Stochastic calculus of variations on the diffeomorphisms group

Transcription

1 Title Stochastic calculus of variations on the diffeomorphisms group Warwick, April 2012 Ana Bela Cruzeiro Dep. Mathematics IST (TUL) Grupo de Física-Matemática Univ. Lisboa ( ) 1 / 19

2 The diffeomorphisms group The diffeomorphisms group M compact finite-dim. Riemannian manifold Here M = T 2 G s = {g H s (M, M) : g bijective, g 1 H s (M, M)} If s > 2, H s C 1 and is a (infinite dim.) Hilbert manifold, locally diffeomorphic to H s g(tm) = {X H s (M; TM) : πox = g}, π : TM M chart at g given by: ϕ : H s g(tm) {diffeom.on M} ϕ(x)(.) = exp ox(.) ( ) 2 / 19

3 The diffeomorphisms group G s is a group for composition of maps Lie algebra: G s = H s (TM)(= He(TM)) s (e = id) On G s we consider the Riemannian metric < X g, Y g > L 2= < X g (x), Y g (x) > dm(x) M for X, Y T g (G s (M)) = H s g(tm) (weak Riemannian structure, Ebin-Marsden) Volume preserving counterparts: G s V = {g Gs : (g) (dm) = dm} G s V = {X Hs : div X = 0} ( ) 3 / 19

4 The diffeomorphisms group G s V submanifold of Gs There exists a right-invariant Levi-Civita connection 0 : 0 X Y = P e( X Y ) where P e orth. projection into the divergence free part in the Hodge decomposition, H s (TM) = div 1 ({0}) L 2 grad H s+1 (M) ( ) 4 / 19

5 The diffeomorphisms group Hydrodynamics: Geodesic equation for 0 = Euler equation equaivalent to u t + (u. u) = p u t + 0 u(u) = p ( ) 5 / 19

6 Brownian motions Brownian motions On the Lie algebra we consider dx(t) = k (A k dx 1 k (t) + B kdx 2 k (t)) x i k i.i.d. real valued Br. motions, A k, B k L 2 o.n. basis: k Z 2 {(0, 0)}, k 2 = k k 2 2, i = θ i 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 )] ( ) 6 / 19

7 Brownian motions Brownian motions on the group G V : dg(t) = oρ(x(t)) g(t), g(0) = e for ρ an Hermitian operator that diagonalizes in the basis A k and B k with the same eigenvalues λ k. Theorem. The process is well defined iff k λ2 k < Proof. Use S. Fang s methodology, e.g. Generator: ρ = 1 λ 2 k 2 ( 2 A k + B 2 k ) k Z f (g) = d dɛ ɛ=0f (exp (ɛz )g In particular no canonical L 2 Brownian motion. ( ) 7 / 19

8 Brownian motions Constants of structure: [A k, A l ] = [k, l] 2 k l ( k + l B k+l + k l B k l ) [k, l] [B k, B l ] = 2 k l ( k + l B k+l k l B k l ) [k, l] [A k, B 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 ( ) 7 / 19

9 Brownian motions For α k,l := 1 (l (l + k)) 2 k l k + l β k,l := α k,l = 1 (l (l k)) 2 k l k l Christoffel symbols: [k, l] = k 1 l 2 k 2 l 1 0 A k,a l = [k, l](α k,l B k+l + β k,l B k l ), 0 B k,b l = [k, l]( α k,l B k+l + β k,l B k l ) 0 A k,b l = [k, l]( α k,l A k+l +β k,l A k l ), 0 B k,a l = [k, l]( α k,l A k+l β k,l A k l ) ( ) 8 / 19

10 Brownian motions Remarks: 1. The Christoffel symbols give rise to unbounded antihermitian operators on G. 2. Since 0 A k,a k = 0 B k,b k = 0, Stratanovich = Itô in the equation for g(t). 3. We do not want use the metric < U, V > ρ =< ρ(u), V >. ( ) 9 / 19

11 Lifting to the frame bundle Lifting to the frame bundle Orthonormal frames above G V : r : T g (G V ) G V isometric isomorphism O(G V )= collection of o.n. frames (frame bundle) above G V it can be identified with S = U(G V ) G V where U stands for unitary group. Lie algebra of S= S = su (G V ) G V. Denote (σ, ω) the parallelism in S defined by the Levi-Civita connection. ( ) 10 / 19

12 Lifting to the frame bundle Lift of a vector filed Z : < Z, σ > U,g = UZ,..., < Z, ω >= 0 We have [ Z f ] o π = Z (foπ), π : S G V Lifted Laplacian: ρ = 1 λ 2 k 2 ( Bk ) Ã k k Then [ ρ f ] o π = ρ (foπ) generates the lifted ρ-brownian motion r x (t) π(r x (t)) = g(t) < odr x (t), ω >= 0 ( ) 11 / 19

13 Stochastic calculus of variations Stochastic calculus of variations Derivation of the Itô map x r x (t): Theorem.For r 0 S and given a semimartingale ξ with values in T r0 (S) with an antisymmetric diffusion coefficient, we have, < d dτ τ=0 r x (r 0 + τξ)(t), σ >= ξ x,t < d dτ τ=0 r x (r 0 + τξ)(t), ω >= γ x,t where dξ (t) = (Γ ξ x,t ρ ργ ξ x,t ) dx(t) + γ x,t (ρ dx(t)) with γ x,0 = 0 and ξ x,0 =< ξ, σ > dγ(t) = Ω(ξ (t), ρ dx(t)) ( ) 12 / 19

14 Stochastic calculus of variations Difficulty: We cannot choose ρ = Id and Γ ξ x,t ρ ργ ξ x,t is not antisymmetric. ( ) 13 / 19

15 Truncated diffusions Truncated diffusions Consider the case λ k = 1 for k N and λ k = 0 for k > N, ρ N the corresponding operator; Denote by r N x (t) the S-valued process for such this choice and call it truncated diffusion. ( r N x not finite dimensional projections: still infinite-dimensional, but driven by a finite number of Brownian motions) π( r N x,t) = g N x,t < d r N x,t, ω >= 0 where and dgx,t N = odx N (t)gx,t, N gx,0 N = e dx N (t) = (A k dxk 1 (t) + B kdxk 2 (t)) k N ( ) 14 / 19

16 Truncated diffusions Theorem. For r 0 S and given a semimartingale ξ with values in T r0 (S), with an antisymmetric diffusion coefficient, we have < d dτ τ=0 r N x (r 0 + τξ)(t), σ >= ξ N x,t < d dτ τ=0 r N x (r 0 + τξ)(t), ω >= γ N x,t where, for components k such that k N we have d(ξ N (t)) k = (γ N x,t( dx N (t)) k where γx,0 N = 0 and ξn x,0 =< ξ, σ >. dγ N (t) = Ω(ξ N (t), dx N (t)) Proof. As Γ antisymmetric, matrix (Γρ N ρ N Γ) k,j, with k, j N equal to zero. ( ) 15 / 19

17 Consequences Among the consequences Corollary.(Bismut formula) If Φ is a cylindrical functional on G V, we have d dτ τ=0e(φ(π( r x N,N (r 0 + τξ)))) = E(< [ r x N,N ] 1 (ζx,t), N DΦ > g N ) x,t where ζ N satisfies (dζ N ) k = (γ N x,tdx N (t)) k 1 2 (RicciN (ζ N (t))) k dt k N, and where Ricci N is the operator defined by Ricci N (Z ) = (Ω(A k, Z, A k ) + Ω(B k, Z, B k )) k N ( ) 16 / 19

18 Consequences Expression of the Ricci tensor: Ricci N (A j ) = k N Ricci N (B j ) = k N [k, j] 4 k 2 + j 2 k 2 j 2 k j 2 k + j 2 A j [k, j] 4 k 2 + j 2 k 2 j 2 k j 2 k + j 2 B j ( ) 17 / 19

19 Consequences Weitzenbock formulae: d( f ) l (df ) l Ricci ρ (df ) l = k,i ρ(k) 2 Γ i l,k ( k i f + i k f )+ k,m,j ρ(k) 2 (Γ l k,m [e j, e m ] k +Γ l m,k [e k, e j ] m ) j (e k = A k or B k ). ( ) 18 / 19

20 Consequences Back to Hydrodynamics: The equation is equivalent to u t + 0 u(u) + ν k N 0 k 0 k (u) + νriccin (u) = p u + (u. u) + νc(n) u = p t (Navier-Stokes equation) ( ) 19 / 19

21 References A. B. C., P. Malliavin, Renormalized stochastic calculus of variations for a renormalized infinite-dimensional Brownian motion, Stochastics (2009) A. B. C., Stochastic calculus of variations for the diffeomorphisms group, Bulletin des Sciences Mathématiques (2011) A. B. C., Hydrodynamics, probability and the geometry of the diffeomorphisms group, Seminar on StochasticAnalysis, Random Fields and Applicatios IV, R. C. Dalang. M. Dozzy, F. Russo ed, Birkhauser P.P. 63 (2011) ( ) 19 / 19