Stochastic Euler-Poincaré reduction

28/10/2015
Auteurs : Marc Arnaudon
Publication GSI2015
OAI : oai:www.see.asso.fr:11784:13650
DOI : You do not have permission to access embedded form.

Résumé

We will prove a Euler-Poincaré reduction theorem for stochastic processes taking values in a Lie group, which is a generalization of the Lagrangian version of reduction and its associated variational principles. We will also show examples of its application to the rigid body and to the group of diffeomorphisms, which includes the Navier-Stokes equation on a bounded domain and the Camassa-Holm equation.

Stochastic Euler-Poincaré reduction

Média

Voir la vidéo

Métriques

129
8
613.93 Ko
 application/pdf
bitcache://2c8664180430ee70d18c415c2a2bfd5e73d0d77e

Licence

Creative Commons Attribution-ShareAlike 4.0 International

Sponsors

Organisateurs

logo_see.gif
logocampusparissaclay.png

Sponsors

entropy1-01.png
springer-logo.png
lncs_logo.png
Séminaire Léon Brillouin Logo
logothales.jpg
smai.png
logo_cnrs_2.jpg
gdr-isis.png
logo_gdr-mia.png
logo_x.jpeg
logo-lix.png
logorioniledefrance.jpg
isc-pif_logo.png
logo_telecom_paristech.png
csdcunitwinlogo.jpg
<resource  xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
                xmlns="http://datacite.org/schema/kernel-4"
                xsi:schemaLocation="http://datacite.org/schema/kernel-4 http://schema.datacite.org/meta/kernel-4/metadata.xsd">
        <identifier identifierType="DOI">10.23723/11784/13650</identifier><creators><creator><creatorName>Marc Arnaudon</creatorName></creator></creators><titles>
            <title>Stochastic Euler-Poincaré reduction</title></titles>
        <publisher>SEE</publisher>
        <publicationYear>2015</publicationYear>
        <resourceType resourceTypeGeneral="Text">Text</resourceType><dates>
	    <date dateType="Created">Mon 1 Jun 2015</date>
	    <date dateType="Updated">Wed 31 Aug 2016</date>
            <date dateType="Submitted">Sat 17 Feb 2018</date>
	</dates>
        <alternateIdentifiers>
	    <alternateIdentifier alternateIdentifierType="bitstream">2c8664180430ee70d18c415c2a2bfd5e73d0d77e</alternateIdentifier>
	</alternateIdentifiers>
        <formats>
	    <format>application/pdf</format>
	</formats>
	<version>24676</version>
        <descriptions>
            <description descriptionType="Abstract">
We will prove a Euler-Poincaré reduction theorem for stochastic processes taking values in a Lie group, which is a generalization of the Lagrangian version of reduction and its associated variational principles. We will also show examples of its application to the rigid body and to the group of diffeomorphisms, which includes the Navier-Stokes equation on a bounded domain and the Camassa-Holm equation.

</description>
        </descriptions>
    </resource>
.

Deterministic framework Stochastic framework Stochastic Euler-Poincaré reduction. Marc Arnaudon Université de Bordeaux, France GSI, École Polytechnique, 29 October 2015 Marc Arnaudon Stochastic Euler-Poincaré reduction. Deterministic framework Stochastic framework References Arnaudon, Marc; Chen, Xin; Cruzeiro, Ana Bela; Stochastic Euler-Poincaré reduction. J. Math. Phys. 55 (2014), no. 8, 17pp Chen, Xin; Cruzeiro, Ana Bela; Ratiu, Tudor S.; Constrained and stochastic variational principles for dissipative equations with advected quantities. arXiv:1506.05024 Marc Arnaudon Stochastic Euler-Poincaré reduction. Deterministic framework Stochastic framework 1 Deterministic framework Euler-Poincaré equations Diffeomorphism group on a compact Riemannian manifold Volume preserving diffeomorphism group Lagrangian paths Characterization of the geodesics on Gs V , ·, · 0 Euler-Poincaré equation on Gs V 2 Stochastic framework Semi-martingales in a Lie group Stochastic Euler-Poincaré reduction Group of volume preserving diffeomorphisms Navier-Stokes and Camassa-Holm equations Marc Arnaudon Stochastic Euler-Poincaré reduction. Deterministic framework Stochastic framework Euler-Poincaré equations Diffeomorphism group on a compact Riemannian manifold Volume preserving diffeomorphism group Lagrangian paths Characterization of the geodesics on Gs V , ·, · 0 Euler-Poincaré equation on Gs V Let M be a Riemannian manifold and L : TM × [0, T] → R a Lagrangian on M. Let q ∈ C1 a,b([0, T]; M) := {q ∈ C1([0, T], M), q(0) = a, q(T) = b}. The action functional C : C1 a,b([0, T]; M) → R is defined by C (q(·)) := T 0 L (q(t), ˙q(t), t) dt. The critical points for C satisfy the Euler-Lagrange equation d dt ∂L ∂ ˙q − ∂L ∂q = 0. Marc Arnaudon Stochastic Euler-Poincaré reduction. Deterministic framework Stochastic framework Euler-Poincaré equations Diffeomorphism group on a compact Riemannian manifold Volume preserving diffeomorphism group Lagrangian paths Characterization of the geodesics on Gs V , ·, · 0 Euler-Poincaré equation on Gs V Let M be a Riemannian manifold and L : TM × [0, T] → R a Lagrangian on M. Let q ∈ C1 a,b([0, T]; M) := {q ∈ C1([0, T], M), q(0) = a, q(T) = b}. The action functional C : C1 a,b([0, T]; M) → R is defined by C (q(·)) := T 0 L (q(t), ˙q(t), t) dt. The critical points for C satisfy the Euler-Lagrange equation d dt ∂L ∂ ˙q − ∂L ∂q = 0. Marc Arnaudon Stochastic Euler-Poincaré reduction. Deterministic framework Stochastic framework Euler-Poincaré equations Diffeomorphism group on a compact Riemannian manifold Volume preserving diffeomorphism group Lagrangian paths Characterization of the geodesics on Gs V , ·, · 0 Euler-Poincaré equation on Gs V Let M be a Riemannian manifold and L : TM × [0, T] → R a Lagrangian on M. Let q ∈ C1 a,b([0, T]; M) := {q ∈ C1([0, T], M), q(0) = a, q(T) = b}. The action functional C : C1 a,b([0, T]; M) → R is defined by C (q(·)) := T 0 L (q(t), ˙q(t), t) dt. The critical points for C satisfy the Euler-Lagrange equation d dt ∂L ∂ ˙q − ∂L ∂q = 0. Marc Arnaudon Stochastic Euler-Poincaré reduction. Deterministic framework Stochastic framework Euler-Poincaré equations Diffeomorphism group on a compact Riemannian manifold Volume preserving diffeomorphism group Lagrangian paths Characterization of the geodesics on Gs V , ·, · 0 Euler-Poincaré equation on Gs V Let M be a Riemannian manifold and L : TM × [0, T] → R a Lagrangian on M. Let q ∈ C1 a,b([0, T]; M) := {q ∈ C1([0, T], M), q(0) = a, q(T) = b}. The action functional C : C1 a,b([0, T]; M) → R is defined by C (q(·)) := T 0 L (q(t), ˙q(t), t) dt. The critical points for C satisfy the Euler-Lagrange equation d dt ∂L ∂ ˙q − ∂L ∂q = 0. Marc Arnaudon Stochastic Euler-Poincaré reduction. Deterministic framework Stochastic framework Euler-Poincaré equations Diffeomorphism group on a compact Riemannian manifold Volume preserving diffeomorphism group Lagrangian paths Characterization of the geodesics on Gs V , ·, · 0 Euler-Poincaré equation on Gs V Suppose that the configuration space M = G is a Lie group and L : TG → R is a left invariant Lagrangian: (ξ) := L(e, ξ) = L(g, g · ξ), ∀ξ ∈ TeG, g ∈ G. (here and in the sequel, g · ξ = TeLgξ) The action functional C : C1 a,b([0, T]; G) → R is defined by C (g(·)) := T 0 L (g(t), ˙g(t)) dt = T 0 (ξ(t)) dt, where ξ(t) := g(t)−1 · ˙g(t). [J.E. Marsden, T. Ratiu 1994] [J.E. Marsden, J. Scheurle 1993]: g(·) is a critical point for C if and only if it satisfies the Euler-Poincaré equation on T∗ e G d dt d dξ − ad∗ ξ(t) d dξ = 0, where ad∗ ξ : T∗ e G → T∗ e G is the dual action of adξ : TeG → TeG: ad∗ ξ η, θ = η, adξ θ , η ∈ T∗ e G, θ ∈ TeG. Marc Arnaudon Stochastic Euler-Poincaré reduction. Deterministic framework Stochastic framework Euler-Poincaré equations Diffeomorphism group on a compact Riemannian manifold Volume preserving diffeomorphism group Lagrangian paths Characterization of the geodesics on Gs V , ·, · 0 Euler-Poincaré equation on Gs V Suppose that the configuration space M = G is a Lie group and L : TG → R is a left invariant Lagrangian: (ξ) := L(e, ξ) = L(g, g · ξ), ∀ξ ∈ TeG, g ∈ G. (here and in the sequel, g · ξ = TeLgξ) The action functional C : C1 a,b([0, T]; G) → R is defined by C (g(·)) := T 0 L (g(t), ˙g(t)) dt = T 0 (ξ(t)) dt, where ξ(t) := g(t)−1 · ˙g(t). [J.E. Marsden, T. Ratiu 1994] [J.E. Marsden, J. Scheurle 1993]: g(·) is a critical point for C if and only if it satisfies the Euler-Poincaré equation on T∗ e G d dt d dξ − ad∗ ξ(t) d dξ = 0, where ad∗ ξ : T∗ e G → T∗ e G is the dual action of adξ : TeG → TeG: ad∗ ξ η, θ = η, adξ θ , η ∈ T∗ e G, θ ∈ TeG. Marc Arnaudon Stochastic Euler-Poincaré reduction. Deterministic framework Stochastic framework Euler-Poincaré equations Diffeomorphism group on a compact Riemannian manifold Volume preserving diffeomorphism group Lagrangian paths Characterization of the geodesics on Gs V , ·, · 0 Euler-Poincaré equation on Gs V Suppose that the configuration space M = G is a Lie group and L : TG → R is a left invariant Lagrangian: (ξ) := L(e, ξ) = L(g, g · ξ), ∀ξ ∈ TeG, g ∈ G. (here and in the sequel, g · ξ = TeLgξ) The action functional C : C1 a,b([0, T]; G) → R is defined by C (g(·)) := T 0 L (g(t), ˙g(t)) dt = T 0 (ξ(t)) dt, where ξ(t) := g(t)−1 · ˙g(t). [J.E. Marsden, T. Ratiu 1994] [J.E. Marsden, J. Scheurle 1993]: g(·) is a critical point for C if and only if it satisfies the Euler-Poincaré equation on T∗ e G d dt d dξ − ad∗ ξ(t) d dξ = 0, where ad∗ ξ : T∗ e G → T∗ e G is the dual action of adξ : TeG → TeG: ad∗ ξ η, θ = η, adξ θ , η ∈ T∗ e G, θ ∈ TeG. Marc Arnaudon Stochastic Euler-Poincaré reduction. Deterministic framework Stochastic framework Euler-Poincaré equations Diffeomorphism group on a compact Riemannian manifold Volume preserving diffeomorphism group Lagrangian paths Characterization of the geodesics on Gs V , ·, · 0 Euler-Poincaré equation on Gs V We will be interested in variations ξ(·) satisfying ˙ξ(t) = ˙ν(t) + adξ(t) ν(t) for some ν ∈ C1 ([0, T], TeG), which is equivalent to the variation of g(·) with the perturbation gε(t) = g(t)eε,ν (t), where eε,ν (t) is the unique solution to the following ODE on G: d dt eε,ν (t) = εeε,ν (t) · ˙ν(t), eε,ν (0) = e. Marc Arnaudon Stochastic Euler-Poincaré reduction. Deterministic framework Stochastic framework Euler-Poincaré equations Diffeomorphism group on a compact Riemannian manifold Volume preserving diffeomorphism group Lagrangian paths Characterization of the geodesics on Gs V , ·, · 0 Euler-Poincaré equation on Gs V We will be interested in variations ξ(·) satisfying ˙ξ(t) = ˙ν(t) + adξ(t) ν(t) for some ν ∈ C1 ([0, T], TeG), which is equivalent to the variation of g(·) with the perturbation gε(t) = g(t)eε,ν (t), where eε,ν (t) is the unique solution to the following ODE on G: d dt eε,ν (t) = εeε,ν (t) · ˙ν(t), eε,ν (0) = e. Marc Arnaudon Stochastic Euler-Poincaré reduction. Deterministic framework Stochastic framework Euler-Poincaré equations Diffeomorphism group on a compact Riemannian manifold Volume preserving diffeomorphism group Lagrangian paths Characterization of the geodesics on Gs V , ·, · 0 Euler-Poincaré equation on Gs V Let M be a n-dimensional compact Riemannian manifold. We define Gs := g : M → M a bijection , g, g−1 ∈ Hs (M, M) , where Hs(M, M) denotes the manifold of Sobolev maps of class s > 1 + n 2 from M to itself. If s > 1 + n 2 then Gs is a C∞ Hilbert manifold. Gs is a group under composition between maps, right translation is smooth, left translation and inversion are only continuous. Gs is also a topological group (but not an infinite dimensional Lie group). Marc Arnaudon Stochastic Euler-Poincaré reduction. Deterministic framework Stochastic framework Euler-Poincaré equations Diffeomorphism group on a compact Riemannian manifold Volume preserving diffeomorphism group Lagrangian paths Characterization of the geodesics on Gs V , ·, · 0 Euler-Poincaré equation on Gs V Let M be a n-dimensional compact Riemannian manifold. We define Gs := g : M → M a bijection , g, g−1 ∈ Hs (M, M) , where Hs(M, M) denotes the manifold of Sobolev maps of class s > 1 + n 2 from M to itself. If s > 1 + n 2 then Gs is a C∞ Hilbert manifold. Gs is a group under composition between maps, right translation is smooth, left translation and inversion are only continuous. Gs is also a topological group (but not an infinite dimensional Lie group). Marc Arnaudon Stochastic Euler-Poincaré reduction. Deterministic framework Stochastic framework Euler-Poincaré equations Diffeomorphism group on a compact Riemannian manifold Volume preserving diffeomorphism group Lagrangian paths Characterization of the geodesics on Gs V , ·, · 0 Euler-Poincaré equation on Gs V Let M be a n-dimensional compact Riemannian manifold. We define Gs := g : M → M a bijection , g, g−1 ∈ Hs (M, M) , where Hs(M, M) denotes the manifold of Sobolev maps of class s > 1 + n 2 from M to itself. If s > 1 + n 2 then Gs is a C∞ Hilbert manifold. Gs is a group under composition between maps, right translation is smooth, left translation and inversion are only continuous. Gs is also a topological group (but not an infinite dimensional Lie group). Marc Arnaudon Stochastic Euler-Poincaré reduction. Deterministic framework Stochastic framework Euler-Poincaré equations Diffeomorphism group on a compact Riemannian manifold Volume preserving diffeomorphism group Lagrangian paths Characterization of the geodesics on Gs V , ·, · 0 Euler-Poincaré equation on Gs V The tangent space TηGs at arbitrary η ∈ Gs is TηGs = U : M → TM of class Hs , U(m) ∈ Tη(m)M . The Riemannian structure on M induces the weak L2, or hydrodynamic, metric ·, · 0 on Gs given by U, V 0 η := M Uη(m), Vη(m) m dµg(m), for any η ∈ Gs, U, V ∈ TηGs. Here Uη := U ◦ η−1 ∈ TeGs and µg denotes the Riemannian volume asociated with (M, g). Obviously, ·, · 0 is a right invariant metric on Gs. Marc Arnaudon Stochastic Euler-Poincaré reduction. Deterministic framework Stochastic framework Euler-Poincaré equations Diffeomorphism group on a compact Riemannian manifold Volume preserving diffeomorphism group Lagrangian paths Characterization of the geodesics on Gs V , ·, · 0 Euler-Poincaré equation on Gs V The tangent space TηGs at arbitrary η ∈ Gs is TηGs = U : M → TM of class Hs , U(m) ∈ Tη(m)M . The Riemannian structure on M induces the weak L2, or hydrodynamic, metric ·, · 0 on Gs given by U, V 0 η := M Uη(m), Vη(m) m dµg(m), for any η ∈ Gs, U, V ∈ TηGs. Here Uη := U ◦ η−1 ∈ TeGs and µg denotes the Riemannian volume asociated with (M, g). Obviously, ·, · 0 is a right invariant metric on Gs. Marc Arnaudon Stochastic Euler-Poincaré reduction. Deterministic framework Stochastic framework Euler-Poincaré equations Diffeomorphism group on a compact Riemannian manifold Volume preserving diffeomorphism group Lagrangian paths Characterization of the geodesics on Gs V , ·, · 0 Euler-Poincaré equation on Gs V The tangent space TηGs at arbitrary η ∈ Gs is TηGs = U : M → TM of class Hs , U(m) ∈ Tη(m)M . The Riemannian structure on M induces the weak L2, or hydrodynamic, metric ·, · 0 on Gs given by U, V 0 η := M Uη(m), Vη(m) m dµg(m), for any η ∈ Gs, U, V ∈ TηGs. Here Uη := U ◦ η−1 ∈ TeGs and µg denotes the Riemannian volume asociated with (M, g). Obviously, ·, · 0 is a right invariant metric on Gs. Marc Arnaudon Stochastic Euler-Poincaré reduction. Deterministic framework Stochastic framework Euler-Poincaré equations Diffeomorphism group on a compact Riemannian manifold Volume preserving diffeomorphism group Lagrangian paths Characterization of the geodesics on Gs V , ·, · 0 Euler-Poincaré equation on Gs V Let be the Levi-Civita connection associated with the Riemannian manifold (M, g). We define a right invariant connection 0 on Gs by 0 ˜X ˜Y (η) := ∂ ∂t t=0 ˜Y(ηt ) ◦ η−1 t ◦ η + Xη Yη ◦ η, where ˜X, ˜Y ∈ L (Gs), Xη := ˜X ◦ η−1, Yη := ˜Y ◦ η−1 ∈ L s(M), and η is a C1 curve in Gs such that η0 = η and d dt t=0 ηt = ˜X(η). Here L (Gs) denotes the set of smooth vector fields on Gs. 0 is the Levi-Civita connection associated to Gs, ·, · 0 . Marc Arnaudon Stochastic Euler-Poincaré reduction. Deterministic framework Stochastic framework Euler-Poincaré equations Diffeomorphism group on a compact Riemannian manifold Volume preserving diffeomorphism group Lagrangian paths Characterization of the geodesics on Gs V , ·, · 0 Euler-Poincaré equation on Gs V Let be the Levi-Civita connection associated with the Riemannian manifold (M, g). We define a right invariant connection 0 on Gs by 0 ˜X ˜Y (η) := ∂ ∂t t=0 ˜Y(ηt ) ◦ η−1 t ◦ η + Xη Yη ◦ η, where ˜X, ˜Y ∈ L (Gs), Xη := ˜X ◦ η−1, Yη := ˜Y ◦ η−1 ∈ L s(M), and η is a C1 curve in Gs such that η0 = η and d dt t=0 ηt = ˜X(η). Here L (Gs) denotes the set of smooth vector fields on Gs. 0 is the Levi-Civita connection associated to Gs, ·, · 0 . Marc Arnaudon Stochastic Euler-Poincaré reduction. Deterministic framework Stochastic framework Euler-Poincaré equations Diffeomorphism group on a compact Riemannian manifold Volume preserving diffeomorphism group Lagrangian paths Characterization of the geodesics on Gs V , ·, · 0 Euler-Poincaré equation on Gs V For s > 1 + n 2 , let Gs V := g, g ∈ Gs , g is volume preserving . Gs V is still a topological group. The tangent space TeGs V is G s V = TeGs V = U, U ∈ TeGs , div(U) = 0 . The L2-metric ·, · 0 and its Levi-Civita connection 0,V are defined on Gs V by orthogonal projection. More precisely the Levi Civita connection on Gs V is given by 0,V X Y = Pe( 0 X Y) with Pe the orthogonal projection on G s V : Hs (TM) = G s V ⊕ dHs+1 (M). Marc Arnaudon Stochastic Euler-Poincaré reduction. Deterministic framework Stochastic framework Euler-Poincaré equations Diffeomorphism group on a compact Riemannian manifold Volume preserving diffeomorphism group Lagrangian paths Characterization of the geodesics on Gs V , ·, · 0 Euler-Poincaré equation on Gs V For s > 1 + n 2 , let Gs V := g, g ∈ Gs , g is volume preserving . Gs V is still a topological group. The tangent space TeGs V is G s V = TeGs V = U, U ∈ TeGs , div(U) = 0 . The L2-metric ·, · 0 and its Levi-Civita connection 0,V are defined on Gs V by orthogonal projection. More precisely the Levi Civita connection on Gs V is given by 0,V X Y = Pe( 0 X Y) with Pe the orthogonal projection on G s V : Hs (TM) = G s V ⊕ dHs+1 (M). Marc Arnaudon Stochastic Euler-Poincaré reduction. Deterministic framework Stochastic framework Euler-Poincaré equations Diffeomorphism group on a compact Riemannian manifold Volume preserving diffeomorphism group Lagrangian paths Characterization of the geodesics on Gs V , ·, · 0 Euler-Poincaré equation on Gs V For s > 1 + n 2 , let Gs V := g, g ∈ Gs , g is volume preserving . Gs V is still a topological group. The tangent space TeGs V is G s V = TeGs V = U, U ∈ TeGs , div(U) = 0 . The L2-metric ·, · 0 and its Levi-Civita connection 0,V are defined on Gs V by orthogonal projection. More precisely the Levi Civita connection on Gs V is given by 0,V X Y = Pe( 0 X Y) with Pe the orthogonal projection on G s V : Hs (TM) = G s V ⊕ dHs+1 (M). Marc Arnaudon Stochastic Euler-Poincaré reduction. Deterministic framework Stochastic framework Euler-Poincaré equations Diffeomorphism group on a compact Riemannian manifold Volume preserving diffeomorphism group Lagrangian paths Characterization of the geodesics on Gs V , ·, · 0 Euler-Poincaré equation on Gs V For s > 1 + n 2 , let Gs V := g, g ∈ Gs , g is volume preserving . Gs V is still a topological group. The tangent space TeGs V is G s V = TeGs V = U, U ∈ TeGs , div(U) = 0 . The L2-metric ·, · 0 and its Levi-Civita connection 0,V are defined on Gs V by orthogonal projection. More precisely the Levi Civita connection on Gs V is given by 0,V X Y = Pe( 0 X Y) with Pe the orthogonal projection on G s V : Hs (TM) = G s V ⊕ dHs+1 (M). Marc Arnaudon Stochastic Euler-Poincaré reduction. Deterministic framework Stochastic framework Euler-Poincaré equations Diffeomorphism group on a compact Riemannian manifold Volume preserving diffeomorphism group Lagrangian paths Characterization of the geodesics on Gs V , ·, · 0 Euler-Poincaré equation on Gs V Consider the ODE on M d dt (gt (x)) = u (t, gt (x)) g0(x) = x. Here u(t, ·) ∈ TeGs for every t > 0. For every fixed t > 0, gt (·) ∈ Gs(M). So g ∈ C1([0, T], Gs). If div(u(t)) = 0 for every t then g ∈ C1([0, T], Gs V ) Marc Arnaudon Stochastic Euler-Poincaré reduction. Deterministic framework Stochastic framework Euler-Poincaré equations Diffeomorphism group on a compact Riemannian manifold Volume preserving diffeomorphism group Lagrangian paths Characterization of the geodesics on Gs V , ·, · 0 Euler-Poincaré equation on Gs V Consider the ODE on M d dt (gt (x)) = u (t, gt (x)) g0(x) = x. Here u(t, ·) ∈ TeGs for every t > 0. For every fixed t > 0, gt (·) ∈ Gs(M). So g ∈ C1([0, T], Gs). If div(u(t)) = 0 for every t then g ∈ C1([0, T], Gs V ) Marc Arnaudon Stochastic Euler-Poincaré reduction. Deterministic framework Stochastic framework Euler-Poincaré equations Diffeomorphism group on a compact Riemannian manifold Volume preserving diffeomorphism group Lagrangian paths Characterization of the geodesics on Gs V , ·, · 0 Euler-Poincaré equation on Gs V Consider the ODE on M d dt (gt (x)) = u (t, gt (x)) g0(x) = x. Here u(t, ·) ∈ TeGs for every t > 0. For every fixed t > 0, gt (·) ∈ Gs(M). So g ∈ C1([0, T], Gs). If div(u(t)) = 0 for every t then g ∈ C1([0, T], Gs V ) Marc Arnaudon Stochastic Euler-Poincaré reduction. Deterministic framework Stochastic framework Euler-Poincaré equations Diffeomorphism group on a compact Riemannian manifold Volume preserving diffeomorphism group Lagrangian paths Characterization of the geodesics on Gs V , ·, · 0 Euler-Poincaré equation on Gs V [V.I. Arnold 1966] [D.G. Ebin, J.E. Marsden 1970] A Lagrangian path g ∈ C2([0, T], Gs V ) satisfying the equation above is a geodesic on Gs V , ·, · 0,V (i.e. 0,V ˙g(t) ˙g(t)) if and only of the velocity field u satisfies the Euler equation for incompressible inviscid fluids (E) ∂u ∂t = − uu − p divu = 0 Notice that the term p corresponds to the use of 0 instead of 0,V : the first system rewrites as ∂u ∂t = − 0,V u u divu = 0 Marc Arnaudon Stochastic Euler-Poincaré reduction. Deterministic framework Stochastic framework Euler-Poincaré equations Diffeomorphism group on a compact Riemannian manifold Volume preserving diffeomorphism group Lagrangian paths Characterization of the geodesics on Gs V , ·, · 0 Euler-Poincaré equation on Gs V [V.I. Arnold 1966] [D.G. Ebin, J.E. Marsden 1970] A Lagrangian path g ∈ C2([0, T], Gs V ) satisfying the equation above is a geodesic on Gs V , ·, · 0,V (i.e. 0,V ˙g(t) ˙g(t)) if and only of the velocity field u satisfies the Euler equation for incompressible inviscid fluids (E) ∂u ∂t = − uu − p divu = 0 Notice that the term p corresponds to the use of 0 instead of 0,V : the first system rewrites as ∂u ∂t = − 0,V u u divu = 0 Marc Arnaudon Stochastic Euler-Poincaré reduction. Deterministic framework Stochastic framework Euler-Poincaré equations Diffeomorphism group on a compact Riemannian manifold Volume preserving diffeomorphism group Lagrangian paths Characterization of the geodesics on Gs V , ·, · 0 Euler-Poincaré equation on Gs V If we take : TeGs V → R as (X) := X, X , X ∈ TeGs V , and define the action functional C : C1 e,e([0, T], Gs V ) → R by C (g(·)) := T 0 ˙g(t) · g(t)−1 dt, then a Lagrangian path g ∈ C2([0, T], Gs V ) integral path of u is a critical point of C if and only if u satisfies the Euler equation (E). [J.E. Marsden, T. Ratiu 1994] [J.E. Marsden, J. Scheurle 1993] Marc Arnaudon Stochastic Euler-Poincaré reduction. Deterministic framework Stochastic framework Euler-Poincaré equations Diffeomorphism group on a compact Riemannian manifold Volume preserving diffeomorphism group Lagrangian paths Characterization of the geodesics on Gs V , ·, · 0 Euler-Poincaré equation on Gs V [S. Shkoller 1998] If we take : TeGs V → R as the H1 metric (X) := M X, X m dµg(m) + α2 M X, X m dµg(m), X ∈ TeGs V , and define the action functional C : C1 e,e([0, T], Gs V ) → R in the same way as before, then a Lagrangian path g ∈ C2([0, T], Gs V ) integral path of u is a critical point of C if and only if u satisfies the Camassa-Holm equation    ∂ν ∂t + u · ν + α2 ( u)∗ · ∆ν = p, ν = (1 + α2∆)u, div(u) = 0. Marc Arnaudon Stochastic Euler-Poincaré reduction. Deterministic framework Stochastic framework Semi-martingales in a Lie group Stochastic Euler-Poincaré reduction Group of volume preserving diffeomorphisms Navier-Stokes and Camassa-Holm equations Aim: to establish a stochastic Euler-Poincaré reduction theorem in a general Lie group. To apply it to volume preserving diffeomorphisms of a compact symmetric space. Stochastic term will correspond for Euler equation to introducing viscosity. Marc Arnaudon Stochastic Euler-Poincaré reduction. Deterministic framework Stochastic framework Semi-martingales in a Lie group Stochastic Euler-Poincaré reduction Group of volume preserving diffeomorphisms Navier-Stokes and Camassa-Holm equations Aim: to establish a stochastic Euler-Poincaré reduction theorem in a general Lie group. To apply it to volume preserving diffeomorphisms of a compact symmetric space. Stochastic term will correspond for Euler equation to introducing viscosity. Marc Arnaudon Stochastic Euler-Poincaré reduction. Deterministic framework Stochastic framework Semi-martingales in a Lie group Stochastic Euler-Poincaré reduction Group of volume preserving diffeomorphisms Navier-Stokes and Camassa-Holm equations Aim: to establish a stochastic Euler-Poincaré reduction theorem in a general Lie group. To apply it to volume preserving diffeomorphisms of a compact symmetric space. Stochastic term will correspond for Euler equation to introducing viscosity. Marc Arnaudon Stochastic Euler-Poincaré reduction. Deterministic framework Stochastic framework Semi-martingales in a Lie group Stochastic Euler-Poincaré reduction Group of volume preserving diffeomorphisms Navier-Stokes and Camassa-Holm equations An Rn-valued semimartingale ξt has a decomposition ξt (ω) = Nt (ω) + At (ω) where (Nt ) is a local martingale and (At ) has finite variation. If (Nt ) is a martingale, then E[Nt |Fs] = Ns, t ≥ s. We are interested in semimartingales which furthermore satisfy At (ω) = t 0 as(ω) ds. Defining Dξt dt := lim ε→0 E ξt+ε − ξt ε Ft , we have Dξt dt = at Marc Arnaudon Stochastic Euler-Poincaré reduction. Deterministic framework Stochastic framework Semi-martingales in a Lie group Stochastic Euler-Poincaré reduction Group of volume preserving diffeomorphisms Navier-Stokes and Camassa-Holm equations An Rn-valued semimartingale ξt has a decomposition ξt (ω) = Nt (ω) + At (ω) where (Nt ) is a local martingale and (At ) has finite variation. If (Nt ) is a martingale, then E[Nt |Fs] = Ns, t ≥ s. We are interested in semimartingales which furthermore satisfy At (ω) = t 0 as(ω) ds. Defining Dξt dt := lim ε→0 E ξt+ε − ξt ε Ft , we have Dξt dt = at Marc Arnaudon Stochastic Euler-Poincaré reduction. Deterministic framework Stochastic framework Semi-martingales in a Lie group Stochastic Euler-Poincaré reduction Group of volume preserving diffeomorphisms Navier-Stokes and Camassa-Holm equations An Rn-valued semimartingale ξt has a decomposition ξt (ω) = Nt (ω) + At (ω) where (Nt ) is a local martingale and (At ) has finite variation. If (Nt ) is a martingale, then E[Nt |Fs] = Ns, t ≥ s. We are interested in semimartingales which furthermore satisfy At (ω) = t 0 as(ω) ds. Defining Dξt dt := lim ε→0 E ξt+ε − ξt ε Ft , we have Dξt dt = at Marc Arnaudon Stochastic Euler-Poincaré reduction. Deterministic framework Stochastic framework Semi-martingales in a Lie group Stochastic Euler-Poincaré reduction Group of volume preserving diffeomorphisms Navier-Stokes and Camassa-Holm equations Itô formula : f(ξt ) = f(ξ0) + t 0 df(ξs), dNs + t 0 df(ξs), dAs + 1 2 t 0 Hessf(dξs ⊗ dξs). From this we see that ξt is a local martingale if and only if for all f ∈ C2(Rn), f(ξt ) − f(ξ0) − 1 2 t 0 Hessf(dξs ⊗ dξs) is a real valued local martingale. This property becomes a definition for manifold-valued martingales. Definition Let at ∈ Tξt M an adapted process. If for all f ∈ C2(M) f(ξt )−f(ξ0)− t 0 df(ξs), as ds− 1 2 t 0 Hessf(dξs⊗dξs) is a real valued local martingale then Dξt dt = at . Marc Arnaudon Stochastic Euler-Poincaré reduction. Deterministic framework Stochastic framework Semi-martingales in a Lie group Stochastic Euler-Poincaré reduction Group of volume preserving diffeomorphisms Navier-Stokes and Camassa-Holm equations Itô formula : f(ξt ) = f(ξ0) + t 0 df(ξs), dNs + t 0 df(ξs), dAs + 1 2 t 0 Hessf(dξs ⊗ dξs). From this we see that ξt is a local martingale if and only if for all f ∈ C2(Rn), f(ξt ) − f(ξ0) − 1 2 t 0 Hessf(dξs ⊗ dξs) is a real valued local martingale. This property becomes a definition for manifold-valued martingales. Definition Let at ∈ Tξt M an adapted process. If for all f ∈ C2(M) f(ξt )−f(ξ0)− t 0 df(ξs), as ds− 1 2 t 0 Hessf(dξs⊗dξs) is a real valued local martingale then Dξt dt = at . Marc Arnaudon Stochastic Euler-Poincaré reduction. Deterministic framework Stochastic framework Semi-martingales in a Lie group Stochastic Euler-Poincaré reduction Group of volume preserving diffeomorphisms Navier-Stokes and Camassa-Holm equations Itô formula : f(ξt ) = f(ξ0) + t 0 df(ξs), dNs + t 0 df(ξs), dAs + 1 2 t 0 Hessf(dξs ⊗ dξs). From this we see that ξt is a local martingale if and only if for all f ∈ C2(Rn), f(ξt ) − f(ξ0) − 1 2 t 0 Hessf(dξs ⊗ dξs) is a real valued local martingale. This property becomes a definition for manifold-valued martingales. Definition Let at ∈ Tξt M an adapted process. If for all f ∈ C2(M) f(ξt )−f(ξ0)− t 0 df(ξs), as ds− 1 2 t 0 Hessf(dξs⊗dξs) is a real valued local martingale then Dξt dt = at . Marc Arnaudon Stochastic Euler-Poincaré reduction. Deterministic framework Stochastic framework Semi-martingales in a Lie group Stochastic Euler-Poincaré reduction Group of volume preserving diffeomorphisms Navier-Stokes and Camassa-Holm equations Itô formula : f(ξt ) = f(ξ0) + t 0 df(ξs), dNs + t 0 df(ξs), dAs + 1 2 t 0 Hessf(dξs ⊗ dξs). From this we see that ξt is a local martingale if and only if for all f ∈ C2(Rn), f(ξt ) − f(ξ0) − 1 2 t 0 Hessf(dξs ⊗ dξs) is a real valued local martingale. This property becomes a definition for manifold-valued martingales. Definition Let at ∈ Tξt M an adapted process. If for all f ∈ C2(M) f(ξt )−f(ξ0)− t 0 df(ξs), as ds− 1 2 t 0 Hessf(dξs⊗dξs) is a real valued local martingale then Dξt dt = at . Marc Arnaudon Stochastic Euler-Poincaré reduction. Deterministic framework Stochastic framework Semi-martingales in a Lie group Stochastic Euler-Poincaré reduction Group of volume preserving diffeomorphisms Navier-Stokes and Camassa-Holm equations Let G be a Lie group with right invariant metric ·, · and right invariant connection . Let G := TeG be the Lie algebra of G. Consider a countable family Hi , i ≥ 1, of elements of G , and u ∈ C1([0, T], G ). Consider the Stratonovich equation dgt = i≥1 Hi ◦ dWi t − 1 2 Hi Hi dt + u(t) dt · gt g0 = e where the (Wi t ) are independent real valued Brownian motions. Itô formula writes f(gt ) =f(g0) + i≥1 t 0 df(gs), Hi dWi s + t 0 df(gs), u(s)gs ds + 1 2 i≥1 t 0 Hessf(Hi (gs), Hi (gs)) ds. This implies that Dgt dt = u(t)gt . Particular case If (Hi ) is an orthonormal basis, Hi Hi = 0, is the Levi Civita connection associated to the metric and u ≡ 0, then gt is a Brownian motion in G. Marc Arnaudon Stochastic Euler-Poincaré reduction. Deterministic framework Stochastic framework Semi-martingales in a Lie group Stochastic Euler-Poincaré reduction Group of volume preserving diffeomorphisms Navier-Stokes and Camassa-Holm equations Let G be a Lie group with right invariant metric ·, · and right invariant connection . Let G := TeG be the Lie algebra of G. Consider a countable family Hi , i ≥ 1, of elements of G , and u ∈ C1([0, T], G ). Consider the Stratonovich equation dgt = i≥1 Hi ◦ dWi t − 1 2 Hi Hi dt + u(t) dt · gt g0 = e where the (Wi t ) are independent real valued Brownian motions. Itô formula writes f(gt ) =f(g0) + i≥1 t 0 df(gs), Hi dWi s + t 0 df(gs), u(s)gs ds + 1 2 i≥1 t 0 Hessf(Hi (gs), Hi (gs)) ds. This implies that Dgt dt = u(t)gt . Particular case If (Hi ) is an orthonormal basis, Hi Hi = 0, is the Levi Civita connection associated to the metric and u ≡ 0, then gt is a Brownian motion in G. Marc Arnaudon Stochastic Euler-Poincaré reduction. Deterministic framework Stochastic framework Semi-martingales in a Lie group Stochastic Euler-Poincaré reduction Group of volume preserving diffeomorphisms Navier-Stokes and Camassa-Holm equations Let G be a Lie group with right invariant metric ·, · and right invariant connection . Let G := TeG be the Lie algebra of G. Consider a countable family Hi , i ≥ 1, of elements of G , and u ∈ C1([0, T], G ). Consider the Stratonovich equation dgt = i≥1 Hi ◦ dWi t − 1 2 Hi Hi dt + u(t) dt · gt g0 = e where the (Wi t ) are independent real valued Brownian motions. Itô formula writes f(gt ) =f(g0) + i≥1 t 0 df(gs), Hi dWi s + t 0 df(gs), u(s)gs ds + 1 2 i≥1 t 0 Hessf(Hi (gs), Hi (gs)) ds. This implies that Dgt dt = u(t)gt . Particular case If (Hi ) is an orthonormal basis, Hi Hi = 0, is the Levi Civita connection associated to the metric and u ≡ 0, then gt is a Brownian motion in G. Marc Arnaudon Stochastic Euler-Poincaré reduction. Deterministic framework Stochastic framework Semi-martingales in a Lie group Stochastic Euler-Poincaré reduction Group of volume preserving diffeomorphisms Navier-Stokes and Camassa-Holm equations On the space S (G) of G-valued semimartingales define J(ξ) = 1 2 E T 0 Dξ dt 2 dt . Perturbation: for v ∈ C1([0, T], G ) satisfying v(0) = v(T) = 0 and ε > 0, let eε,v (·) ∈ C1([0, T], G) the flow generated by εv: d dt eε,v (t) = ε ˙v(t) · eε,v (t) eε,v (0) = e Definition We say that g ∈ S (G) is a critical point of J if for all v ∈ C1([0, T], G ) satisfying v(0) = v(T) = 0, dJ dε ε=0 gε,v = 0 where gε,v (t) = eε,v (t)g(t). Marc Arnaudon Stochastic Euler-Poincaré reduction. Deterministic framework Stochastic framework Semi-martingales in a Lie group Stochastic Euler-Poincaré reduction Group of volume preserving diffeomorphisms Navier-Stokes and Camassa-Holm equations On the space S (G) of G-valued semimartingales define J(ξ) = 1 2 E T 0 Dξ dt 2 dt . Perturbation: for v ∈ C1([0, T], G ) satisfying v(0) = v(T) = 0 and ε > 0, let eε,v (·) ∈ C1([0, T], G) the flow generated by εv: d dt eε,v (t) = ε ˙v(t) · eε,v (t) eε,v (0) = e Definition We say that g ∈ S (G) is a critical point of J if for all v ∈ C1([0, T], G ) satisfying v(0) = v(T) = 0, dJ dε ε=0 gε,v = 0 where gε,v (t) = eε,v (t)g(t). Marc Arnaudon Stochastic Euler-Poincaré reduction. Deterministic framework Stochastic framework Semi-martingales in a Lie group Stochastic Euler-Poincaré reduction Group of volume preserving diffeomorphisms Navier-Stokes and Camassa-Holm equations On the space S (G) of G-valued semimartingales define J(ξ) = 1 2 E T 0 Dξ dt 2 dt . Perturbation: for v ∈ C1([0, T], G ) satisfying v(0) = v(T) = 0 and ε > 0, let eε,v (·) ∈ C1([0, T], G) the flow generated by εv: d dt eε,v (t) = ε ˙v(t) · eε,v (t) eε,v (0) = e Definition We say that g ∈ S (G) is a critical point of J if for all v ∈ C1([0, T], G ) satisfying v(0) = v(T) = 0, dJ dε ε=0 gε,v = 0 where gε,v (t) = eε,v (t)g(t). Marc Arnaudon Stochastic Euler-Poincaré reduction. Deterministic framework Stochastic framework Semi-martingales in a Lie group Stochastic Euler-Poincaré reduction Group of volume preserving diffeomorphisms Navier-Stokes and Camassa-Holm equations Theorem g is a critical point of J if and only if du(t) dt = −ad∗ ˜u(t)u(t) − K(u(t)) with ˜u(t) = u(t) − 1 2 i≥1 Hi Hi , ad∗ u v, w = v, aduv and K : G → G satisfies K(u), v = − u, 1 2 i≥1 adv Hi Hi + Hi (adv (Hi )) Remark 1 If for all i ≥ 1, Hi = 0, or uv = 0 for all u, v ∈ G , then K(u) = 0 and we get the standard Euler-Poincaré equation. Marc Arnaudon Stochastic Euler-Poincaré reduction. Deterministic framework Stochastic framework Semi-martingales in a Lie group Stochastic Euler-Poincaré reduction Group of volume preserving diffeomorphisms Navier-Stokes and Camassa-Holm equations Theorem g is a critical point of J if and only if du(t) dt = −ad∗ ˜u(t)u(t) − K(u(t)) with ˜u(t) = u(t) − 1 2 i≥1 Hi Hi , ad∗ u v, w = v, aduv and K : G → G satisfies K(u), v = − u, 1 2 i≥1 adv Hi Hi + Hi (adv (Hi )) Remark 1 If for all i ≥ 1, Hi = 0, or uv = 0 for all u, v ∈ G , then K(u) = 0 and we get the standard Euler-Poincaré equation. Marc Arnaudon Stochastic Euler-Poincaré reduction. Deterministic framework Stochastic framework Semi-martingales in a Lie group Stochastic Euler-Poincaré reduction Group of volume preserving diffeomorphisms Navier-Stokes and Camassa-Holm equations Proposition If for all i ≥ 1, Hi Hi = 0 then K(u) = − 1 2 i≥1 Hi · Hi u + R(u, Hi )Hi . In particular if (Hi ) is an o.n.b. of G then K(u) = − 1 2 u = − 1 2 ∆u + 1 2 Ric u the Hodge Laplacian. Marc Arnaudon Stochastic Euler-Poincaré reduction. Deterministic framework Stochastic framework Semi-martingales in a Lie group Stochastic Euler-Poincaré reduction Group of volume preserving diffeomorphisms Navier-Stokes and Camassa-Holm equations Let Gs v = {g : M → M volume preserving bijection, such that g, g−1 ∈ Hs }. Assume s > 1 + dimM 2 . Then Gs V is a C∞ smooth manifold. Lie algebra G s V = TeGs V = {X : Hs (M, TM), π(X) = e, div(X) = 0}. Notice that π(X) = e means that X is a vector field on M: X(x) ∈ Tx M. On G s V consider the two scalar products X, Y 0 = M X(x), Y(x) dx and X, Y 1 = M X(x), Y(x) dx + M X(x), Y(x) dx. The Levi Civita connection on Gs V is given by 0V X Y = Pe( 0 X Y) with 0 the Levi Civita connection of ·, · 0 on Gs and Pe the orthogonal projection on G s V : Hs (TM) = G s V ⊕ dHs+1 (M). One can find (Hi )i≥1 such that for all i ≥ 1, Hi Hi = 0, div(Hi ) = 0, and i≥1 H2 i f = ν∆f, f ∈ C2 (M). Marc Arnaudon Stochastic Euler-Poincaré reduction. Deterministic framework Stochastic framework Semi-martingales in a Lie group Stochastic Euler-Poincaré reduction Group of volume preserving diffeomorphisms Navier-Stokes and Camassa-Holm equations Let Gs v = {g : M → M volume preserving bijection, such that g, g−1 ∈ Hs }. Assume s > 1 + dimM 2 . Then Gs V is a C∞ smooth manifold. Lie algebra G s V = TeGs V = {X : Hs (M, TM), π(X) = e, div(X) = 0}. Notice that π(X) = e means that X is a vector field on M: X(x) ∈ Tx M. On G s V consider the two scalar products X, Y 0 = M X(x), Y(x) dx and X, Y 1 = M X(x), Y(x) dx + M X(x), Y(x) dx. The Levi Civita connection on Gs V is given by 0V X Y = Pe( 0 X Y) with 0 the Levi Civita connection of ·, · 0 on Gs and Pe the orthogonal projection on G s V : Hs (TM) = G s V ⊕ dHs+1 (M). One can find (Hi )i≥1 such that for all i ≥ 1, Hi Hi = 0, div(Hi ) = 0, and i≥1 H2 i f = ν∆f, f ∈ C2 (M). Marc Arnaudon Stochastic Euler-Poincaré reduction. Deterministic framework Stochastic framework Semi-martingales in a Lie group Stochastic Euler-Poincaré reduction Group of volume preserving diffeomorphisms Navier-Stokes and Camassa-Holm equations Let Gs v = {g : M → M volume preserving bijection, such that g, g−1 ∈ Hs }. Assume s > 1 + dimM 2 . Then Gs V is a C∞ smooth manifold. Lie algebra G s V = TeGs V = {X : Hs (M, TM), π(X) = e, div(X) = 0}. Notice that π(X) = e means that X is a vector field on M: X(x) ∈ Tx M. On G s V consider the two scalar products X, Y 0 = M X(x), Y(x) dx and X, Y 1 = M X(x), Y(x) dx + M X(x), Y(x) dx. The Levi Civita connection on Gs V is given by 0V X Y = Pe( 0 X Y) with 0 the Levi Civita connection of ·, · 0 on Gs and Pe the orthogonal projection on G s V : Hs (TM) = G s V ⊕ dHs+1 (M). One can find (Hi )i≥1 such that for all i ≥ 1, Hi Hi = 0, div(Hi ) = 0, and i≥1 H2 i f = ν∆f, f ∈ C2 (M). Marc Arnaudon Stochastic Euler-Poincaré reduction. Deterministic framework Stochastic framework Semi-martingales in a Lie group Stochastic Euler-Poincaré reduction Group of volume preserving diffeomorphisms Navier-Stokes and Camassa-Holm equations Corollary (1) g is a critical point of J ·,· 0 if and only if u solves Navier-Stokes equation ∂u ∂t = − uu + ν 2 ∆u − p divu = 0 (2) Assume M = T2 the 2-dimensional torus. Then g is a critical point of J ·,· 1 if and only if u solves Camassa-Holm equation    ∂u ∂t = − uv − 2 j=1 vj uj + ν 2 ∆v − p v = u − ∆u divu = 0 For the proof, use Itô formula and compute in different situations ad∗ v (u) and K(u). Marc Arnaudon Stochastic Euler-Poincaré reduction. Deterministic framework Stochastic framework Semi-martingales in a Lie group Stochastic Euler-Poincaré reduction Group of volume preserving diffeomorphisms Navier-Stokes and Camassa-Holm equations Corollary (1) g is a critical point of J ·,· 0 if and only if u solves Navier-Stokes equation ∂u ∂t = − uu + ν 2 ∆u − p divu = 0 (2) Assume M = T2 the 2-dimensional torus. Then g is a critical point of J ·,· 1 if and only if u solves Camassa-Holm equation    ∂u ∂t = − uv − 2 j=1 vj uj + ν 2 ∆v − p v = u − ∆u divu = 0 For the proof, use Itô formula and compute in different situations ad∗ v (u) and K(u). Marc Arnaudon Stochastic Euler-Poincaré reduction. Deterministic framework Stochastic framework Semi-martingales in a Lie group Stochastic Euler-Poincaré reduction Group of volume preserving diffeomorphisms Navier-Stokes and Camassa-Holm equations Corollary (1) g is a critical point of J ·,· 0 if and only if u solves Navier-Stokes equation ∂u ∂t = − uu + ν 2 ∆u − p divu = 0 (2) Assume M = T2 the 2-dimensional torus. Then g is a critical point of J ·,· 1 if and only if u solves Camassa-Holm equation    ∂u ∂t = − uv − 2 j=1 vj uj + ν 2 ∆v − p v = u − ∆u divu = 0 For the proof, use Itô formula and compute in different situations ad∗ v (u) and K(u). Marc Arnaudon Stochastic Euler-Poincaré reduction.