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61
Meanfield backward stochastic differential equations and related patial differential equations
, 2007
"... In [5] the authors obtained MeanField backward stochastic differential equations (BSDE) associated with a Meanfield stochastic differential equation (SDE) in a natural way as limit of some highly dimensional system of forward and backward SDEs, corresponding to a large number of “particles” (or “a ..."
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Cited by 181 (14 self)
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In [5] the authors obtained MeanField backward stochastic differential equations (BSDE) associated with a Meanfield stochastic differential equation (SDE) in a natural way as limit of some highly dimensional system of forward and backward SDEs, corresponding to a large number of “particles” (or “agents”). The objective of the present paper is to deepen the investigation of such MeanField BSDEs by studying them in a more general framework, with general driver, and to discuss comparison results for them. In a second step we are interested in partial differential equations (PDE) whose solutions can be stochastically interpreted in terms of MeanField BSDEs. For this we study a MeanField BSDE in a Markovian framework, associated with a MeanField forward equation. By combining classical BSDE methods, in particular that of “backward semigroups” introduced by Peng [14], with specific arguments for MeanField BSDEs we prove that this MeanField BSDE describes the viscosity solution of a nonlocal PDE. The uniqueness of this viscosity solution is obtained for the space of continuous functions with polynomial growth. With the help of an example it is shown that for the nonlocal PDEs associated to MeanField BSDEs
LARGE POPULATION STOCHASTIC DYNAMIC GAMES: CLOSEDLOOP MCKEANVLASOV SYSTEMS AND THE NASH Certainty Equivalence Principle
, 2006
"... We consider stochastic dynamic games in large population conditions where multiclass agents are weakly coupled via their individual dynamics and costs. We approach this large population game problem by the socalled Nash Certainty Equivalence (NCE) Principle which leads to a decentralized control ..."
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Cited by 74 (9 self)
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We consider stochastic dynamic games in large population conditions where multiclass agents are weakly coupled via their individual dynamics and costs. We approach this large population game problem by the socalled Nash Certainty Equivalence (NCE) Principle which leads to a decentralized control synthesis. The McKeanVlasov NCE method presented in this paper has a close connection with the statistical physics of large particle systems: both identify a consistency relationship between the individual agent (or particle) at the microscopic level and the mass of individuals (or particles) at the macroscopic level. The overall game is decomposed into (i) an optimal control problem whose HamiltonJacobiBellman (HJB) equation determines the optimal control for each individual and which involves a measure corresponding to the mass effect, and (ii) a family of McKeanVlasov (MV) equations which also depend upon this measure. We designate the NCE Principle as the property that the resulting scheme is consistent (or soluble), i.e. the prescribed control laws produce sample paths which produce the mass effect measure. By construction, the overall closedloop behaviour is such that each agent’s behaviour is optimal with respect to all other agents in the game theoretic Nash sense.
Mean field dynamics of fermions and the timedependent HartreeFock equation
, 2002
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Non uniqueness of stationary measures for selfstabilizing diffusions
 in "Stochastic Processes and their Applications
"... We investigate the existence of invariant measures for selfstabilizing diffusions. These stochastic processes represent roughly the behavior of some Brownian particle moving in a doublewell landscape and attracted by its own law. This specific selfinteraction leads to nonlinear stochastic differe ..."
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Cited by 15 (9 self)
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We investigate the existence of invariant measures for selfstabilizing diffusions. These stochastic processes represent roughly the behavior of some Brownian particle moving in a doublewell landscape and attracted by its own law. This specific selfinteraction leads to nonlinear stochastic differential equations and permits to point out singular phenomenons like non uniqueness of associated stationary measures. The existence of several invariant measures is essentially based on the non convex environment and requires generalized Laplace’s method approximations. Key words and phrases: selfinteracting diffusion; stationary measures; double well potential; perturbed dynamical system; Laplace’s method; fixed point theorem.
Large deviations and a Kramers’ type law for selfstabilizing diffusions, in "Ann
 Appl. Probab
"... We investigate exit times from domains of attraction for the motion of a selfstabilized particle traveling in a geometric (potential type) landscape and perturbed by Brownian noise of small amplitude. Selfstabilization is the effect of including an ensembleaverage attraction in addition to the us ..."
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Cited by 13 (4 self)
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We investigate exit times from domains of attraction for the motion of a selfstabilized particle traveling in a geometric (potential type) landscape and perturbed by Brownian noise of small amplitude. Selfstabilization is the effect of including an ensembleaverage attraction in addition to the usual statedependent drift, where the particle is supposed to be suspended in a large population of identical ones. A Kramers ’ type law for the particle’s exit from the potential’s domains of attraction and a large deviations principle for the selfstabilizing diffusion are proved. It turns out that the exit law for the selfstabilizing diffusion coincides with the exit law of a potential diffusion without selfstabilization and a drift component perturbed by average attraction. We show that selfstabilization may substantially delay the exit from domains of attraction, and that the exit location may be completely different. 1. Introduction. We
FORWARDBACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS AND CONTROLLED MCKEAN VLASOV DYNAMICS
"... ABSTRACT. The purpose of this paper is to provide a detailed probabilistic analysis of the optimal control of nonlinear stochastic dynamical systems of the McKean Vlasov type. Motivated by the recent interest in mean field games, we highlight the connection and the differences between the two sets o ..."
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Cited by 10 (5 self)
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ABSTRACT. The purpose of this paper is to provide a detailed probabilistic analysis of the optimal control of nonlinear stochastic dynamical systems of the McKean Vlasov type. Motivated by the recent interest in mean field games, we highlight the connection and the differences between the two sets of problems. We prove a new version of the stochastic maximum principle and give sufficient conditions for existence of an optimal control. We also provide examples for which our sufficient conditions for existence of an optimal solution are satisfied. Finally we show that our solution to the control problem provides approximate equilibria for large stochastic games with mean field interactions. 1.
Nonlinear Markov Semigroups and Interacting Lévy Type Processes
 Journ. Stat. Physics
"... Semigroups of positivity preserving linear operators on measures of a measurable space X describe the evolutions of probability distributions of Markov processes on X. Their dual semigroups of positivity preserving linear operators on the space of measurable bounded functions B(X) on X describe the ..."
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Cited by 10 (8 self)
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Semigroups of positivity preserving linear operators on measures of a measurable space X describe the evolutions of probability distributions of Markov processes on X. Their dual semigroups of positivity preserving linear operators on the space of measurable bounded functions B(X) on X describe the evolutions of averages over the trajectories of these Markov processes. In this paper we introduce and study the general class of semigroups of nonlinear positivity preserving transformations on measures that is nonlinear Markov or Feller semigroups. An explicit structure of generators of such groups is given in case when X is the Euclidean space R d (or more generally, a manifold) showing how these semigroups arise from the general kinetic equations of statistical mechanics and evolutionary biology that describe the dynamic law of large numbers for Markov models of interacting particles. Well posedness results for these equations are given together with applications to interacting particles: dynamic law of large numbers and central limit theorem, the latter being new already for the standard coagulationfragmentation models. Key words. Positivity preserving measurevalued evolutions, conditionally positive operators, Markov models of interacting particles, dynamic law of large numbers, normal fluctuations, rate of convergence, kinetic equations, interacting stable jumpdiffusions, Lévy type processes, coagulationfragmentation.
Convergence to the equilibria for selfstabilizing processes in doublewell landscape
 Ann. Probab
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