Diffusion by optimal transport in the Heisenberg group
Diffusion by optimal transport in the Heisenberg group Perspectives in Optimal transportation
Nicolas JUILLET
IRMA, Strasbourg
Munich, December 2011
Diffusion by optimal transport in the Heisenberg group The Heisenberg group
The geometry of H .
A basis for left-invariant vector fields is
X = E 1 − y
2 E 3 , Y = E 2 + x
2 E 3 and U = [ X , Y ] = E 3 . A curve is horizontal if γ ˙ ∈ Vect( X , Y ) for any t. Actually
˙
γ( t ) = a ( t ) X (γ( t )) + b ( t ) Y (γ( t )).
It has norm |˙ γ| = p
a 2 ( t ) + b 2 ( t ) .
The diffusion operator is ∆ = X 2 + Y 2 .
Diffusion by optimal transport in the Heisenberg group The Heisenberg group
The Riemannian Heisenberg group H ε
Let ε > 0 and H ε := ( R 3 , d ε , L 3 ) .
An orthonormal basis at point ( x , y , u ) is
X = E 1 − y
2 E 3 , Y = E 2 + x
2 E 3 and ε U = ε[ X , Y ] = ε E 3 . If
˙
γ( t ) = ( a ( t ) X + b ( t ) Y + c ( t ) U )(γ( t )) then |˙ γ( t )| ε =
q
a 2 + b 2 + c
2ε
2.
The diffusion operator is ∆ ε = ∆ + ε 2 U 2 .
Diffusion by optimal transport in the Heisenberg group The Heisenberg group
Geodesics of H .
A curve is horizontal if and only if the third coordinate evolves like the algebraic area swept by the complex projection.
The length of the horizontal curves is exactly the length of the projection in C .
The geodesics of H are the horizontal curves whose projection is an arc of circle
or a line.
Diffusion by optimal transport in the Heisenberg group The Heisenberg group
Geodesics of H .
A curve is horizontal if and only if the third coordinate evolves like the algebraic area swept by the complex projection.
The length of the horizontal curves is exactly the length of the projection in C .
The geodesics of H are the horizontal curves whose projection is an arc of circle
or a line.
Diffusion by optimal transport in the Heisenberg group The Heisenberg group
Geodesics of H .
A curve is horizontal if and only if the third coordinate evolves like the algebraic area swept by the complex projection.
The length of the horizontal curves is exactly the length of the projection in C .
The geodesics of H are the horizontal curves whose projection is an arc of circle
or a line.
Diffusion by optimal transport in the Heisenberg group Notations and définitions of transport optimal
Wasserstein metrics
Let W be the L 2 -minimal metric with respect to the Carnot-Carathéodory metric d and W ε with respect to d ε .
W ε ≤ W ,
P 2 ( H ) = P 2 ( H ε ) as topological spaces.
Lipschitz curves (resp. absolutely continuous) of P 2 ( H ) are Lipschitz (resp.
absolutely continuous) in P 2 ( H ε ) .
Diffusion by optimal transport in the Heisenberg group Notations and définitions of transport optimal
The relative entropy
The relative entropy of µ = ρ L is given by
H (µ) = H (µ | L ) = Z
ρ ln (ρ)( x ) d L ( x ).
Big entropy: µ concentrated on a few space.
Small entropy: µ take a lot of space.
Diffusion by optimal transport in the Heisenberg group Results
Synthetic Ricci curvature – definition
A space ( X , d ,ν) satisfies CD ( 0 ,+∞) if
∀µ 0 ,µ 1 ∈ P 2 ( X ) absolutely continuous, there exists a geodesic (µ t ) t ∈[ 0 , 1 ] such that
t ∈ [ 0 , 1 ] → Ent (µ t | ν) ∈ R is convex.
Diffusion by optimal transport in the Heisenberg group Results
For H ε = ( R 3 , d ε , L 3 ) , the curvature-dimension condition CD (− 2 1
ε
2, 3 ) is satisfied.
Theorem (J.)
For H = ( R 3 , d c , L 3 ) , the curvature-dimension condition CD ( K , N ) is not satisfied for
any N ∈ [ 1 ,+∞[ and K ∈ R .
Diffusion by optimal transport in the Heisenberg group Results
Gradient flow of the relative entropy.
An absolutely continuous curve (µ t ) t ≥ 0 is a gradient flow of H if t 7→ H (µ t ) is decreasing.
for almost every t > 0,
∂ t H (µ t ) = − Speed (µ t ) · Slope ( H )(µ t ) Speed (µ t ) = Slope ( H )(µ t ).
Theorem (J.)
The curve (µ t ) t > 0 is a gradient flow of H if and only if µ t = ρ t L 3 with
∂
∂ t ρ t = ∆ρ t .
Diffusion by optimal transport in the Heisenberg group Results
HWI inequalities
From
H (µ) ≥ H (ν) − p
I ε (ν) W ε (µ,ν) + 1 2 ·
− 1 2 ε 2
W ε (µ, ν) 2
to
H (µ) ≥ H (ν) − p
I (ν) W (µ,ν) − C (µ) W (µ,ν) 3 / 2 ,
but only if I ε (ν) = I (ν) + ε J (ν) is finite.
Diffusion by optimal transport in the Heisenberg group Results
