Non ergodicity of diffusions on the diffeomorphisms group - PDF document

Non ergodicity of diffusions on the diffeomorphisms group Hong-Kong, July 2009 Geometry of the diffeomorphisms group on the torus G V - group of volume preserving diffeomorphisms on a manifold ( here for simplicity, T 2 , 2-dim. torus) G V - its Lie algebra (vector fields with zero divergence) The L 2 norm defines on G a canonical Hilbert structure. More generally, G α = { g : T 2 → T 2 bijection , g, g − 1 ∈ H α } G α is a C ∞ infinite dimensional Hilbert mani- For α > 2 fold. We are interested in Brownian motions on the group G V . k E 2 Brownian motions will have generators of type L = � k with E k solutions of Euler equation. 1

Motivation Euler equation V. I. Arnold (1966) showed that the solution of Euler equa- 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 � g ( t ) || 2 || ˙ L 2 dt 2 where g are measure preserving diffeomorphisms; i.e., Euler equation = geodesic equation for the L 2 metric. d � Γ i,j u i u j dtu = − i,j From the geometry (e.g. curvature) one can derive properties of the motion (like stability) 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 L F ( g )( θ ) = c ∆ f ( g ( θ )) for F ( g ) = fog . 2

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 1 � T 2 f ( θ ) exp i ( k.θ ) dθ 1 ⊗ dθ 2 ˆ f ( k ) = (2 π ) d then if f ∈ L 2 , � ˆ f ( θ ) = f ( k ) exp i ( k. θ ) k ∈ Z 2 f ( − k ) = ¯ ˆ ˆ f real iff f ( k ) Z 2 subset of Z 2 such that each equivalence class of the equiv- ˜ alence relation defined by k ≃ k ′ if k + k ′ = 0 has a unique representative in ˜ Z 2 . Then � ℜ ˆ f ( k ) cos( k.θ ) − ℑ ˆ f ( θ ) = 2 f ( k ) sin( k.θ ) k ∈ ˜ Z 2 3

Orthonormal basis of G : A k = 1 | k | [( k 2 cos k.θ ) ∂ 1 − ( k 1 cos k.θ ) ∂ 2 )] B k = 1 | k | [( k 2 sin k.θ ) ∂ 1 − ( k 1 sin k.θ ) ∂ 2 )] Z 2 − { (0 , 0) } , k ∈ ˜ | k | 2 = k 2 1 + k 2 2 , ∂ ∂ i = ∂θ i Constants of structure of G l c l Recall: [ e k , e s ] = � k,s e l They are given by [ A k , A l ] = [ k, l ] 2 | k || l | ( | k + l | B k + l + | k − l | B k − l ) [ B k , B l ] = − [ k, l ] 2 | k || l | ( | k + l | B k + l − | k − l | B k − l ) [ A k , B l ] = − [ k, l ] 2 | k || l | ( | k + l | A k + l − | k − l | A k − l ) [ ∂ i , A k ] = − k i B k [ ∂ i , B k ] = k i A k 4

Define the following functions 1 2 | k || l || k + l | ( l | ( l + k )) α k,l := 1 β k,l := α − k,l = 2 | k || l || k − l | ( l | ( l − k )) [ k, l ] = k 1 l 2 − k 2 l 1 The Christoffel symbols k,s = 1 Recall: Γ l 2 ( c l k,s − c k s,l + c s l,k ) Γ A k ,A l = [ k, l ]( α k,l B k + l + β k,l B k − l ) Γ B k ,B l = [ k, l ]( − α k,l B k + l + β k,l B k − l ) Γ A k ,B l = [ k, l ]( − α k,l A k + l + β k,l A k − l ) Γ B k ,A l = [ k, l ]( − α k,l A k + l − β k,l A k − l ) Christofell symbols give rise to unbounded antihermitian operators on G . The Ricci curvature is negative (and divergent). 5

On the Lie algebra G V define the Brownian motion on G � ρ 2 ( k )( A k dx k ( t ) + B k dy k ( t )) dx ( t ) = k � =0 where x k , y k are independent copies of real Brownian motions, � ρ 2 ( k ) < ∞ . The stochastic flow dg ( t ) = ( odx ( t ))( g ( t )) , g (0) = Id is well defined and is a continuous process with values in G 0 V . 1 If ρ ( k ) = | k | α , α > 2 we can replace homeomorphisms by diffeomorphisms. References for definition and regularity: P. Malliavin (1999) and S. Fang (2002). Generator: L = 1 1 1 � � | k | 2 α ∂ A k ∂ A k F ( g ) + | k | 2 α ∂ B k ∂ B k F ( g ) 2 k k Some properties: 1 • When ρ ( k ) = | k | α the process g ( t ) coincides with the Brownian motion associated to the metric H α − 1 . k ρ 2 ( k ) k 2 • When ρ ( k ) = ρ ( | k | ) and c = � i < ∞ , 2 L F ( g )( θ ) = c ∆ f ( g ( θ )) if F ( g )( θ ) = f ( g ( θ )). 6

Theorem. (A.B. Cruzeiro, P. Malliavin 2008) Assume that there exists k 0 , k 1 with [ k 0 , k 1 ] � = 0 such that the corresponding ρ ( k 0 ) , ρ ( k 1 ) 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 g t does not exist. Lack of compactness comes from the energy dissipation from low to high Fourier modes. Proof. 1. Regular representation of diffeomorphism group Let U be the unitary group of L 2 ( T 2 ), the Hilbert space of complex valued square integrable functions. The multiplica- tive unitary subgroup: let U m be the subgroup of the unitary group U defined as U m := � � U ∈ U ; U ( f 1 f 2 ) = U ( f 1 ) × U ( f 2 ) , Define regular representation as the map j : ˜ G �→ U m that associates to g ∈ G the operator U g ( f ) = f ◦ g, ∀ f ∈ L 2 Theorem. The regular representation is a surjective iso- morphism. 7

The representation j induces a morphism j ′ of Lie algebras; define A k = j ′ ( A k ), A k F ( U g ) = D A k F ( U g ), B k = j ′ ( B k ). Then U t := j ( g t ), � A k odx k ( t ) + B k ody k ( t )) dU t = U t ( k Parametrize U g by c q s ( g ) = ( U g ( e s ) | e q ), where e s = exp i ( s.θ ) 1 � c q T 2 exp {− iq.θ + is.g ( θ ) } dθ then s ( g ) := (2 π ) 2 � e q [ A k c q � � � s ]( g ) = ( D A k U ∗ ) g ( e s ) i � = T 2 exp( − i ( q.θ − s.g ( θ )) × ( s.A k )( g ( θ )) dθ (2 π ) d Then from A k dx k ( t ) + B k dy k ( t ) − ρ 2 ( k ) � 2 | k | 2 [ s, k ] 2 ) dU t = U t ( k or from direct computation of Itˆ o formula with exp i ( s.g t ( θ )) we deduce, 8

ρ ( k )[ s, k ] s ( g t ) = i � | k | ( c q s + k ( g t ) + c q dc q s − k ( g t )) dx k ( t ) 2 k +1 ρ ( k )[ s, k ] � | k | ( c q s + k ( g t ) − c q s − k ( g t )) dy k ( t ) 2 k ρ 2 ( k )[ s, k ] 2 − 1 � 2 c q s ( g t ) | k | 2 k 2. Transfer energy matrix Consider the coefficients of U t c q s ( x, t ) = ( U x,t e s | e q ) q, s ∈ Z 2 with fixed q ; then the energy functional ξ t ( s ) := E ( | c q s ( x, t ) | 2 ) satisfies the o.d.e. dξ t dt = M ( ξ t ) where M is a real symmetric negative definite matrix which has for associated quadratic form ρ 2 ( k )[ s, k ] 2 ( M ( ξ ) | ξ ) = − 1 | k | 2 (( ξ s − ξ s + k ) 2 + ( ξ s − ξ s − k ) 2 ) � 2 k,s 9

3. Jump process associated to a Dirichlet form 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 Z 2 . 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 1 � � × Λ n ≤ t < × Λ n , M l M l l l n ≤ ϕ ( t ) n ≤ ϕ ( t )+1 and where { Λ k } 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. 10

4. Escape the energy towards high modes 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 s 0 fixed, lim t →∞ ξ t ( s 0 ) = 0 Consider the semigroup P t ( f )( k 0 ) = E k 0 ( f ( η ( t ))) Let µ be the uniform measure defined on Z 2 Q ( φ ) := ( M ( φ ) | φ ); consider � φ � 2 := Q ( φ ) + � φ � 2 L 2 µ D the associated Hilbert space constructed by completing the C ∞ functions with compact support on Z 2 Operator N , the closure of M in L 2 µ , is selfadjoint and � f � 2 µ + || f || 2 D := ( N f | f ) L 2 L 2 µ By the Spectral Theorem � 0 N = λ d Π( λ ) −∞ where Π( λ ) is an orthogonal projection operator in L 2 µ , the map λ �→ Image(Π( λ )) being an increasing function with val- ues in the closed subspaces of L 2 µ ; 11

� 0 exp( tλ ) d (Π( λ ) f ) , ∀ f ∈ L 2 P t f = µ . −∞ � 0 � P t f � 2 exp( tλ ) d ( � Π( λ ) f � 2 ) l 2 = −∞ This integral does not converge to 0 if and only if the measure d ( � Π( λ ) f � 2 ) has a Dirac mass at the origin, wich means that ∃ ψ ∈ L 2 ν , ψ � = 0 , such that Q ( ψ ) = 0 impossible by hypothesis − → lim t →∞ � P t f � L 2 ν = 0 ∀ f ∈ ˜ G and ξ t ( j 0 ) = ( P t ( δ j . 0 ) | δ q . ) ≤ || P t ( δ j . 0 ) || L 2 ν Finally, from the invariance of the measure and the uni- tarity of the operators involved we can deduce that � s ( g x ( t )) | 2 ) ≥ c > 0 | c l E ( which contradicts the last convergence. 12