Let E be a finite set, {F i } i∈E a family of vector fields on R d leaving positively invariant a... more Let E be a finite set, {F i } i∈E a family of vector fields on R d leaving positively invariant a compact set M and having a common zero p ∈ M. We consider a piecewise deterministic Markov process (X, I) on M × E defined by Ẋt = F It (X t ) where I is a jump process controlled by X : We show that the behavior of (X, I) is mainly determined by the behavior of the linearized process (Y, J) where Ẏt = A Jt Y t , A i is the Jacobian matrix of F i at p and J is the jump process with rates (a ij (p)). We introduce two quantities Λ -and Λ + respectively defined as the minimal (respectively maximal) growth rate of Y t , where the minimum (respectively maximum) is taken over all the ergodic measures of the angular process (Θ, J) with Θ t = Yt Yt . It is shown that Λ + coincides with the top Lyapunov exponent (in the sense of ergodic theory) of (Y, J) and that under general assumptions Λ -= Λ + . We then prove that, under certain irreducibility conditions, X t → p exponentially fast when Λ + < 0 and (X, I) converges in distribution at an exponential rate toward a (unique) invariant measure supported by M \ {p} × E when Λ -> 0. Some applications to certain epidemic models in a fluctuating environment are discussed and illustrate our results.
HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific r... more HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Journal of Dynamics and Differential Equations, 1996
We present a general framework to study compact limit sets of trajectories for a class of nonauto... more We present a general framework to study compact limit sets of trajectories for a class of nonautonomous systems, including asymptotically autonomous differential equations, certain stochastic differential equations, stochastic approximarion processes with decreasing gain, and fictitious plays in game theory. Such limit sets are shown to be internally chain recurrent, and conversely,
Probability Theory and Related Fields, May 6, 2016
Let M be a compact connected oriented Riemannian manifold. The purpose of this paper is to invest... more Let M be a compact connected oriented Riemannian manifold. The purpose of this paper is to investigate the long time behavior of a degenerate stochastic differential equation on the state space M × R n ; which is obtained via a natural change of variable from a self-repelling diffusion taking the form where {B t } is a Brownian vector field on M , σ > 0 and V x (y) = V (x, y) is a diagonal Mercer kernel. We prove that the induced semi-group enjoys the strong Feller property and has a unique invariant probability µ given as the product of the normalized Riemannian measure on M and a Gaussian measure on R n . We then prove an exponential decay to this invariant probability in L 2 (µ) and in total variation.
Stochastics And Partial Differential Equations: Analysis And Computations, Sep 8, 2015
In the present work we study self-interacting diffusions following an infinite dimensional approa... more In the present work we study self-interacting diffusions following an infinite dimensional approach. First we prove existence and uniqueness of a solution with Markov property. Then we study the corresponding transition semigroup and, more precisely, we prove that it has Feller property and we give an explicit form of an invariant probability of the system.
Reinforced random walks, random perturbations of dynamical systems, chain recurrence, attractors.... more Reinforced random walks, random perturbations of dynamical systems, chain recurrence, attractors. ½ 5 j 5 5 1 d 5 5 d < < We let и denote the L norm on R defined by v s Ý v . We may is1 i denote by d the induced distance.
We consider random switching between finitely many vector fields leaving positively invariant a c... more We consider random switching between finitely many vector fields leaving positively invariant a compact set. Recently, Li, Liu and Cui showed in [12] that if one the vector fields has a globally asymptotically stable (G.A.S.) equilibrium from which one can reach a point satisfying a weak Hörmander-bracket condition, then the process converges in total variation to a unique invariant probability measure. In this note, adapting the proof in and using results of , the assumption of a G.A.S. equilibrium is weakened to the existence of an accessible point at which a barycentric combination of the vector fields vanishes. Some examples are given which demonstrate the usefulness of this condition.
We propose a new concept for the analysis of games, the TASP, which gives a precise prediction ab... more We propose a new concept for the analysis of games, the TASP, which gives a precise prediction about non-equilibrium play in games whose Nash equilibria are mixed and are unstable under fictitious play-like learning processes. We show that, when players learn using weighted stochastic fictitious play and so place greater weight on more recent experience, the time average of play often converges in these "unstable" games, even while mixed strategies and beliefs continue to cycle. This time average, the TASP, is related to the best response cycle first identified by . Though conceptually distinct from Nash equilibrium, for many games the TASP is close enough to Nash to create the appearance of convergence to equilibrium. In other games, the TASP may be quite distant from Nash.
HAL (Le Centre pour la Communication Scientifique Directe), Aug 9, 2018
HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific r... more HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific r... more HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
We consider models of N interacting objects, where the interaction is via a common resource and t... more We consider models of N interacting objects, where the interaction is via a common resource and the distribution of states of all objects. We consider the case where the number of transitions per time slot per object vanishes as N grows. We show that, under mild assumptions and for large N , the occupancy measure converges, in probability and in mean square over any finite horizon, to a deterministic dynamical system. Our method of proof is inspired by stochastic approximation algorithms. The convergence results allow us to derive properties valid in the stationary regime. We use this to develop a critique of the fixed point method sometimes used in conjunction with the decoupling assumption.
This article fills a gap in the mathematical analysis of Adaptive Biasing algorithms, which are e... more This article fills a gap in the mathematical analysis of Adaptive Biasing algorithms, which are extensively used in molecular dynamics computations. Given a reaction coordinate, ideally, the bias in the overdamped Langevin dynamics would be given by the gradient of the associated free energy function, which is unknown. We consider an adaptive biased version of the overdamped dynamics, where the bias depends on the past of the trajectory and is designed to approximate the free energy. The main result of this article is the consistency and efficiency of this approach. More precisely we prove the almost sure convergence of the bias as time goes to infinity, and that the limit is close to the ideal bias, as an auxiliary parameter of the algorithm goes to 0. The proof is based on interpreting the process as a self-interacting dynamics, and on the study of a non-trivial fixed point problem for the limiting flow obtained using the ODE method.
Let E be a finite set, {F i } i∈E a family of vector fields on R d leaving positively invariant a... more Let E be a finite set, {F i } i∈E a family of vector fields on R d leaving positively invariant a compact set M and having a common zero p ∈ M. We consider a piecewise deterministic Markov process (X, I) on M × E defined by Ẋt = F It (X t ) where I is a jump process controlled by X : We show that the behavior of (X, I) is mainly determined by the behavior of the linearized process (Y, J) where Ẏt = A Jt Y t , A i is the Jacobian matrix of F i at p and J is the jump process with rates (a ij (p)). We introduce two quantities Λ -and Λ + respectively defined as the minimal (respectively maximal) growth rate of Y t , where the minimum (respectively maximum) is taken over all the ergodic measures of the angular process (Θ, J) with Θ t = Yt Yt . It is shown that Λ + coincides with the top Lyapunov exponent (in the sense of ergodic theory) of (Y, J) and that under general assumptions Λ -= Λ + . We then prove that, under certain irreducibility conditions, X t → p exponentially fast when Λ + < 0 and (X, I) converges in distribution at an exponential rate toward a (unique) invariant measure supported by M \ {p} × E when Λ -> 0. Some applications to certain epidemic models in a fluctuating environment are discussed and illustrate our results.
HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific r... more HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Journal of Dynamics and Differential Equations, 1996
We present a general framework to study compact limit sets of trajectories for a class of nonauto... more We present a general framework to study compact limit sets of trajectories for a class of nonautonomous systems, including asymptotically autonomous differential equations, certain stochastic differential equations, stochastic approximarion processes with decreasing gain, and fictitious plays in game theory. Such limit sets are shown to be internally chain recurrent, and conversely,
Probability Theory and Related Fields, May 6, 2016
Let M be a compact connected oriented Riemannian manifold. The purpose of this paper is to invest... more Let M be a compact connected oriented Riemannian manifold. The purpose of this paper is to investigate the long time behavior of a degenerate stochastic differential equation on the state space M × R n ; which is obtained via a natural change of variable from a self-repelling diffusion taking the form where {B t } is a Brownian vector field on M , σ > 0 and V x (y) = V (x, y) is a diagonal Mercer kernel. We prove that the induced semi-group enjoys the strong Feller property and has a unique invariant probability µ given as the product of the normalized Riemannian measure on M and a Gaussian measure on R n . We then prove an exponential decay to this invariant probability in L 2 (µ) and in total variation.
Stochastics And Partial Differential Equations: Analysis And Computations, Sep 8, 2015
In the present work we study self-interacting diffusions following an infinite dimensional approa... more In the present work we study self-interacting diffusions following an infinite dimensional approach. First we prove existence and uniqueness of a solution with Markov property. Then we study the corresponding transition semigroup and, more precisely, we prove that it has Feller property and we give an explicit form of an invariant probability of the system.
Reinforced random walks, random perturbations of dynamical systems, chain recurrence, attractors.... more Reinforced random walks, random perturbations of dynamical systems, chain recurrence, attractors. ½ 5 j 5 5 1 d 5 5 d < < We let и denote the L norm on R defined by v s Ý v . We may is1 i denote by d the induced distance.
We consider random switching between finitely many vector fields leaving positively invariant a c... more We consider random switching between finitely many vector fields leaving positively invariant a compact set. Recently, Li, Liu and Cui showed in [12] that if one the vector fields has a globally asymptotically stable (G.A.S.) equilibrium from which one can reach a point satisfying a weak Hörmander-bracket condition, then the process converges in total variation to a unique invariant probability measure. In this note, adapting the proof in and using results of , the assumption of a G.A.S. equilibrium is weakened to the existence of an accessible point at which a barycentric combination of the vector fields vanishes. Some examples are given which demonstrate the usefulness of this condition.
We propose a new concept for the analysis of games, the TASP, which gives a precise prediction ab... more We propose a new concept for the analysis of games, the TASP, which gives a precise prediction about non-equilibrium play in games whose Nash equilibria are mixed and are unstable under fictitious play-like learning processes. We show that, when players learn using weighted stochastic fictitious play and so place greater weight on more recent experience, the time average of play often converges in these "unstable" games, even while mixed strategies and beliefs continue to cycle. This time average, the TASP, is related to the best response cycle first identified by . Though conceptually distinct from Nash equilibrium, for many games the TASP is close enough to Nash to create the appearance of convergence to equilibrium. In other games, the TASP may be quite distant from Nash.
HAL (Le Centre pour la Communication Scientifique Directe), Aug 9, 2018
HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific r... more HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific r... more HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
We consider models of N interacting objects, where the interaction is via a common resource and t... more We consider models of N interacting objects, where the interaction is via a common resource and the distribution of states of all objects. We consider the case where the number of transitions per time slot per object vanishes as N grows. We show that, under mild assumptions and for large N , the occupancy measure converges, in probability and in mean square over any finite horizon, to a deterministic dynamical system. Our method of proof is inspired by stochastic approximation algorithms. The convergence results allow us to derive properties valid in the stationary regime. We use this to develop a critique of the fixed point method sometimes used in conjunction with the decoupling assumption.
This article fills a gap in the mathematical analysis of Adaptive Biasing algorithms, which are e... more This article fills a gap in the mathematical analysis of Adaptive Biasing algorithms, which are extensively used in molecular dynamics computations. Given a reaction coordinate, ideally, the bias in the overdamped Langevin dynamics would be given by the gradient of the associated free energy function, which is unknown. We consider an adaptive biased version of the overdamped dynamics, where the bias depends on the past of the trajectory and is designed to approximate the free energy. The main result of this article is the consistency and efficiency of this approach. More precisely we prove the almost sure convergence of the bias as time goes to infinity, and that the limit is close to the ideal bias, as an auxiliary parameter of the algorithm goes to 0. The proof is based on interpreting the process as a self-interacting dynamics, and on the study of a non-trivial fixed point problem for the limiting flow obtained using the ODE method.
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Papers by Michel Benaïm