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Non ergodicity of diffusions on the diffeomorphisms group

Geometry of the diffeomorphisms group on the torus GV - group of volume preserving diffeomorphisms on a manifold (here for simplicity, T 2, 2-dim. torus) GV - its Lie algebra (vector fields with zero divergence) The L2 norm defines on G a canonical Hilbert structure. More generally, Gα = {g : T 2 → T 2 bijection, g, g−1 ∈ Hα} For α > 2 Gα is a C∞ infinite dimensional Hilbert mani- fold. We are interested in Brownian motions on the group GV . Brownian motions will have generators of type L =

with Ek solutions of Euler equation.

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Motivation Euler equation

tion ∂u ∂t + u.∇u = −∇p, div u = 0 corresponds to the velocity of a flow which is critical for the action functional S[g] = 1 2

g(t)||2

where g are measure preserving diffeomorphisms; i.e., Euler equation = geodesic equation for the L2 metric. d dtu = −

Γi,juiuj From the geometry (e.g. curvature) one can derive properties

Navier-Stokes equation ∂u ∂t + u.∇u = ν∆u − ∇p, div u = 0 We can regard u(t, .) as the drift (mean velocity) of a diffusion process on the diffeomorphism group; the Lapla- cian being the second order term in the generator. We have LF(g)(θ) = c∆f(g(θ)) for F(g) = fog.

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We consider Fourier developments. The collection exp i(k.θ) constitutes an o.n. basis of the space of complex -valued func- tions on the torus. Fourier transform is defined as ˆ f(k) = 1 (2π)d

then if f ∈ L2, f(θ) =

ˆ f(k) exp i(k. θ) f real iff ˆ f(−k) = ¯ ˆ f(k) ˜ Z2 subset of Z2 such that each equivalence class of the equiv- alence relation defined by k ≃ k′ if k + k′ = 0 has a unique representative in ˜

f(θ) = 2

ℜ ˆ f(k) cos(k.θ) − ℑ ˆ f(k) sin(k.θ)

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Orthonormal basis of G: Ak = 1 |k|[(k2 cos k.θ)∂1 − (k1 cos k.θ)∂2)] Bk = 1 |k|[(k2 sin k.θ)∂1 − (k1 sin k.θ)∂2)] k ∈ ˜ Z2 − {(0, 0)}, |k|2 = k2

∂i =

Constants of structure of G Recall: [ek, es] =

They are given by [Ak, Al] = [k, l] 2|k||l|(|k + l|Bk+l + |k − l|Bk−l) [Bk, Bl] = − [k, l] 2|k||l|(|k + l|Bk+l − |k − l|Bk−l) [Ak, Bl] = − [k, l] 2|k||l|(|k + l|Ak+l − |k − l|Ak−l) [∂i, Ak] = −kiBk [∂i, Bk] = kiAk

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Define the following functions αk,l := 1 2|k||l||k + l|(l | (l + k)) βk,l := α−k,l = 1 2|k||l||k − l|(l | (l − k)) [k, l] = k1l2 − k2l1 The Christoffel symbols Recall: Γl

ΓAk,Al = [k, l](αk,lBk+l + βk,lBk−l) ΓBk,Bl = [k, l](−αk,lBk+l + βk,lBk−l) ΓAk,Bl = [k, l](−αk,lAk+l + βk,lAk−l) ΓBk,Al = [k, l](−αk,lAk+l − βk,lAk−l) Christofell symbols give rise to unbounded antihermitian

The Ricci curvature is negative (and divergent).

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On the Lie algebra GV define the Brownian motion on G dx(t) =

ρ2(k)(Ak dxk(t) + Bk dyk(t)) where xk, yk are independent copies of real Brownian motions, ρ2(k) < ∞. The stochastic flow dg(t) = (odx(t))(g(t)), g(0) = Id is well defined and is a continuous process with values in G0

If ρ(k) =

diffeomorphisms. References for definition and regularity: P. Malliavin (1999) and S. Fang (2002). Generator: L = 1 2

1 |k|2α∂Ak∂AkF(g) +

1 |k|2α∂Bk∂BkF(g) Some properties:

Brownian motion associated to the metric Hα−1.

2LF(g)(θ) = c∆f(g(θ)) if F(g)(θ) = f(g(θ)).

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Assume that there exists k0, k1 with [k0, k1] = 0 such that the corresponding ρ(k0), ρ(k1) are not zero. Then a probabil- ity measure carried by the group ˜ G (Borel measurable volume preserving maps on the torus) which is invariant for the Brow- nian motion gt does not exist. Lack of compactness comes from the energy dissipation from low to high Fourier modes. Proof.

Let U be the unitary group of L2(T 2), the Hilbert space of complex valued square integrable functions. The multiplica- tive unitary subgroup: let Um be the subgroup of the unitary group U defined as Um :=

G → Um that associates to g ∈ G the operator Ug(f) = f ◦ g, ∀f ∈ L2

morphism.

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The representation j induces a morphism j′ of Lie algebras; define Ak = j′(Ak), AkF(Ug) = DAkF(Ug), Bk = j′(Bk). Then Ut := j(gt), dUt = Ut (

Akodxk(t) + Bkodyk(t)) Parametrize Ug by cq

es = exp i(s.θ) cq

1 (2π)2

[Akcq

i (2π)d

Then from dUt = Ut (

Akdxk(t) + Bkdyk(t) − ρ2(k) 2|k|2 [s, k]2)

we deduce,

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dcq

2

ρ(k)[s, k] |k| (cq

+1 2

ρ(k)[s, k] |k| (cq

−1 2cq

ρ2(k)[s, k]2 |k|2

Consider the coefficients of Ut cq

with fixed q; then the energy functional ξt(s) := E(|cq

satisfies the o.d.e. dξt dt = M(ξt) where M is a real symmetric negative definite matrix which has for associated quadratic form (M(ξ) | ξ) = −1 2

ρ2(k)[s, k]2 |k|2 ((ξs − ξs+k)2 + (ξs − ξs−k)2)

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Rescale the s column of the matrix M by dividing each term by −|s|2; then we obtain a probability measure carried by the complement of s; making this construction for all s we define a random walk X(n) on Z2. The jump process is defined as η(t) := X(ϕ(t)) where the change of clock ϕ(t) is the integer valued function ϕ(t):

1 Ml

× Λn ≤ t <

1 Ml

× Λn, and where {Λk} is a sequence of independent exponential times. The infinitesimal generator of the process η(t) is M The jumps can appear at −k, k where ρ(k) = 0. This jump process is conservative (it cannot go to infinity in a finite time). As a consequence: existence of the process.

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Energy dissipation from low to high Fourier modes: rate computed via asymptotics of jump processes on Fourier modes in P. Malliavin - J. Ren 2008 For all s0 fixed, lim

Consider the semigroup Pt(f)(k0) = Ek0(f(η(t))) Let µ be the uniform measure defined on Z2 Q(φ) := (M(φ) | φ); consider φ2 := Q(φ) + φ2

µ

D the associated Hilbert space constructed by completing the C∞ functions with compact support on Z2 Operator N, the closure of M in L2

f2

µ + ||f||2

µ

By the Spectral Theorem N =

λ dΠ(λ) where Π(λ) is an orthogonal projection operator in L2

map λ → Image(Π(λ)) being an increasing function with val- ues in the closed subspaces of L2

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Ptf =

exp(tλ)d(Π(λ)f), ∀ f ∈ L2

Ptf2

exp(tλ)d(Π(λ)f2) This integral does not converge to 0 if and only if the measure d(Π(λ)f2) has a Dirac mass at the origin, wich means that ∃ ψ ∈ L2

impossible by hypothesis − → limt→∞ PtfL2

ν =

0 ∀f ∈ ˜ G and ξt(j0) = (Pt(δj.

0)|δq.) ≤ ||Pt(δj. 0)||L2 ν

Finally, from the invariance of the measure and the uni- tarity of the operators involved we can deduce that E(

which contradicts the last convergence.

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Stability (work in progress with M. Arnaudon ) We follow the point of view initiated by Ebin and Mars- den of considering the L2 metric and the corresponding weak Riemannian structure in Gα. In particular Ebin and Marsden proved the (local) existence

corresponding Levi-Civita connection. We also have existence

Curvature satisfies < R(X, Y, Z), W >L2≤ C||X||Hα||Y ||Hα||Z||Hα||W||H1 Let g and ˜ g be two Brownian motions satisfying dg(t) = (odx(t))(g(t)) with g(t0) = ϕ, g(t0) = ψ. Then the following Kendall coupling formula holds, EdL2(g(t), ˜ g(t)) = dL2(φ, ψ) +E t ˜

(|∇TW k|2

for t < σC (coupling time), where T denotes the tangent vector to the unit speed geodesic joinning ϕ and ψ and W k are Jacobi fields along this geodesic. From the negativity of the Ricci tensor we deduce that, with positive probability, there are unstable stochastic paths. (applications to Navier-Stokes Lagrangian flows in view)