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
()
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The diffeomorphisms group
The diffeomorphisms group
M compact finite-dim. Riemannian manifold
Here M = T2
Gs = {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
Hgs (TM) = {X ∈ H s (M; TM) : πoX = g}, π : TM → M
chart at g given by:
ϕ : Hgs (TM) → {diffeom.on M}
ϕ(X )(.) = exp oX (.)
()
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The diffeomorphisms group
Gs is a group for composition of maps
Lie algebra:
G s = H s (TM)(= Hes (TM))
(e = id)
On Gs we consider the Riemannian metric
Z
< Xg , Yg >L2 =
< Xg (x), Yg (x) > dm(x)
(Gs (M))
for X , Y ∈ Tg
Ebin-Marsden)
=
M
s
Hg (TM)
(weak Riemannian structure,
Volume preserving counterparts:
GVs = {g ∈ Gs : (g)∗ (dm) = dm}
GVs = {X ∈ H s : div X = 0}
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The diffeomorphisms group
GVs submanifold of Gs
There exists a right-invariant Levi-Civita connection ∇0 :
∇0X Y = Pe (∇X Y )
where Pe orth. projection into the divergence free part in the Hodge
decomposition,
H s (TM) = div −1 ({0}) ⊕L2 grad H s+1 (M)
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The diffeomorphisms group
Hydrodynamics:
Geodesic equation for ∇0 = Euler equation
∂u
+ (u.∇u) = −∇p
∂t
equaivalent to
∂u
+ ∇0u (u) = −∇p
∂t
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Brownian motions
Brownian motions
On the Lie algebra we consider
dx(t) =
X
(Ak dxk1 (t) + Bk dxk2 (t))
k
xki
i.i.d. real valued Br. motions, Ak , Bk L2 o.n. basis:
Ak =
1
[(k2 cos k .θ)∂1 − (k1 cos k .θ)∂2 )]
|k |
Bk =
1
[(k2 sin k .θ)∂1 − (k1 sin k .θ)∂2 )]
|k |
k ∈ Z̃ 2 − {(0, 0)},
|k |2 = k12 + k22 ,
∂
∂i = ∂θ
i
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Brownian motions
Brownian motions on the group GV :
dg(t) = oρ(x(t)) g(t),
g(0) = e
for ρ an Hermitian operator that diagonalizes in the basis Ak and Bk
with the same eigenvalues λk .
Theorem.
P
The process is well defined iff k λ2k < ∞
Proof. Use S. Fang’s methodology, e.g.
Generator:
∆ρ =
1X 2 2
λk (∂Ak + ∂B2k )
2
k
∂Z f (g) =
d
|=0 f (exp (Z )g
d
In particular no canonical L2 Brownian motion.
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Brownian motions
Constants of structure:
[Ak , Al ] =
[k , l]
(|k + l|Bk +l + |k − l|Bk −l )
2|k ||l|
[Bk , Bl ] = −
[k , l]
(|k + l|Bk +l − |k − l|Bk −l )
2|k ||l|
[Ak , Bl ] = −
[k , l]
(|k + l|Ak +l − |k − l|Ak −l )
2|k ||l|
[∂i , Ak ] = −ki Bk
[∂i , Bk ] = ki Ak
()
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Brownian motions
For
αk ,l :=
1
(l | (l + k ))
2|k ||l||k + l|
βk ,l := α−k ,l =
1
(l | (l − k ))
2|k ||l||k − l|
[k , l] = k1 l2 − k2 l1
Christoffel symbols:
∇0Ak ,Al = [k , l](αk ,l Bk +l + βk ,l Bk −l ), ∇0Bk ,Bl = [k , l](−αk ,l Bk +l + βk ,l Bk −l )
∇0Ak ,Bl = [k , l](−αk ,l Ak +l +βk ,l Ak −l ), ∇0Bk ,Al = [k , l](−αk ,l Ak +l −βk ,l Ak −l )
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Brownian motions
Remarks:
1. The Christoffel symbols give rise to unbounded antihermitian operators on
G.
2. Since ∇0Ak ,Ak = ∇0Bk ,Bk = 0, Stratanovich = Itô in the equation for g(t).
3. We do not want use the metric < U, V >ρ =< ρ(U), V >.
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Lifting to the frame bundle
Lifting to the frame bundle
Orthonormal frames above GV :
r : Tg (GV ) → GV isometric isomorphism
O(GV )= collection of o.n. frames (frame bundle) above GV
it can be identified with S = U(GV ) × GV
where U stands for unitary group.
Lie algebra of S= S = su (GV ) × GV .
Denote (σ, ω) the parallelism in S defined by the Levi-Civita connection.
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Lifting to the frame bundle
Lift of a vector filed Z :
< Z̃ , σ >U,g = UZ , ...
, < Z̃ , ω >= 0
We have [∂Z f ] o π = ∂Z̃ (foπ), π : S → GV
Lifted Laplacian:
X
˜ρ = 1
∆
λ2k (∂Ã2 + ∂B̃2 )
k
k
2
k
˜ ρ (foπ) generates the lifted ρ-Brownian motion rx (t)
Then [∆ρ f ] o π = ∆
π(rx (t)) = g(t)
< odrx (t), ω >= 0
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Stochastic calculus of variations
Stochastic calculus of variations
Derivation of the Itô map x → rx (t):
Theorem.For r0 ∈ S and given a semimartingale ξ with values in Tr0 (S)
with an antisymmetric diffusion coefficient, we have,
<
d
0
rx (r0 + τ ξ)(t), σ >= ξx,t
dτ |τ =0
<
d
rx (r0 + τ ξ)(t), ω >= γx,t
dτ |τ =0
where
0 ρ − ρΓξ 0 ) ◦ dx(t) + γx,t (ρ ◦ dx(t))
dξ 0 (t) = (Γξx,t
x,t
dγ(t) = Ω(ξ 0 (t), ρ ◦ dx(t))
0
=< ξ, σ >
with γx,0 = 0 and ξx,0
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Stochastic calculus of variations
Difficulty:
We cannot choose ρ = Id and
0 ρ − ρΓξ 0
Γξx,t
x,t
is not antisymmetric.
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Truncated diffusions
Truncated diffusions
Consider the case λk = 1 for |k | ≤ N and λk = 0 for |k | > N, ρN the
corresponding operator;
Denote by rxN (t) the S-valued process for such this choice and call it
truncated diffusion.
( rxN not finite dimensional projections: still infinite-dimensional, but driven
by a finite number of Brownian motions)
N
N
π(r̃x,t
) = gx,t
N
< ◦d r̃x,t
, ω >= 0
where
N
N
dgx,t
= odx N (t)gx,t
,
N
gx,0
=e
and
dx N (t) =
X
(Ak dxk1 (t) + Bk dxk2 (t))
|k |≤N
()
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Truncated diffusions
Theorem.
For r0 ∈ S and given a semimartingale ξ with values in Tr0 (S), with an
antisymmetric diffusion coefficient, we have
<
d
N
r N (r0 + τ ξ)(t), σ >= ξx,t
dτ |τ =0 x
d
N
r N (r0 + τ ξ)(t), ω >= γx,t
dτ |τ =0 x
where, for components k such that |k | ≤ N we have
<
k
N
d(ξ N (t))k = (γx,t
(◦dx N (t))
dγ N (t) = Ω(ξ N (t), ◦dx N (t))
N = 0 and ξ N =< ξ, σ >.
where γx,0
x,0
Proof.
As Γ antisymmetric, matrix (ΓρN − ρN Γ)k ,j , with |k |, |j| ≤ N equal to zero.
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Consequences
Among the consequences
Corollary.(Bismut formula)
If Φ is a cylindrical functional on GV , we have
d
N
|τ =0 E(Φ(π(r̃xN,N (r0 + τ ξ)))) = E(< [r̃xN,N ]−1 (ζx,t
), DΦ >g N )
x,t
dτ
where ζ N satisfies
1
N
(dζ N )k = (γx,t
dx N (t))k − (Ricci N (ζ N (t)))k dt
2
|k | ≤ N, and where Ricci N is the operator defined by
Ricci N (Z ) = −
X
(Ω(Ak , Z , Ak ) + Ω(Bk , Z , Bk ))
|k |≤N
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Consequences
Expression of the Ricci tensor:
Ricci N (Aj ) = −
X
[k , j]4
|k |2 + |j|2
Aj
|k |2 |j|2 |k − j|2 |k + j|2
[k , j]4
|k |2 + |j|2
Bj
|k |2 |j|2 |k − j|2 |k + j|2
|k |≤N
Ricci N (Bj ) = −
X
|k |≤N
()
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Consequences
Weitzenbock formulae:
=
X
d(∆f )l − ∆(df )l − Ricciρ (df )l
X
ρ(k )2 Γil,k (∂k ∂i f +∂i ∂k f )+
ρ(k )2 (Γlk ,m [ej , em ]k +Γlm,k [ek , ej ]m )∂j
k ,i
k ,m,j
(ek = Ak or Bk ).
()
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Consequences
Back to Hydrodynamics:
The equation
X
∂u
+ ∇0u (u) + ν
∇0k ∇0k (u) + νRicci N (u) = −∇p
∂t
|k |≤N
is equivalent to
∂u
+ (u.∇u) + νc(N)∆u = −∇p
∂t
(Navier-Stokes equation)
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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)
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