Isma¨el Bailleul Bonn, October 9, 2007. · Isma¨el Bailleul Poisson boundary of a relativistic...
Transcript of Isma¨el Bailleul Bonn, October 9, 2007. · Isma¨el Bailleul Poisson boundary of a relativistic...
Explanation of titleResults
How can we show these results?Where is the boundary?
Poisson boundary of a relativistic diffusion
Ismael Bailleul
Bonn, October 9, 2007.
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
Layout of the talk
1. Explanation of title
2. Results
3. How can we show these results?
4. Where is the boundary?
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
Minkowski spacetimeRandom timelike pathsAsymptotic behaviour
Poisson boundary︸ ︷︷ ︸
Analysis
of a relativistic︸ ︷︷ ︸
Geometry
diffusion︸ ︷︷ ︸
Probability
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
Minkowski spacetimeRandom timelike pathsAsymptotic behaviour
Poisson boundary of a relativistic diffusion
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
Minkowski spacetimeRandom timelike pathsAsymptotic behaviour
Minkowski spacetime
time
space
ǫ0
ǫ1ǫ2
ǫ3
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
Minkowski spacetimeRandom timelike pathsAsymptotic behaviour
Minkowski spacetime
time
space
ξ
x
t
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
Minkowski spacetimeRandom timelike pathsAsymptotic behaviour
Minkowski spacetime
t1
ξ2
ξ1
x1
x2
t2
ξ = (t, x) ∈ R × R3
“Signal” traveling at a speedstrictly less than the speed oflight:
|x2 − x1| < c(t2 − t1)
Trajectory of a “signal” traveling at a speed strictly less than thespeed of light = timelike path
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
Minkowski spacetimeRandom timelike pathsAsymptotic behaviour
Minkowski spacetime
t1
ξ2
ξ1
x1
x2
t2
ξ = (t, x) ∈ R × R3
• Speed of light = 1• If a timelike path joins ξ1 andξ2:
|x2 − x1| < t2 − t1
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
Minkowski spacetimeRandom timelike pathsAsymptotic behaviour
Minkowski spacetime
t1
ξ2
ξ1
x1
x2
t2
ξ = (t, x) ∈ R × R3
• q(ξ) = t2 − |x |2
• If a timelike path joins ξ1
and ξ2: q(ξ2 − ξ1) > 0.
(R × R
3, q)
is denoted R1,3
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
Minkowski spacetimeRandom timelike pathsAsymptotic behaviour
Causality
ξ = (x , t)
Past of ξ : ζ = (y , s) ∈ R × R3 ; q(ζ − ξ) > 0 , s < t
Future of ξ : ζ = (y , s) ∈ R × R3 ; q(ζ − ξ) > 0 , s > t
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
Minkowski spacetimeRandom timelike pathsAsymptotic behaviour
Causality
ξ
ξ′
ξ′′
ξ is in the future of ξ′
ξ is not in the future of ξ′′
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
Minkowski spacetimeRandom timelike pathsAsymptotic behaviour
Hyperbolic space : H
0
ξ
ξq(ξ) < 0
H
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
Minkowski spacetimeRandom timelike pathsAsymptotic behaviour
Poisson boundary of a relativistic diffusion
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
Minkowski spacetimeRandom timelike pathsAsymptotic behaviour
Random timelike paths
How can we construct random timelike paths?
A recipe:ξr ∈ H, random,
ξs = ξ0 +
∫ s
0
ξr dr .ξ0
ξs
ξ1
ξ1
ξ0
ξs
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
Minkowski spacetimeRandom timelike pathsAsymptotic behaviour
Random timelike paths
Numerous kinds of randomness: from simple, to complicated...
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
Minkowski spacetimeRandom timelike pathsAsymptotic behaviour
Random timelike paths
Numerous kinds of randomness: from simple, to complicated...
Markov process, enjoying the Strong Markov property
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
Minkowski spacetimeRandom timelike pathsAsymptotic behaviour
Random timelike paths
Numerous kinds of randomness: from simple, to complicated...
Markov process, enjoying the Strong Markov property
• Where geometry and probabilitymeet:For any isometry ϕ of H, and any
point ξ of H, the image by ϕ of a
trajectory started from ξ has the
same law as a trajectory started
from ϕ(ξ).
ξξs
ϕ(ξ)
ϕ(ξs )
ϕ
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
Minkowski spacetimeRandom timelike pathsAsymptotic behaviour
Dudley’s theorem
Theorem (Description of strong Markov processes on H × R1,3, with a
law invariant by the action of affine isometries of spacetime)
These are the processes (ξs , ξs) ∈ H × R1,3 such that
• ξrr>0 is a Markov process on H, invariant under the action of
isometries of H,
• and ξs = ξ0 +∫ s
0 ξr dr.
ξr can have different types of behaviour:
• continuous : ξr is a Brownian motion on H,
• jump process: ξr is a Poisson process on H,
• “mixings” of jump and continuous trajectories.
Analogous description as that of Levy processes on R.
Relativistic diffusion: process on H × R1,3: ξs Brownian motion on H
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
Minkowski spacetimeRandom timelike pathsAsymptotic behaviour
Poisson boundary of a relativistic diffusion
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
Minkowski spacetimeRandom timelike pathsAsymptotic behaviour
Analytical description
Generator: infinitesimal characteristics of the random motion
x
V (x)xt ≃ x + tV (v)
Example: vector field V :
∀ f ,f (xt )−f (x)
t−→tց0
(V .f
)(x) : first order
differential operator.
If V .f = 0, f constant along trajectories.
∀ f ,E(ξ,ξ)
[f (ξt ,ξt)−f (ξ,ξ)
]
t−→tց0
Lf (ξ, ξ) =H
ξf
2 + ∂ξf (ξ): second
order differential operator.
If Lf = 0, then E(ξ,ξ)
[f (ξt , ξt)
]= f (ξ, ξ).
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
Minkowski spacetimeRandom timelike pathsAsymptotic behaviour
Analytical description
Lf = 0: L-harmonic function
Poisson boundary: set of all bounded L-harmonic functions.
Correspondence: L-harmonic functions 0 6 f 6 1 ⇔ “invariantevents”
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
Minkowski spacetimeRandom timelike pathsAsymptotic behaviour
Probabilistic description
Ω is made up of trajectories (ξs , ξs)s>0 with values in H × R1,3
ξ0
ξs ξs
ξ0
Event: collection of trajectories with the same properties
“Invariant event”
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
Minkowski spacetimeRandom timelike pathsAsymptotic behaviour
Si ∈ A
∈ A
∈ A
Examples
BIf f (ξs , ξs) a.s. converges, theevent lims→+∞
f (ξs , ξs) ∈ B ⊂ R
is an “invariant event”.
The σ-algebra of invariant events isgenerated by the sets of thepreceding form.
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
1. Title explanation
2. Results
3. How can we show these results?
4. Where is the boundary?
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
Notation
H × R1,3
ξ
ξ
ǫ0
σ ∈ S2
H
ξ =(chρ, (shρ)σ
)
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
Asymptotic behaviour of the relativistic diffusion
Theorem (Invariant σ-algebra of the relativistic diffusion)
(ξs , ξs) ∈ H × R1,3: relativistic diffusion
P(ξ,ξ) its law when started from (ξ, ξ)
The following limits exist P(ξ,ξ)-almost surely:
lims→+∞
σs ≡ σ∞,
lims→+∞
q(ξs , ε0 + σ∞) ≡ Rσ∞
∞ .
The σ-algebra of invariant events is generated by the events of the
form
σ∞ ∈ A, Rσ∞
∞ ∈ B,
where A ⊂ S2, B ⊂ R.
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
Asymptotic behaviour of the relativistic diffusion
σ∞
σ∞
θs ξs
σ∞
Rσ∞
∞
ε0 + σ∞
Rσ∞∞ε0
σ∞
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
Poisson boundary of the relativistic diffusion
Theorem (Poisson boundary of L)
One has for all A ⊂ S2, B ⊂ R,
P(ξ,ξ)
(σ∞ ∈ A, Rσ∞
∞ ∈ B)
=
∫
A×B
hσ(ξ, ξ)hσℓ (ξ, ξ)dσdℓ,
with explicit functions hσ and hσℓ .
Every bounded L-harmonic function is of the form
∫
H(σ, ℓ)hσ(ξ, ξ)hσℓ (ξ, ξ)dσdℓ,
where H(σ, ℓ) is a bounded function.
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
ApproachCoupling
1. Explanation of title
2. Results
3. How can we show these results?
4. Where is the boundary?
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
ApproachCoupling
Approach
One looks for converging quantities:
the direction σs of the speed → σ∞
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
ApproachCoupling
Approach
One looks for converging quantities:
the direction σs of the speed → σ∞
After conditioning, we find:
q(ξs , ε0 + σ∞) converges
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
ApproachCoupling
Approach
One looks for converging quantities:
the direction σs of the speed → σ∞
After conditioning, we find:
q(ξs , ε0 + σ∞) converges
1 =∫
S2 hσdσ,
1 =∫
S2×Rhσhσ
ℓ dℓdσ
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
ApproachCoupling
Choquet’s theorem on a convex compact set
Convex compacta
• Caratheodory (finitedimension),
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
ApproachCoupling
Choquet’s theorem on a convex compact set
Convex compacta
• Caratheodory (finitedimension),
Theorem (Choquet)
Every point of compact convex metric space, K, is the barycenter
of a probability with support in the set of extremal points of K.
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
ApproachCoupling
Choquet’s theorem on a convex compact set
Convex compacta
• Caratheodory (finitedimension),
Theorem (Choquet)
Every point of compact convex metric space, K, is the barycenter
of a probability with support in the set of extremal points of K.
Elliptic framework : LaplacianK = f > 0 ; ,f = 0, f (O) = 1, elliptic Harnack principle =⇒K compact (unif. convergence on compact sets)
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
ApproachCoupling
Choquet’s theorem on cones
g
0
fE(H ∩ C)
Cone of f > 0, Lf = 0,R
fdµ > 0
H K = f > 0 ; Lf = 0,∫
fdµ 6 1 iscompact (unif. cv. on compacta),C = f > 0 ; Lf = 0,
∫fdµ < ∞ is a
well-capped cone.
Theorem (Choquet’s theorem on well-capped cones)
If f =∫
E(H∩C) h µ(dh) and g 6 f , then there exists
0 6 G 6 1, g =∫
E(H∩C)hG(h)µ(dh).
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
ApproachCoupling
Choquet’s theorem on cones
• 1 =∫
hσhσℓ dσdℓ
• hσhσℓ minimal
=⇒Every L-harmonic function 0 6 g 6 1 is of theform
∫G (σ, ℓ)hσhσ
ℓ dσdℓ.
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
ApproachCoupling
Choquet’s theorem on cones
• 1 =∫
hσhσℓ dσdℓ
• hσhσℓ minimal
=⇒Every L-harmonic function 0 6 g 6 1 is of theform
∫G (σ, ℓ)hσhσ
ℓ dσdℓ.
• hσhσℓ minimal iff the only bounded functions f such that
Lhσhσℓ f =
L(hσhσℓf )
hσhσℓ
= 0 are constants.
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
ApproachCoupling
Choquet’s theorem on cones
• 1 =∫
hσhσℓ dσdℓ
• hσhσℓ minimal
=⇒Every L-harmonic function 0 6 g 6 1 is of theform
∫G (σ, ℓ)hσhσ
ℓ dσdℓ.
• hσhσℓ minimal iff the only bounded functions f such that
Lhσhσℓ f =
L(hσhσℓf )
hσhσℓ
= 0 are constants.
• Lhσhσℓ → random motion in H × R
1,3: conditioned diffusion(ξt , ξt = ξ0 +
∫ t
0 ξr dr) ∈ H × R1,3,
ξs a diffusion in H
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
ApproachCoupling
Coupling
If Lhσhσℓ f = 0, one has E(ξ,ξ)
[f (ξT , ξT )
]= f (ξ, ξ).
(ξ, ξ)
(ξT , ξT )
(ξ, ξ)
(ξT ′
, ξT ′
)
f (ξ, ξ) = E(ξ,ξ)
[f (ξT , ξT )
]
= E(ξ,ξ)
[f (ξ
T ′, ξ
T ′)]
= f (ξ, ξ)
Difficulty: two independent trajectories have no reason to meet! =⇒
coupling
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
ApproachCoupling
Coupling
Construct form a (random) trajectory, started from point x , a (random)trajectory, started form point y
the trajectory started from y has the good law
both trajectories meet
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
ApproachCoupling
Coupling
Construct form a (random) trajectory, started from point x , a (random)trajectory, started form point y
x y
the trajectory started from y has the good law
both trajectories meet
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
ApproachCoupling
Coupling
Construct form a (random) trajectory, started from point x , a (random)trajectory, started form point y
x y
the trajectory started from y has the good law
both trajectories meet
• Dimension: 3 // 3 random parameters in Brownian motion
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
ApproachCoupling
Coupling
Construct form a (random) trajectory, started from point x , a (random)trajectory, started form point y
x y
the trajectory started from y has the good law
both trajectories meet
• Dimension: 3 // 3 random parameters in Brownian motion
• Hypoelliptic framework:(
ξs ,∫ s
0 ξr dr)
∈ H × R1,3
dimension: 7 // 3 random parameters: ξs ∈ H
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
ApproachCoupling
Sketch of proof
Theorem
Every bounded Lhσhσℓ -harmonic function is constant.
Proof
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
ApproachCoupling
Sketch of proof
Theorem
Every bounded Lhσhσℓ -harmonic function is constant.
Proof
We bring back the situation to a 2 dimensional problem showingthat every bounded Lhσhσ
ℓ -harmonic function only depends on twocoordinates.
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
ApproachCoupling
Sketch of proof
Theorem
Every bounded Lhσhσℓ -harmonic function is constant.
Proof
We bring back the situation to a 2 dimensional problem showingthat every bounded Lhσhσ
ℓ -harmonic function only depends on twocoordinates.
Hypoelliptic coupling =⇒ they are constant
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
Causal boundaryResult
1. Explanation of title
2. How can we show such a result?
3. Where is the boundary?
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
Causal boundaryResult
Causal boundary
Where do timelike paths go?
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
Causal boundaryResult
Causal boundary
Where do timelike paths go?
Equivalence relation: two timelikepaths γtt>0 and γ′
tt>0 converge
towards the same point if they have
the same past:
⋃
t>0
I−(γt) =⋃
t>0
I−(γ′t)
Two infinitely far points are identifiedif they have the same past
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
Causal boundaryResult
Causal boundary
Where do timelike paths go?
γt
I−(γt)past of γt
Figure: Past of a trajectory
Equivalence relation: two timelikepaths γtt>0 and γ′
tt>0 converge
towards the same point if they have
the same past:
⋃
t>0
I−(γt) =⋃
t>0
I−(γ′t)
Two infinitely far points are identifiedif they have the same past
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
Causal boundaryResult
Causal boundary
Figure: The path convergestowards the same point asthe lightlike rays
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
Causal boundaryResult
Causal boundary
Figure: The path convergestowards the same point asthe lightlike rays
Every lightlike ray of some hyperplanconverges towards the same point,
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
Causal boundaryResult
Causal boundary
Figure: The path convergestowards the same point asthe lightlike rays
Every lightlike ray of some hyperplanconverges towards the same point,
Every trajectory that approaches thishyperplan as it goes to the infiniteconverge toward that point
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
Causal boundaryResult
Causal boundary
Figure: The path convergestowards the same point asthe lightlike rays
Every lightlike ray of some hyperplanconverges towards the same point,
Every trajectory that approaches thishyperplan as it goes to the infiniteconverge toward that point
The boundary can be identified
with S2 × R
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
Causal boundaryResult
Asymptotic behaviour, geometric version
Theorem
ξss>0 almost surely converges towards a point ξ∞ of the causal
boundary.
The invariant σ-algebra is generated by the events of the formξ∞ ∈ A
.
ε0 + σ∞
Rσ∞∞
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
Causal boundaryResult
Conclusion, prospects
Figure: Relativistic diffusionon a Lorentzian manifold
Franchi, Le Jan (2006) : Schwarzchild,Franchi (2007) Godel’s universe
Ad hoc methods for the Lorentzian
manifold framework:• stochastic calculus
• coupling
• causal boundary...
Everything remains to be done
Ismael Bailleul Poisson boundary of a relativistic diffusion
Explanation of titleResults
How can we show these results?Where is the boundary?
Causal boundaryResult
Conclusion, prospects
Figure: Relativistic diffusionon a Lorentzian manifold
Franchi, Le Jan (2006) : Schwarzchild,Franchi (2007) Godel’s universe
Ad hoc methods for the Lorentzian
manifold framework:• stochastic calculus
• coupling
• causal boundary...
Everything remains to be done
Asymptotic behaviour – Associated with any geometrical object?
Lifetime – Under which conditions (of local and global nature) doesthe process have an almost surely finite life time? Is thisprobabilistic incompleteness linked with lightlike/timelike geodesicincompleteness?
Ismael Bailleul Poisson boundary of a relativistic diffusion