Equilibrium large deviations for mean-field systems with translation - PowerPoint PPT Presentation

Equilibrium large deviations for mean-field systems with translation invariance Julien Reygner CERMICS – École des Ponts ParisTech

The Dawson-Gärtner Theory for confined systems Translation invariant systems Application to capital distribution Motivation and outline Study of the fluctuations of large systems with mean-field interactions, from Statistical Physics ... ◮ Large deviation theory Freidlin, Wentzell – ’79 ◮ McKean-Vlasov models and propagation of chaos, Dawson, Gärtner – Mem. AMS ’89 ...to Stochastic Portfolio Theory . ◮ Atlas and first-order models Fernholz – ’02 , Banner, Fernholz, Karatzas – ’05 ◮ with mean-field interactions Shkolnikov – SPA ’12 , Jourdain, R. – AF ’15 , Bruggeman – PhD Thesis Julien Reygner Equilibrium large deviations

The Dawson-Gärtner Theory for confined systems McKean-Vlasov particle system Translation invariant systems LDP for the stationary measure Application to capital distribution Outline The Dawson-Gärtner Theory for confined systems Translation invariant systems Application to capital distribution Julien Reygner Equilibrium large deviations

The Dawson-Gärtner Theory for confined systems McKean-Vlasov particle system Translation invariant systems LDP for the stationary measure Application to capital distribution Definition of the particle system Consider the system of n SDEs n � d X i ( t ) = −∇ V ( X i ( t ))d t − 1 in R d , ∇ W ( X i ( t ) − X j ( t ))d t + σ d β i ( t ) n j =1 with: ◮ V : R d → [0 , + ∞ ) external potential ; ◮ W : R d → [0 , + ∞ ) even interaction potential ; ◮ σ 2 > 0 a temperature parameter. The interactions between particles are of mean-field type, and the configuration is encoded by the empirical measure n � µ n ( t ) = 1 δ X i ( t ) ∈ P ( R d ) . n i =1 Natural questions: ◮ large-scale ( n → + ∞ ) and long time ( t → + ∞ ) behaviour; ◮ both at the level of typical behaviour and fluctuations . Dawson, Gärtner – Mem. AMS ’89 as a continuous version of Curie-Weiss model, Garnier, Papanicolaou, Yang – SIFIN ’13 for an application to systemic risk. Julien Reygner Equilibrium large deviations

The Dawson-Gärtner Theory for confined systems McKean-Vlasov particle system Translation invariant systems LDP for the stationary measure Application to capital distribution The Dawson-Gärtner Theory Dawson, Gärtner – Mem. AMS ’89 : write the evolution of n � µ n ( t ) = 1 δ X i ( t ) ∈ P ( R d ) . n i =1 as a formal infinite-dimensional SDE σ in P ( R d ) , d µ n ( t ) = − Grad F [ µ n ( t )]d t + √ n d β ( t ) where: ◮ F is the free energy defined on P ( R d ) by � � � σ 2 V µ + 1 F [ µ ] = µ log µ + ( W ∗ µ ) µ 2 2 σ 2 = S [ µ ] + V [ µ ] + W [ µ ] . 2 ���� � �� � Entropy Energy ◮ Grad is the gradient with respect to some ‘Riemannian metric’ on P ( R d ) adapted to the covariance of the noise β ( t ) . (related with quadratic Wasserstein distance by Jordan-Kinderlehrer-Otto, Carrillo-McCann-Villani, Ambrosio-Gigli-Savaré ...) Julien Reygner Equilibrium large deviations

The Dawson-Gärtner Theory for confined systems McKean-Vlasov particle system Translation invariant systems LDP for the stationary measure Application to capital distribution The Dawson-Gärtner Theory σ Formal infinite-dimensional SDE d µ n ( t ) = − Grad F [ µ n ( t )]d t + √ n d β ( t ) . When n → + ∞ : ◮ LLN : µ n converges to the solution of the McKean-Vlasov PDE ∂ t µ = − Grad F [ µ ] = σ 2 2 ∆ µ + div ( µ ( ∇ V + ∇ W ∗ µ )) , which is also a propagation of chaos result. � �  − 2 n   writes exp , (formal) σ 2 F ◮ The invariant measure satisfies a LDP with rate function 2   σ 2 F + Cte . ◮ Extension of the Freidlin-Wentzell theory : definition of an action functional , identification of the free energy as a quasipotential . Main message ◮ The dynamical behaviour of the large-scale system, both typical (LLN) and atypical (LDP), is described the free energy . ◮ The latter quantity is only derived from the stationary distribution. Julien Reygner Equilibrium large deviations

The Dawson-Gärtner Theory for confined systems McKean-Vlasov particle system Translation invariant systems LDP for the stationary measure Application to capital distribution Stationary measure for the particle system The particle system X ( t ) = ( X 1 ( t ) , . . . , X n ( t )) ∈ ( R d ) n defined by n � d X i ( t ) = −∇ V ( X i ( t ))d t − 1 ∇ W ( X i ( t ) − X j ( t ))d t + σ d β i ( t ) n j =1 can be rewritten d X ( t ) = − n ∇ U n ( X ( t ))d t + σ d β ( t ) n � where, for x = ( x 1 , . . . , x n ) ∈ ( R d ) n and µ n ( x ) = 1 δ x i , n i =1 n n � � U n ( x ) = 1 1 V ( x i ) + W ( x i − x j ) = V [ µ n ( x )] + W [ µ n ( x )] . n 2 n 2 i =1 i,j =1 Assume and define � � � � � − 2 V ( x ) d ν ( x ) = 1 − 2 V ( x ) z = x ∈ R d exp d x < + ∞ , z exp d x. σ 2 σ 2 ◮ The process X has a unique stationary distribution P n on ( R d ) n . � � ◮ Letting Q n = ν ⊗ n , we have d P n − 2 n on ( R d ) n . [ x ] ∝ exp σ 2 W [ µ n ( x )] d Q n Julien Reygner Equilibrium large deviations

The Dawson-Gärtner Theory for confined systems McKean-Vlasov particle system Translation invariant systems LDP for the stationary measure Application to capital distribution Equilibrium large deviations for the empirical measure Let P n = P n ◦ µ − 1 and Q n = Q n ◦ µ − 1 be probability measures on P ( R d ) . n n Then � � � � d P n − 2 n d P n − 2 n [ x ] ∝ exp σ 2 W [ µ n ( x )] ⇒ [ µ ] ∝ exp σ 2 W [ µ ] , d Q n d Q n so that � � � � − 2 n − 2 n d P n [ µ ] ∝ exp σ 2 W [ µ ] d Q n [ µ ] ≍ exp σ 2 W [ µ ] − n R [ µ | ν ] where, by Sanov’s Theorem , R [ µ | ν ] is the relative entropy � d µ � � = S [ µ ] + 2 R [ µ | ν ] = R d d µ log σ 2 V [ µ ] + Cte . d ν As a consequence, P n satisfies a LDP on P ( R d ) with rate function I [ µ ] = R [ µ | ν ] + 2 σ 2 W [ µ ] + Cte = 2 σ 2 F [ µ ] + Cte . ◮ Rigorous formulation based on the Laplace-Varadhan Lemma , see Léonard – SPA ’87 , Dawson-Gärtner – Mem. AMS ’89 ; ◮ variations on topology and assumptions on the regularity and integrability of V and W , culminating in Dupuis, Laschos, Ramanan – arXiv:1511.06928 . Julien Reygner Equilibrium large deviations

The Dawson-Gärtner Theory for confined systems McKean-Vlasov particle system Translation invariant systems LDP for the stationary measure Application to capital distribution Partial conclusion For mean-field particle systems with an equilibrium Gibbs measure : ◮ both the dynamical and static behaviour at large scales are described by the free energy , ◮ which can be derived from the equilibrium distribution by an elementary ‘Sanov+Laplace-Varadhan’ procedure. Preview of the sequel of the talk : ◮ Robert Fernholz’ talk : systems of rank-based interacting diffusions (equivalently: first-order models , competing particles ) allow to recover empirical capital distribution curves; ◮ for large markets , it can be argued that mean-field interactions provide a correct approximation of such models through propagation of chaos ; ◮ it is therefore natural to look for a free energy for such models! Main technical issue: lack of equilibrium due to translation invariance . Julien Reygner Equilibrium large deviations

The Dawson-Gärtner Theory for confined systems McKean-Vlasov systems without external potential Translation invariant systems Systems of rank-based interacting diffusions Application to capital distribution Outline The Dawson-Gärtner Theory for confined systems Translation invariant systems Application to capital distribution Julien Reygner Equilibrium large deviations

The Dawson-Gärtner Theory for confined systems McKean-Vlasov systems without external potential Translation invariant systems Systems of rank-based interacting diffusions Application to capital distribution Systems without external potential We want to address the situation where V ≡ 0 , i.e. n � d X i ( t ) = − 1 in R d . ∇ W ( X i ( t ) − X j ( t ))d t + σ d β i ( t ) n j =1 Malrieu – AAP ’03 , Cattiaux, Guillin, Malrieu – PTRF ’08 : link with granular media equation. Fouque, Sun – ’13 : model of inter-bank borrowing and lending. ◮ Trajectorial LLN and LDP on [0 , T ] remain valid, the associated free energy writes F [ µ ] = σ 2 2 S [ µ ] + W [ µ ] . ◮ The drift is invariant by translation , and the centre of mass n � Ξ( t ) = 1 X i ( t ) n i =1 is a Brownian motion: no equilibrium! Malrieu – AAP ’03 : the system seen from its centre of mass is ergodic under suitable assumptions on W . Julien Reygner Equilibrium large deviations