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Liste des exposés

I. Nouvelles tendances en mécanique des fluides

II. Problèmes à frontière libre

III. Modèles mathèmatiques et méthodes numériques en mécanique des milieux continus

VI. Analyse et contrôle des EDP

VIII. Analyse non-lisse et optimisation

(cette liste est en processus d'actualisation)

**List des exposés**

1.

CIMPEAN Iulian iulian.cimpean@imar.ro

Simion Stoilow Institute of the Romanian Academy, Romania

Titre: From excessive functions to semimartingales on Dirichlet spaces (details)**Résumé:**

We extend to semi-Dirichlet forms a well known semimartingale characterization for functionals associated to symmetric forms, due to M. Fukushima. Our approach is new and it is based on a semigroup description for differences of excessive functions.

Simion Stoilow Institute of the Romanian Academy, Romania

Titre: From excessive functions to semimartingales on Dirichlet spaces (details)

We extend to semi-Dirichlet forms a well known semimartingale characterization for functionals associated to symmetric forms, due to M. Fukushima. Our approach is new and it is based on a semigroup description for differences of excessive functions.

2.

DEACONU Madalina Madalina.Deaconu@inria.fr

Inria Centre de Recherche Nancy - Grand Est & IECL, France

Titre: Stochastic approach of rupture phenomena - application to avalanches (details)**Résumé:**

We introduce first a new interpretation of the fragmentation equation by means of branching processes. We focus afterwards on an unifying method for constructing continuous time fragmentation ~-~ branching processes on the space of all fragmentation sizes, induced either by continuous fragmentation kernels or by a discontinuous one which leads to a stochastic model for the fragmentation phase of an avalanche. For these processes, solutions of some particular stochastic differential equations of fragmentation, we construct a numerical approximation scheme. We analyse in particular this numerical results on our model of avalanches and emphasize the fractal property observed theoretically in our model. This is a joint work with Lucian Beznea (IMAR) and Oana Lupa\c scu (Bucharest).

Inria Centre de Recherche Nancy - Grand Est & IECL, France

Titre: Stochastic approach of rupture phenomena - application to avalanches (details)

We introduce first a new interpretation of the fragmentation equation by means of branching processes. We focus afterwards on an unifying method for constructing continuous time fragmentation ~-~ branching processes on the space of all fragmentation sizes, induced either by continuous fragmentation kernels or by a discontinuous one which leads to a stochastic model for the fragmentation phase of an avalanche. For these processes, solutions of some particular stochastic differential equations of fragmentation, we construct a numerical approximation scheme. We analyse in particular this numerical results on our model of avalanches and emphasize the fractal property observed theoretically in our model. This is a joint work with Lucian Beznea (IMAR) and Oana Lupa\c scu (Bucharest).

3.

GOREAC Dan Dan.Goreac@u-pem.fr

Université Paris-Est, UMR8050, France

Titre: Control-Based Design for Stochastic Gene Networks (details)**Résumé:**

We present some targeted-behaviour based issues in the hybrid modelling of networks. The common method is derived from the theory of backward stochastic systems (backward stochastic differential equations/ backward difference schemes) by interpreting the reaction speeds as externally controlled (thus, modifiable) parameters. In the case of first-order (linear) models, we give explicit (algebraic) conditions on the sets of parameters leading to "controllability" (targeted behavior). Differences between continuous and discrete modeling are emphasized. For more general nonlinear systems, if the time allows it, we give an intuition on how parameters can be chosen by using reflected backward equations and embedding in spaces of measures. This talk is based on joint works with M. Martinez (UPEM), T. Diallo (UPEM), E. Rotenstein (UAIC), C. Grosu (UAIC).

Université Paris-Est, UMR8050, France

Titre: Control-Based Design for Stochastic Gene Networks (details)

We present some targeted-behaviour based issues in the hybrid modelling of networks. The common method is derived from the theory of backward stochastic systems (backward stochastic differential equations/ backward difference schemes) by interpreting the reaction speeds as externally controlled (thus, modifiable) parameters. In the case of first-order (linear) models, we give explicit (algebraic) conditions on the sets of parameters leading to "controllability" (targeted behavior). Differences between continuous and discrete modeling are emphasized. For more general nonlinear systems, if the time allows it, we give an intuition on how parameters can be chosen by using reflected backward equations and embedding in spaces of measures. This talk is based on joint works with M. Martinez (UPEM), T. Diallo (UPEM), E. Rotenstein (UAIC), C. Grosu (UAIC).

4.

GROSU Alexandra Claudia grosu_alexandra_claudia@yahoo.com

ALEXANDRU IOAN CUZA UNIVERSITY OF IASI, Romania

Titre: Approximate (null-)controllability; Controlled Markov switch process; Invariance; Stochastic gene networks (details)**Résumé:**

We propose an explicit, easily-computable algebraic criterion for approximate null-controllability of a class of general piecewise linear switch systems with multiplicative noise.

ALEXANDRU IOAN CUZA UNIVERSITY OF IASI, Romania

Titre: Approximate (null-)controllability; Controlled Markov switch process; Invariance; Stochastic gene networks (details)

We propose an explicit, easily-computable algebraic criterion for approximate null-controllability of a class of general piecewise linear switch systems with multiplicative noise.

5.

LAZARI Alexandru lazarialexandru@mail.md

Moldova State University, Republic of Moldova

Titre: Stationary Stochastic Games with Final Sequence of States (details)**Résumé:**

A stochastic system with final sequence of states represents a Markov process \(L\) with finite set of states \(V\) that stops its evolution as soon as given final sequence of states \(X=(x_1,x_2,\ldots,x_m)\in V^m\) is reached, i.e. the states \(x_1,\ x_2,\ \ldots,\ x_m\) are reached consecutively in the given order. The stochastic system \(L\) starts its evolution from the state \(v\in V\) with the probability \(p^*(v)\), where \(\sum\limits_{v\in V}p^*(v)\)=1. The transition time of the system from an arbitrary state \(u\in V\) to the next state \(v\in V\) is equal to 1 and the transition probability from \(u\) to \(v\) is \(p(u,v)\) at every discrete moment of time \(t\), where \(\sum\limits_{v\in V}p(u,v)=1\), \(\forall u\in V\). We consider the following game. Initially, each player defines his transition matrix, which represents his game strategy. The initial state of the system is established according to the initial distribution of the states \((p^*(v))_{v\in V}\). After that, the stochastic system passes to the next state according to the strategy of the first player. Next, consecutively, each player acts on the system according to the own game strategy. After the last player, the first player acts on the system evolution and the game continues in this way until the given final sequence of states is achieved. The player who acts the last on the system, is considered the winner of the game. Our goal is to study the duration of this game and the win probabilities, knowing the stationary strategy established by each player, the initial state distribution and the final sequence of states of the stochastic system. We prove that the distribution of the game duration and the win distribution are homogeneous linear recurrent sequences and develop a polynomial algorithm to determine the initial state and the generating vector of these recurrences. Using the generating function, the main probabilistic characteristics are determined.

Moldova State University, Republic of Moldova

Titre: Stationary Stochastic Games with Final Sequence of States (details)

A stochastic system with final sequence of states represents a Markov process \(L\) with finite set of states \(V\) that stops its evolution as soon as given final sequence of states \(X=(x_1,x_2,\ldots,x_m)\in V^m\) is reached, i.e. the states \(x_1,\ x_2,\ \ldots,\ x_m\) are reached consecutively in the given order. The stochastic system \(L\) starts its evolution from the state \(v\in V\) with the probability \(p^*(v)\), where \(\sum\limits_{v\in V}p^*(v)\)=1. The transition time of the system from an arbitrary state \(u\in V\) to the next state \(v\in V\) is equal to 1 and the transition probability from \(u\) to \(v\) is \(p(u,v)\) at every discrete moment of time \(t\), where \(\sum\limits_{v\in V}p(u,v)=1\), \(\forall u\in V\). We consider the following game. Initially, each player defines his transition matrix, which represents his game strategy. The initial state of the system is established according to the initial distribution of the states \((p^*(v))_{v\in V}\). After that, the stochastic system passes to the next state according to the strategy of the first player. Next, consecutively, each player acts on the system according to the own game strategy. After the last player, the first player acts on the system evolution and the game continues in this way until the given final sequence of states is achieved. The player who acts the last on the system, is considered the winner of the game. Our goal is to study the duration of this game and the win probabilities, knowing the stationary strategy established by each player, the initial state distribution and the final sequence of states of the stochastic system. We prove that the distribution of the game duration and the win distribution are homogeneous linear recurrent sequences and develop a polynomial algorithm to determine the initial state and the generating vector of these recurrences. Using the generating function, the main probabilistic characteristics are determined.

6.

LOPUSANSCHI Olga olga.lopusanschi@upmc.fr

LPMA (Paris VI), France

Titre: Une construction de l’aire de Lévy avec drift comme limite renormalisée des chaînes de Markov sur graphes périodiques (details)**Résumé:**

Dans la théorie des chemins rugueux, l’aire de Lévy joue un rôle important non seulement en tant que composante du mouvement brownien, mais aussi dans l’étude de la convergence des solutions des EDS, et c’est là où l’absence ou la présence d’un drift à la limite est cruciale. Le but de cet exposé est de construire explicitement une aire de Lévy avec drift comme limite renormalisée d’une chaîne de Markov sur un graphe périodique, d’en donner quelques propriétés et d’illustrer le tout par quelques exemples de modèles issus de la physique quantique.

LPMA (Paris VI), France

Titre: Une construction de l’aire de Lévy avec drift comme limite renormalisée des chaînes de Markov sur graphes périodiques (details)

Dans la théorie des chemins rugueux, l’aire de Lévy joue un rôle important non seulement en tant que composante du mouvement brownien, mais aussi dans l’étude de la convergence des solutions des EDS, et c’est là où l’absence ou la présence d’un drift à la limite est cruciale. Le but de cet exposé est de construire explicitement une aire de Lévy avec drift comme limite renormalisée d’une chaîne de Markov sur un graphe périodique, d’en donner quelques propriétés et d’illustrer le tout par quelques exemples de modèles issus de la physique quantique.

7.

MARZOUK Cyril marzouk.cyril@gmail.com

LPMA, Universités Paris 6 & 7, France

Titre: Geometry of large random non-crossing partitions (details)**Résumé:**

A partition of \(\{1, \dots, n\}\) can be represented in the unit disk of the complex plane if we identify each integer \(1 \le k \le n\) with the complex number \(\exp(-2\mathrm{i}\pi k/n)\) and represent the blocks of the partition by the corresponding convex polygon. The partition is said to be non-crossing when these polygons do not intersect; the subset of the disk formed by the union of the chords then defines a (geodesic) lamination of the disk. I will present some limit theorems for such laminations when the partitions are random, sampled according to a Boltzmann distribution, and their size \(n \to \infty\).

LPMA, Universités Paris 6 & 7, France

Titre: Geometry of large random non-crossing partitions (details)

A partition of \(\{1, \dots, n\}\) can be represented in the unit disk of the complex plane if we identify each integer \(1 \le k \le n\) with the complex number \(\exp(-2\mathrm{i}\pi k/n)\) and represent the blocks of the partition by the corresponding convex polygon. The partition is said to be non-crossing when these polygons do not intersect; the subset of the disk formed by the union of the chords then defines a (geodesic) lamination of the disk. I will present some limit theorems for such laminations when the partitions are random, sampled according to a Boltzmann distribution, and their size \(n \to \infty\).

8.

MATICIUC Lucian lucian.maticiuc@tuiasi.ro

Department of Mathematics, "Gheorghe Asachi" Technical University, Iasi, Romania

Titre: A Generalized Skorokhod Problem with Càdlàg Discontinuities (details)**Résumé:**

We show the existence and uniqueness of a solution for a Skorohod type problem with jumps and driven by a maximal monotone operator.

Department of Mathematics, "Gheorghe Asachi" Technical University, Iasi, Romania

Titre: A Generalized Skorokhod Problem with Càdlàg Discontinuities (details)

We show the existence and uniqueness of a solution for a Skorohod type problem with jumps and driven by a maximal monotone operator.

9.

PASCU Mihai mihai.pascu@unitbv.ro

Transilvania University of Brasov, Faculty of Mathematics and Computer Science, Romania

Titre: Brownian couplings on constant curvature manifolds (details)**Résumé:**

\documentclass[12pt]{amsart} \usepackage{amsfonts} \usepackage[margin=2cm]{geometry} \usepackage{hyperref} \usepackage{palatino} \begin{document} \title{Brownian couplings on constant curvature manifolds} \author{Mihai N. Pascu} \address{"Transilvania" University of Bra\c sov \\ Faculty of Mathematics and Computer Science \\ Str. Iuliu Maniu Nr. 50 \\ Brasov -- 500091 \\ ROMANIA } \email{mihai.pascu@unitbv.ro} \maketitle This talk is based on the recent results obtained with Ionel Popescu in \cite% {Pascu-Popescu} and \cite{Pascu-Popescu SPA}. This research was initially motivated by a stochastic modification of Rado's \emph{Lion and Man problem} (see \cite{Littlewood}), in which a Brownian Lion chases a Brownian Man on the surface of the unit sphere. The question was here whether one can construct Brownian couplings for which the distance between the Lion and the Man is fixed, i.e. fixed-distance Brownian couplings on \(\mathcal{S}^{2}\subset \mathbb{R}^{3}\). After solving this problem, we investigated similar problems in the context of the model spaces \(\mathbb{M}_{K}^{n}\) of constant curvature \(K\) and dimension \(n\geq 1\) (\cite{Pascu-Popescu}), and also in the case of general manifolds without boundary (\cite{Pascu-Popescu SPA}). In this talk we will present the results obtain in \cite{Pascu-Popescu}. The results can be summarized to the following: one can construct Brownian couplings on the model space(s) \(\mathbb{M}^{n}_{K}\) for which the distance between the processes is a deterministic function \(\rho :[0,\infty )\rightarrow \lbrack 0,\infty )\) if and only if the function \(\rho \) is continuous and satisfies almost everywhere the differential inequality \begin{equation*} -(n-1)\sqrt{K}\tan\left(\tfrac{\sqrt{K}\rho(t)}{2}\right)\le \rho^{\prime}(t)\le -(n-1)\sqrt{K}\tan\left(\tfrac{\sqrt{K}\rho(t)}{2}% \right)+\tfrac{2(n-1)\sqrt{K}}{\sin(\sqrt{K}\rho(t))}. \end{equation*} The above represents a characterization of all co-adapted couplings of Brownian motions on the model space \(\mathbb{M}^{n}_{K}\) for which the distance between the processes is deterministic. In addition, the construction of the coupling is explicit for every choice of \(\rho\) satisfying the above hypotheses. Time depending, I will also present some related results. \begin{thebibliography}{9} \bibitem{Littlewood} John~E. Littlewood, \emph{Littlewood's miscellany}, Cambridge University Press, Cambridge, 1986, Edited and with a foreword by B% {\'e}la Bollob{\'a}s. \bibitem{Pascu-Popescu} M. N. Pascu, I. Popescu, \emph{Couplings of Brownian motions of deterministic distance in model spaces of constant curvature} (preprint available on Arxiv) \bibitem{Pascu-Popescu SPA} M. N. Pascu, I. Popescu, \emph{Shy and fixed-distance couplings of Brownian motions on manifolds}, Stochastic Process. Appl. 126 (2016), No. 2, pp. 628--650. \end{thebibliography} \end{document}

Transilvania University of Brasov, Faculty of Mathematics and Computer Science, Romania

Titre: Brownian couplings on constant curvature manifolds (details)

\documentclass[12pt]{amsart} \usepackage{amsfonts} \usepackage[margin=2cm]{geometry} \usepackage{hyperref} \usepackage{palatino} \begin{document} \title{Brownian couplings on constant curvature manifolds} \author{Mihai N. Pascu} \address{"Transilvania" University of Bra\c sov \\ Faculty of Mathematics and Computer Science \\ Str. Iuliu Maniu Nr. 50 \\ Brasov -- 500091 \\ ROMANIA } \email{mihai.pascu@unitbv.ro} \maketitle This talk is based on the recent results obtained with Ionel Popescu in \cite% {Pascu-Popescu} and \cite{Pascu-Popescu SPA}. This research was initially motivated by a stochastic modification of Rado's \emph{Lion and Man problem} (see \cite{Littlewood}), in which a Brownian Lion chases a Brownian Man on the surface of the unit sphere. The question was here whether one can construct Brownian couplings for which the distance between the Lion and the Man is fixed, i.e. fixed-distance Brownian couplings on \(\mathcal{S}^{2}\subset \mathbb{R}^{3}\). After solving this problem, we investigated similar problems in the context of the model spaces \(\mathbb{M}_{K}^{n}\) of constant curvature \(K\) and dimension \(n\geq 1\) (\cite{Pascu-Popescu}), and also in the case of general manifolds without boundary (\cite{Pascu-Popescu SPA}). In this talk we will present the results obtain in \cite{Pascu-Popescu}. The results can be summarized to the following: one can construct Brownian couplings on the model space(s) \(\mathbb{M}^{n}_{K}\) for which the distance between the processes is a deterministic function \(\rho :[0,\infty )\rightarrow \lbrack 0,\infty )\) if and only if the function \(\rho \) is continuous and satisfies almost everywhere the differential inequality \begin{equation*} -(n-1)\sqrt{K}\tan\left(\tfrac{\sqrt{K}\rho(t)}{2}\right)\le \rho^{\prime}(t)\le -(n-1)\sqrt{K}\tan\left(\tfrac{\sqrt{K}\rho(t)}{2}% \right)+\tfrac{2(n-1)\sqrt{K}}{\sin(\sqrt{K}\rho(t))}. \end{equation*} The above represents a characterization of all co-adapted couplings of Brownian motions on the model space \(\mathbb{M}^{n}_{K}\) for which the distance between the processes is deterministic. In addition, the construction of the coupling is explicit for every choice of \(\rho\) satisfying the above hypotheses. Time depending, I will also present some related results. \begin{thebibliography}{9} \bibitem{Littlewood} John~E. Littlewood, \emph{Littlewood's miscellany}, Cambridge University Press, Cambridge, 1986, Edited and with a foreword by B% {\'e}la Bollob{\'a}s. \bibitem{Pascu-Popescu} M. N. Pascu, I. Popescu, \emph{Couplings of Brownian motions of deterministic distance in model spaces of constant curvature} (preprint available on Arxiv) \bibitem{Pascu-Popescu SPA} M. N. Pascu, I. Popescu, \emph{Shy and fixed-distance couplings of Brownian motions on manifolds}, Stochastic Process. Appl. 126 (2016), No. 2, pp. 628--650. \end{thebibliography} \end{document}

10.

RASCANU Aurel aurel.rascanu@uaic.ro

”Octav Mayer” Mathematics Institute of the Romanian Academy, Romania

Titre: On the continuity of the Feynman–Kac formula (details)**Résumé:**

It is well--known since the work of Pardoux and Peng : \textit{Backward stochastic differential equations and quasilinear parabolic partial differential equations}, LNCIS \textbf{176}, Springer (1992), 200--217, that Backward Stochastic Differential Equations provide probabilistic formulae for the solution of (systems of) second order elliptic and parabolic equations, thus providing an extension of the Feynman--Kac formula to semilinear PDEs. This method was applied to the class of PDEs with a nonlinear Neumann boundary condition first by Pardoux and Zhang: \textit{Generalized BSDEs and nonlinear Neumann boundary value problems}, Probab. Theory Related Fields 110 (1998), 535--558, and extended to variational inequalities by Maticiuc and Rascanu: \textit{A stochastic approach to a multivalued Dirichlet--Neumann problem}, Stochastic Process. Appl. 120 (2010), 777--800. Here we arrive, by two different methods, to solve the gap in the proofs on the continuity with respect to x (resp. to (t,x)) of the Feynman--Kac formula extended to Neumann boundary value problems.

”Octav Mayer” Mathematics Institute of the Romanian Academy, Romania

Titre: On the continuity of the Feynman–Kac formula (details)

It is well--known since the work of Pardoux and Peng : \textit{Backward stochastic differential equations and quasilinear parabolic partial differential equations}, LNCIS \textbf{176}, Springer (1992), 200--217, that Backward Stochastic Differential Equations provide probabilistic formulae for the solution of (systems of) second order elliptic and parabolic equations, thus providing an extension of the Feynman--Kac formula to semilinear PDEs. This method was applied to the class of PDEs with a nonlinear Neumann boundary condition first by Pardoux and Zhang: \textit{Generalized BSDEs and nonlinear Neumann boundary value problems}, Probab. Theory Related Fields 110 (1998), 535--558, and extended to variational inequalities by Maticiuc and Rascanu: \textit{A stochastic approach to a multivalued Dirichlet--Neumann problem}, Stochastic Process. Appl. 120 (2010), 777--800. Here we arrive, by two different methods, to solve the gap in the proofs on the continuity with respect to x (resp. to (t,x)) of the Feynman--Kac formula extended to Neumann boundary value problems.

11.

ROTENSTEIN Eduard eduard.rotenstein@uaic.ro

"Alexandru Ioan Cuza" University of Iasi, Romania

Titre: Infection Time in Multi-Stable Gene Networks. A BSVI With Non-Convex, Switch-Dependent Reflection Model (details)**Résumé:**

We investigate a mathematical model associated to the infection time in multistable gene networks. The mathematical processes are of hybrid switch type. The switch is governed by pure jump modes and linked to DNA bindings. The differential component follows backward stochastic dynamics (of PDMP type) and is reflected in some mode-dependent non-convex domains. First, we study the existence of solution to these backward stochastic variational inclusions (BSVI) by reducing them to a family of ordinary variational inclusions with generalized reflection in semiconvex domains. Second, by considering control-dependent drivers, we hint to some model-selection approach by embedding the (controlled) BSVI in a family of regular measures. Regularity, support and structure properties of these sets are given.

"Alexandru Ioan Cuza" University of Iasi, Romania

Titre: Infection Time in Multi-Stable Gene Networks. A BSVI With Non-Convex, Switch-Dependent Reflection Model (details)

We investigate a mathematical model associated to the infection time in multistable gene networks. The mathematical processes are of hybrid switch type. The switch is governed by pure jump modes and linked to DNA bindings. The differential component follows backward stochastic dynamics (of PDMP type) and is reflected in some mode-dependent non-convex domains. First, we study the existence of solution to these backward stochastic variational inclusions (BSVI) by reducing them to a family of ordinary variational inclusions with generalized reflection in semiconvex domains. Second, by considering control-dependent drivers, we hint to some model-selection approach by embedding the (controlled) BSVI in a family of regular measures. Regularity, support and structure properties of these sets are given.

12.

VILLEMONAIS Denis denis.villemonais@univ-lorraine.fr

Université de Lorraine, France

Titre: Exponential convergence to a quasi-stationary distribution (details)**Résumé:**

Our aim during this talk is to present a new framework of assumptions that entails the non-uniform exponential convergence of conditioned Markov processes to a quasi-stationary distribution. In this new framework, most of the difficulties usually involved in proving such a convergence can be overcome. Applications to multi-dimensional diffusions conditioned not to hit a boundary and to population processes conditionned not to be extinct will be provided as illustrations of our results. This work is a collaboration with Nicolas Champagnat.

Université de Lorraine, France

Titre: Exponential convergence to a quasi-stationary distribution (details)

Our aim during this talk is to present a new framework of assumptions that entails the non-uniform exponential convergence of conditioned Markov processes to a quasi-stationary distribution. In this new framework, most of the difficulties usually involved in proving such a convergence can be overcome. Applications to multi-dimensional diffusions conditioned not to hit a boundary and to population processes conditionned not to be extinct will be provided as illustrations of our results. This work is a collaboration with Nicolas Champagnat.

13.

ZALINESCU Adrian adrian.zalinescu@gmail.com

"O. Mayer" Institute of Mathematics, Romanian Academy, Iasi, Romania

Titre: Jump diffusions with oblique subgradients (details)**Résumé:**

We study a stochastic differential equation with a Wiener--Poisson driving term which is reflected along a multivalued vector field of directions.

"O. Mayer" Institute of Mathematics, Romanian Academy, Iasi, Romania

Titre: Jump diffusions with oblique subgradients (details)

We study a stochastic differential equation with a Wiener--Poisson driving term which is reflected along a multivalued vector field of directions.

**Participants**

1.

ANICULAESEI Gheorghe gani@uaic.ro

Al.I.Cuza University, Romania

Al.I.Cuza University, Romania

2.

DHERSIN Jean - Stephane dhersin@math.univ-paris13.fr

Université Paris 13, France

Université Paris 13, France

3.

LUPU Titus titus.lupu@eth-its.ethz.ch

Institute of Theoretical Studies, ETH Zürich, Switzerland

Institute of Theoretical Studies, ETH Zürich, Switzerland