Dynamics of geodesic °ows with random forcing on Lie ...web.mst.edu/~huwen/slides_Dynamics_of_geodesic... · Classical mechanics for general group G of the symmetry of motion. I

Dynamics of geodesic flows with randomforcing on Lie groups with left–invariant

metrics.

Wenqing Hu. 1

(Joint work with Vladimir Sverak. 2)

1School of Mathematics, University of Minnesota, Twin Cities.2School of Mathematics, University of Minnesota, Twin Cities.

The classical Langevin equation.

I Langevin, P. (1908). “Sur la theorie du mouvementbrownien”. C. R. Acad. Sci. (Paris) 146 : pp. 530–533.

I Random movement of a particle in a fluid due to collisionswith the molecules of the fluid.


I In modern terminology :

µqµt = b(qµ

t )− λqµt + σ(qµ

t )Wt ,

where qµ0 = q0 ∈ Rn , qµ

0 = p0 ∈ Rn .

I qµt is the position of the particle suspended in a fluid ; µ > 0 is

the mass of the particle ; b(•) ∈ C 1 is an external forcing ;λ > 0 is the friction coefficient ; Wt is a standard Brownianmotion in Rn ; σ(•) ∈ C 1 is the diffusion matrix, so thata(q) = σ(q)σT (q) is non–degenerate.


I Second order stochastic differential equation can be turnedinto a first order one : Let the momentum pµ

t = µqµt , and we

have {µqµ

t = pµt ,

pµt = b(qµ

t )− λqµt + σ(qµ

t )Wt .

I (qµt , pµ

t ) ∈ Rn × Rn = R2n, (qµ0 , pµ

0 ) = (q0, µp0).

I Phase space = Rn × Rn = R2n.

The classical Langevin equation : Newtonian mechanicsfrom Lagrangian/Hamiltonian point of view.

I Suppose the external forcing b(•) = 0, mass µ = 1.

I We think of the noise as a perturbation σ(•) = ε · id , ε > 0.

I Langevin equation looks like

qt = −λqt + εWt .

I Without dissipation term −λqt and fluctuation term εWt theequation is just Newtonian equation of a free particle :

qt = 0 .

The classical Langevin equation : Newtonian mechanicsfrom Lagrangian/Hamiltonian point of view.

I Classical mechanics :

Newtonian ⇐⇒ Lagrangian ⇐⇒ Hamiltonian

I From the Lagrangian mechanics point of view the equation

qt = 0

is describing a geodesic flow on the group of symmetry ofmotion Rn, the curvature = 0.

I From the Hamiltonian mechanics point of view the equation

qt = 0

is the equation {qt = pt

pt = 0

where (qt , pt) ∈ Rn × Rn on the phase space R2n.

Classical mechanics for general group G of the symmetryof motion.

I Other than the trivial group Rn, we can consider much moregeneral group G describing the symmetry of motion. Thegroup G is a Lie group with a left–invariant metrics.

I Lagrangian mechanics is the geodesic flow on the group G .

I Let a(t) ∈ G be the trajectory of the geodesic flow.

I Schematically we can write

a = 0

for the free motion on G and

a = −λa + εW

for the Langevin equation on G .

I But we have to interpret the above two equations in a rightway, which we will do it later.

Classical mechanics for general group G of the symmetryof motion : Physical Relevancy.

I The more general group G describing the symmetry of motionprovides us with a tool to understand more complicatedmechanical structure.

I If G = Rn as in the classical Langevin equation, then themechanics is the motion of a free particle. The Langevinequation then describes the dynamics of this particle subjectto dissipation and random fluctuation.

I If G = SO(3), then the mechanics is the motion of a rigidbody with a fixed overhanging point. The Langevin equationthen describes the dynamics of the rigid body subject todissipation and random fluctuation.


I We can pick an even more sophisticated group.

I Let the group G be the volume–preserving diffeomorphismgroup of a certain domain M.

I This group G models the motion of ideal incompressible fluidwithin that domain M. The original picture is from Arnold’sclassical work in 1966 3.

I The equation of the “free particle” is now an Euler’s equation

ut + (u · ∇)u +∇p = 0 , divu = 0 ,

and the “Langevin equation” is now a stochasticNavier–Stokes equation

ut + (u · ∇)u +∇p = ∆u + εW , divu = 0 .

3Arnold, V.I., Sur la geometrie differentielle des groupes de Lie dedimension infinie et ses applications a l’hydrodynamique des fluides parfaits,Ann. Inst. Fourier (Grenoble), 16, 1966 fasc. 1, pp. 319–361.


I What type of problems are people usually interested in thesestochastic mechanical models ?

I Long time evolution, ergodicity, invariant measure ...

I For example, in the classical work of Hairer–Mattingly 4, theyestablished the ergodicity for a stochastically forced 2–dNavier–Stokes equation. The stochastic forcing is degenerate.

I Another example : problems about turbulent mixing 5.

4Hairer, M., Mattingly, J.C., Ergodicity of the 2D Navier–Stokes equationswith degenerate stochastic forcing, Annals of Mathematics, (2) 164 (2006), no.3, 993–1032.

5Komorowski, T., Papanicolaou, G., Motion in a Gaussian incompressibleflow, Annals of Applied Probability, 7, 1, 1997, pp. 229–264.

Dynamics of geodesic flow on finite–dimensional group G .

I Recall our equation of a “free particle” on the group G :

a(t) = 0 .

I We have to interpret the derivative a as a covariant derivativeof a along the curve a(t).

I Need a metric on G .

Dynamics of geodesic flow on finite–dimensional group G .

I G : n–dimensional Lie group ; g = TeG : Lie algebra.

I On g we introduce an inner product 〈•, •〉, and a basise1, ..., en.

6

I Left–invariant frame : carry the basis e1, ..., en to any TaG byleft–translation b → ab : ek(a) = aek .

I Left–invariant metrics on G is given by 〈•, •〉 :〈ξ, η〉a = 〈ξkek , ηkek〉 for ξ = ξkek(a), η = ηkek(a) ∈ TaG .

I Kinetic energy : T (a, a) =1

2〈a, a〉a.

6Remark : Actually the dual space g∗ and dual basis e1, ..., en are involvedin the Hamiltonian formalism, but for simplicity and easy understanding we willidentify g∗ with g via 〈•, •〉.

Fig. 1: Configuration space : Left–invariant frame on G .

Dynamics of geodesic flow on finite–dimensional group G :Lagrangian formalism.

I Let a(t) be a geodesic flow on G with respect to the metric〈•, •〉•. That is to say, The first variation of the action withrespect to the kinetic energy

S0,t(a) =1

2

∫ t

0〈a(s), a(s)〉a(s)ds

vanishes at a(t) (”principle of least action”).

I a(t) is the trajectory in classical mechanics.

I In the sense of covariant derivative we have a(t) = 0.

Dynamics of geodesic flow on finite–dimensional group G :Hamiltonian formalism.

I Recall that the equation

qt = 0

is the equation {qt = pt

pt = 0

where (qt , pt) ∈ Rn × Rn on the phase space R2n.

I In the case of a symmetry group G the Hamiltonian formalismis a first order differential equation on TG ∼= G × g. 7

I Coordinate on TG : (a, z) ∈ G × g → zkek(a) ∈ TaG .

7Remark : It is actually the space T ∗G = G × g∗, but we identified g withg∗.

Fig. 2: Configuration space : G × g.

Dynamics of geodesic flow on finite–dimensional group G :Hamiltonian formalism.

I Hamiltonian H(a, a) = T (a, a).

I Hamiltonian equation

{a−1a = zz = q(z , z) .

where (a, z) ∈ G × g.

I q(z1, z2) is a quadratic, bilinear form that can be calculatedfrom the structure constants of the Lie algebra g.

I z = q(z , z) is the so called Euler–Arnold equation. 8

8Tao, T., The Euler–Arnold equation. available online athttps ://terrytao.wordpress.com/2010/06/07/the-euler-arnold-equation/

Example : Rigid body.

I G = SO(3).

I “kinematic equation” : a−1a = z .

I “dynamic equation” : for z = z1e1 + z2e2 + z3e3 we have

z1 =I2 − I3

I1z2z3 ,

z2 =I3 − I1

I2z3z1 ,

z3 =I1 − I2

I3z1z2 .

I Euler’s equation for a rigid body.

Langevin equation on finite–dimensional group G .

I Langevin equation for the dynamics of geodesic flow onfinite–dimensional group G can be written as

{a−1a = z ,

z = q(z , z)− λz + εσW .

I (a, z) ∈ G × g, ε > 0 and σ = (σ1, ..., σr ) is an n × r matrix,σk ∈ Rn, k = 1, ..., r , Wt = (W 1

t , ...,W rt )T is the standard

Brownian motion in Rr .

I We want degenerate noise so we usually pick r < n :stochastic forcing through a few degrees of freedom given bythe vectors σ1,...,σr .

Ergodic theory for Langevin equation on finite–dimensionalgroup G .

I Our major interest is the ergodic theory of the equation

{a−1a = z ,

z = q(z , z)− λz + εσW .

I Existence and uniqueness of invariant measure ? Long–timeconvergence to the invariant measure ? Structure of invariantmeasure ?

I The classical Langevin equation

{qt = pt

pt = −pt + εWt

has an invariant measure with density ∝ exp(− 1ε2 |p|2Rn), that

is the Boltzmann distribution.


I Think of the Langevin equation as a stochastic dynamicalsystem on G × g :

(az

)=

(az

q(z , z)− λz

)+ ε

(0

σW

). (∗)

I Even if σ is non–degenerate, the noise is degenerate for theLangevin dynamics on G × g.

I (∗) gives a Markov process (at , zt) on G × g.

I PDE perspective : the Fokker–Planck equation is for theevolution of the density f = f (a, z ; t) of the process (at , zt) :

ft + zkek f + qk(z , z)∂f

∂zk+

∂

∂zk

(−λzk f − ε2

2hkl ∂f

∂z l

)= 0 .

Here h = σσT .

Ergodic theory for Markov processes.

I Ergodic theory for Markov processes ≈ irreducibility +smoothing .

I Smoothing = “loss of memory” = “essentially stochastic” = “locally spread stochasticity to all directions” = “ aperiodicity” (for Markov chains) = “Hypoellipticity” (for diffusions).

I Roughly speaking, the process injects stochasticity to alldirections via the interaction of the noise and thedeterministic drift term, so that the process locally will reachan open set around the solution of the deterministic system 9.

9Hairer, M., On Malliavin’s proof of Hormander’s theorem. Bulletin dessciences mathematiques, 135(6), August 2011.

Ergodic theory for Markov processes.

I Consider the SDE

dx = X0(x)dt +r∑

k=1

Xk(x) ◦ dW k

on a manifold M.

I Let X0 = {Xk , k ≥ 1} and recursively defineXk+1 = Xk ∪ {[X ,Xj ],X ∈ Xk and j ≥ 0}.

I Lie bracket :[X1,X2] = ∇X1∇X2 −∇X2∇X1 = DX2X1 − DX1X2.

I Hormander’s parabolic hypoellipticity condition : ∪k≥1Xk

spans the whole tangent space TxM.

I This condition ensures the existence of a smooth density forthe corresponding Markov process xt , and for the solution tothe Fokker–Planck equation.


I Back to the Langevin equation :

(az

)=

(az

q(z , z)− λz

)+ ε

(0

σW

).

I X0(a, z) =

(z

qk(z , z)∂

∂zk− λz

), Xk(a, z) =

(0σk

),

k = 1, 2, ..., r , σk ∈ g as constant vector fields.

I What is the bracket

[(AU

),

(BV

)]?

Fig. 3: Lie bracket of

(AU

)and

(BV

).


I Let the vector field Q(z , z) = qk(z , z)∂

∂zk.

I Let Σ0 = {σj , j = 1, 2, ..., r}, and for k = 0, 1, 2, ... werecursively defineΣk+1 = Σk ∪ {Q(σj , σ), σ ∈ Σk , j = 1, 2, ..., r}. LetΣ = ∪∞k=1Σk .

I Theorem 1. The Langevin system

(az

)=

(az

q(z , z)− λz

)+ ε

(0

σW

)

satisfies the Hormander’s parabolic hypo–elliptic condition ifand only if Σ spans g. In this case if G is compact, theinvariant density for the a–process on G is constant withrespect to the Haar measure on G.

More conservation laws.

I The Langevin equation

{a−1a = z ,

z = q(z , z)−λz+εσW

is incorporated with a dissipative structure : the friction term−λz dissipates the energy.

I This energy dissipation is compensated by the noise termεσW , and when a balance is reached we approach aninvariant measure.

I We cannot remove −λz unless we make use of a moreconservative noise.

More conservation laws.

I What are the conservation laws of the “free” equation ? Recallthat when we remove the friction and the noise in theLangevin equation we come back to the Hamiltonian equation

{a−1a = z ,z = q(z , z) .

I Conservation of energy (Hamiltonian) H(z) =1

2〈z , z〉.

I Conservation of angular momentum : the equation z = q(z , z)moves the variable z only on a submanifold O(η) ⊂ g.

I O∗(η) = {aηa−1, a ∈ G , η ∈ g∗} : co–adjoint orbit.

I Z = {H = const} ∩ O(η) is the manifold on which thez–variable moves.

Fig. 4: Configuration space : G × Z .

Constrained Brownian motion compatible with theconservation law.

I Configuration space is now G × Z .

I The Langevin equation

{a−1a = z ,

z = q(z , z)−λz+εσW .

I When we remove the dissipation −λz , we have to replace thenoise +εσW by a conservative noise εξ restricted to Z .

I {a−1a = z ,z = q(z , z)+εξ .

I Consideration from canonical ensemble : the new noise ξ hasto be adapted to the invariant measure of the Hamiltoniandynamics z = q(z , z).

Constrained Brownian motion compatible with theconservation law.

I Work with constrained Brownian motion 10.

10Freidlin, M., Wentzell, A., On the Neumann problem for PDE’s with asmall parameter and the corresponding diffusion processes, Probability Theoryand Related Fields, 152(1), pp. 101–140, January 2012.

Fig. 5: Constrained Brownian motion compatible with the canonicalensemble.

Ergodic theory for the conservative stochastic equation.

I Conservative stochastic perturbations of geodesic flow onfinite–dimensional group G :

{a−1a = z ,z = q(z , z) + εξ .

I Markov process on G × Z defined by the equation

(az

)=

(az

q(z , z)

)+ ε

(0ξ

).

I Apply Hormander’s parabolic hypoellipticity condition again.


I Recall that the phase space is now G × Z .

Fig. 6: Configuration space : G × Z .


I Our stochastic dynamical system is

(az

)=

(az

q(z , z)

)+ ε

(0ξ

).

I We can think of X0 =

(az

q(z , z)

)and Xk =

(0Vk

),

k = 1, ..., dimZ , where {Vk}dimZk=1 spans TZ .

I These vectors are on the tangent space T (G × Z ) = g× TZ .

I We need the brackets to generate the whole g = TeG .

Fig. 7: T (G × Z ) = g× TZ .


I What type of Lie brackets we will have ?

Fig. 8: Type 1 Lie bracket.


I Type 1 Lie bracket parallelly moves the tangent vectors on TZto tangent vectors on g = TeG .

Fig. 9: From the tangent space TZ to the whole manifold Z − Z .


I By iterating type 1 Lie brackets we finally get all vectors ofthe form ∇mφ(s).

I Taylor expansion :

φ(s) = φ(s0) +∇φ(s0)(s − s0) +1

2∇2φ(s0)(s − s0)

2 + ...

+1

m!∇mφ(s0)(s − s0, ..., s − s0) + ... .

I The full parabolic Lie algebra hull contains Z − Z .

Fig. 10: Type 2 Lie bracket.


I Fix a vector z0 ∈ Z , type 2 Lie brackets generate all vectors ofthe form [z0, z ] for all z ∈ Z .

I The full parabolic Lie algebra hull is invariant under themapping : z → [z0, z ] for all z ∈ Z .


I Theorem 2. The conservative stochastic system

(az

)=

(az

q(z , z)

)+ ε

(0ξ

)

satisfies the Hormander’s parabolic hypo–elliptic condition ifand only if the Lie algebra hull containing Z − Z and invariantunder the mapping z → [z0, z ] for all z ∈ Z coincides with g.In this case if G is compact, then the long–term dynamics forthe a–process will approach an invariant measure withconstant density with respect to the Haar measure on G.

Non–compact case.

I We also consider an example of a non–compact groupG = Rn and a one–dimensional submanifold γ : R→ Z ⊂ Rn.

I {a = γ(s) ;s = εw .

I Variable s is arc–length parameter on Z ; γ(s) is aparametrization of Z ; w(t) is Brownian motion on Z .

I Center of mass ∫ `

0γ(s)ds = 0 .

Non–compact case.

I Take a bounded C(2) function ϕ : R→ Rn, so thatϕ′′(s) = γ(s).

I

a(t)− a(0) =

∫ t

0γ(εw(t ′))dt ′ =

∫ t

0ϕ′′(εw(t ′))dt ′ .

I Apply Ito formula

ϕ(εw(t))−ϕ(εw(0)) =

∫ t

0εϕ′(εw(t ′))dw(t ′)+

∫ t

0

ε2

2ϕ′′(εw(t ′))dt ′ .

Non–compact case.

I Therefore

1√t(a(t)− a(0))

=

∫ t

0

2

ε√

tϕ′(w(t ′))(−dw(t ′))

− 2

ε2√

t(ϕ(a(t))− ϕ(a(0))) .

Ia(t)− a(0)√

t

d .⇀ N

(0,

4

ε2Σk`

).

I

Σk` =1

`

∫ `

0ϕ′k(s)ϕ′`(s)ds .

Thank you for your attention !

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Dynamics of geodesic °ows with random forcing on Lie ...web.mst.edu/~huwen/slides_Dynamics_of_geodesic... · Classical mechanics for general group G of the symmetry of motion. I