Brownian Motion and lorentzian manifolds

Transcription

1 The case of Jürgen Angst Institut de Recherche Mathématique Avancée Université Louis Pasteur, Strasbourg École d été de Probabilités de Saint-Flour juillet 2008, Saint-Flour

2 1 Construction of the diffusion, geometrical framework 2 Two natural sub-diffusions Asymptotic behavior of the diffusion

3 Dudley s diffusion

4 Minkowski space-time and hyperbolic space Let R 1,d := {ξ = (ξ 0, ξ i ) R R d } denote the Minkowski spacetime of special relativity, equipped with the pseudo-metric : and d ξ, ξ := ξ 0 2 ξ i 2, i=1 H d := {ξ R 1,d ξ 0 > 0 and ξ, ξ = 1}. If (e 0, e 1,..., e d ) is the canonical basis in R 1,d, (e j ) its dual basis, the matrices E j = e 0 e j + e j e 0 generate the hyperbolic rotations.

5 Let ξ s be an hyperbolic brownian motion on H d and ξ s := ξ 0 + s 0 ξ u du. Then (ξ s, ξ s ) is a diffusion on R 1,d H d. Its law is invariant under the action of the Lorentz group.

6 If ξ s = (ξs, 0 ξ s ) and s(t) is defined as ξs(t) 0 = t, then the euclidian trajectory Z t := ξ s(t) satisfies dz(t) dt < 1. The velocity of Z(t) has norm < 1 (the velocity of the light).

7 The relativistic diffusion of Franchi and Le Jan

8 Construction of the diffusion, geometrical framework The general framework G(M) pseudo-orthonormal frame bundle with first element in T 1 M π 1 T 1 M positive part of the unitary tangent bundle M oriented lorentzian manifold of dimension d + 1, equipped with the Levi-Civita connexion

9 The infinitesimal generator of the diffusion Let V j be the canonical vertical vector field associated to the matrix E j and H 0 the first horizontal vector field. We define d L := H 0 + σ2 2 V, where V := Vj 2. If L 0 is the infinitesimal generator of the geodesic flow on T 1 M and V the vertical laplacian, for all F C 2 (T 1 M), one has on G(M) : j=1 (L 0 F ) π 1 = H 0 (F π 1 ), ( V F ) π 1 = V(F π 1 ). On T 1 M, the operator L induce the operator : G := L 0 + σ2 2 V.

10 Theorem (Franchi and Le Jan) The stochastic differential equation (stratonovich form) d ( ) dψ s = H 0 (Ψ s ) ds + σ V j (Ψ s ) dws j defines a diffusion (ξ s, ξ s ) := π 1 (Ψ s ) on T 1 M, whose infinitesimal generator is G = L 0 + σ2 2 V. If j=1 ξ (s) : Tξs M T ξ0 M denote the inverse parallel transport along the curves C 1 (ξ s 0 s s), then ζ s := ξ (s) ξ s is an hyperbolic brownian motion on T ξ0 M.

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12 They are the lorentzian manifolds M of the type : M = I M, where I is an interval in R and M = S 3, R 3, or H 3, equipped with pseudo-metric : ds 2 = dt 2 α 2 (t)dl 2. (1) where dl 2 is the standard riemannian metric on M.

13 Examples of scale factors α

14 Consider the following coordinates system ξ µ = (t, r, θ) on M : t ]0, T [, with 0 < T +, r [0, 1] if k = 1 and r R + if k = 1 ou k = 0, θ = sin(φ) cos(ψ) sin(φ) sin(ψ) cos(φ) S 2, φ [0, π], ψ R/2πZ. In this chart, the pseudo-metric (1) writes : ( ) dr ds 2 = dt 2 α 2 2 (t) 1 kr 2 + r2 dφ 2 + r 2 sin 2 (φ)dψ 2. (2)

15 Integration of geodesics Lemma Let (ξ s, ξ s ) T 1 M be a time-like geodesic, parametrized by its arc-length s. Then, the functions a := α(t s ) ṫ 2 s 1 and b := α 2 (t s )rs 2 θ s θ s are constant along (ξ s, ξ s ). Lemma Let (ξ u, ξ u ) T 1 M be a light-like (or null) geodesic. Then, the functions a := α(t u )ṫ u are constant along (ξ u, ξ u ). and b := α 2 (t u )r 2 u θ u θ u

16 Light-like geodesics Let (ξ u, ξ u ) be a light-like geodesic. As the function a := α(t u )ṫ u is constant, by integrating, one gets : tu t 0 α(v)dv = a u.

17 Light-like geodesics Let (ξ u, ξ u ) be a light-like geodesic. We define ρ := b /a and ε := sign(ṙ 0 ) ; then r u is given by : tu r 2 u ρ 2 = r0 2 ρ2 + ε t 0 arcsin ( ) ru 2 ρ 2 1 ρ 2 = arcsin dv α(v), r 2 0 ρ2 1 ρ 2 + ε tu t 0 dv α(v), argsh ( ) ru 2 ρ ρ 2 = argsh r 2 0 ρ2 1 + ρ 2 + ε tu t 0 dv α(v).

18 Light-like geodesics Let (ξ u, ξ u ) be a light-like geodesic, then θ u v where v := θ 0 θ 0 θ 0 θ 0. In the frame R := ( u := θ 0, v, w := u v ), one has φ u φ 0 = π/2, ψ u ψ 0 = sign( ψ 0 ) F ( tu t 0 ) dv. α(v)

19 Asymptotic behavior of light-like geodesics Let (ξ u, ξ u ) be a light-like geodesic ; define the time τ := sup{u, t u T }. Proposition If T dv/α(v) < +, then, when u goes to τ, the functions r u and ψ u converge to r and ψ (finite). If T dv/α(v) = +, then, when u goes to τ, r u goes to infinity when k = 0 or 1, and goes to 1 when k = 1. Moreover, ψ u converge when k = 0 or 1, and tends to ± when k = 1.

20 Two natural sub-diffusions Asymptotic behavior of the diffusion The relativistic diffusion in

21 The SDE s satisfied by (ξ s, ξ s ) Two natural sub-diffusions Asymptotic behavior of the diffusion If Ψ s is a solution of ( ) and (ξ s, ξ s ) = π 1 (Ψ s ) T 1 M, then : dt s = ṫ s ds, dr s = ṙ s ds, dφ s = φ s ds, dψ s = ψ s ds dṫ s = [ ( ) ] α(t s ) α ṙ2 (t s ) s 1 krs 2 + rs 2 θ s 2 + 3σ2 2 ṫs ds + dm ṫs, [ dṙ s = 2 α (t s )ṫ s ṙ s kr sṙs 2 α(t s ) 1 krs 2 [ ( dφ α ) (t s )ṫ s s = 2 + ṙs α(t s ) r s ] + r s (1 krs) 2 θ s 2 + 3σ2 2 ṙs ds + dm ṙs, φ s + sin(φ s ) cos(φ s ) 2 3σ 2 ] ψ s + φ s ds + dm φ 2 s, [ ( dψ α ) ] (t s )ṫ s s = 2 + ṙs ψ s 2 cot(φ s ) φ α(t s ) r s ψ s + 3σ2 ψ s ds + dm ψ s 2 s.

22 Two natural sub-diffusions Two natural sub-diffusions Asymptotic behavior of the diffusion The 2d-diffusion (t s, ṫ s ) satisfies : dt s = ṫ s ds, dṫ s = [ H(t s ) ( ṫ 2s 1 ) ] + 3σ2 2 ṫs ds + dm ṫs, where H = α /α is the Hubble function.

23 Two natural sub-diffusions Two natural sub-diffusions Asymptotic behavior of the diffusion The radial diffusion (t s, r s, a s, b s, c s ) with a 2 s = b 2 s/rs 2 + c 2 s : dt s = 1 + a 2 s/α 2 (t s )ds, db s = 3σ2 2 b sds + σ2 α 2 (t s )rs 2 ds + dms b, 2b s dr s = 1 kr 2 s α 2 (t s ) c s ds, dc s = 3σ2 2 c sds + b2 s 1 kr 2 s α 2 (t s )rs 3 ds + dms c, da s = 3σ2 2 a sds + σ 2 α2 (t s ) a s + dm a s, where a s := α(t s ) ṫ 2 s 1, b s := α 2 (t s )rs 2 θ s, c s := α2 (t s)ṙ s. 1 kr 2 s

24 Two natural sub-diffusions Asymptotic behavior of the diffusion Asymptotic behavior of the 2d-diffusion Proposition Suppose that t H(t) is decreasing and non-negative on R +. Define H = lim + H(t). If H > 0, then the process ṫ s is recurrent in ]1 + [. If H = 0, then ṫ s is transient, and if we define ( ts ) 1 Z s := α(t s )ṫ s α(u)du, then t 0 0 < lim inf Z s < lim sup Z s < + a.s. s + Z s s + d σ 2 /2 Γ(2).

25 Two natural sub-diffusions Asymptotic behavior of the diffusion Asymptotic behavior of the diffusion Let (ξ s, ξ s ) be the diffusion of Franchi and Le Jan and as before τ := sup{s, t s T }. Proposition If T dv/α(v) < +, then, when s goes to τ, the processes r s and θ s converge a.s., to some finite random variables r R + and θ S 2. If T dv/α(v) = +, then, when s goes to τ, r s goes to plus infinity a.s. when k = 0 ou 1, and it goes to 1 when k = 1. Moreover the point θ s converge a.s. when k = 0 or 1, and has an oscillatory behavior in S 2 when k = 1.

26 Two natural sub-diffusions Asymptotic behavior of the diffusion Thanks for your attention...