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Propagation of chaos for interacting particles subject to environmental noise Michele Coghi 23/10/2014 Propagation of chaos for interacting particles subject to environmental noise 23/10/2014 1 / 24

Introduction 1 The model 2 Existence and limit theorem 3 Propagation of Chaos 4 Summary 5 Propagation of chaos for interacting particles subject to environmental noise 23/10/2014 2 / 24

Part I Introduction Propagation of chaos for interacting particles subject to environmental noise 23/10/2014 3 / 24

Filtered pobability space, (Ω , F , ( F t ) t ≥ 0 , P ) . Interacting particle system � � � � � � N dX i , N X i , N − X j , N dt + � ∞ X i , N = 1 ◦ dW k j = 1 K k = 1 σ k t t t t t N i = 1 , ..., N Empirical Measure N t = 1 � S N δ X i , N → µ t N t i = 1 Limit Equation d µ t + div ( b µ t µ t ) d t + � ∞ k = 1 div ( σ k ( x ) µ t ) ◦ d W k � t = 0 b µ t = K ∗ µ t Propagation of chaos for interacting particles subject to environmental noise 23/10/2014 4 / 24

Independent noise [Sznitman] N = 1 � � − X j , N d X i , N � X i , N d t + d W i K i = 1 , ..., N t t t t N j = 1 ∂ t µ t + div ( b µ t µ t ) − 1 2 ∆ µ t = 0 Deterministic model [Dobrushin] N d = 1 � � d t X i , N � X i , N − X j , N K i = 1 , ..., N t t t N j = 1 ∂ t µ t + div ( b µ t µ t ) = 0 Propagation of chaos for interacting particles subject to environmental noise 23/10/2014 5 / 24

Part II The Model Propagation of chaos for interacting particles subject to environmental noise 23/10/2014 6 / 24

Lipschitz continuity of coefficients K , σ k : R d → R d , k ∈ N | K ( x ) − K ( y ) | ≤ L K | x − y | ∞ | σ k ( x ) − σ k ( y ) | 2 ≤ L σ | x − y | 2 � k = 1 Covariance k = 1 | σ k ( x ) | 2 < ∞ , � ∞ Q : R d × R d → R d × d Q ij ( x , y ) = � ∞ k ( x ) σ j k = 1 σ i k ( y ) . Q : R d → R d × d Q ( x , y ) = Q ( x − y ) Q ( 0 ) = Id . Propagation of chaos for interacting particles subject to environmental noise 23/10/2014 7 / 24

From Stratonovich to Ito, � t � t � t ∞ s + 1 � � � � � � X i , N ◦ dB k X i , N dB k � X i , N σ k s = σ k ( D σ k · σ k ) ds s s s 2 0 0 0 k = 1 where ( D σ k · σ k ) i ( x ) = � d j = 1 σ j k ( x ) ∂ j σ i k ( x ) . Further assume div σ k = 0 for each k ∈ N Along with the previous assumptions on Q , it implies d ∞ � � σ j k ( x ) ∂ j σ i 0 = k ( x ) . k = 1 j = 1 Stratonovich and Itô formulations coincide. Propagation of chaos for interacting particles subject to environmental noise 23/10/2014 8 / 24

SDE - Itô formulation N ∞ = 1 � � � � dX i , N � X i , N − X j , N � X i , N dB k K dt + σ k t t t t t N j = 1 k = 1 i = 1 , ..., N . SPDE - Itô formulation ∞ t = 1 � div ( σ k ( x ) µ t ) dB k d µ t + div ( b µ t µ t ) dt + 2 ∆ µ t . k = 1 Propagation of chaos for interacting particles subject to environmental noise 23/10/2014 9 / 24

� � � � P 1 ( R d ) = probability on R d R d | x | d µ ( x ) < + ∞ µ � � � µ, ν ∈ P 1 ( R d ) W 1 ( ν, µ ) = R 2 d | x − y | m ( d x , d y ) , inf m ∈ Γ( µ,ν ) m ∈ Γ( µ, ν ) iff, Γ( µ, ν ) = { m ∈ P 1 ( R 2 d ) : m ( A × R d ) = µ ( A ) , m ( R d × A ) = ν ( A ) , ∀ A ∈ B ( R d ) } Remark The metric space ( P 1 ( R d ) , W 1 ) is complete and separable. Propagation of chaos for interacting particles subject to environmental noise 23/10/2014 10 / 24

X ∞ space of the stochastic processes, µ : [ 0 , T ] × Ω → P 1 ( R d ) such that � � � sup R d | x | d µ t ( x ) < + ∞ E t ∈ [ 0 , T ] � � d ∞ ( µ, ν ) := E sup t ∈ [ 0 , T ] W 1 ( µ t , ν t ) Remark It follows by the completeness of ( P 1 ( R d ) , W 1 ) that ( X ∞ , d ∞ ) is a complete metric space. Propagation of chaos for interacting particles subject to environmental noise 23/10/2014 11 / 24

Initial Condition Concerning the initial condition µ 0 : Ω → P 1 ( R d ) of the SPDE we shall always assume that i) µ 0 is F 0 -measurable; �� � ii) E R d | x | d µ 0 ( x ) < ∞ . Definition µ ∈ X ∞ is a solution of the SPDE with initial condition µ 0 if, for all φ ∈ C 2 R d � � , b b ( R d ) � µ t , φ � is F t -adapted for every test function φ ∈ C ∞ weak formulation � t � t � µ s , b µ s · ∇ φ � ds + 1 � µ t , φ � = � µ 0 , φ � + � µ s , ∆ φ � ds 2 0 0 � t ∞ � � µ s , σ k · ∇ φ � dB k + s . 0 k = 1 Propagation of chaos for interacting particles subject to environmental noise 23/10/2014 12 / 24

Part III Existence and limit Theorem Propagation of chaos for interacting particles subject to environmental noise 23/10/2014 13 / 24

Theorem Given T ≥ 0 and µ 0 : Ω → P 1 ( R d ) , there exists a solution µ = ( µ t ) t ∈ [ 0 , T ] of the SPDE starting from µ 0 and defined up to time T. Moreover, if µ N 0 → µ 0 , as N → ∞ , then µ N → µ, as N → ∞ , in the metric d ∞ . Propagation of chaos for interacting particles subject to environmental noise 23/10/2014 14 / 24

Linear SPDE ∞ t = 1 � div ( σ k ( x ) µ t ) dB k d µ t + div ( b µ t ) dt + 2 ∆ µ t . k = 1 b = b ( x , t , ω ) , F t -adapted process, continuous in t , Lipschitz continuous in x . � d X t = b ( t , X t ) d t + � ∞ k σ k ( X t ) d B k t = x ∈ R d X 0 Proposition Given µ 0 , the push forward µ t ( ω ) = X ( t , ., ω ) # µ 0 ( ω ) is in X ∞ and solves the linear SPDE. Propagation of chaos for interacting particles subject to environmental noise 23/10/2014 15 / 24

Given µ = ( µ t ) t ∈ [ 0 , T ] ∈ X ∞ , � b µ ( t , x , ω ) := R d K ( x − y ) µ t ( ω, d y ) . t ) d B k d X µ = b µ ( X µ t ) d t + � k σ k ( X µ t t X µ = x 0 Φ µ 0 : X ∞ → X ∞ (Φ µ 0 µ ) t ( ω ) := X µ ( t , ., ω ) # µ 0 ( ω ) ω -a.s. . Propagation of chaos for interacting particles subject to environmental noise 23/10/2014 16 / 24

Theorem The operator Φ µ 0 has a unique fixed point µ = { µ t } in X ∞ , this fixed point is a solution of the SPDE. Φ µ 0 is a contraction, ∀ µ, ν ∈ X ∞ d ∞ (Φ µ 0 µ, Φ µ 0 ν ) ≤ γ T d ∞ ( µ, ν ) Propagation of chaos for interacting particles subject to environmental noise 23/10/2014 17 / 24

Let ω ∈ Ω and t ∈ [ 0 , T ] be fixed. m = ( X µ ( t , ., ω ) , X ν ( t , ., ω )) # µ 0 ∈ Γ((Φ µ 0 µ ) t ( ω ) , (Φ µ 0 ν ) t ( ω )) . Indeed, A ∈ B ( R d ) , m ( A × R d ) = µ 0 { x ∈ R d : X µ t ∈ A } = ( X µ t ) # µ 0 ( A ) = (Φ µ 0 µ ) t ( A ) . In the same way, m ( R d × A ) = (Φ µ 0 ν ) t ( ω )( A ) . From the definition of the Wasserstein metric W 1 , � � � d ∞ (Φ µ 0 µ, Φ µ 0 ν ) ≤ E sup R d | X µ ( t , x ) − X ν ( t , x ) | d µ 0 . t ∈ [ 0 , T ] Propagation of chaos for interacting particles subject to environmental noise 23/10/2014 18 / 24

Given µ = { µ t } t ≥ 0 , ν = { ν t } t ≥ 0 ∈ X ∞ , � � � � � � E sup | X µ ( t , x ) − X ν ( t , x ) | � F 0 ≤ γ T E sup W 1 ( µ t , ν t ) � F 0 � � t ∈ [ 0 , T ] t ∈ [ 0 , T ] By the definition of b µ : � � � � � R d K ( x − y ′ ) d ν t ( y ′ ) � | b µ ( t , x ) − b ν ( t , x ) | = R d K ( x − y ) d µ t ( y ) − � � � � Given ω ∈ Ω a.s. and t ∈ [ 0 , T ] , for every m ∈ Γ( µ t ( ω ) , ν t ( ω )) � � � � � R d × R d K ( x − y ) d m ( y , y ′ ) − R d × R d K ( x − y ′ ) d m ( y , y ′ ) � | b µ ( s , x ) − b ν ( s , x ) | = � � � � � R d × R d | K ( x − y ) − K ( x − y ′ ) | d m ( y , y ′ ) ≤ � R d × R d | y − y ′ | d m ( y , y ′ ) ≤ L K Propagation of chaos for interacting particles subject to environmental noise 23/10/2014 19 / 24

Lemma Let T > 0 . Let µ 0 , ν 0 : Ω → P 1 ( R d ) be two initial conditions, and let µ, ν ∈ X ∞ be the respective solutions of the SPDE given by the contraction method described before, then there exists a constant C T > 0 , such that d ∞ ( µ, ν ) ≤ C T E [ W 1 ( µ 0 , ν 0 )] t ( ω ) = X S N S N ( ., ω ) # S N 0 ( ω ) , t ω -a.s. t Propagation of chaos for interacting particles subject to environmental noise 23/10/2014 20 / 24

Part IV Propagation of Chaos Propagation of chaos for interacting particles subject to environmental noise 23/10/2014 21 / 24

(Ω , F , F t , P ) filtered probability space, ( X i ) i ∈ N , sequence of i.i.d., R d -valued, F 0 -measurable r.v.s, B k t , k ≥ 1, independent Brownian motions, which are independent from the X i , ( F B t ) t ≥ 0 the filtration generated by ( B k t ) k ≥ 1 . � N S N 0 := 1 i = 0 δ X i → µ 0 , in the metric E [ W 1 ( · , · )] , as N → ∞ . N Theorem � R d � Given r ∈ N and φ 1 , ..., φ r ∈ C b , we have r � � � � � � � X 1 , N X r , N � � F B N →∞ E lim φ 1 · . . . · φ r = � µ t , φ i � � t t t i = 1 Propagation of chaos for interacting particles subject to environmental noise 23/10/2014 22 / 24

correlated stochastic particles existence result for a non linear SPDE convergence of the empirical measure S N t to the solution of the SPDE µ t propagation of chaos Propagation of chaos for interacting particles subject to environmental noise 23/10/2014 23 / 24