A -convergence result and an application to the derivation of the - PowerPoint PPT Presentation

A Γ -convergence result and an application to the derivation of the Monge-Ampère gravitational model Luigi Ambrosio Scuola Normale Superiore, Pisa luigi.ambrosio@sns.it http://cvgmt.sns.it Luigi Ambrosio (SNS) A Γ -convergence result and... Fields Symposium, 2020 1 / 23

Overview [1] L.A., A YMERIC B ARADAT , Y ANN B RENIER : Monge-Ampère gravitation as a Γ -limit of good rate functions. Preprint, 2020. [2] Y ANN B RENIER : A double large deviation principle for Monge- Ampère gravitation. Bull. Inst. Math. Acad. Sin., 11 (2016), 23–41. We derive the discrete version of the Vlasov-Monge-Ampère system starting from a stochastic model of a Brownian point cloud η → 0 X ε,η , dX ε,η = v ε ( t , X ε,η ) dt + η ( t ) dB t , ε → 0 lim lim t where the inner limit is based on the Freidlin-Wentzell theorem and the outer limit relies on Γ -convergence. Compared to the paper [2] the new contribution is on the Γ -convergence result, which makes the ǫ -limit more rigorous. Luigi Ambrosio (SNS) A Γ -convergence result and... Fields Symposium, 2020 2 / 23

Plan Action functionals induced by convex functions 1 The Γ -convergence result 2 The Vlasov-Monge-Ampère gravitational model 3 Derivation of VMA via large deviations and Γ -convergence 4 Luigi Ambrosio (SNS) A Γ -convergence result and... Fields Symposium, 2020 3 / 23

Action functionals induced by convex functions Let H be a Hilbert space, λ ∈ R , f : H → R ∪ { + ∞} λ -convex, l.s.c. (and proper). For h 0 , h 1 ∈ H we consider the action functional Λ f ( h 0 , h 1 ) : C ([ 0 , 1 ]; H ) → R ∪ { + ∞} defined by � 1 0 | x ′ ( t ) | 2 + |∇ f ( x ( t )) | 2 dt  if x ∈ AC 2 ([ 0 , 1 ]; H ) , x ( i ) = h i , i = 0 , 1    + ∞ otherwise.  The goal is to analyze the stability of Λ f ( h 0 , h 1 ) w.r.t. variational convergence of f and convergence of the endpoints h i . Luigi Ambrosio (SNS) A Γ -convergence result and... Fields Symposium, 2020 4 / 23

Meaning of ∇ f In the case λ ≥ 0, for x ∈ D ( f ) = { f < + ∞} , ∇ f ( x ) is the element with minimal norm in the subdifferential ∂ f ( x ) : ∂ f ( x ) := { p ∈ H : f ( y ) ≥ f ( x ) + � p , y − x � ∀ y ∈ H } . However, a "variational” characterization of |∇ f ( x ) | can be provided [ f ( x ) − f ( y )] + |∇ f ( x ) | = sup . | x − y | y � = x It yields that x �→ |∇ f ( x ) | is lower semicontinuous in H , a very useful property also in non-Hilbertian contexts. Luigi Ambrosio (SNS) A Γ -convergence result and... Fields Symposium, 2020 5 / 23

Lack of continuity of L In general terms, the Lagrangian L ( x , p ) = | p | 2 + |∇ f ( x ) | 2 is only l.s.c. w.r.t. x , even in finite dimensions. So, the regularity of minimizers of Λ (ensured, e.g., by a coercitivity assumption on f ) is problematic. One can prove that | x ′ | ∈ L ∞ ( 0 , 1 ) f Lipschitz on bounded sets = ⇒ thanks to the Du Bois-Reymond equation d x ′ ( t ) L p ( x ( t ) , x ′ ( t )) − L ( x ( t ) , x ′ ( t )) � � = 0 dt that can be obtained just performing variations in the independent variable. Can we derive a EL equation, formally x ′′ ( t ) = ∇ 2 f ( x ( t )) ∇ f ( x ( t )) , or get higher regularity? Luigi Ambrosio (SNS) A Γ -convergence result and... Fields Symposium, 2020 6 / 23

A particular case In the case of the application to VMA, H = R Nd and, for some A = ( a 1 , . . . , a N ) ∈ H , f is the semiconvex function f ( x ) = − 1 | x − A σ | 2 , 2 min σ ∈ Σ N with the notation A σ = ( a σ ( 1 ) , . . . , a σ ( N ) ) . Notice that, out of singularities of the distance, one has |∇ f | 2 = | f | 2 , hence we may replace Λ f by the simpler functional � 1 | x ′ ( t ) | 2 + | f ( x ( t )) | 2 dt . Λ ′ f ( x ) := 0 However, the “effective” functional will be the more difficult one with |∇ f ( x ( t )) | 2 ! Luigi Ambrosio (SNS) A Γ -convergence result and... Fields Symposium, 2020 7 / 23

Resolvent map J τ f For − 2 τλ < 1, J τ f ( x ) = ( Id + τ∂ f ) − 1 ( x ) is the unique minimizer of the map y �→ f ( y ) + 1 2 τ | y − x | 2 and the minimal value f τ ( x ) is the Moreau-Yosida approximation of f . Theorem. ( λ ≥ 0 ) J τ f is a contraction, f τ is convex and f τ ∈ C 1 , 1 ( H ) with Lip ( ∇ f τ ) ≤ τ − 1 . Moreover p ∈ ∂ f ( x ) ⇐ ⇒ p = ∇ f τ ( x + τ p ) . In particular, choosing p = ∇ f ( x ) gives |∇ f ( x ) | = |∇ f τ | ( x + τ ∇ f ( x )) . Luigi Ambrosio (SNS) A Γ -convergence result and... Fields Symposium, 2020 8 / 23

Variants of Λ It is not hard to include non-autonomous variants of Λ or even replace the action by � 1 � 2 dt . � � x ′ ( t ) − ∇ f ( x ( t )) � 0 This is due to the fact that a nonsmooth chain rule (for instance Thm. 1.2.5 [AGS]) gives that t �→ f ( x ( t )) is absolutely continuous in [ 0 , 1 ] (in particular h i ∈ D ( f )) ) whenever | x ′ | and |∇ f ( x ) | belong to L 2 ( 0 , 1 ) , with d dt f ( x ( t )) = �∇ f ( x ( t )) , x ′ ( t ) � a.e. in ( 0 , 1 ) . Therefore, the product term is a null Lagrangian. Playing with the convexity parameter λ one can consider also � 1 � 2 dt . � x ′ ( t ) − ( λ x ( t ) − ∇ f ( x ( t ))) � � 0 Luigi Ambrosio (SNS) A Γ -convergence result and... Fields Symposium, 2020 9 / 23

Mosco convergence versus Γ -convergence Definition. We say that f n → f Mosco-converge to f if: (i) lim inf n f n ( x n ) ≥ f ( x ) whenever x n → x weakly in H; (ii) for all x ∈ H there exist x n → x strongly, with lim sup n f n ( x n ) ≤ f ( x ) . • In finite dimensions, no difference w.r.t. the usual version of Γ -convergence. In infinite dimensions, it is more appropriate, as it ensures strong convergence of resolvents: J τ f n ( x ) → J τ f ( x ) . • Under an equi-coercitivity assumption w.r.t. the strong topology of H , again the two versions of Γ -convergence become equivalent and, in addition, the infimum of Λ f ( h 0 , h 1 ) is always attained. Luigi Ambrosio (SNS) A Γ -convergence result and... Fields Symposium, 2020 10 / 23

Main Γ -convergence result Theorem. If f n : H → R ∪ { + ∞} are λ -convex and l.s.c., with f n → f w.r.t. Mosco convergence, and if h n , i → h i strongly , sup n |∇ f n ( h n , i ) | < ∞ , i = 0 , 1 , then Λ f n ( h n , 0 , h n , 1 ) Γ -converge to Λ f ( h 0 , h 1 ) in the C ([ 0 , 1 ]; H ) topology. Luigi Ambrosio (SNS) A Γ -convergence result and... Fields Symposium, 2020 11 / 23

Sketch of proof ( λ = 0): Γ − lim inf The Γ − lim inf inequality follows immediately from the variational characterization of |∇ f ( x ) | , which yields the joint lower semicontinuity n →∞ |∇ f n ( x n ) | 2 ≥ |∇ f ( x ) | 2 lim inf whenever x n → x strongly. This would not work if the weak convergence of the x n were weak and this fact forces the use of the C ([ 0 , 1 ]; H ) topology. The proof of the Γ − lim sup inequality (construction of the recovery sequence) uses the strong convergence of resolvents, and for this reason Mosco convergence is needed. Luigi Ambrosio (SNS) A Γ -convergence result and... Fields Symposium, 2020 12 / 23

Sketch of proof ( λ = 0): Γ − lim sup Fix x ( t ) with Λ f ( h 0 , h 1 )( x ( · )) < ∞ , τ > 0 and set x τ x τ ( t ) = J τ f ( x ( t )) . n ( t ) = J τ f n ( x ( t )) , The monotonicity properties |∇ f ( J τ f ( h )) | ≤ | h − J τ f ( h ) | ≤ |∇ f ( h ) | h ∈ H τ together with the contractivity of J τ f yield � 1 � 1 n | 2 | x ′ | 2 + | x − x τ n ) ′ | 2 + |∇ f n ( x τ n ) | 2 dt | ( x τ lim sup ≤ lim sup dt τ 2 n →∞ n →∞ 0 0 � 1 | x ′ | 2 + | x − x τ | 2 = dt τ 2 0 � 1 | x ′ | 2 + |∇ f | 2 ( x ) dt . ≤ 0 Luigi Ambrosio (SNS) A Γ -convergence result and... Fields Symposium, 2020 13 / 23

Adjustement of the endpoints The recovery sequence x n ( t ) = x τ ( n ) ( t ) can be obtained by a diagonal n argument, except for the fact that the endpoint conditions x n ( 0 ) = h n , 0 , x n ( 1 ) = h n , 1 a priori are not satisfied. However, calling h ∗ n , i := x n ( i ) = J τ ( n ) f n ( h i ) the “wrong” terminal values, we at least have h ∗ n , i → h i . In addition, the monotonicity properties of the resolvent grant n , i ) | 2 < ∞ n →∞ |∇ f n ( h ∗ lim sup provided τ ( n ) → 0 sufficiently slowly. This, combined with the assumption lim sup n |∇ f n ( h n , i ) | 2 < ∞ , grant the possibility to interpolate, in small intervals, [ − δ n , 0 ] , [ 1 , 1 + δ n ] between h n , i and h ∗ n , i with a small cost. Finally, a rescaling of [ − δ n , 1 + δ n ] to [ 0 , 1 ] gives the result. Luigi Ambrosio (SNS) A Γ -convergence result and... Fields Symposium, 2020 14 / 23