Stochastically perturbed geodesic flows onLie groups.

Wenqing Hu. 1

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

The classical Langevin equation.

I Random movement of a particle in a fluid due to collisionswith the molecules of the fluid. 1

Fig. 1: P. Langevin (1872–1946).

1Langevin, P. (1908). “Sur la theorie du mouvement brownien”. C. R. Acad.Sci. (Paris) 146 : pp. 530–533.

The classical Langevin equation.

I Motion of a free particle :

qt = 0 .

I Hamiltonian equation :

{qt = pt ,pt = 0 .

I q0 = q0 ∈ Rn, q0 = p0 ∈ Rn.

Fig. 2: Motion of a free particle in Rn.

The classical Langevin equation.

I Motion of a damped free particle :

qt = −λqt .

I Hamiltonian equation :

{qt = pt ,pt = −λpt .

I λ > 0, q0 = q0 ∈ Rn, q0 = p0 ∈ Rn.

Fig. 3: Motion of a damped free particle in Rn.

The classical Langevin equation.

I Motion of a damped free particle subject to randomfluctuation :

qt = −λqt+εWt .

I Hamiltonian equation :

{qt = pt ,

pt = −λpt+εWt .

I λ > 0, ε > 0, q0 = q0 ∈ Rn, q0 = p0 ∈ Rn.

I Wt is a standard Brownian motion in Rn.

Fig. 4: Motion of a damped free particle subject to random fluctuationsin Rn.

The classical Langevin equation.

I Random fluctuation : pt = −λpt+εWt .

I The process pt is an Ornstein–Uhlenbeck process.

I What is Wt ?

I Stochastic differential equation

dpt = −λptdt+εdWt

can be understood as an integral equation

pt2 − pt1 = −λ

∫ t2

t1

psds+ε(Wt2 −Wt1)

for 0 ≤ t1 < t2 ≤ t.

The classical Langevin equation.

I Wt is a standard Brownian motion (Wiener process).

Fig. 5: A.Einstein (1879–1955).

Fig. 6: N.Wiener (1894–1964).

The classical Langevin equation.

I Einstein–Wiener definition :(a) For any t ≥ 0 and s > 0, Wt+s −Wt ∼ N (0, s) ;(b) For any 0 < t1 < ... < tn, Wt1 , Wt2 −Wt1 ,...,Wtn −Wtn−1 are independent ;(c) With probability 1 the process Wt has continuoustrajectories.

Fig. 7: A sample path of Wt in dimension 2.

The classical Langevin equation.

I The Brownian component is only in the momentum variable :

{qt = pt ,

pt = −λpt+εWt .

I “physical Brownian motion” 2.

2Einstein, A., Uber die von der molekularkinetischen Theorie der Warmegeforderte Bewegung von in ruhenden Flussigkeiten suspendierten Teilchen,Annalen der Physik, 322(8), pp.549–560.

The classical Langevin equation.

I

Fig. 8: Motion of a damped free particle subject to randomfluctuations in Rn.

Equation of a “free” rigid body with fixed overhangingpoint.

I Motion of a free particle :

qt = 0 .

I Configuration space G = Rn.

I Hamiltonian equation :

{qt = pt ,pt = 0 .

I Phase space (qt , pt) ∈ Rn × Rn.

Fig. 9: Motion of a free particle in Rn.

Equation of a “free” rigid body with fixed overhangingpoint.

I Motion of a rigid body with a fixed overhanging point.

I Configuration space G = SO(3).

I “kinematic equation” : a(t) ∈ SO(3) such that

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 z ∈ TeG = g.

I Euler’s equation for a rigid body.

Fig. 10: Motion of a rigid body.

Equation of a “free” rigid body with fixed overhangingpoint.

I Configuration space G = SO(3).

I Phase space (a, z) ∈ G × TeG = G × g. 3

3Remark : In fact the phase space is the (co)tangent bundle T ∗G ∼= TG ,but since G = SO(3) is a Lie group, the bundle T ∗G ∼= TG is parallizable, andcan be viewed as G × g.

Fig. 11: Phase space : G × g.

Langevin equation of a rigid body.

I Configuration space G = SO(3).

I Phase space (a, z) ∈ G × TeG = G × g.

I “kinematic equation” :

a−1a = z .

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

z1 =I2 − I3

I1z2z3−λz1+εW 1

t ,

z2 =I3 − I1

I2z3z1−λz2+εW 2

t ,

z3 =I1 − I2

I3z1z2−λz3+εW 3

t .

Langevin equation of a rigid body.

I In short form{

a−1a = z ,

z = q(z , z)− λz + εW .

I q(z1, z2) is a quadratic, bilinear form.

I Process (at , zt) lives in G × g, G = SO(3).

Fig. 12: Phase space : G × g.

Motion of ideal and viscous incompressible fluid.

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 4.

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 .

4Arnold, 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.

Classical mechanics from group theoretic point of view.

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

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

5

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.

5Remark : 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. 13: Configuration space : Left–invariant frame on G .

Classical mechanics from group theoretic point of view.

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

I Hamiltonian equation

{a−1a = zz = q(z , z) .

where (a, z) ∈ G × g.

I The solution a(t) gives a geodesic flow on the group G withrespect to its left–invariant metric.

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. 6

6Tao, T., The Euler–Arnold equation. available online athttps ://terrytao.wordpress.com/2010/06/07/the-euler-arnold-equation/

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 .

Stochastic mechanical models : ergodic theory.

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 7, theyestablished the ergodicity for a stochastically forced 2–dNavier–Stokes equation. The stochastic forcing is degenerate.

I Another example : problems about turbulent mixing 8.

7Hairer, M., Mattingly, J.C., Ergodicity of the 2D Navier–Stokes equationswith degenerate stochastic forcing, Annals of Mathematics, (2) 164 (2006), no.3, 993–1032.

8Komorowski, T., Papanicolaou, G., Motion in a Gaussian incompressibleflow, Annals of Applied Probability, 7, 1, 1997, pp. 229–264.

Stochastic mechanical models : ergodic theory.

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 = −λpt + εWt

has an invariant measure with density ∝ exp(− λε2 |p|2Rn), that

is the Boltzmann distribution.

Ergodic theory : deterministic vs. stochastic.

I Ergodic theory for the classical Hamiltonian equation

{a−1a = zz = q(z , z) .

is usually very hard.

I It is much easier for the stochastic equation

{a−1a = z ,

z = q(z , z)− λz + εσW .

Fig. 14: Ergodic theory.

Markov process associated with Langevin equation onfinite–dimensional group G .

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.

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 spreads stochasticity to alldirections via the interaction of the injection of noise in a fewdirections and the deterministic drift term, so that the processlocally will reach an open set around the solution of thedeterministic system 9.

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

Fig. 15: Ergodic theory for Markov processes.

Hypoellipticity for the classical Langevin equation.

I Let us look at the system of stochastic differential equationssatisfied by the process (qt , pt) starting from (q0, p0) ∈ R2n :

{qt = pt ,

pt = −λpt+εWt .

Fig. 16: A.N.Kolmogorov (1903–1987).

Hypoellipticity for the classical Langevin equation.

I Suppose (q0, p0) ∈ R2n and

P((qt , pt) ∈ dqdp) = g(t; q, p)dqdp .

I Basic principle proposed by Kolmogorov : The probabilitydensity g(t; q, p) is a fundamental solution to theFokker–Planck equation (forward Kolmogorov equation)10 :

∂g

∂t= −

n∑i=1

pi∂g

∂qi+ λ

n∑i=1

∂

∂pi(pig) +

ε2

2

n∑i=1

∂2g

∂p2i

;

g(0; q, p) = δ(q0, p0) .

10Kolmogorov, A.N., Uber die analytischen Methoden in derWahrscheinlichkeitsrechnung, Math. Ann., 104, 1931, pp. 415–458.

Hypoellipticity for the classical Langevin equation.

I The closed–form solution when n = 1, λ = 0 is also given byKolmogorov in a small paper on Brownian motion 11 :

g(t; q, p)

=8√

3

πε4t2exp

{− 1

ε2

[(p − p0)

2

2t+

6(q − q0 − p+p02 t)2

t3

]}.

I In general when λ > 0 the solution g(t; q, p) will again besmooth.

I “Smoothing”= full regularity !

I Why degenerate PDE yields smooth solution ?

11Kolmogorov, A.N., Zur Theorie Brownschen Bewegung, Annals ofMathematics, 35, 1934, pp. 116–117.

Hypoellipticity for the classical Langevin equation.

I Think of the classical Langevin equation as a stochasticdynamical system on R2n :

(qp

)=

(p−λp

)+

(0

εW

).

I Noise W is only injected in the p–direction.

I It is carried to the q–direction via interaction of the noise W

and the drift term p∂

∂q.

I Local smoothing.

Fig. 17: Hypoellipticity in Kolmogorov’s example.

Fig. 18: L.Hormander (1931–2012).

Hypoellipticity for general diffusion process.

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.

Ergodic theory for Langevin equation on finite–dimensionalgroup G .

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. 19: Lie bracket of

(AU

)and

(BV

).

Ergodic theory for Langevin equation on finite–dimensionalgroup G .

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. (Hu–Sverak, 2015) 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. 20: Phase space : G × Z .

Constrained Brownian motion compatible with theconservation law.

I Phase 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 12.

12Freidlin, 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. 21: 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.

Ergodic theory for the conservative stochastic equation.

I Theorem 2. (Hu–Sverak, 2015) The conservative stochasticsystem (

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.

Ergodic theory for the conservative stochastic equation.

I The Lie algebra hull containing Z − Z and invariant under themapping z → [z0, z ] for all z ∈ Z coincides with g is a purelyalgebraic property in terms of finite dimensional objectsrelated to the group’s Lie algebra.

I The original Hormander’s condition is related to iterated Liebrackets of vector fields on G × g. It is usually hard to check.

I We made use of algebraic as well as geometric observations tofind these optimal conditions.

I The conditions we found also shed some light in infinitedimensions. It can be thought of as a characterization of howthe non–linearity amplifies the effect of noise, which can onlyact in a few directions. This is exactly what we see in fluidflows.

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 .

Non–compact case.

I Imagine the motion of a satellite in space.

I We interpret Z as a “control curve”.

I The motion of the satellite is only random along directionspointed out by Z .

Fig. 22: Controlled random motion of a satellite.

Non–compact case.

I Conclusion : The position a(t) of the satellite, after a longtime, satisfies a central limit theorem.

I It is non trivial because random forcing is “controlled”.

I A classical method of auxiliary functions using Ito’s formulawas used to obtain a central limit theorem for the a–process.

Thank you for your attention !