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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., AYMERIC BARADAT, YANN BRENIER: Monge-Ampère gravitation as a Γ-limit of good rate functions. Preprint, 2020. [2] YANN BRENIER: 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 lim

ε→0 lim η→0 X ε,η,

dX ε,η

t

= vε(t, X ε,η)dt + η(t)dBt, where the inner limit is based on the Freidlin-Wentzell theorem and the

• uter 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

1

Action functionals induced by convex functions

2

The Γ-convergence result

3

The Vlasov-Monge-Ampère gravitational model

4

Derivation of VMA via large deviations and Γ-convergence

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 h0, h1 ∈ H we consider the action functional Λf(h0, h1) : C([0, 1]; H) → R ∪ {+∞} defined by      1

0 |x′(t)|2 + |∇f(x(t))|2 dt

if x ∈ AC2([0, 1]; H), x(i) = hi, i = 0, 1 +∞

• therwise.

The goal is to analyze the stability of Λf(h0, h1) w.r.t. variational convergence of f and convergence of the endpoints hi.

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)| = sup

y=x

[f(x) − f(y)]+ |x − y| . 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 f Lipschitz on bounded sets = ⇒ |x′| ∈ L∞(0, 1) thanks to the Du Bois-Reymond equation d dt

• x′(t)Lp(x(t), x′(t)) − L(x(t), x′(t))
• = 0

that can be obtained just performing variations in the independent variable. Can we derive a EL equation, formally x′′(t) = ∇2f(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 = RNd and, for some A = (a1, . . . , aN) ∈ H, f is the semiconvex function f(x) = −1 2 min

σ∈ΣN

|x − Aσ|2, with the notation Aσ = (aσ(1), . . . , aσ(N)). Notice that, out of singularities of the distance, one has |∇f|2 = |f|2, hence we may replace Λf by the simpler functional Λ′

f(x) :=

1 |x′(t)|2 + |f(x(t))|2 dt. 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 Jτf

For −2τλ < 1, Jτf(x) = (Id + τ∂f)−1(x) is the unique minimizer of the map y → f(y) + 1 2τ |y − x|2 and the minimal value fτ(x) is the Moreau-Yosida approximation of f.

• Theorem. (λ ≥ 0) Jτf is a contraction, fτ is convex and fτ ∈ C1,1(H)

with Lip(∇fτ) ≤ τ −1. Moreover p ∈ ∂f(x) ⇐ ⇒ p = ∇fτ(x + τp). In particular, choosing p = ∇f(x) gives |∇f(x)| = |∇fτ|(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

• x′(t) − ∇f(x(t))
• 2 dt.

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 hi ∈ D(f))) whenever |x′| and |∇f(x)| belong to L2(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

• x′(t) − (λx(t) − ∇f(x(t)))
• 2 dt.

Luigi Ambrosio (SNS) A Γ-convergence result and... Fields Symposium, 2020 9 / 23

Mosco convergence versus Γ-convergence

• Definition. We say that fn → f Mosco-converge to f if:

(i) lim infn fn(xn) ≥ f(x) whenever xn → x weakly in H; (ii) for all x ∈ H there exist xn → x strongly, with lim supn fn(xn) ≤ 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: Jτfn(x) → Jτf(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 Λf(h0, h1) is always attained.

Luigi Ambrosio (SNS) A Γ-convergence result and... Fields Symposium, 2020 10 / 23

Main Γ-convergence result

• Theorem. If fn : H → R ∪ {+∞} are λ-convex and l.s.c., with fn → f

w.r.t. Mosco convergence, and if hn,i → hi strongly, sup

n |∇fn(hn,i)| < ∞,

i = 0, 1, then Λfn(hn,0, hn,1) Γ-converge to Λf(h0, h1) 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 lim inf

n→∞ |∇fn(xn)|2 ≥ |∇f(x)|2

whenever xn → x strongly. This would not work if the weak convergence of the xn 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 Λf(h0, h1)(x(·)) < ∞, τ > 0 and set xτ

n (t) = Jτfn(x(t)),

xτ(t) = Jτf(x(t)). The monotonicity properties |∇f(Jτf(h))| ≤ |h − Jτf(h)| τ ≤ |∇f(h)| h ∈ H together with the contractivity of Jτf yield lim sup

n→∞

1 |(xτ

n )′|2 + |∇fn(xτ n )|2 dt

≤ lim sup

n→∞

1 |x′|2 + |x − xτ

n |2

τ 2 dt = 1 |x′|2 + |x − xτ|2 τ 2 dt ≤ 1 |x′|2 + |∇f|2(x) dt.

Luigi Ambrosio (SNS) A Γ-convergence result and... Fields Symposium, 2020 13 / 23

Adjustement of the endpoints

The recovery sequence xn(t) = xτ(n)

n

(t) can be obtained by a diagonal argument, except for the fact that the endpoint conditions xn(0) = hn,0, xn(1) = hn,1 a priori are not satisfied. However, calling h∗

n,i := xn(i) = Jτ(n)fn(hi) the “wrong” terminal values,

we at least have h∗

n,i → hi. In addition, the monotonicity properties of

the resolvent grant lim sup

n→∞ |∇fn(h∗ n,i)|2 < ∞

provided τ(n) → 0 sufficiently slowly. This, combined with the assumption lim supn |∇fn(hn,i)|2 < ∞, grant the possibility to interpolate, in small intervals, [−δn, 0], [1, 1 + δn] between hn,i and h∗

n,i with a small cost.

Finally, a rescaling of [−δn, 1 + δn] to [0, 1] gives the result.

Luigi Ambrosio (SNS) A Γ-convergence result and... Fields Symposium, 2020 14 / 23

A nonlinear interpolation lemma

• Lemma. Let f : H → R ∪ {+∞} be convex, l.s.c. Then for all τ > 0

and all q0, q1 ∈ D(∂f) the infimum of Λf(q0, q1) can be estimated from above by 2|∇f(q0)|2 +

• 8 + 4

τ 2

• |q1 − q0|2 +
• 4 + 8τ 2

|∇f(q0) − ∇f(q1)|2. Sketch of proof. Having in mind that p ∈ ∂f(x) iff p = ∇fτ(x + τp), we first interpolate linearly between qi + τ∇f(qi), i = 0, 1 γ(t) := (1 − t)

• q0 + τ∇f(q0)
• + t
• q1 + τ∇f(q1)
• and then we go back to the “original variables” to get an admissible

curve x(t) from q0 to q1: x(t) := γ(t) − τ∇fτ(γ(t)). The Lipschitz bound on ∇fτ and the equivalence ∇f(x) = ∇fτ(γ) give the result.

Luigi Ambrosio (SNS) A Γ-convergence result and... Fields Symposium, 2020 15 / 23

The VMA system

The Vlasov-Monge-Ampère system, introduced by Y.Brenier in 2011 ∂tf(t, x, ξ) + divx

• ξf(t, x, ξ)
• − divξ
• ∇φ(t, x)f(t, x, ξ)
• = 0

det

• I + ∇2φ(t, x)
• = ̺(t, x),

̺(t, x) =

• f(t, x, ξ) dξ

can be viewed as a nonlinear variant of the classical Vlasov-Poisson system, as det

• I + ∇2φ(t, x)
• ∼ 1 + ∆φ(t, x).

Formally, it can also be viewed (A.-Gangbo) as an Hamiltonian ODE in the Wasserstein space of probabilities in phase space, with velocity vt(x, ξ) :=

• ξ, −∇φ(t, x)
• and 1

2|x|2 + φ(t, ·) is the K-potential of the optimal transport problem

from ̺(t, ·) to the uniform measure.

Luigi Ambrosio (SNS) A Γ-convergence result and... Fields Symposium, 2020 16 / 23

Discrete version of the VMA system

If we replace (say on the torus) the uniform measure by a family of N points A = (a1, . . . , aN) with N large, and the continuous densities by discrete ones (x1, . . . , xN), formally VMA corresponds to (∗) x′′

i (t) = xi(t) − aσopt(i)

i = 1, . . . , N because the dynamic is ruled by the discrete optimal transport problem min

σ∈ΣN N

• i=1

|xi − aσ(i)|2. Strongly inspired by Brenier’s paper (Bull. Inst. Mat. Sin. 2016), we want to derive a more rigorous version of (*) starting from a Brownian point cloud, by applying LDP and Γ-convergence. Problems arise from the lack of well-posedness of the ODE, that can be attacked within the DiPerna-Lions theory, but only in the a.e. sense.

Luigi Ambrosio (SNS) A Γ-convergence result and... Fields Symposium, 2020 17 / 23

First step: adding noise to the ai

By adding noise to the ai, ai → ai + √εBi

t, and viewing the evolution

problem modulo permutations (or, equivalently, in the space of empirical measures), the probability density ̺ǫ(t, x) for the point cloud is given in Rn by 1 N! √ 2πǫt

dN

• σ∈ΣN

exp

• −

N

• i=1

|xi − aσ(i)|2 2εt

• .

In PDE terms, it is a continuity equation ∂t̺ε + div(vε̺ε) = 0, where the driving vector field vε is representable by vε(t, x) := x − ∇fε(t, x) 2t , fε(t, x) = εt log 1 N!

• σ∈ΣN

exp x, Aσ εt

• .

Luigi Ambrosio (SNS) A Γ-convergence result and... Fields Symposium, 2020 18 / 23

Laplace and large deviation principles

By Laplace’s principle, fε(t, ·) converge to the convex function f(x) := max

σ∈ΣN

x, Aσ, so that (at least at differentiability points x of f) lim

ε→0 vε(t, x) = x − ∇f(x)

2t = 1 2t ∇D2(x), D2(x) := 1 2 min

σ∈ΣN

|x − Aσ|2.

• Definition. A family (ηn) of probability measures in X satisfies the LDP

with speed (an) and rate functional I if lim inf

n→∞ an log ηn(A) ≥ − inf x∈A I(x)

for A ⊂ X open, and lim sup

n→∞ an log ηn(C) ≤ − inf x∈C I(x)

for C ⊂ X closed.

Luigi Ambrosio (SNS) A Γ-convergence result and... Fields Symposium, 2020 19 / 23

Second step: adding noise to the ODE driven by vε

Now, by adding noise not only to the lattice, but also the points dX ε,η

t

= vε(t, X ε,η)dt + √η √ t d ˜ Bt the Freidlin-Wentzell theorem ensures that, with ǫ fixed, the conditioned laws of X ε,η in [t0, t1] ⊂ (0, ∞) satisfy a LDP principle in X = C([t0, t1]; RN) with speed η and rate functional Iε(z) := t1

t0

t|z′(t) − vε(t, z(t))|2 dt set to +∞ if z / ∈ H1([t0, t1]; RNd). Remembering the limit of vε, we may take the limit also as ε → 0.

Luigi Ambrosio (SNS) A Γ-convergence result and... Fields Symposium, 2020 20 / 23

Convergence to an effective functional

A time-dependent version of the general Γ-convergence result, gives:

• Theorem. The functionals Iε(z), even with prescribed boundary

conditions, Γ-converge in X = C([t0, t1]; RNd) to I(z) = t1

t0

t

• z′(t) − z(t) − ∇f(z(t))

2t

• 2 dt.

If z(t) does touch singularities of f, minimizers of I satisfy, after an exponential rescaling, exactly the discrete VMA system. The advantage of this “variational” derivation is that it makes sense regardless of this assumption, so that we may consider minimizers of I as the “true” solutions to the discrete VMA which, as stated, is ill posed.

Luigi Ambrosio (SNS) A Γ-convergence result and... Fields Symposium, 2020 21 / 23

A 1-dimensional regularity result

Set d = 1, let, as usual, f(x) := max

σ∈ΣN

Aσ, x x ∈ RN and let π(x) the equivalence relation in {1, . . . , N} induced by i ∼ j iff xi = xj.

• Theorem. For any minimizer z(t) of the functional

T |z′(t)|2 + |z(t) − ∇f(z(t))|2 dt with endpoint conditions z(0) ∈ {P}σ, z(T) ∈ {Q}σ, there exist 0 < t1 < · · · < tk < T such that z is smooth and π(z(t)) is constant in the intervals (ti, ti+1). The d-dimensional case is open.

Luigi Ambrosio (SNS) A Γ-convergence result and... Fields Symposium, 2020 22 / 23

Thank you for the attention!

Slides available upon request

Luigi Ambrosio (SNS) A Γ-convergence result and... Fields Symposium, 2020 23 / 23