Ismaël Bailleul Bonn, October 9, 2007. · Ismaël Bailleul Poisson boundary of a relativistic diffusion. Explanation of title Results How can we show these results? Where is the

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



Layout of the talk

1. Explanation of title

2. Results

3. How can we show these results?

4. Where is the boundary?




Minkowski spacetimeRandom timelike pathsAsymptotic behaviour

Poisson boundary︸︷︷︸

Analysis

of a relativistic︸︷︷︸

Geometry

diffusion︸︷︷︸

Probability










Minkowski spacetime

time

space

ǫ0

ǫ1ǫ2

ǫ3





Minkowski spacetime

time

space

ξ

x

t





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





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





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





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





Causality

ξ

ξ′

ξ′′

ξ is in the future of ξ′

ξ is not in the future of ξ′′





Hyperbolic space : H

0

ξ

ξq(ξ) < 0

H










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






Numerous kinds of randomness: from simple, to complicated...







Markov process, enjoying the Strong Markov property







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





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










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).





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”





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”





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.




1. Title explanation

2. Results






Notation

H × R1,3

ξ

ξ

ǫ0

σ ∈ S2

H

ξ =(chρ, (shρ)σ

)




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.




Asymptotic behaviour of the relativistic diffusion

σ∞

σ∞

θs ξs

σ∞

Rσ∞

∞

ε0 + σ∞

Rσ∞∞ε0

σ∞




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.




ApproachCoupling


2. Results






ApproachCoupling

Approach

One looks for converging quantities:

the direction σs of the speed → σ∞




ApproachCoupling

Approach



After conditioning, we find:

q(ξs , ε0 + σ∞) converges




ApproachCoupling

Approach



After conditioning, we find:

q(ξs , ε0 + σ∞) converges

1 =∫

S2 hσdσ,

1 =∫

S2×Rhσhσ

ℓ dℓdσ




ApproachCoupling

Choquet’s theorem on a convex compact set

Convex compacta

• Caratheodory (finitedimension),




ApproachCoupling


Convex compacta


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.




ApproachCoupling


Convex compacta


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)




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).




ApproachCoupling


• 1 =∫

hσhσℓ dσdℓ

• hσhσℓ minimal

=⇒Every L-harmonic function 0 6 g 6 1 is of theform

∫G (σ, ℓ)hσhσ

ℓ dσdℓ.




ApproachCoupling


• 1 =∫

hσhσℓ dσdℓ




ℓ dσdℓ.

• hσhσℓ minimal iff the only bounded functions f such that

Lhσhσℓ f =

L(hσhσℓf )

hσhσℓ

= 0 are constants.




ApproachCoupling


• 1 =∫

hσhσℓ dσdℓ




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




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




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




ApproachCoupling

Coupling


x y






ApproachCoupling

Coupling


x y



• Dimension: 3 // 3 random parameters in Brownian motion




ApproachCoupling

Coupling


x y



• 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




ApproachCoupling

Sketch of proof

Theorem

Every bounded Lhσhσℓ -harmonic function is constant.

Proof




ApproachCoupling

Sketch of proof

Theorem


Proof

We bring back the situation to a 2 dimensional problem showingthat every bounded Lhσhσ

ℓ -harmonic function only depends on twocoordinates.




ApproachCoupling

Sketch of proof

Theorem


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




Causal boundaryResult


2. How can we show such a result?






Causal boundary

Where do timelike paths go?





Causal boundary


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





Causal boundary


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





Causal boundary

Figure: The path convergestowards the same point asthe lightlike rays





Causal boundary


Every lightlike ray of some hyperplanconverges towards the same point,





Causal boundary



Every trajectory that approaches thishyperplan as it goes to the infiniteconverge toward that point





Causal boundary



Every trajectory that approaches thishyperplan as it goes to the infiniteconverge toward that point

The boundary can be identified

with S2 × R





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∞∞





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





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?


Ismaël Bailleul Bonn, October 9, 2007. · Ismaël Bailleul Poisson boundary of a relativistic diffusion. Explanation of title Results How can we show these results? Where is the

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