On mean discounted numbers of passage times in small balls of Itô processes observed at discrete times

In collaboration with F. Bernardin (CETE de Lyon) and M. Martinez (Université Paris-Est – Marne-la-Vallée), M. Bossy and D. Talay obtained in [15] estimates on mean values of the number of times that Itô processes observed at discrete times visit small balls in . This technique, in the infinite horizon case, is inspired by Krylov's arguments [58] . In the finite horizon case, motivated by an application in stochastic numerics, the number of visits is discounted by a locally exploding coefficient, and the proof involves accurate properties of last passage times at 0 of one dimensional semimartingales.

#### Approximation of equilibrium distributions of some stochastic systems with McKean-Vlasov interactions

M. Bossy and D. Talay have continued their collaboration with Á. Ganz (Pontificia Universidad Católica de Chile) after her Ph.D. thesis under their supervision. They consider a McKean-Vlasov model admitting an equilibrium measure. This year, they have established a quasi-optimal rate of convergence for the approximation by interacting particles systems of the integral of any test function with respect to the equilibrium measure.

#### Mathematical analysis of Lagrangian stochastic models and their confinement

In collaboration with J.-F. Jabir (CMM, Santiago de Chile), M. Bossy has studied theoretical problems on Lagrangian stochastic models in simplified cases where, in particular, the pressure term is neglected.

Motivated by the Stochastic Downscaling Model (SDM) in meteorology and based on Lagrangian stochastic models (see Section 7.1 ), they constructed a Lagrangian system confined within a regular domain of and satisfying the mean no-permeability boundary condition:

where is the outward normal unit vector related to . This year, the problem of
the existence of traces associated to (2 ) have been studied in the case
. Under suitable hypotheses, the distribution _{t}(x, u)
of the corresponding confined process has been shown to admit a strong trace
()(t, x, u) for satisfying the specular boundary condition

#### Interacting particle systems in Lagrangian modeling and simulation of turbulent flows

M. Bossy continued her collaboration initiated in 2006 with A. Rousseau (MOISE INRIA Grenoble – Rhône-Alpes) and F. Bernardin (CETE Clermont-Ferrand), on the Stochastic Lagrangian simulation methodology and the software development of Stochastic Downscaling Method (SDM, see section 7.1 ). SDM refines the characteristics of the wind (velocity and variability) inside a mesh given by a coarse Numerical Weather Prediction (NWP) Model. The wind velocity is simulated by fluid particles, living inside a domain , whose evolution is governed by stochastic differential equations. Such a model raises several original problems which require a strong interaction between mathematics, scientific computation and physics.

This year, improvements in SDM concern mainly the simulation of the meteorological forcing at the boundary of the computational domain. The numerical scheme now approximates the SDM Lagrangian dynamics submitted to non homogeneous Dirichlet boundary conditions and/or periodic conditions on the Eulerian velocities. In [48] , we present a numerical validation of the confinement scheme used for SDM on a simple confined one-dimensional Langevin Ornstein Uhlenbeck process. Numerical tests in the meteorological context and comparison with a LES method are in progress and presented in Section 7.1 .

#### Hamilton Jacobi equations with constraints in population dynamics

P.-E. Jabin and N. Champagnat have obtained existence and uniqueness results for Hamilton-Jacobi PDEs with a maximum constraint that are naturally obtained from models of evolutionary dynamics belonging to the biological framework of “adaptive dynamics”. These equations were alreary formally obtained [56] , but the well-posedness of the problem appeared to be delicate. The results they obtained give the good conditions to impose on the solution in order to ensure uniqueness in a simplified model without mutation. This work is currently being written.

#### Large deviation principle for adaptive dynamics diffusion models

In [19] , N. Champagnat has obtained large deviation results on diffusions models of adaptive dynamics. He has also obtained results on the problem of exit from a domain, which can be biologically interpreted as the phenomenon of evolution by punctuated equilibria. The main difficulty in this work comes from the degeneracy of the diffusion coefficient of the model and the discontinuity of the drift coefficient.

#### On Dirichlet eigenvectors for two-dimensional neutral birth and death processes

In collaboration with L. Miclo (CNRS, Univ. Paul Sabatier, Toulouse), N. Champagnat obtained the full spectral decomposition of the transition matrices of neutral two-dimensional birth and death Markov chains. Because of the specific form of the eigenvectors, they were also able to characterize the maximal Dirichlet eigenvalues in a family of subdomains of the original state space of the process. These results have then been applied to determine the limiting quasi-stationary distribution of nearly neutral two-dimensional birth and death processes. This work is currently being written.

#### Study of particular self-stabilizing diffusions

The well-known Kramers-Eyring law characterizes the time needed by classical diffusions to exit from some bounded domain. This description was extended by S. Herrmann, P. Imkeller and D. Peithmann [57] to particular self-stabilizing diffusions using the large deviation theory. These diffusions represent typically the motion of a particle subject to three sources of forcing. Firstly, it wanders in a landscape whose geometry is determined by a convex potential. Secondly, its trajectories are perturbed by Brownian noise of small amplitude. The third source of forcing can be thought of as self-stabilization: the particle is attracted by its own distribution. The convexity of the potential is essential in this previous study. S. Herrmann and J. Tugaut investigate non-convex situations namely symmetric double-well potentials. They focus their attention on invariant measures for selfstabilizing diffusions living in these potentials. This part is in fact the first step in the study of large deviations phenomenons and exit problems. The authors point out the existence of several stationary measures [49] , one symmetric and two others “outlying”. Asymptotic analysis permits to give more informations about the set of all invariant measures as the noise intensity of the global stochastic system becomes small [51] . Finally the uniqueness problem concerning these measures is analysed and the main result concerns the uniqueness of the symmetric invariant measure [50] . All these detailled studies permit finally to emphasize large deviations principles for self-stabilizing diffusions in non convex landscapes.