Momentum Maps and Stochastic Clebsch Action Principles

Momentum Maps and Stochastic Clebsch Action Principles Ana Bela Cruzeiro1 , Darryl D. Holm2 , Tudor S. Ratiu3 (1) GFMUL and Mathematics Department, Instituto Superior T´ ecnico, Lisboa, Portugal

arXiv:1604.04554v1 [math-ph] 15 Apr 2016

(2) Mathematics Department, Imperial College, London, United Kingdom (3) Department of Mathematics, Jiao Tong University, Shanghai, China and Section de Math´ ematiques, Ecole Polytechnique F´ ed´ erale de Lausanne, Switzerland

Draft 14 April 2016 Keywords: Geometric mechanics; stochastic processes; Clebsch variational principles

Abstract We develop the Clebsch variational method of deriving stochastic differential equations whose solutions follow the flow of a stochastic nonlinear Lie algebra operation on a configuration manifold. In the stochastic Clebsch action principle presented here, the noise couples to the phase space variables through a momentum map. This special coupling simplifies the structure of the resulting stochastic Hamilton equations. In particular, these stochastic Hamilton equations collectivize for Hamiltonians which depend only on the momentum map variable. The Stratonovich equations are derived from the Clebsch variational principle and then converted into Itˆ o form. We compare the Stratonovich and Itˆ o forms of the equations governing the components of the momentum map. The Itˆ o contraction term turns out to be a double Poisson bracket. Finally, we present the stochastic Hamiltonian formulation of the collectivized momentum map dynamics and derive the corresponding Kolmogorov forward and backward equations.

Contents 1 Background and motivation

2

2 Variational principle for Stratonovich stochastic coadjoint motion

3

3 Itˆ o formulation of stochastic coadjoint motion

10

4 Stochastic Hamiltonian formulation

14

4.1

The deterministic Hamilton equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

4.2

The stochastic Hamilton equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

4.3

The Kolmogorov equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

1

AB Cruzeiro, DD Holm, TS Ratiu

1

Stochastic Clebsch Action Principles – Draft

13 April 2016

2

Background and motivation

In seeking to develop a stochastic version of geometric mechanics, one may do well to build on the framework laid out in Poincaré’s famous short paper [22]. This fundamental source of geometric mechanics was carefully reviewed recently from a modern perspective in [18]. Likewise, a primary source of stochastic Hamilton equations is [3], which was recently reviewed and developed further from the geometric mechanics viewpoint in [17]. The purpose of the present work is to continue the theme of these earlier developments by revisiting Poincaré’s starting point [22] and continuing along that line by augmenting Hamilton’s principle to introduce stochasticity into Poincaré’s original framework. Although Poincaré [22] used a version of what we would now call reduction by symmetry, here we use an earlier approach due to Clebsch [7], which introduces constrained variations into Hamilton’s principle by imposing velocity maps corresponding in the deterministic case to the infinitesimal transformations of a Lie group. (For up to date applications to mechanics of the Clebsch method, see [8].) In a certain sense, Clebsch [7] presages the Pontryagin maximum principle in optimal control theory. In the present paper, however, the velocity maps are made stochastic. Thus, we consider stochastic Clebsch action principles whose variables are stochastic. The equations of motion derived are stochastic ordinary (or partial) differential equations (SDEs, or SPDEs) for motion on coadjoint orbits of (finite or infinite dimensional) Lie algebras. We comment now on the distinction between the stochastic Clebsch and reduced Lagrangian approaches. A stochastic Lagrangian symmetry reduction process was developed in [2, 6]. In that case, the Lagrangian curves in the configuration space are stochastic diffusion processes, which are critical states of the action functional. In these works, the drift of the stochastic processes is regarded as its (mean, generalized) time derivative and the action functional is defined in terms of this velocity. The corresponding Euler-Poincaré equations of motion, satisfied by the velocity, are deterministic (ordinary differential equations when the configuration space is finite-dimensional, or partial differential equations in the infinite-dimensional case). In the present paper, as in [13], the stochastic Clebsch approach is not equivalent to the reduced stochastic Lagrangian processes approach employed in [2, 6]. In particular, the action functionals considered in [2, 6] are not random. Instead, they are defined as expectations of the classical Lagrangians computed on stochastic processes, whereas in the present work, and in [13], the action functionals are taken to be random. In addition, the velocities in the reduction approach of [2, 6] are identified with the drift of the underlying diffusion processes, which, as is well known, requires the computation of a conditional expectation. Finally, in the reduced stochastic Lagrangian approach of [2, 6], it is not possible to take arbitrary variations; instead, a particular form for the variations is required and the final resulting equations of motion depend on this choice. However, in the present work and in [13], the variations are arbitrary. Therefore, the present stochastic Clebsch action principle cannot be regarded as a formulation of the EulerPoincaré variational principle obtained in [6, Theorems 3.2 and 3.4]. In order to consider the present variational principle approach from the viewpoint of reduction by symmetry, one would need to interpret the velocity as an Itˆ o derivative of the underlying stochastic curves, in which case the resulting stochastic action functional would be divergent. This divergence was avoided in [2, 6] via the “renormalization” achieved by taking conditional expectations. Outline of the paper. Following the Clebsch approach to the Euler-Poincaré equations, in Section 2 we introduce a stochastic velocity map in the Stratonovich sense as a constraint in Hamilton’s principle for motion on a manifold acted upon by infinitesimal transformations of a Lie algebra. With hindsight, we see that the stochasticity in the velocity map is coupled to the motion by the momentum map which arises from the variation of the Lagrangian function and the deterministic part of the velocity map. The resulting stationarity conditions generalize the classical deterministic formulations of motion on coadjoint orbits of Lie algebras in Poincaré [22] and Hamel [11], by making them stochastic. In Section 3, we present the Itˆ o formulations of the stationary variational conditions. Three alternative routes are taken in calculating the Itˆ o double-bracket forms of the variational equations for stochastic coadjoint motion. In Section 4, we discuss the Poisson structure of the Stratonovich-Hamiltonian formulation of the



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stochastic motion equations. We also give the Itˆ o interpretation of the Casimir functions for the Lie-Poisson part of the bracket in this formulation, and derive the associated Lie-Poisson Fokker-Planck equation for the motion of the probability density function on the level sets of Casimir functions. Finally, we close by pointing out some avenues for further research in stochastic geometric mechanics. For the sake of simplicity and practicality, and to make this paper easily accessible to physicists and engineers, we will work exclusively in local coordinates. An intrinsic formulation of the results in this paper will also be sketched; although we will defer its complete formulation to a future paper.

2

Variational principle for Stratonovich stochastic coadjoint motion

In [22], Poincaré begins by considering the transitive action of a Lie group of smooth transformations of a manifold, whose points in local coordinates are written as q = (q 1 , . . . , q n ) and whose infinitesimal transformations are represented by the vector field obtained at linear order in the Taylor series. Let α = 1, . . . , n, be the indices of the local coordinates of the Lie algebra of this group. Denote by Aα [f ] any infinitesimal transformation of this group, and express its action on a smooth function f as Aα [f ] :=

n X

Aiα

i=1

∂f ∂f ∂f ∂f = A1α 1 + A2α 2 + · · · + Anα n , ∂q i ∂q ∂q ∂q

(2.1)

where Aiα are functions of (q 1 , . . . , q n ). Throughout this paper, Greek indices enumerate Lie algebra basis elements, Latin indices denote coordinates on the manifold, and the standard Einstein summation convention is assumed. Since these transformations form a Lie algebra, Poincaré remarks that Aα [Aβ ] − Aβ [Aα ] =

r X

γ=1

cαβ γ Aγ

⇐⇒

Asα

∂Akβ ∂Ak − Asβ sα = cαβ γ Akγ , ∀k = 1, . . . , n, α, β = 1, . . . , r, s ∂q ∂q

(2.2)

where cαβ γ ∈ R are the structure constants of the Lie algebra in the basis attached to the Greek indices. Geometric setup. We give now a glimpse of the global formulation. Poincaré [22] does not really use a transformation group, only its associated Lie algebra action, i.e., he takes a configuration n-manifold Q of a mechanical system and a Lie algebra morphism g ∋ u 7→ uQ ∈ X(Q) of a given Lie algebra g, dim g =: r < ∞, to the Lie algebra X(Q) of vector fields on Q, endowed with the usual Lie bracket [X, Y ][f ] := X[Y [f ]] − Y [X[f ]], where X, Y ∈ X(Q), f ∈ C ∞ (Q), and X[f ] is the differential of f in the direction X, given in coordinates by (2.1). The coordinate expression ∂ ∂ uQ (q) =: uiQ (q) i =: Aiα (q)uα i (2.3) ∂q ∂q of uQ ∈ X(Q), relative to a coordinate system (q 1 , . . . , q n ) on the chart domain U ⊂ Q and a basis {e1 , . . . , er } of g, is thus determined by the functions Aiα ∈ C ∞ (U ) and the basis expansion u =: uα eα of u ∈ g. Since [uQ , vQ ] = [u, v]Q for any u, v ∈ g, the functions Aα := Aiα ∈ C ∞ (U, Rn ), defined by (eα )Q =: Aiα ∂q∂ i , satisfy (2.2)1 , which is equivalent to saying that the local vector fields Aα , Aβ ∈ X(U ) satisfy (2.2)

[Aα , Aβ ] = cαβ γ Aγ

(2.4)

The action is assumed to be transitive in [22], which means that any tangent vector vq ∈ Tq Q can be written locally as vq = aα (eα )Q (q) = aα Aiα (q) ∂q∂ i , for some aα ∈ R. If (q 1 , . . . , q n ) are local coordinates on Q, the corresponding standard coordinates on the tangent bundle T Q and the cotangent bundle T ∗ Q are (q 1 , . . . , q n , q˙1 , . . . , q˙n ) and (q 1 , . . . , q n , p1 , . . . , pn ), respectively, where vq = q˙i ∂q∂ i 1 Thus, Poincar´ e works with a right action of the underlying Lie group on the manifold; we adopt his index conventions in [22], also used in[5]. For left actions, g → X(Q) is a Lie algebra anti-homomorphism, i.e., [uQ , vQ ] = −[u, v]Q .



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and pq = pi dq i for any vq ∈ Tq Q and pq ∈ Tq∗ Q (the cotangent space at q ∈ Q, the dual of Tq Q). Throughout the paper, we use these naturally induced coordinates. The sign convention for the canonical Poisson bracket on T ∗ Q adopted in this paper is, in standard coordinates, {f, g} =

∂g ∂f ∂f ∂g − k , ∂q k ∂pk ∂q ∂pk

for any

f, g ∈ C ∞ (T ∗ Q).

(2.5)

∗

If h ∈ C ∞ (T ∗ Q), its Hamiltonian vector field is denoted by XhT Q ∈ X(T ∗ Q). Hamilton’s equations for a curve d f (c(t)) = {f, h}(c(t)) for any f ∈ C ∞ (T ∗ Q). c(t) ∈ T ∗ Q in Poisson bracket form are dt When working with a general Poisson manifold (P, {·, ·}), the Hamiltonian vector field XhP ∈ X(P ) of h ∈ C ∞ (P ) is defined by df XhP := {f, h}. Pairing notation. For any manifold Q, finite or infinite dimensional, we denote by h·, ·i : T ∗ Q × T Q → R the natural (weakly, in the infinite-dimensional case) non-degenerate fiberwise duality pairing. Given a Lie algebra g, which is always finite dimensional in this paper, the non-degenerate duality pairing between its dual g∗ and g is denoted by h·, ·ig : g∗ × g → R. Given f ∈ C ∞ (g∗ ), the functional derivative

The momentum map.

δf δµ

∈ g of f evaluated at µ ∈ g∗ is defined by

δf d f (µ + ǫδµ) = δµ, , dǫ ǫ=0 δµ g

for all δµ ∈ g.

(2.6)

The momentum map JT ∗ Q : T ∗ Q → g∗ of the lifted g-action to T ∗ Q is defined by ∗

uT ∗ Q = XJTu Q ,

for any u ∈ g,

(2.7)

where JuT ∗ Q (pq ) := hJT ∗ Q (pq ), uig . Its expression is (see, e.g., [19, §12.1, formula (12.1.15)]) JuT ∗ Q (pq ) = hpq , uQ (q)i ,

pq ∈ T ∗ Q, u ∈ g,

or, in coordinates JT ∗ Q (q i , pi ) = pj Ajα (q i )eα ,

(2.8)

where {e1 , . . . , er } is the basis of g∗ dual to the basis {e1 , . . . , er } of g. This momentum map is infinitesimally equiv[u,v] ariant, i.e.,2 JT ∗ Q = −{JuT ∗ Q , JvT ∗ Q }, for all u, v ∈ g. A useful equivalent statement of infinitesimal equivariance is (see, e.g., [19, §11.5, formula (11.5.6)] with a sign change because we work with right actions) Tpq JT ∗ Q (uT ∗ Q (pq )) = ad∗u JT ∗ Q (pq )

(2.9)

[e ,e ]

If we denote m := JT ∗ Q ∈ C ∞ (T ∗ Q, g∗ ), mα := JeTα∗ Q , m[α,β] := JT ∗αQ β ∈ C ∞ (T ∗ Q), we have m = mα eα , mα (pq ) = pi Aiα (q)

(2.10)

and the infinitesimal equivariance is expressed in coordinates as (2.5)

(2.4)

− m[α,β] = {mα , mβ } = −pj [Aα , Aβ ]j = −pj cαβ γ Ajγ

(2.10)

= −cαβ γ mγ .

(2.11)

If G is a Lie group with Lie algebra g acting on the right on Q, then the g-action on Q is given by the infinitesimal d q · exp(tu) ∈ Tq Q, where q · g denotes generator vector field uQ ∈ X(Q) defined at every q ∈ Q by uQ (q) := dǫ ǫ=0 ∗ the action of g ∈ G on the point q ∈ Q. The momentum map J : T Q → g∗ given in (2.8) is equivariant relative to the given right G-action on P and the right coadjoint G-action on g∗ , i.e., JT ∗ Q (pq · g) = Ad∗g JT ∗ Q (pq ) for all g ∈ G. 2 This

is the infinitesimal equivariance relation for right actions. For left actions, the sign in the right hand side changes.



13 April 2016

5

The Lie-Poisson bracket. The dual g∗ of any finite dimensional Lie algebra g is endowed with the the LiePoisson bracket (see, e.g., [19, §13.1, p.416]) δf δh {f, h}± (µ) = ± µ, , f, h ∈ C ∞ (g∗ ), µ ∈ g∗ . (2.12) , δµ δµ We denote by g∗± the vector space g∗ endowed with the Poisson bracket (2.12). The Hamiltonian vector field of h ∈ C ∞ (g∗ ) defined by the equation f˙ = {f, h} for any f ∈ C ∞ (g∗ ) has the expression Xh± (µ) = ∓ ad∗δh µ (the signs correspond). Given ξ ∈ g, ad∗ξ : g∗ → g∗ is the dual of the linear map g ∋ η 7→ adξ η := [ξ, η] ∈ g.

δµ

If G is a Lie group with Lie algebra g, denote by G ∋ h 7→ Lg (h) := gh ∈ G left and G ∋ h 7→ Rg (h) := hg ∈ G right translation by g ∈ G. Then the momentum map JR : T ∗ G → g∗− , JR (αg ) = Te∗ Lg (αg ), αg ∈ Tg∗ G, of the lifted right translation on G (i.e., JR equals JT ∗ G given in (2.8) for the action of G on itself given by right translations) is a Poisson map, i.e., {f, h}− ◦ JR = {f ◦ JR , h ◦ JR }. Similarly, the momentum map JL : T ∗ G → g∗+ , JL (αg ) = Te∗ Rg (αg ), αg ∈ Tg∗ G, of left translation is a Poisson map, i.e., {f, h}+ ◦ JL = {f ◦ JL , h ◦ JL }. (For the proof see, e.g., [19, §13.3].) More generally, the momentum map J : T ∗ Q → g∗− of the lifted right g-action to T ∗ Q is a Poisson map; the coordinate expression of this statement is (2.11). A function k ∈ C ∞ (g∗ ) such that {k, f }± = 0, for all f ∈ C ∞ (g∗ ), or, equivalently, Xk± = 0, is called a Casimir function. This definition is valid for any Poisson manifold, not just g∗ . Introducing stochasticity into the Clebsch methodology. We assume that all stochastic processes are defined in the same filtered probability space (Ω, P, Pt ). Let t 7→ Wtk (ω), k = 1, . . . N , ω ∈ Ω, be N independent real-valued Brownian motions, ξ1 , . . . , ξN ∈ g, and Ω ∋ ω 7→ (pq )ω (t) ∈ T ∗ Q be random variables for every t. The induced random variable on Q, the foot point of (pq )ω (t), is denoted by Ω ∋ ω 7→ qω (t) ∈ Q. For simplicity we will remove the probability variable ω in the notation. Stratonovich differentiation is denoted by X ◦ dY and Itˆ o differentiation simply by XdY . Then, given ξ1 , . . . , ξN ∈ g and a g-valued random curve

t 7−→ (pq )ω (t), ◦dqω (t) − uω (t)Q (qω (t))dt − (ξk )Q (qω (t)) ◦ dWtk (ω)

is a process whose coordinate expression is

i α k (pi )ω (t) ◦dqωi (t) − Aiα (qω (t))uα ω (t)dt − Aα (qω (t))ξk ◦ dWt . Remark on notation. For simplicity, we no longer write the probability variable ω in the notation and, instead, we use symbols u, p, q, etc., to denote semimartingales. Given the Lagrangian ℓ ∈ C ∞ (g × Q), introduce the stochastic action, defined for semimartingales u ∈ C([0, T ], g), q ∈ C([0, T ], Q), (pq ) ∈ C([0, T ], T ∗Q), and define the constrained stochastic action integral S( u, pq) by Z T ℓ( u(t), q(t))dt + hpq (t) , ◦d q(t) − u(t)Q ( q(t))dt − (ξk )Q ( q(t)) ◦ dWtk i , (2.13) S( u, pq) = 0

where the semimartingales ( u, pq) are assumed to be regular enough for the above integrals to be finite. Indeed, all stochastic processes considered in this paper will be continuous semimartingales with regular coefficients. In local coordinates, the stochastic action integral (2.13) may be recognized as the sum of a Lebesgue integral and a Stratonovich integral Z T Z T i α S( u, pq) = ℓ( u(t), q(t))dt − pi (t)Aα ( q(t)) u (t) dt + pi ◦d qi (t) − Aiα ( q(t))ξkα ◦ dWtk . (2.14) {z } |0 {z } |0 Lebesque integral Stratonovich integral For notational convenience, we introduce for every t ∈ [0, T ] the stochastic Lie algebra element dxtα := uα (t)dt + ξkα ◦ dWtk ,

(2.15)



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as a convenient way to say that, when we integrate some stochastic process Xt with respect to dxtα , we mean Z

0

Z

T

Xt dxtα :=

T

Xt uα (t)dt +

0

Z

0

T

Xt ξkα ◦ dWtk ,

α = 1, . . . , dim g.

Thus, we rewrite the action integral in (2.14) in the abbreviated form S( u, pq) =

Z

0

T

ℓ( u(t), q(t))dt + pi (t) ◦d qi (t) − Aiα ( q(t))dxtα .

(2.16)

We assume that the Lagrangian ℓ(u, q), viewed as a function ℓ : g × Q → R, is hyperregular, i.e., for every q ∈ Q, δℓ ∈ g∗ × {q} is a diffeomorphism. In particular, n is a function of (u, q) and, the map g × {q} ∋ (u, q) 7→ n := δu conversely, u is a function of (n, q). Thus, replacing the variables u ∈ g and q ∈ Q by the semimartingales u(t), q(t), we get the semimartingale n( u(t), q(t)). d We consider variations of the semimartingales ( u(t), pq(t)), i.e., dǫ ( u(t) + ǫδ u(t), pi (t) + ǫδpi (t), qj (t) + ǫ=0 j ǫδ q (t)) where δ u(t), δpi (t), δ q(t) are arbitrary local random curves of bounded variation with bounded smooth coefficients and δ q(t) vanishes at the endpoints in time: δ q(0) = 0 = δ q(T ). More generally (and according to [9, page 55]), we can define, for a semimartingale Xt with regular coefficients, dXt = adWt + bt dt, the mean square derivative 1 ǫ Xt − Xt , δ Xt := lim (2.17) ǫ→0 ǫ where Xtǫ is a family of regular semimartingales of the form dXtǫ = atǫ ◦ dWt + btǫ dt, such that Xt0 = Xt and where the limit is taken in the L2 (Ω) sense.

Consider a random point (qω , pω ) in the manifold T ∗ Q and f ∈ C ∞ (T ∗ Q). The differential of f in the direction of the (deterministic) vector field Z ∈ X(T ∗ Q) is given by d f (γω (ǫ)), hdf, Zi (qω , pω ) = dǫ ǫ=0

where γω is a curve starting from (qω , pω ) with initial velocity Z(qω , pω ) and the limit is taken in L2 (Ω). Therefore hdf, Zi (qω , pω ) consists in evaluating hdf, Zi in the random point (qω , pω ). Consider now a semimartingale of the form Z t Z t α k Yt ( q, p) = Y0 + ψ( q(s), p(s))ds, φα ( q(s), p(s))ξk ◦ dWs +

(2.18)

0

0

where q(t), p(t) are Q-, respectively, T ∗ Q-valued semimartingales, with q(t) the footpoint of p(t), φα , ψ ∈ C ∞ (T ∗ Q) are deterministic smooth functions, and ξk = ξkα eα ∈ g are given (constant) elements. The (Stratonovich) stochastic Poisson bracket of f ( q(t), p(t)) is defined by {f ( q(s), p(s)), ◦ds Y} := hdf, Xφα i ( q(s), p(s))ξkα ◦ dWsk + hdf, Xψ i ( q(s), p(s))ds

(2.19)

where Xφ , Xψ denote the Hamiltonian vector fields of φ and ψ. In standard cotangent bundle local coordinates, it reads, (2.20) {f ( q(t), p(t)), ◦dt Y} := δ qk (f ( q(t), p(t)) ◦ dt δpk Y − δpk (f ( q(t), p(t)) ◦ dt δ qk Y, where the notation δ qk Y (resp. δpk Y) means that we are simply varying qk (t) 7→ qk (t)+ǫ (resp. p( t) 7→ pk (t)+ǫ). These can be regarded as derivatives with respect to the initial conditions of the semimartingales q(t) and p(t) and, therefore, δ qk (f ( q(t), p(t))) =

∂f ( q(t), p(t)), ∂q k

δpk (f ( q(t), p(t))) =

∂f ( q(t), p(t)). ∂pk

(2.21)

Given the form (2.18) of the semimartingale Yt , we have {f ( q(t), p(t)), ◦dt Y} = {f, φα }( q(t), p(t))ξkα ◦ dWtk + {f, ψ}( q(t), p(t))dt.

(2.22)



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7

The constraint imposed by the pairing with the Lagrange multipliers pi (t) defines the ith component of the stochastic velocity map, d qi (t) = Aiα ( q(t))dxtα .

(2.23)

With these definitions and preparatory formulas, the following theorem holds. Theorem 1. [Stratonovich stochastic variational principle] The stochastic variational principle δS = 0 with action integral S given in equation (2.23), with dxtα defined in (2.15), yields the following stochastic dynamical equations ∂ℓ ( u(t), q(t))Aiα ( q(t))dt, ∂q i ∂ℓ dt pi (t) = {pi (t), nβ ( q(t), p(t))}dxtβ + i ( u(t), q(t))dt , ∂q

dt nα ( q(t), p(t)) = {nα ( q(t), p(t)) , nβ ( q(t), p(t))}dxtβ + dt qi (t) = { qi (t), nβ ( q(t), p(t))}dxtβ ,

(2.24)

δℓ ∂ℓ for all α = 1, . . . , dim g, i = 1, . . . , dim Q, where n := δu ∈ g∗ , i.e., nα = ∂u Moreover, n( q(t), p(t)) = α. α ∗ i ∗ m( q(t), p(t)) a.s., where m(pq ) = JT Q (pq ) = mα (pq )e ∈ g , mα = pi Aα (q) and hence mα ( q, p) = pi Aiα ( q).

The Poisson brackets in this formula need interpretation, first, because they are not of the form (2.18), and, second, since nα depends only on the variables u and q i . First, the Poisson brackets on the right hand side of (2.24) are taken in the sense of (2.20) or its global version. Second, as will be shown below, the stationarity δℓ i condition δS = 0 yields the relation δu α ( q(t), p(t)) = pi (t)Aα ( q(t)) almost surely, which says that n( q(t), p(t)) = m( q(t), p(t)) almost surely and that the Lagrange multipliers pi (t) also depend on the semimartingales u(t) and qi (t), as expected. Pretending now that mα = pi Aiα (q) depends on the variables q i and pi , as if they were independent T ∗ Q-chart variables, one computes the Poisson brackets in (2.24) using the derivative of the semimartingale {f, g}( q(t), p( t)) as defined in (2.17) with variations as in (2.21). Remark 2. Looking back at the Stratonovich integral in the stochastic action functional (2.14), we see that the ∂ℓ stochasticity couples to the phase space variables through the momentum map via the relations ∂u α ( u(t), q(t)) = i pi (t)Aα ( q(t)). ♦ Proof. The first step in the proof of Theorem 1 is to take the variations of the action integral (2.14), thereby finding the following equations, which hold almost surely, δℓ ( u(t), q(t)) − pi (t)Aiα ( q(t)) = 0 , δuα δp(t) : dt qi (t) − Aiα ( q(t))dxtα = 0 ,

δ u(t) :

(2.25)

∂ℓ ∂Aj δ q(t) : dt pi (t) + pj (t) αi ( q(t))dxtα − ( u(t), q(t))dt = 0 , ∂q ∂q i after integrations by parts using the vanishing of the term (pi (t) δ qi (t))|T0 at the endpoints in time, which folδℓ lows from the assumption δ qi (0) = 0 = δ qi (T ). In particular, we have nα ( q(t), p(t)) = δu α ( u(t), q(t)) = mα ( q(t), p(t)). Therefore, taking the stochastic differential of the first equation, then using the second and third



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equations in (2.25), we get, dropping the t-dependence notation in the semimartingales, ∂Ai δℓ = d pi Aiα ( q) = (◦dpi )Aiα ( q) + pi jα ( q) ◦ d qj dt mα ( q, p) = d α δu ∂q ! j ∂Aβ ∂Ajα ∂ℓ (2.15) i β ( ( ( q)Akβ ( q)dxtβ = Aα ( q) −pj q )d x u , q )dt + p + j t ∂q i ∂q i ∂q k ! j j ∂A ∂A ∂ℓ β = pj Akβ ( q) kα ( q) − Akα ( q) k ( q) dxtβ + i ( u, q)Aiα ( q)dt ∂q ∂q ∂q ∂ℓ i A ( q)dt ∂q i α ∂ℓ = −cαβ γ mγ ( q, p) dxtβ + i ( u, q)Aiα ( q)dt ∂q ∂ℓ (2.11) = {mα ( q, p), mβ ( q, p)} dxtβ + i ( u, q)Aiα ( q)dt ∂q (2.2)

= pj cβα γ pi Ajγ ( q) dxtβ +

(2.26)

which is the first equation in (2.24). The second equation dt qi = Aiα ( q)dxtα in (2.25) and the identity { qi , mβ ( q, p)} = Aiβ ( q) yield the second equation in (2.24). Finally, the third equation dt pi = −pj {pi , mβ ( q, p)} = −pj

∂Ajβ ∂qi

∂Ajα α ∂qi ( q)dxt

+

∂ℓ ∂qi ( u,

q)dt in (2.25) and the identity

( q) yield the third equation in (2.24).

The first variational equation in (2.25) captures the momentum map relation (2.10), and the latter two equations in (2.25) produce the corresponding equations in (2.24), when expressed in terms of the canonical Poisson bracket { · , · } on T ∗ Q. The second equation in (2.25) recovers the velocity map in (2.23), and the third equation determines the evolution of the dual canonical momentum variable, the Lagrange multiplier pi . The penultimate equality in (2.26) yields the following result. Corollary 3. Hamilton’s principle δS = 0 for the constrained action integral in (2.23) recovers stochastic coadjoint motion equation in the following form,

dt

∂ℓ ∂ℓ ∗ ∂ℓ + i ( u(t), q(t))Aiα ( q(t))dt, ( u(t), q(t)) = addxt ( u(t), q(t)) ∂uα ∂u ∂q α

(2.27)

β ∂ℓ ∂ℓ where ad∗dxt ∂u ( u(t), q(t)) α := − cαβ γ ∂u γ ( u(t), q(t))dxt .

The stochastic equations of motion on g∗ × Q. The presence of the Poisson brackets in (2.24) suggests the existence of a Hamiltonian version of these equations. This will be explored in detail in Section 4. Here we just introduce a stochastic version of the Legendre transform and derive certain equations on g∗ × Q whose geometric structure will be investigated in Section 4. In the classical deterministic case, recall that the Legendre transform of a Lagrangian L : g → R to a Hamiltonian ∗ δL ∗ d δL d H : g∗ → R, mapping the Euler-Poincaré equations dt δξ = − adξ δξ to the Lie-Poisson equations dt µ = − ad δH µ δµ

(and, conversely, if the map is a diffeomorphism), is given by (see, e.g., [19, §13.5, p. 437]) µ :=

δL , δξ

H(µ) := hµ, ξig − L(ξ),

ξ ∈ g,

µ ∈ g∗ .

∗ If the map g ∋ ξ 7→ µ = δL δξ ∈ g is a diffeomorphism, the Lagrangian L and Hamiltonian H given above, are called hyperregular. We define below a stochastic version of this Legendre transform, depending on a parameter, replacing the Lie algebra element by the stochastic vector field (2.15) and the element in the dual of the Lie algebra by a semimartingale.



13 April 2016

9

We proceed in the following way. We say that the Lagrangian ℓ : g × Q → R is hyperregular if the function ℓ(·, q) : g → R is hyperregular for every q ∈ Q. We work with hyperregular Lagrangians from now on. Define, as δℓ ∈ g∗ , invert this relation for every q ∈ Q to get u = u(n, q), and introduce the in Theorem 1, n := n(u, q) := δu Hamiltonian function h(n, q) := hn, u(n, q)ig − ℓ(u(n, q), q). Next, recalling that m = JT ∗ Q (pq ) = pi Aiα (q)eα ∈ g∗ , consider the function h(m, q), i.e., we replace the first variable n of h by the expression m. Now, replace the variables (u, q) ∈ g×Q by semimartingales ( u(t), q(t)) and form a semimartingale h(m( q(t), p(t)), q(t)) (which corresponds to a stochastic Hamiltonian, as explained in subsection 4.2), by imposing, in analogy with the deterministic case, the stochastic derivative of the semimartingale h(m( q(t), p(t)), q(t), q) to equal (2.10)

dt h(m( q(t), p(t)), q(t)) = pi (t)Aiα ( q(t))dxtα − ℓ( u(t), q(t)) dt = mα ( q(t), p(t)) dxtα − ℓ( u(t), q(t))dt, (2.28) where dxtα = uα dt + ξkα ◦ dWtk . This semimartingale is of the form (2.18), namely dh(m( q(t), p(t)), q(t)) = (h1 )α (m( q(t), p(t)), q(t))ξkα ◦ dWtk + h2 (m( q(t), p(t)), q(t))dt. In agreement with our previous definitions, we shall use the notation

∂h ∂(h1 )α ∂h2 dt (m( q(t), p(t)), q(t)) = (m( q(t), p(t)), q(t))ξkα ◦ dWtk + ((m( q(t), p(t)), q(t))dt, (2.29) ∂mβ ∂mβ ∂mα ∂(h1 )α ∂h2 ∂h α k q (t), p (t)), q (t)) = q (t), p (t)), q (t))ξ ◦ dW + (2.30) (m( (m( (m( q(t), p(t)), q(t))dt. dt k t ∂q j ∂q j ∂q j Theorem 4. The stochastic variational principle δS = 0, with action integral defined in (2.23) and semimartingale h(m( q(t), p(t)), q(t)) introduced above, implies the equations ∂h dt mα ( q(t), p(t)) = {mα ( q(t), p(t)) , mβ ( q(t), p(t))}dxtβ − Ajα ( q(t)) j (m( q(t), p(t)), q(t)) dt ∂q ∂h (m( q(t), p(t)), q(t)) = {mα ( q(t), p(t)), mβ ( q(t), p(t))} ◦ dt ∂mβ (2.31) ∂h + {mα ( q(t), p(t)), qj } j (m( q(t), p(t)), q(t))dt , ∂q ∂h dt qi (t) = { qi , mβ ( q(t), p(t))}dxtβ = Aiβ ( q(t))dxtβ = Aiβ ( q(t)) ◦ dt (m( q(t), p(t)), q(t)). ∂mβ with the convention that the Poisson brackets are computed as in Theorem 1. Proof. We drop in the computations below the notational t-dependence of the semimartingales. By Theorem 1, we know that n( q, p) = m( q, p) a.s. and that (2.24) hold. Next, we take the differential of condition (2.28). Thus, if δmα ( q, p) and δ qi are arbitrary variations (namely random curves of bounded variation in t) of the semimartingales mα ( q, p) and qi , respectively, we get ∂h ∂h (m( q, p), q)δ qi (m( q, p), q)δmα ( q, p) + d d ∂mα ∂q i ∂ℓ ∂ℓ i u , q ) dt − δ q ( u, q)dt, ( = δmα (m( q, p))dxtα + δ uα mα (m( q, p)) − ∂uα ∂q i which is equivalent a.s. to ∂ℓ ∂h (m( q, p), q) = dxtα , ( mα ( q, p) − u , q ) = 0 , d ∂uα ∂mα ∂h ∂ℓ (m( ( u, q) dt . d q , p ), q ) = − ∂q i ∂q i

and (2.32)



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10

Note that the first equation implies, as expected from Theorem 1, the a.s. equality of the semimartingales n( q, p) = m( q, p). Using the identities (2.32) and the equations (2.24), we compute dmα ( q, p) and d qi and get ∂h dmα ( q, p) = {mα ( q, p), mβ ( q, p)}dxtβ − Ajα ( q) j (( q, p), q)dt ∂q ∂h ∂h (2.32) (( q, p), q) + {mα ( q, p), qj } j (( q, p), q)dt , = {mα ( q, p), mβ ( q, p)} ◦ d ∂mβ ∂q ∂h β β (2.32) i i i i (( q, p), q) , d q = { q , mβ ( q, p)}dxt = Aβ ( q)dxt = Aβ ( q) ◦ d ∂mβ which are the equations (2.31). Remark 5. The defining relation mα ( q, p) = pi Aiα ( q) and the second equation in (2.24) imply ∂Ai dmα ( q, p) = d pi Aiα ( q) = (◦dpi )Aiα ( q) + pi jα ( q) ◦ d qj ∂q i ∂A (2.24) = (◦dpi )Aiα ( q) + pi jα ( q)Ajβ ( q)dxtβ . ∂q By Theorem 1, we know that n( q, p) = m( q, p) a.s. and hence the first equation in (2.24) yields ∂ℓ ( u, q)Aiα ( q)dt ∂q i ! ∂Ajβ ∂Ajα ∂ℓ k k Aβ ( q) k ( q) − Aα ( q) k ( q) dxtβ + i ( u, q)Aiα ( q)dt. ∂q ∂q ∂q

dmα ( q, p) = {mα ( q, p), mβ ( q, p)} dxtβ + (2.26)

= pj

Comparing these two expressions, we conclude the a.s. equality (◦dpi )Aiα ( q)

=

−pj Akα ( q)

∂Ajβ

( q)dxtβ + ∂q k

∂ℓ ( u, q)Aiα ( q)dt = ∂q i

∂Ajβ

−pj ( q)dxtβ ∂q i

! ∂ℓ + i ( u, q)dt Aiα ( q). (2.33) ∂q

Note that this identity is clearly implied by the third equation in (2.24).

3

♦

Itˆ o formulation of stochastic coadjoint motion

As before, t 7→ Wtk (ω), k = 1, . . . N , ω ∈ Ω, are N independent real-valued Brownian motions and ξ1 , . . . , ξN ∈ g. For each ξk , k = 1, . . . , N , define the Hamiltonian vector field Xξk ∈ X(T ∗ Q) by (2.25)

Xξk := { · , mα (q, p)ξkα } = { · , pi Aiα (q)ξkα } ,

(3.1)

i.e., Xξk is the Hamiltonian vector field on T ∗ Q with Hamiltonian function T ∗ Q ∋ pq 7→ hJT ∗ Q (pq ), ξk i ∈ R, k = 1, . . . , N . As in the previous section, we denote interchangeably points in T ∗ Q by pq or (q, p). Define the operator on semimartingales of the form f ( q(t), p(t)), where f ∈ C ∞ (T ∗ Q) by (Xξk f )( q(t), p(t)) := {f ( q(t), p(t)), mα ( q(t), p(t))ξkα } = f ( q(t), p(t)), pi (t)Aiα ( q(t))ξki ,

(3.2)

where the brackets in the right hand side are those of semimartingales, as defined in (2.17), with variations as in (2.20). Note that the result of the operation (Xξk f )( q(t), p(t)), defined in (3.2), is again a semimartingale.

xtβ ∈ g by In analogy with (2.15), define the Itˆ o stochastic element db db xtα = uα (t)dt + ξkα dWtk .

(3.3)



11

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The Itˆ o stochastic Hamiltonian vector field Xdbxt is also defined by the Poisson bracket operation

xtβ := {f ( q(t), p(t)), pi (t)Aiβ ( q(t))}db xtβ , (Xdbxt f )( q(t), p(t)) := {f ( q(t), p(t)), mβ ( q(t), p(t))}db

(3.4)

for any f ∈ C ∞ (T ∗ Q), where, again, the brackets in the right hand side are those of semimartingales, as defined in (2.17), with variations as in (2.20). The result of the operation (Xdbxt f )( q(t), p(t)), defined in (3.4), is again a semimartingale. In both (3.2) and (3.4), in agreement with the conventions in Section 2, we define the semimartingales qi (t) := q i ( q(t), p(t)) and pi (t) := pi ( q(t), p(t)). With these notations, we have the following result. Corollary 6. [Itˆ o stochastic variational conditions] The corresponding Itˆ o forms of the Stratonovich stochastic variational equations (2.24) are given by N

1X ∂ℓ dt mα ( q(t), p(t)) = (Xdbxt mα ) ( q(t), p(t)) + ( Xξk (Xξk mα )) ( q(t), p(t))dt + i ( u(t), q(t))Aiα ( q(t))dt , (3.5) 2 ∂q k=1

dt qi (t) = Xdbxt qi (t) +

N 1X

2

Xξk (Xξk qi )(t) dt ,

(3.6)

k=1 N

dtpi (t) = Xdbxt pi (t) +

1X ∂ℓ Xξk (Xξk pi )(t) dt + i ( u(t), q(t))dt . 2 ∂q

(3.7)

k=1

Remark 7. Remarkably, the Itˆ o interpretation for the coadjoint dynamics of the momentum map defined by ∂ℓ i mα ( q(t), p(t)) := ∂u α ( u(t), q(t)) = pi (t)Aα ( q(t)) has the same double bracket structure as the individual equations for the phase space variable (q, p). Several perspectives of how this preservation of structure in Corollary 6 occurs, can be seen by considering three different direct proofs of it. ♦ First proof. In all the proofs below, we ignore the t-dependence notation on the semimartingales. The first proof of Corollary 6 begins by streamlining the notation in the Stratonovich stochastic equations (2.24) of Theorem 1, to write them simply as dmα ( q, p) = (Xdxt mα )( q, p) +

∂ℓ ( u, q)Aiα ( q) dt , ∂q i

d qi = Xdxt qi ,

dpi = Xdxt pi +

∂ℓ ( u, q)dt , ∂q i

(3.8)

in terms of the following Poisson bracket operator (analogous to (3.4))

Xdxt := { · , mβ ( q, p)}dxtβ := { · , pi Aiβ ( q)}dxtβ .

(3.9)

We want to write these expressions in Itˆ o form. For this, we recall Itˆ o’s formula: if Xt is a semimartingale with regular coefficients and f a smooth function (c.f., for example, [15]), then 1 2 dt f (Xt ) = ∂i f (Xt ) ◦ dt Xti = ∂i f (Xt )dt Xti + ∂i,j f (Xt )dt Xti .dt Xtj . 2

(3.10)

The corresponding Itˆ o forms of the latter Stratonovich expressions in (2.24) are then written equivalently as N

d qi = Xdbxt qi +

1X Xξk (Xξk qi ) dt , 2 k=1 N

1X ∂ℓ dpi = Xdbxt pi + Xξk (Xξk pi ) dt + i ( u, q)dt . 2 ∂q

(3.11)

k=1

We prove the first one, as the other is similarly derived. To simplify notation, we write simply Aiα instead of Aiα ( q). Recall that d qi = Aiα uα dt + Aiα ξkα ◦ dWtk



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12

By Itˆ o’s formula (3.10), the only term which is not of bounded variation in the expression for dAiα is equal to j β ∂ i k ∂qj (Aα )Aβ ξk dWt and we conclude that d(Aiα ξkα ξkα ).dWtk

=

N X ∂Ai

α

k=1

∂q j

Ajβ ξkβ ξkα dt.

Thus, (3.10) yields N

d qi = Aiα uα dt + (Aiα ξkα )dWtk +

1 X ∂Aiα j β α A ξ ξ dt, 2 ∂q j β k k k=1

which is the expanded version of the first equation in (3.11).

Having introduced this streamlined notation for d qi and dpi in the Itˆ o equations (3.11), we calculate the Itˆ o o rule for the derivative equation for the components of the momentum map mα ( q, p) := pi Aiα ( q), by using the Itˆ of a product of a pair of Itˆ o semimartingales, X and Y, given by d(XY) = Xd Y + YdX + dX.d Y ,

and for

dX = σdW ,

dY = σ ˜ dW

we have dX.d Y = σ˜ σ dt ,

(3.12)

where dX.d Y is the co-variation, or Itˆ o contraction. According to the Itˆ o product rule, the Itˆ o contraction in computing dmα ( q, p) = d(pi Aiα ( q)) from equation (3.11) is dpi . dAiα =

N X

(Xξk pi ) . (Xξk Aiα ( q)) .

(3.13)

k=1

Indeed, this Itˆ o contraction expression comes from the fact that the martingale parts of the processes pi and Aiα ( q) are given, respectively, by dpi ≃ −pj

∂Ajα ( q)ξkα dWtk ∂q i

and

dAiα ( q) ≃

∂Aiα ( q)Ajβ ( q)ξkβ dWtk ∂q j

where Φ ≃ Ψ means that Φ − Ψ is a process of bounded variation. Remarkably, this Itˆ o contraction (3.13) turns out to be exactly what we need to show by direct calculation from (3.11) that N

dmα ( q, p) = ( Xdbxt mα ) ( q, p) +

1X ∂ℓ ( Xξk (Xξk mα )) ( q, p)dt + i ( u, q)Aiα ( q)dt 2 ∂q k=1

N

= {mα ( q, p),

mβ ( q, p)}db xtβ

X 1 + {{mα ( q, p), mβ ( q, p)} , mγ ( q, p)} ξkβ ξkγ dt 2

(3.14)

k=1

∂ℓ + i ( u, q)Aiα ( q) dt . ∂q

In the direct calculation, the Itˆ o contraction is cancelled by a cross term arising from applying the second-order 1 derivative operator 2 Xξk (Xξk · ) from equation (3.11) to the quadratic product mα ( q, p) = pi Aiα ( q). This completes the first proof of Corollary 6. Second proof of the first equation in (3.7). In the statement of the Corollary 6, the first Itˆ o equation in (3.7) may also be verified by an even more direct calculation than in (3.14), as follows. To simplify notations in the computations ∂ℓ below, we temporarily suppress the dependence of Aiα on the semimartingale q, of ∂q i on the semimartingales ( u, q), and of mα ( q, p) on the semimartingales ( q, p). We also suppress the k-index. By equations (2.11), (2.24), the definition (3.3) of db xtβ , the Itô product rule in (3.12), and Theorem 1, we have, ∂ℓ i A dt , ∂q i α ∂ℓ 1 d(pi Aiα ) = − pj [ Aα , Aβ ]j db xtβ − d pj [ Aα , Aβ ]j ξ β .dWt + i Aiα dt , 2 ∂q dmα = {mα , mβ }dxtβ +

(3.15)



13 April 2016

where we have substituted the momentum map definition mα = pi Aiα from equation (2.10). We have, d pj [ Aα , Aβ ]j ξ β . dWt = (dpj )[ Aα , Aβ ]j + pj d[ Aα , Aβ ]j ξ β .dWt ∂ = (dpj )[ Aα , Aβ ]j + pj l [ Aα , Aβ ]j (d ql ) ξ β .dWt ∂q l ∂Aγ ∂ (2.25) = − pl j [ Aα , Aβ ]j ξ γ + pj l [ Aα , Aβ ]j Alγ ξ γ ξ β dt ∂q ∂q j ∂Aγ ∂ = − pj [ Aα , Aβ ]l l + pj Alγ l [ Aα , Aβ ]j ξ β ξ γ dt ∂q ∂q j β γ = pj Aγ , [Aα , Aβ ] ξ ξ dt (2.11) = mγ , { mα , mβ } ξ β ξ γ dt .

Consequently, we may write the entire equation (3.15) as (reinstating the k-indices) ! N X 1 ∂ℓ β β γ xt − mγ , {mα , mβ } ξk ξk dt + i Aiα dt dmα = {mα , mβ }db 2 ∂q k=1 ! N X 1 ∂ℓ β β γ {mα , mβ } , mγ ξk ξk dt + i Aiα dt = {mα , mβ }( uβ dt + ξk dWtk ) + 2 ∂q k=1 ! N X 1 ∂ℓ = Xdbxt mα + Xξk (Xξk mα ) dt + i Aiα dt using the notations (3.1) and (3.9) , 2 ∂q

13

(3.16)

(3.17)

k=1

in agreement with the first equation in (3.7).

Third proof of the first equation in (3.7). Let η : [0, T ] → g be an arbitrary random curve of bounded variation. We begin the third proof by computing δℓ ( u, q) , η = d pi Aiα ( q) η α d δu ∂Ai = Aiα ( q)η α ◦ dpi + pi jα ( q)η α ◦ d qj ∂q j ∂Aα ∂Ajα (2.15) α i β α i β = − pj ◦ dWtk q ) u A ( q )η dt − p q )ξ A ( q )η ( ( j k β β (2.25) ∂q i ∂q i ∂ℓ + Aiβ ( q)η β i ( u, q)dt ∂q ∂Aiα ∂Aiα β β α j α j + pi j ( q)η Aβ ( q) u dt + pi j ( q)η Aβ ( q)ξk ◦ dWtk ∂q ∂q ! ! j j j j ∂η ∂η ∂u ∂(ξ ) k (2.3) Q Q Q Q i i ( q) dt + pj (ξk )iQ i − ηQ ( q) ◦ dWtk = pj uiQ i − ηQ ∂q ∂q i ∂q ∂q i + Aiβ ( q)η β j

∂ℓ ( u, q)dt ∂q i j

= pj [uQ , ηQ ] ( q)dt + pj [(ξk )Q , ηQ ] ( q) ◦ dWtk + Aiβ ( q)η β = pj [u, η]jQ ( q)dt + pj [ξk , η]jQ ( q) ◦ dWtk + Aiβ ( q)η β

∂ℓ ( u, q)dt ∂q i

∂ℓ ( u, q)dt . ∂q i

Next, we compute the Itˆ o contraction term, namely the difference between the Stratonovich integral above and the corresponding Itˆ o one. Consequently, we find ∂Ai dpj ≃ − pi jα ( q)ξkα dWtk ∂q



and d[ξk , η]jQ ( q) = d(Ajα ( q)[ξk , η]α ( q)) ≃

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14

∂Ajα β α i q )[ξ , η] ( q )A ( q )ξ dWtk . ( k β k ∂q i

Therefore, d(pj [ξk , η]jQ ( q))

∂Aiα ∂Ajα β j k β α α i ≃ − pi Aβ ( q)[ξk , η] ( q) j ( q)ξk dWt + pj ( q)[ξk , η] ( q)Aβ ( q)ξk dWtk ∂q ∂q i ! ∂[ξk , η]iQ ∂(ξk )iQ j j = pi (ξk )Q ( q) ( q) − [ξk , η]Q ( q) ( q) dWtk ∂q j ∂q j h ii = pi (ξk )Q , [ξk , η]Q ( q)dWtk i

= pi [ξk , [ξk , η]]Q ( q)dWtk

and we obtain the Itˆ o contraction term N N X X j d pj [ξk , η]jQ ( q) . dWtk = pj ξk , [ξk , η] Q ( q)dt = pj (adξk adξk η)jQ ( q)dt . k=1

k=1

Thus, the first Stratonovich equation in (2.24) reads, in the Itˆ o version, δl δl δl d ( u, q) = − ad∗u ( u, q) dt − ad∗ξk ( u, q) dWtk δu δu δu N ∂ℓ 1X ∗ δl + ( u, q) dt + i ( u, q)Aiα ( q)eα ∈ g∗ , adξk ad∗ξk 2 δu ∂q k=1

which is an explicit version of the first equation in (3.7). This finishes the third proof of Corollary 6.

4

Stochastic Hamiltonian formulation

The goal of this section is to present the Hamiltonian version of Theorem 1 and analyze its consequences. In Section 2, we found the stochastic equations of motion (2.24) on g∗ × T ∗ Q and (2.31) on g∗ × Q. We want to deduce these equations in a purely Hamiltonian manner, without any reference to variational principles or the Lagrangian formulation of Sections 2 and 3. Thus, we need stochastic Hamiltonians e h( m, pq ) and h( m, q) (this latter one was already defined in the last paragraph of Section 2), as well as Poisson brackets on g∗ × T ∗ Q and g∗ × Q.

4.1

The deterministic Hamilton equations

We first recall the Poisson structure on g∗ × P introduced in [16], where the Lie group G, whose Lie algebra is g, acts on the right on the Poisson manifold P by Poisson diffeomorphisms. The Poisson manifold g∗ × P . In this paragraph, the entire discussion is non-stochastic. We recall below the results in [16] relevant to our development and expand on it in certain directions needed later on. The framework studied in [16], adapted to our situation, is the following. Let a Lie group G act on the right by Poisson diffeomorphisms on the Poisson manifold P . Endow T ∗ G×P with the Poisson bracket equal to the sum of the canonical bracket {·, ·} on T ∗ G and the given Poisson bracket {·, ·} P on P . Define the free proper left G-action by Poisson diffeomorphisms

∗ on (T ∗ G×P, {·, ·}+ {·, ·}P ) by h·(αg , p) := Thg Lh−1 (αg ), p · h−1 , where g, h ∈ G, αg ∈ Tg∗ G, p ∈ P , and p·h−1 de-

notes the given right action of h−1 on the point p. Then the map φ : T ∗ G× P ∋ (αg , p) 7→ (Te∗ Lg (αg ), p · g) ∈ g∗ × P



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15

is G-invariant and induces a diffeomorphism φ/G : (T ∗ G × P )/G → g∗ × P . The push forward of the quotient Poisson bracket on (T ∗ G × P )/G by φ/G yields the Poisson bracket p p δf δh (p) − dhµ (p), (p) + {f µ , hµ }P (p) , (4.1) {f, h}g∗×P (µ, p) = {f p , hp }− (µ) + df µ (p), δµ P δµ P for all f, h ∈ C ∞ (g∗ × P ), where f µ , hµ ∈ C ∞ (P ) and f p , hp ∈ C ∞ (g∗ ) are defined by f µ (p) := f p (µ) := f (µ, p), for all µ ∈ g∗ , p ∈ P , and similarly for h ([16, Proposition 2.1]). Thus, Hamilton’s equations for h ∈ C ∞ (g∗ × P ) are p d δh d ∗ µ (p) + XhPµ (p), (4.2) µ = ad δhp µ − JT ∗ P (dh (p)) , p= δµ dt dt δµ P where XhPµ denotes the Hamiltonian vector field of hµ ∈ C ∞ (P ) on the Poisson manifold P and JT ∗ P : T ∗ P → g∗ is the momentum map of the cotangent lifted action (see (2.8) with Q replaced by P ). now that the right g-action on P has a momentum map JP : P → g∗ , which means that ξP [f ] = n Suppose o f, JξP for all f ∈ C ∞ (P ) and all ξ ∈ g, where JξP (p) := hJP (p), ξig . Suppose also that JP is infinitesimally P n o [ξ,η] equivariant i.e., JP = − JξP , JηP , for all ξ, η ∈ g. We recall that the existence of a momentum map on P P

for a connected Lie group action forces the group orbits to be included in the symplectic leaves of P , which is a rather stringent condition. There are many examples of Poisson Lie group actions that do not admit a momentum map (see, e.g., [21, Chapters 4 and 5] for a discussion of this problem). However, in the presence of an equivariant momentum map JP : P → g∗ , the diffeomorphism ψ : g∗ × P ∋ (µ, p) 7→ (µ − JP (p), p) ∈ g∗ × P pushes forward the Poisson bracket {·, ·}g∗ ×P , given by (4.1), to the sum Poisson bracket {f, h}sum (µ, p) := {f p , hp }− (µ) + {f µ , hµ }P (p)

(4.3)

on g∗ × P . This is proved for left actions in [16, Proposition 2.2]; our formulas in (4.1) and the definition of ψ have relative sign changes because we work with a right G-action on P . The proof is a direct verification. Hamilton’s d f = {f, h} for the sum Poisson bracket (4.3) are equations dt d µ = ad∗δhp µ, δµ dt

d p = XhPµ (p). dt

(4.4)

Using (4.3), it follows that if k ∈ C ∞ (g∗ × P ) is a Casimir function on (g∗ × P, {·, ·}sum ), then k ◦ ψ is a Casimir function on (g∗ × P, {·, ·}g∗ ×P ). In particular, if kP ∈ C ∞ (P ) is a Casimir function, then the function (µ, p) 7→ kP (p) is a Casimir function for (g∗ × P, {·, ·}g∗ ×P ). This can also be easily checked directly using (4.2). More interestingly, if kg∗ ∈ C ∞ (g∗ ) is a Casimir function on g∗ , then (µ, p) 7→ kg∗ (µ − JP (p)) is a Casimir function for (g∗ × P, {·, ·}g∗ ×P ) ([16, Corollary 2.3]). Since the projections πg∗ : (g∗ × P, {·, ·}sum ) → g∗− and πP : (g∗ × P, {·, ·}sum ) → P are Poisson maps, their compositions πg∗ ◦ ψ : (g∗ × P, {·, ·}g∗ ×P ) ∋ (µ, p) 7−→ µ − JP (p) ∈ g∗− ,

πP ◦ ψ : (g∗ × P, {·, ·}g∗ ×P ) ∋ (µ, p) 7−→ p ∈ P

(4.5)

with the Poisson diffeomorphism ψ : (g∗ × P, {·, ·}g∗ ×P ) → (g∗ × P, {·, ·}sum ) are also Poisson maps. Remarkably, the projection πg∗ : (g∗ × P, {·, ·}g∗ ×P ) → g∗− is also a Poisson map, as an easy direct verification shows, using for f ∈ C ∞ (g∗ ) the identities (f ◦ πg∗ )p = f for every p ∈ P and (f ◦ πg∗ )µ = f(µ), a constant function on P , for every µ ∈ g∗ . In particular, this says that Hamilton’s equations (4.2) for a Hamiltonian of the form h := h ◦ πg∗ , where h ∈ C ∞ (g∗ ) (i.e., h does not depend on p ∈ P ), are the Lie-Poisson equations for h on g∗− which completely decouple from the second equation in (4.2). This second equation is given by an infinitesimal generator d at every instance of time, namely, if µ(t) is a solution of the Lie-Poisson equation dt µ = ad∗δh µ, then the second δµ

equation in (4.2) is the time-dependent infinitesimal generator equation δh d (p(t)). p(t) = dt δµ(t) P

(4.6)



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Similarly, the projection πP : g∗ × P → P is a Poisson map relative to both Poisson brackets {·, ·}g∗ ×P and {·, ·}sum , because if f ∈ C ∞ (P ), then (f ◦ πP )p = f(p), a constant on g∗ , and (f ◦ πP )µ = f, for any µ ∈ g∗ . Hamilton’s equations (4.2) and (4.4) show that the manifolds {µ} × P and g∗− × {p} for any µ ∈ g∗ , p ∈ P , are not Poisson submanifolds of g∗ × P endowed with either Poisson bracket {·, ·}g∗ ×P or {·, ·}sum . The Poisson brackets on g∗ × Q and g∗ × T ∗ Q. We specialize the results of the previous paragraph to the following Poisson manifold: Q, endowed with the zero Poisson structure, and T ∗ Q, endowed with the canonical Poisson structure (whose local expression is (2.5)). We continue to work in the non-stochastic context. For any f, h ∈ C ∞ (g∗ × Q), the Poisson bracket (4.1) reads T − cαβ γ mγ − Ajα ∂h/∂mβ ∂f /∂mα {f, h}g∗− ×Q (m, q) : = Aiβ 0 ∂h/∂q j ∂f /∂q i * + * + q q δh δf q q m m = {f , h }− (m) + df (q), (q) − dh (q), (q) , δm Q δm Q

(4.7)

where f q ∈ C ∞ (g∗ ) and f µ ∈ C ∞ (Q) are defined by f q (µ) := f m (q) := f (m, q), for all m ∈ g∗ , q ∈ Q, and {·, ·}− is the minus Lie-Poisson bracket (2.12) on g∗− . Similarly, for any fe, e h ∈ C ∞ (g∗ × T ∗ Q), the Poisson bracket (4.1) reads   T   ∂Ak j α e e − m − A p ∂ h/∂m ∂ f /∂mα  k ∂qj [α,β ] α β n o    j  e Aiβ 0 δji  fe, e h ∗ ∗ (m, pq ) :=  ∂ fe/∂q i      ∂ h/∂q  g− ×T Q k ∂A ∂e h/∂pj ∂ fe/∂pi − pk ∂qiβ − δij 0 + * * ! ! o n pq epq e δ f δ h (pq ) − dhm (pq ), (m) + df m (pq ), = fepq , e hpq δm δm − ∗ T Q

+

(pq ) T ∗Q

n o + fem , e hm (pq ), (4.8)

where fepq ∈ C ∞ (g∗ ) and fem ∈ C ∞ (T ∗ Q) are defined by fepq (m) := fem (pq ) := fe(m, pq ), for all m ∈ g∗ , pq ∈ T ∗ Q, and {·, ·} is the canonical Poisson bracket (2.5) on T ∗ Q. This proves the first statement in the following theorem. Theorem 8. The brackets (4.7) and (4.8) are Poisson brackets on g∗ × Q and g∗ × T ∗ Q, respectively. Hamilton’s h ∈ C ∞ (g∗ × T ∗ Q), equations on g∗ × T ∗ Q are given by (4.2) with P replaced by T ∗ Q. In standard coordinates, for e these equations are d h ∂Ak ∂ e ∂e h ∂e h − Ajα j + pk jα , mα = −m[α,β] dt ∂mβ ∂q ∂q ∂pj

d i ∂e h ∂e h + , q = Aiβ dt ∂mβ ∂pi

∂Akβ ∂ e h d ∂e h − i . (4.9) pi = −pk i dt ∂q ∂mβ ∂q

The diffeomorphism ψ : g∗ × T ∗ Q ∋ (µ, pq ) 7→ (µ − JT ∗ Q (p), pq ) ∈ g∗ × T ∗ Q pushes forward the Poisson bracket (4.8) to the sum Poisson bracket. If kg∗ ∈ C ∞ (g∗ ) is a Casimir function on g∗ , then (µ, pq ) 7→ kg∗ (µ − JT ∗ Q (pq )) is a Casimir function for (g∗ × T ∗ Q, {·, ·}g∗ ×T ∗ Q ).

Hamilton’s equations (4.2) on g∗ × Q for h ∈ C ∞ (g∗ × Q) (with P replaced by the trivial Poisson manifold Q) are Hamel’s equations ([11]; for a modern formulation, see, e.g., [4, §3.8, p.144] or [5]): q δh d d (q) ⇐⇒ m = ad∗δhq m − JT ∗ Q (dhm (q)), q= δm dt dt δm Q (4.10) d ∂h ∂h d i ∂h − Ajα j , . mα = −cαβ γ mγ q = Aiβ dt ∂mβ ∂q dt ∂mβ If e h does not depend on pq ∈ T ∗ Q (respectively, h does not depend on q ∈ Q), then Hamilton’s equations (4.9) (respectively, (4.10)) decouple into the Lie-Poisson equations on g∗− and the time-dependent infinitesimal generator equations for

δe h δm(t)

∈ g on T ∗ Q (respectively,

δh δm(t)

∈ g on Q).

The four projections of g∗ ×Q and g∗ ×T ∗ Q on every factor are Poisson (g∗ has the minus Lie-Poisson structure). The map g∗ × T ∗ Q ∋ (µ, pq ) 7→ µ− JT ∗ Q (pq ) ∈ g∗− is Poisson. The embedding g∗ × Q ∋ (m, q) 7→ (m, 0q ) ∈ g∗ × T ∗ Q is not Poisson. The map ρ : g∗ × T ∗ Q ∋ (m, pq ) 7−→ (m, q) ∈ g∗ × Q is Poisson.



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Proof. Formulas (4.9) and (4.10) are obtained by calculating (4.2) for these two cases. The statements about the Poisson character of the five projections and the diffeomorphism ψ, the decoupling of the equations for Hamiltonians depending only on m ∈ g∗ , as well as the assertion about the Casimir functions, were proved in the previous paragraph for a general Poisson manifold P . Setting all coordinates pi = 0 in (4.9) does not yield (4.10), i.e., a Hamiltonian vector field on the Poisson manifold g∗ × T ∗ Q, restricted to g∗ × Q is, in general, not tangent to g∗ × Q. This proves that g∗ × Q is not a Poisson submanifold of g∗ × T ∗ Q. Let π : T ∗ Q → Q be the cotangent bundle projection. The map, ρ : g∗ × T ∗ Q → g∗ × Q is Poisson. This is a direct verification using the formulas (f ◦ ρ)m = f m ◦ π, d(f ◦ ρ)m (pq ) = df m (q) ◦ Tpq π, (f ◦ ρ)pq = f q , q δ(f ◦ρ)pq ∗ ∗ = δf δm δm , and the fact that the infinitesimal generators uT Q (of the lifted G-action on T Q) and uQ (of the G-action on Q) are π-related. The last statement is obtained by setting ∂h/∂pi = 0 in (4.9). Remark 9. [Collective Lie-Poisson momentum map dynamics] In (4.10), note that if the Hamiltonian depends only on m ∈ g∗ , i.e., the Hamiltonian is of the form h := h ◦ πg∗ , where h ∈ C ∞ (g∗ ) and πg∗ : (g∗ × Q, {·, ·}g∗ ×Q ) ∋ (m, q) 7→ m ∈ (g∗ , {·, ·}− ), Hamel’s equations become the Lie-Poisson equations on g∗− . In this case, we say the ♦ motion collectivizes (see [10]) since πg∗ is a Poisson map. Remark 10. The manifold Q endowed with the zero Poisson structure does not admit a momentum map JQ : n o ξ ∗ Q → g . Indeed, if JQ existed, we would have ξQ [f ] = f, JQ = 0 for all f ∈ C ∞ (Q) and all ξ ∈ g which Q

implies the false statement that all smooth functions on Q are g-invariant. As a consequence, the statement in the previous paragraph about the Poisson bracket on g∗ × Q being isomorphic to the sum Poisson bracket, which in this case would be just the minus Lie-Poisson bracket, does not apply. Similarly, Casimir functions on g∗ do not induce Casimir functions on g∗ × Q. ♦ For the statement of the next corollary, we need to introduce the fiber translation vector field Tα ∈ X(T ∗ Q) associated to a one-form α ∈ Ω1 (Q). The map T ∗ Q ∋ pq 7→ pq − tα(q) ∈ T ∗ Q, t ∈ R, is a one-parameter group. Define Tα to be the vector field with this flow, i.e., d (pq − tα(q)) ∈ Tpq (T ∗ Q). Tα (pq ) := dt t=0 This vector field is identical to the vertical lift operation by −α ∈ Ω1 (T ∗ Q).

Corollary 11. Hamilton’s equations (4.2) (with P = T ∗ Q) on g∗ × T ∗ Q for e h := h ◦ ρ, where h ∈ C ∞ (g∗ × Q), e i.e., h(m, pq ) := h(m, q), take the form q d d δh (4.11) + Tdhm (pq ). m = ad∗δhq m − JT ∗ Q (dhm (q)) , pq = δm dt dt δm T ∗ Q In addition, (4.11) imply (4.10) and the non-homogeneous Lie-Poisson equations d JT ∗ Q (pq (t)) = ad∗δhq(t) JT ∗ Q (pq (t)) − JT ∗ Q dhm(t) (q(t) dt δm(t)

(4.12)

for JT ∗ Q (pq (t)), where (m(t), q(t)) is the solution of Hamel’s equations (4.10). hm = hm ◦ π ∈ C ∞ (T ∗ Q), where π : T ∗ Q → Q is the cotangent bundle Proof. We have e hpq = hq ∈ C ∞ (g∗ ) and e projection.

We compute JT ∗ (T ∗ Q) (d(hm ◦ π)(pq )), the second summand on the right hand side of the first equation in (4.2) for P = T ∗ Q. To do this, we note that since the G-action on T ∗ Q is the cotangent lifted G-action on Q, the



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cotangent bundle projection π : T ∗ Q → Q is equivariant and thus the infinitesimal generators of the two actions for the same Lie algebra element are π-related, i.e., T π ◦ vT ∗ Q = vQ ◦ π for any v ∈ g. Therefore,

(2.8) JT ∗ (T ∗ Q) (d(hm ◦ π)(pq )) , v g = hd(hm ◦ π)(pq ), vT ∗ Q (pq )ig = dhm (q), Tpq π (vT ∗ Q (pq )) g (2.8)

= hdhm (q), vQ (q)ig = hJT ∗ Q (dhm (q)) , vig .

∗

(4.13)

∗

Next, we compute XeTmQ (pq ) = XhTmQ ◦π (pq ), the second summand on the right hand side of the second equation h in (4.2) for P = T ∗ Q. Since this affects only the dynamics on T ∗ Q, we prove, in general, that ∗

T Q Xk◦π = Tdk ,

for all k ∈ C ∞ (Q).

(4.14)

To prove (4.14), it is easier to work in local coordinates. Hamilton’s equations for k ◦ π are dq i ∂(k ◦ π) = 0, = dt ∂pi

dpi ∂k ∂(k ◦ π) =− i =− dt ∂q i ∂q

∂k 1 n 0 0 whose solution is q i (t) = q0i , pi (t) = p0i − t ∂q i (q0 ), where (q0 , . . . , q0 , p1 , . . . pn ) is the initial condition. Thus, the ∗

T Q flow of Xk◦π is pq 7→ pq − tdk(q) which coincides with the flow of Tdk , thus proving (4.14).

Using the identities (4.13) and (4.14), equations (4.2) become (4.11). Since ρ : (g∗ × T ∗ Q, {·, ·}g∗ ×T ∗ Q ) ∋ (m, pq ) 7−→ (m, q) ∈ (g∗ × Q, {·, ·}g∗×Q ) is a Poisson map by Theorem 8, we ∗ g∗ ×T ∗ Q have T ρ ◦ Xh◦ρ = Xhg ×Q ◦ ρ for any h ∈ C ∞ (g∗ × Q), which is equivalent to saying that (4.11) (the equations ∗

g ×T of motion defined by the Hamiltonian vector field Xh◦ρ

the Hamiltonian vector field

∗

Q

) project to (4.10) (the equations of motion defined by

∗

Xhg ×Q ).

Finally, we prove (4.12). We have ! q(t) d d δh (4.11) (pq (t)) + Tpq (t) JT ∗ Q (Tdhm(t) (pq (t))) JT ∗ Q (pq (t)) = Tpq (t) JT ∗ Q pq (t) = Tpq (t) JT ∗ Q dt dt δm(t) T ∗ Q ∗ (2.9) Q = ad∗δhq(t) JT ∗ Q (pq (t)) + Tpq (t) JT ∗ Q XhTm(t) (p (t)) . (4.15) q ◦π (4.14)

δm(t)

To compute the second summand, pair it with any v ∈ g to get D E D ∗ E ∗ Q Q (pq (t)) = {JvT ∗ Q , hm(t) ◦ π}(pq (t)) Tpq (t) JT ∗ Q XhTm(t) (p (t)) , v = dJvT ∗ Q (pq (t)), XhTm(t) ◦π ◦π q g D E ∗ = −{hm(t) ◦ π, JvT ∗ Q }(pq (t)) = − d(hm(t) ◦ π)(pq (t)), XJTv ∗Q (pq (t)) T Q D E (2.7) m(t) = − d(h ◦ π)(pq (t)), vT ∗ Q (pq (t)) D E (2.9) = − dhm(t) (q(t)), vQ (q(t)) E D (2.8) (4.16) = − JT ∗ Q dhm(t) (q(t) , v . g

Formulas (4.15) and (4.16) yield (4.12).

Corollary 12. If the Hamiltonian e h ∈ C ∞ (g∗ × T ∗ Q) is of the form e h = h ◦ ρ and h ∈ C ∞ (g∗ × Q) isq hyperregular, , q ∈ g∗ × Q i.e., the parameter dependent reduced Legendre transformation g × Q ∋ (u, q) 7→ (m(u, q), q) := δℓ δu

is a diffeomorphism, where ℓ ∈ C ∞ (g × Q) and h(m, q) := hm, u(m, q)ig − ℓ(u(m, q), q), equation (4.12) takes the form d JT ∗ Q (pq (t)) = ad∗u(t) JT ∗ Q (pq (t)) + JT ∗ Q (dℓu(t) (q(t)), (4.17) dt where (u(t), q(t)) is the solution of the Lagrangian version of Hamel’s equations q

q

d δℓ δℓ u = ad∗u + JT ∗ Q (dℓ (q)), dt δu δu q q ∂ℓ d ∂ℓ γ ∂ℓ = −c uβ + Ajα j , αβ α γ dt ∂u ∂u ∂q

d q = uQ (q) dt d i q = Aiβ uβ . dt

⇐⇒ (4.18)



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Proof. By hyperregularity, we can solve for u to get u(m, q) ∈ g and we have * + q δhq δℓ δm, = hδm, u(m, q)ig + hm, Duq (m) · δmig − , Duq (m) · δm = hδm, u(m, q)ig δm g δu(m, q) g

q

q

δℓ because δu(m,q) = m by definition of m and hyperregularity. Thus u(m, q) = δh δm . Since ℓ(u, q) = hm(u, q), uig − h(m(u, q), q), we get δhq u u u dℓ (q) = hdm (q), uig − dm (q), − dhm(u,q) (q) = − dhm(u,q) (q), δm(u, q) g

since, as we just saw and invoking hyperregularity, we have from (4.12) and (4.10), respectively.

4.2

δhq δm(u,q)

= u. Equations (4.17) and (4.18) now follow

The stochastic Hamilton equations

The Stratonovich stochastic Hamilton equations for semimartingales. We begin by defining Stratonovich stochastic Hamilton equations. Let (P, {·, ·}) be a Poisson manifold. For any f ∈ C ∞ (P ), form the semimartingale f (p(t)) obtained by replacing the point p ∈ P by a P -valued semimartingale p(t). Consider a semimartingale Rt Rt Yt (p) := Y0 + 0 φα (p(s))ξkα ◦ dWsk + 0 ψ(p(s))ds, where φα , ψ ∈ C ∞ (P ) are deterministic smooth functions and ξk := ξkα eα ∈ g are constant elements. In analogy with Section 2, the (Stratonovich) stochastic Poisson bracket is defined by {f (p(t)), ◦dt Yt } := {f (p(t)), φα (p(t))}ξkα ◦ dWtk + {f (p(t)), ψ(p(t))}dt.

(4.19)

The Poisson brackets on the right hand side of (4.19) are defined in the following way. In coordinates (p1 , . . . , pn ) on P , the Poisson bracket has the expression {f, g}(p) = Πij (p)

∂f ∂g , ∂pi ∂pj

(4.20)

where Π is the Poisson 2-tensor.

d f (pi (t) + ǫ), the limit being taken in L2 (Ω). Then we vary pi (t) 7→ pi (t) + ǫ and define δpi f (p(t)) := dǫ ǫ=0 Hence we have, ∂f (4.21) δpi f (p(t)) = i (p(t)) ∂p and the the bracket of the two semimartingales f (p(t)) and g(p(t)) is defined by: {f (p(t)), g(p(t))} := Πij ((p(t))δpi f (p(t))δpj g(p(t)) = Πij ((p(t))

∂f ∂g (p(t)) j (p(t)). i ∂p ∂p

(4.22)

Therefore, this is the same as taking the function p 7→ {f, g}(p) and replacing p ∈ P by the semimartingale p(t), i.e., we have {f (p(t)), g(p(t))} = {f, g}(p(t)) and we shall use subsequently interchangeably these two notations. Definition 13. The (Stratonovich) stochastic Hamilton equations for P -valued semimartingales with stochastic Rt Rt semimartingale Hamiltonian Yt (p) := Y0 + 0 φα (p(s))ξkα ◦ dWsk + 0 ψ(p(s))ds are dt f (p(t)) = {f (p(t)), ◦dt Yt } := {f, φα }(p(t))ξkα ◦ dWtk + {f, ψ}(p(t))dt,

for any

f ∈ C ∞ (P ),

(4.23)

where the Poisson bracket semimartingales on the right hand side are defined in (2.17) for variations as in (2.20). If (p1 , . . . , pn ) are coordinates on P , the Stratonovich stochastic Hamilton equations thus take the form dtpi (t) = {pi (t), ◦dt Yt } = {pi , φα }(p(t))ξkα ◦ dWtk + {pi , ψ}(p(t))dt.

(4.24)

o’s formula, Let k ∈ C ∞ (P ) be a Casimir function. Then, for the semimartingale k(p(t)) we have, by Itˆ dt k(p(t)) = {k(p(t)), ◦dt Y} = {k, φα }(p(t))ξkα ◦ dWtk + {k, ψ}(p(t))dt = 0, i.e., the semimartingale k(p(t)) is conserved along the stochastic flow of the stochastic Hamiltonian semimartingale Yt (p). Clearly, k(p(t)) is also conserved in the Itô representation.



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The Stratonovich stochastic Hamilton equations on g∗ × Q and g∗ × T ∗ Q. We continue to denote the semimartingales qi (t) := q i ( q(t), p(t)) and pi (t) := pi ( q(t), p(t)). With this definition, the information in Theorem 8, in particular, having the Poisson bracket (4.7) on g∗ × Q, we form the semimartingale dt ht = (h1 )α (m( q(t), p(t)), q(t))ξkα ◦ dWtk + h2 (m( q(t), p(t)), q(t))dt, where (h1 )α , h2 ∈ C ∞ (g∗ × Q) and ξk = ξkα eα ∈ g. By (4.23), the Stratonovich stochastic Hamilton equations are dt mα ( q(t), p(t)) = {mα ( q(t), p(t)), ◦dt ht (m( q(t), p(t)), q(t))}g∗ ×Q , dt qi (t) = { qi (t), ◦dt ht (m( q(t), p(t)), q(t))}g∗ ×Q , i.e., dt mα ( q(t), p(t)) = {mα ( q(t), p(t)), ◦dt ht (m( q(t), p(t)), q(t))}g∗ ×Q ∂ ht (( q(t), p(t)), q(t)) = {mα ( q(t), p(t)), mβ ( q(t), p(t))}g∗ ×Q ◦ dt ∂mβ ∂ ht + {mα ( q(t), p(t)), qj (t)}g∗ ×Q ◦ dt (( q(t), p(t)), q(t)) ∂q j ∂ ht (( q(t), p(t)), q(t)) = − cαβ γ mγ ( q(t), p(t)) ◦ dt ∂mβ ∂ ht − Ajα ( q(t)) ◦ dt (( q(t), p(t)), q(t)) ∂q j h i ∂ ht ∗ j (( q(t), p(t)), q(t)) , =: add( δht )(( q(t),p(t)), q(t)) m( q(t), p(t)) − Aα ( q(t)) ◦ dt δm ∂q j α

(4.25)

dt qi (t) = { qi (t), ◦dt ht (m( q(t), p(t)), q(t))}g∗ ×Q ∂ ht i (( q(t), p(t)), q(t)) = { q (t), mβ ( q(t), p(t))}g∗ ×Q ◦ d ∂mβ ∂ ht = Aiβ ( q(t)) ◦ d (( q(t), p(t)), q(t)) , ∂mβ

which are identical to the stochastic equations of motion (2.31) in Theorem 4, once we observe that for the functional ht considered there, q-dependence comes only from its bounded variation part (defined by h2 ) and the explicit therefore ◦dt ∂∂qhjt = ∂∂qhjt dt. Note that (4.25) are the stochastic version of Hamel’s equations (4.10). As in the deterministic case (see Remark 9), note that if ht depends only on the g∗ -valued semimartingale m( q(t), p(t)), equations (4.25) decouple into the stochastic Lie-Poisson equations on g∗− and the stochastic infinitesimal generator equation for δ ht δm (( q(t), p(t)), q(t)) ∈ g.

Our goal is to derive (4.25) purely from a Hamiltonian point of view and, similarly, Stratonovich stochastic Hamilton equations on g∗ to T ∗ Q. In particular this means that the semimartingale m( q(t), p(t)) needs to be replaced by a semimartingale m in order not to appeal to the Legendre transformation of the Lagrangian ℓ. So, the setup is the following general situation. Let ht be a semimartingale of the form dt ht = (h1 )α ( m(t), q(t))ξkα ◦ dWtk + h2 ( m(t), q(t))dt , where (h1 )α , h2 are (deterministic) smooth functions evaluated on (g∗ × Q)-valued semimartingales ( m(t), q(t)). et a semimartingale of the form Similarly, denote by h et = (e h1 )α ( m(t), pq (t))ξkα ◦ dWtk + e h2 ( m(t), pq (t))dt , dt h

where (e h1 )α , e h2 ∈ C ∞ (g∗ × T ∗ Q) are evaluated on (g∗ × T ∗ Q)-valued semimartingales ( m(t), pq (t)). Consider the Poisson brackets defined in (4.7) and (4.8). According to Definition 13, the corresponding stochastic Hamilton



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equations on g∗ × Q are defined to be dt f ( m(t), q(t)) = {f ( m(t), q(t)), ◦dt h}g∗− ×Q , for any f ∈ C ∞ (g∗ × Q), respectively on g∗ × T ∗ Q, e }g∗ ×T ∗ Q , dt fe( m(t), pq (t)) = {fe( m(t), pq (t)), ◦dt h −

e , these equations are for any fe ∈ C ∞ (g∗ × T ∗ Q). Notice that, by the form of the Hamiltonian functionals h and h equivalent, respectively, to dt f ( m(t), q(t)) = {f ( m(t), q(t)), (h1 )α ( m(t), q(t))}g∗− ×Q ξkα ◦ dWtk + {f ( m(t), q(t)), h2 ( m(t), q(t))}g∗− ×Q dt

h1 )α ( m(t), pq (t))}g∗− ×T ∗ Q ξkα ◦ dWtk dt fe( m(t), pq (t)) = {fe( m(t), pq (t)), (e h2 ( m(t), pq (t))}g∗− ×T ∗ Q dt. + {fe( m(t), pq (t)), e

We define now the right hand sides of these equations involving the Poisson bracket. For f ∈ C ∞ (g∗ × Q), the Poisson bracket (4.7) of the two semimartingales f ( m(t), q(t)) and ht then reads    T ∂f ◦dt ∂ ht j γ ( m (t), q (t)) ∂mβ  ∂mα  − cαβ mγ (t) − Aα ( q(t))  {f ( m(t), q(t)), ◦dt ht }g∗− ×Q :=    i ∂f A ( q (t)) 0 β ◦dt ∂∂qhjt ∂qi ( m(t), q(t)) * + q h δ t = {f q ( m(t), q(t)), ◦dt htq }− + df m ( m(t), q(t)), ◦dt ( m(t), q(t)) δm Q * + q δf m ( m(t), q(t)) . − ◦dt dht ( m(t), q(t)), (4.26) δm Q In this formula, ∂f /∂mα and ∂f /∂q i are evaluated on the semimartingales m(t) and q(t) and, according to (2.29) and (2.30), ∂(h1 )α ∂ ht ∂h2 := ( m(t), q(t))ξkα ◦ dWtk + ( m(t), q(t))dt, dt ∂mβ ∂mβ ∂mβ ∂h2 ∂(h1 )α ∂ ht α k m (t), q (t))ξ ◦ dW + := ( ( m(t), q(t))dt, dt k t ∂q j ∂q j ∂q j 2 m d(h1 )m m(t), q(t)), df m ( m(t), q(t)), α ( m(t), q(t)), d(h ) (

and

δ(h1 )qα δm

( m(t), q(t)),

Q Tq∗ Q,

δ(h2 )q δm

( m(t), q(t)),

Q δ(h1 )qα /δm,

δf q δm

δ(h1 )qα δm ( m(t),

Q q

q(t)),

δ(h2 )q δm ( m(t),

( m(t), q(t)) are the covectors

δf q δm ( m(t), q(t)), 1 m d(h )α (q), d(h2 )m (q),

q(t)),

df (q) ∈ the elements δ(h ) /δm, δf /δm ∈ g, and the tangent vectors δ(h1 )qα /δm Q (q), δ(h2 )q /δm Q (q), (δf q /δm)Q (q) ∈ Tq Q with the variables (m, q) replaced by the semimartingales ( m(t), q(t)), m

2 q

α k 2 m dt dhtm ( m(t), q(t)) := d(h1 )m α ( m(t), q(t))ξk ◦ dWt + d(h ) ( m(t), q(t))dt q δ ht δ(h1 )qα δ(h2 )q α k ◦d ( m(t), q(t)) := ( m(t), q(t))ξk ◦ dWt + ( m(t), q(t))dt, δm δm δm Q Q Q {f q ( m(t), q(t)), dt htq }− := f q ( m(t), q(t)), (h1 )qα ( m(t), q(t)) − ξkα ◦ dWtk + f q ( m(t), q(t)), (h2 )q ( m(t), q(t)) − dt.



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Similarly, for fe ∈ C ∞ (g∗ × T ∗ Q), the Poisson bracket (4.8) computed for these semimartingales is given by n o et fe( m(t), pq (t)) , ◦dh ∗ g∗ − ×T Q   T   e et  ∂h k ∂f ◦d ∂A t γ j ( m(t), pq (t)) ∂mβ  − Aα ( q(t)) pk (t) ∂qjα ( q(t))    −cαβ mγ (t)  ∂mα  et     ∂ fe i ∂h i    Aβ ( q(t)) 0 δj :=  ∂qi ( m(t), pq (t))   ◦d j t ∂q     k   ∂Aβ j et ∂ fe ( p q (t)) − δ 0 − (t) ∂ h i k i m (t), p (t)) ( ∂q ◦d q t ∂pj ∂pi !! + * p q o n e h δ p q t m p q et ( m(t), pq (t)) = fe ( m(t), pq (t)) , ◦dt h + df ( m(t), pq (t)) , ◦dt δm − T ∗Q + ! * n o δ fepq m em , e (4.27) − ◦dt dht ( m(t), pq (t)) , ( m(t), pq (t)) + fem ( m(t), pq (t)) , ◦dt h t δm ∗ T Q

with the same notational conventions as for the bracket (4.26) and where the last Poisson bracket of semimartingales is defined in (2.19). D q E m Since dt f ( m(t), q(t)) = δf δm ( m(t), q(t)), ◦dt m(t) + hdf ( m(t), q(t)), ◦dt q(t)i, the stochastic Hamilton

equations (i.e., the stochastic versions of equations (4.10) and (4.9)) are, respectively, the stochastic Hamel equations q δ ht m(t) − J ∗ (◦d d hm ( m(t), q(t))), d q (t) = ◦d ( m(t), q(t)) ⇐⇒ dt m(t) = ad∗ δhq T Q t t t t δm ◦dt δmt ( m(t), q(t)) Q ∂ ht ∂ ht ∂ ht i i ( (t) = A dt mα (t) = −cαβ γ mγ (t) ◦ dt , d − Ajα ( q(t)) ◦ dt (4.28) q q (t)) ◦ d t t β ∂mβ ∂q j ∂mβ and ! ! ! et et et ∂h ∂h ∂h ∂Akα j dt mα (t) = −cαβ mγ (t) ◦ dt + pk (t) j ( q(t)) ◦ dt − Aα ( q(t)) ◦ dt , ∂mβ ∂q j ∂q ∂pj ! ! ! et et et ∂Akβ ∂h ∂h ∂h i i + dt , dt pi (t) = −pk (t) i ( q(t)) ◦ dt − dt dt q (t) = Aβ ( q(t)) ◦ dt ∂mβ ∂pi ∂q ∂mβ γ

et ∂h ∂q i

! (4.29) .

These last equations can be written intrinsically as dt m(t) = ad∗

◦dt

eq δh t δm

( m(t), q(t)) p

dtpq (t) =

e q δh t δm

◦dt

!!

T ∗Q

e m ( m(t), pq (t))) , m(t) − JT ∗ (T ∗ Q) ◦dt (dh t

(4.30)

∗

T Q ( m(t), pq (t)) + X◦d em ( m(t), pq (t)) h t

t

By repeating the arguments of the proof of Corollary 11, we derive the following stochastic Hamilton equations (with P = T ∗ Q) on g∗ × T ∗ Q for these type of Hamiltonian functionals: dt m(t) = ad∗

◦dt

q δ ht δm

( m(t), q(t))

m(t) − JT ∗ Q (◦dt dhtm ( m(t), q(t))),

q δ ht ( m(t), pq (t)) + Tdt dhtm ( m(t),pq (t)) ( m(t), pq (t)), dtpq (t) = dt δm T ∗Q where

d (dtpq − ǫ dt dhtm ( m(t), pq (t))) Tdt dhtm ( m(t),pq(t)) (pq ) = dǫ ǫ=0

and where the limit is taken in L2 (Ω).

(4.31)



13 April 2016

In addition, we have the non-homogeneous stochastic Lie-Poisson equations m(t) dt JT ∗ Q (pq (t)) = ad∗ δhq(t) ! JT ∗ Q (pq (t)) − JT ∗ Q ◦dt dht ( m(t), q(t) ◦dt

t δm(t)

23

(4.32)

( m(t), q(t))

for JT ∗ Q (pq (t)), where ( m(t), q(t)) is the solution of the stochastic Hamel equations (4.28). Remark 14. [The analog of the stochastic rigid body] In particular, the stochastic free rigid body dynamics on g∗ is obtained from (4.25) by assuming that the Hamiltonian h : g∗ × Q → R is of the form h(m, q) := 12 mα K αβ mβ , where K αβ are the components of the inner product on g∗ induced by an inner product on g. This Hamiltonian h is computed on semimartingales of the form d mα = rα dt+(Φk )α ◦dWtk with (Φk )α K αβ = ξkβ (notice that K := K αβ is an invertible matrix). Define uβ := rα K αβ . In particular

∂h ∂qi

= 0 and d

∂h ∂mβ

= rα K αβ dt + (Φk )α K αβ ◦ dWtk =

uβ dt + ξkβ ◦ dWtk . Thus, from (4.25), the stochastic free rigid body equations are

d mα = { mα , mβ }( uβ dt + ξkβ ◦ dWtk ) = − cαβ γ mγ dxβt .

4.3

♦

(4.33)

The Kolmogorov equations

We start from the equations (4.24) for the Poisson manifold valued stochastic process p(t) written in coordinates, namely, dtpi (t) = {pi , φα }(p(t))ξkα ◦ dWtk + {pi , ψ}(p(t))dt. Theorem 15. The generator of the process p(t) is the operator Lf := {f, ψ} +

1X {φα ξkα , {φβ ξkβ , f }}. 2

(4.34)

k

Proof. We first compute the difference between the Itˆ o and the Stratonovich differential in the process p(t). Since ∂φα i ij {p , φα }(p(t)) = Π (p(t)) ∂pj (p(t)), this difference is equal to 1 2

2 ∂φα ∂Πij ij ∂ φα + Π ∂pj ∂pl ∂pl ∂pj

lm

(p(t))Π

X ∂φβ (p(t)) m (p(t)) ξkα ξkβ ∂p k

!

dt =: B i (p(t))dt.

The generator then reads 1 Lf (p) = 2

! X ∂f ∂2f im jn ∂φα ∂φβ ij ∂ψ i α β Π Π (p) + Π + B (p) i ξk ξk m n i j j ∂p ∂p ∂p ∂p ∂p ∂p k

which is precisely the expression (4.34). Defining v(t, p) := Ep (f (p(t))) where p(0) = p, the function v satisfies Kolmogorov’s backward equation, namely, ∂v = Lv, ∂t

v(0, p) = f (p).

(4.35)

If the generator L is a hypoelliptic operator then there exists a probability density function ρ(t, p, p′ ), defined by R Ep (f (p(t))) = P f (p′ )ρ(t, p, p′ )dp′ ; here we assume that the Poisson manifold P has a volume form dp relative to which this integration is carried out. This function satisfies the forward Kolmogorov (or Fokker-Planck) equation: ∂ρ (t, p, p′ ) = L∗p′ ρ(t, p, p′ ) ∂t

(4.36)

with ρ(0, p, p′ ) equal to the Dirac measure δ(p′ − p) and where L∗ denotes the adjoint of L. Next, we give a sufficient condition, in terms of the measure on P used to define the probability density function ρ(t, p, p′ ), ensuring a nice formula for the formal adjoint of the operator L defined in (4.34). We take a measure



13 April 2016

24

on P which is induced by a volume form Λ ∈ Ωdim P (P ). We say that a volume form Λ on P is Hamiltonian, if 0 = £Xg Λ = diXg Λ + iXg dΛ = diXg Λ, for all g ∈ C ∞ (P ). Therefore div(f Xg )Λ = £f Xg Λ = if Xg dΛ + dif Xg Λ = d(f iXg Λ)

= df ∧ iXg Λ + f diXg Λ = −iXg (df ∧ Λ) + iXg df Λ = Xg [f ]Λ = {f, g}Λ

which shows that div(f Xg ) = {f, g} for any f, g ∈ C ∞ (P ). Hence, by the Stokes Theorem, Z Z Z Z dif Xg Λ = div(f Xg )Λ = {f, g}Λ = P

Z

f iXg Λ = −

Z

giXf Λ;

(4.37)

∂P

∂P

∂P

P

P

if Xg Λ =

the last equality follows by skew-symmetry of the Poisson bracket. Now let f, g, h ∈ C ∞ (P ) and integrate the identity {hf, g} = h{f, g} + f {h, g} to get Z Z Z f {h, g}Λ = {hf, g}Λ. h{f, g}Λ + P

P

P

By (4.37), the term on the right hand side vanishes if ∂P = ∅ or if at least one of f or g vanish on ∂P . In these cases, we have Z Z {f, g}hΛ = f {g, h}Λ. (4.38) P

P

n(n−1)/2

If (P, ω) is a 2n-dimensional symplectic manifold, the Liouville volume Λ := (−1) n! ω ∧ . . . ∧ ω (n times) is Hamiltonian. Indeed, since £Xg ω = 0 for any g ∈ C ∞ (P ), it immediately follows that £Xg Λ = 0. Corollary 16. Let (P, {·, ·}) be a boundaryless Poisson manifold and Λ a Hamiltonian volume form on P . Relative to the L2 -inner product on P defined by Λ, the formal adjoint of the linear operator L defined in (4.34) has the expression 1X {φβ ξkβ , {φα ξkα , f }}. (4.39) L∗ f = −{f, ψ} + 2 k

This follows directly from (4.34) and (4.38). Consider the Poisson manifold g∗ × Q and the stochastic Hamiltonian (2.28). Define the semimartingale u(t) := u( m( q(t), p(t)), q(t)), where u ∈ C ∞ (g∗ × Q). In this case, Kolmogorov’s backward equation for v(t, m, q) := E(m,q) (f ( m( q(t), p(t))), q(t))) takes the form 1X ∂v = {v, mα uα − ℓ(u, q)} + {mα ξkα , {mβ ξkβ , v}} ∂t 2

(4.40)

k

with v(0, m, q) = f (m, q). Now choose a Hamiltonian volume form Λ on the Poisson manifold g∗ × T ∗ Q. Using the measure defined by Λ, and computing the formal adjoint of L (the right hand side of (4.40)) given by (4.39), we get Kolmogorov’s forward, or Fokker-Planck, equation ∂ρ 1X = −{v, mα uα − ℓ(u, q)} + {mα ξkα , {mβ ξkβ , v}} ∂t 2

(4.41)

k

with ρ(0, (m, q), (m′ , q ′ )) = δ((m′ , q ′ ) − (m, q)). Remark 17. Assume we work on g∗ , where g is a compact Lie algebra, for simplicity. Then there is an invariant inner product on g and, using it, we define an invariant inner product on g∗ whose norm is denoted by k · k. In this case, m 7→ kmk2 is a Casimir function. As Casimirs are conserved along the stochastic flows of the stochastic



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Hamiltonian semimartingales, we have dt kmk2 ( q(t), p(t)) = 0. As a consequence, there exists (c.f. [20]) an invariant probability measure µ on g∗ × Q for the motion. Namely, the measure satisfies Z P(m,q) ( m( q(t), p(t)), q(t)) ∈ B)dµ(m, q) = µ(B) for all Borel set B ⊂ g∗ × Q. This measure desintegrates along the level sets of the Casimir kmk2 .

♦

Acknowledgements We are enormously grateful to our colleagues for their helpful encouraging remarks and interesting enjoyable discussions: A. Arnaudon, S. Albeverio, J.-M. Bismut, N. Bou-Rabee, A. L. Castro, M. D. Chekroun, G. Chirikjian, D. O. Crisan, J. Eldering, M. Engel, F. Gay-Balmaz, A. Grandchamp, P. Lynch, J.-P. Ortega, G. Pavliotis, V. Putkaradze, and C. Tronci. We also acknowledge the Bernoulli Center at EPFL where parts of this work were initiated. ABC was partially supported by Portuguese FCT grant SFRH/BSAB/105789/2014, DDH by ERC Advanced Grant 267382 FCCA, and TSR by NCCR SwissMAP grant of the Swiss NSF.

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