The stochastic differential equation approach toanalysis on path space

Xue-Mei Li

AbstractWe discuss the use of techniques from stochastic differential equations in analysis onpath spaces.

1 Introduction

A. Infinite dimensional geometry received attention particularly due to motivation fromphysics. Infinite dimensional spaces which arise in geometry, topology and analysis,are usually spaces of maps from one finite dimensional manifold to another, of whichthe simplest are the space of paths. Since the 1960’s, a rigorous theory on mappingspaces, and more generally infinite dimensional manifolds modelled on Banach spaces,has been developed by global non-linear analysts, see [12, 13]. In their global approachthe Banach spaces of linear analysis were replaced by Banach manifolds. Anotherremarkable development in around the same period is integration theory on infinitedimensional spaces see e.g. [29], motivated by quantum field theory. The combinationof this with Ito’s stochastic calculus in the development of Malliavin calculus [40] ledto analysis on spaces of paths which are continuous but not smooth, whose elementsinclude sample paths of suitable stochastic processes such as Brownian motions andBrownian bridges on manifolds.

B. LetM be a smooth Riemannian manifold. Fix a point x0 inM and a time T > 0define the path space Cx0M over M to be,

Cx0M = σ : [0, T ]→M : σ(0) = x0 and σ is continuous. (1.1)

Two fundamental problems in analysis on infinite dimensional spaces are the de-velopment of a Sobolev Calculus, in analogous to that on Rn and on finite dimensionalmanifold, and an L2 DeRham-Hodge theory. As stochastic integration of a continu-ous time stochastic process against a Brownian motion is in general at most Holdercontinuous of order α < 1/2, we are obliged to use H-derivatives in the sense ofMalliavin calculus for the construction of a gradient operator (although by now roughpath structure can and has been used). For this there are two approaches: the stochas-tic development approach and the stochastic differential equation approach. Let //(σ)denotes stochastic parallel translation along a path σ. The first approach leads to theBismut tangent space

Hσ = t→ //t(σ)ht, where h ∈ H, t ∈ [0, T ]

INTRODUCTION 2

forH ≡ L1,2

0 (Rn) = h : [0, T ]→ Rn continuous with , h(0) = 0,

the Sobolev space of finite energy. See e.g.[7, 8, 31].It is endowed with the inner product fromL1,2

0 by the stochastic parallel translation:

〈//(σ)h1, //(σ)h2〉Hσ = 〈h1, h2〉H =∫ T

0

〈h(s), h2(s)〉ds. (1.2)

If v1, v2 belong to Hσ , denote Dds = //s

dds//

−1s , then

〈v1, v2〉 =∫ T

0

〈Ddsv1s ,D

dsv2s〉σsds. (1.3)

C. The second approach involves a choice of a stochastic differential equation(SDE) which induces the linear connection, based on which the manifold and otherstructures on the path spaces are defined, c.f. §3. See e.g.[3, 18] for the use of gra-dient stochastic differential equations, [27] for left invariant SDEs on Lie groups and[14, 19] etc. for general elliptic and semi-elliptic SDEs. In the last case the main tool isthe theory of connection associated to the stochastic differential equation as developedin [15, 16]: every metric connection can be induced by a stochastic differential equa-tion and this connection leads to decomposition of noise and filtering out redundantnoise and alternative Sobolev spaces. The Sobolev spaces induced by the Ito map ofSDEs are shown to be independent of the choices of the stochastic differential equa-tions. LetX0 be a smooth vector field,X : Rm×M → TM a smooth bundle map andBt a m-dimensional Brownian motion on a probability space (Ω,F ,Ft, P ). We maychoose this probability space to be the canonical probability spaces, the Wiener spaceover Rm. Let

dxt = X(xt) dBt +X0(xt)dt (1.4)

be the corresponding stochastic differential equation. Its solution from x0 is denotedby xt. Assume that the Markov generator of the SDE is given by 1

2∆ +LZ where Z isa vector field given by 1

2

∑i∇Xi(Xi) +X0 for Xi(x) = X(x)(ei) and ei an o.n.b.

of Rm. Assume that the SDE is conservative. Denote by xt the solution from x0 andξt(x, ω) the solution flow to (1.4) and so xt = ξt(x0). Let Tξt be its derivative flowand I : C0Rm → Cx0M the Ito map defined by:

I(ω)(t) = xt(ω)

which is differentiable in the direction of Cameron-Martin direction, see §4, and

TI(h)(t) = Tξt

∫ t

0

Tξ−1s (X(ξs)(hs)) ds. (1.5)

This discussion is rigorous for compact manifolds. Otherwise we refer to [5, 37, 38, 43]for related discussions. Define

Hσ = ETI(h)|σ(I) : h ∈ L1,20 (Rm).

INTRODUCTION 3

Let ∇ be the LW connection induced by X , see (3.1), and let ∇ be its adjoint con-nection, see §3.2. Let //· be the stochastic parallel transport corresponding to ∇ andfor a vector field Vt along a path σ let D

dtVt = //tddt//t

−1Vt. Consider the stochastic

covariant differential equation

D

dtVt = −1

2Ric

#

xt (Vt) + ∇Z(Vt), ≤ t ≤ T (1.6)

where ˘Ric#x : TxM → TxM corresponds to the Ricci tensor of∇: for u1, u2 ∈ TxM ,

〈Ric#

x (u1), u2〉 = Ric(u1, u2). The solution induces the linear map Wt : Tσ0M →TσtM , the damped stochastic parallel transports. The tangent spaces induced by ∇are:

Hσ =Wt

∫ t

0

W−1s (X(σs)(hs))ds, where h ∈ H, t ∈ [0, T ]

(1.7)

These are Hilbert spaces with inner product induced by the damped parallel translation:

〈v1, v2〉Hσ =∫ T

0

⟨IDdsv1s ,

IDdsv2s

⟩ds (1.8)

where IDdt = Wt

DdtW

−1t . It follows that

IDdt

=D

dt+

12

Ric# − ∇Z. (1.9)

D. For M compact, Hσ in part B and Hσ in part C coincide as sets. Define H =∪Hσ , H∗ = ∪H∗σ , H = ∪Hσ , H∗ = ∪H∗σ be the tensor sub-bundles of the tangentand cotangent space of Cx0M . The two linear maps //(σ) and W (σ) induce differentbut equivalent inner products in the case of non-zero Ricci curvature. In both case thereare integration by parts formulae and the exterior differential operator

d : L2(Cx0M ; R)→L2ΓH∗, ( or L2ΓH∗)

are closable linear operator whose closure is again denoted by d. For non-comapctbase manifold, the integration by parts formula holds under suitable conditions and weassume that this holds whenever d is used. If the measure involved satisfies a Poincareinequality, the Poincare constant, when the second inner product is used, is the same asthat for a Gaussian measure.

E. Consider the space of H differential q-forms ∧Hq∗ := ∪σ ∧q (Hσ)∗, their L2

sections L2Γ ∧q H∗, and the complex

L2(Cx0M ; R) d→ L2Γ ∧1 (H∗) . . . d→ L2Γ ∧q H∗ d→ L2Γ ∧q+1 H∗d→ . . . . (1.10)

However the non-integrability of the vector bundle ∪σHσ , in the general case andPalais’s formula means that d does not map q forms to q + 1 forms [9, 20]. In [19]an alternative family of tensor spaces, H(q), are introduced for which an restricted L2

EXAMPES 4

Hodge decomposition theorem has been proved to hold for the usual differentiationexterior operator:

L2(Cx0M ; R) d→ L2Γ(H∗) . . . d→ L2Γ(Hq)∗ d→ L2Γ(Hq+1)∗ d→ . . . .

The full Hodge theory remains to be proved.As the path space is modelled on the Banach space Ω = C0(Rn), the first approach

could be considered as unwrapping the path space using the stochastic developmentmap and the second approach by the Ito map induced by the SDE. In this article weput together developments through the second approach and make necessary compar-ison whenever needed. See [39] for a discussion on Ito map by the stochastic anti-development map.

Most of the results here works for semi-elliptic diffusion measures. But for trans-parency, we will be mainly discussing the case of an elliptic diffusion. We would liketo emphasize that apriori the geometry of the path space depend on the choice of themeasure, the Riemannian structure and the linear connection. Our aim is to establishan L2 theory which reflects the topology of the path space and its sub-spaces, espe-cially loop spaces. The smooth cohomologies on Cx0M , which are trivial on the pathspace [33, 36], may differs from an Lp cohomology. The Lp cohomologies could bedifferent from the smooth cohomologies even when the space under discussion is fi-nite dimensional. In our case the Lp cohomologies may differ from each other withdifferent constructions. See also [2, 35, 44, 45] for related developments.

If M is simply connected then Cx0M is simply connected as every path on thebased path space is homotopic to the constant loop at x0: Set σ0 = σ, σ0(t) ≡ x0 andσs(t) = σ((1 − s)t). For the free path space the topology of CM is the same as M asevery path is homotopic to its initial point, a point on M . This situation changes forthe loop space. Most of the problem discussed in this article are open on loops spaces.

2 Exampes

The relevance of this analysis can be illustrated by a number of basic examples. Inmost of the examples the inner product for the Bismut tangent space is defined by D

ds ,see (1.8), unless otherwise mentioned.

Example 2.1 The simple task of solving a Poisson equation, in the context of analysison path spaces, involves a series of theorems on path spaces. Let ∆ = −d∗d be theinfinite dimensional Laplacian with d∗ : LΓH∗ → L2(Cx0M ; R), or more generallyd∗ : L2Γ(Hq)∗ → LΓ(Hq−1)∗, the L2 adjoint of d. Let g be an L2 function withrespect to our probability measure on CxM . Is the Poisson equation ∆f = g solvablein the domain of ∆? By a Theorem in [21], the domain of d contains BC2 functionsif the initial domain contains the set of smooth cylindrical functions with compactsupport and consists of BC1 functions, the constant 1 is in the domain of d and hencethe domain of ∆. We may integrate both sides of the Poisson equation to see that∫g = 0 necessarily. By the Hodge Decomposition Theorem in [19], for q = 1, 2,

L2Ωq = Im(d)⊕ Im(d∗)⊕Hq

EXAMPES 5

where Ωp stands for an element of L2Γ(Hq)∗, anH-valued differential q-forms andHq

the space of L2 harmonic 1-forms. Take q = 0, L2Ω0 = Dom(d∗)⊕H1 which meansthat g = d∗φ + h for a 1-form φ and a harmonic function h. The harmonic functionh has zero mean if g has zero mean. If we have irreducibility for the correspondingDirichlet form, then dh = 0 which implies that h is a constant (irreducibility of theDirichlet form) and is hence zero. Such irreducibility follows from a Clark-Oconeformula.

For q = 0, we assume that d has a spectral gap, which means that Poincare in-equality holds. We comment in a later section on the validity of such inequalities. Ithas closed range and

L2Ωp = Dom(d)⊕Dom(d∗)⊕Hp

It follows thatφ = d∗ψ + dF + k

where ψ is a 2-form, F a function, k a harmonic 1-form. Now

∆f = g = d∗φ+ h = −∆F.

Hence F is a solution to the Poisson equation. We refer to the following on discussionson irreducibility of Dirichlet forms, Clark-Ocone formula and Poincare inequalities onpath and loop spaces: [1, 3, 4, 6, 26, 28, 30].

Example 2.2 We interpret integral representation theorem ofL2 functions by the Hodgedecomposition theorem. The Hodge decomposition theorem we mentioned earliershows that

f =∫fdµ+ d∗φ

Now

(d∗φ)# = divφ# =∫ 1

0

〈 IDdsφ#s , X(xs)dBs〉

This gives an integral representation of the L2 function by stochastic integrals. Notethat if X induced the connection on TM then X(xs)dBs is the martingale part of thestochastic development map of xs [16].

Example 2.3 It was shown in [17] Bismut type formula is equivalent to the Driver’s in-tegration by parts formula, the foundation formula to Sobolev calculus on path spaces,by induction and the Markov property. Let f be a BC1 function on a manifold andxt an elliptic diffusion. Ito’s formula leads to the following integral representation forf (xT ), which can be considered a function on the path space depending on 1 time,

f (xT ) = f (x0) +∫ T

0

〈∇PT−sf (xs), X(xs)dBs〉.

EXAMPES 6

Hence

Ef (xT )∫ T

0

〈vs, dxs〉 = E∫ T

0

〈∇PT−sf (xs), X(xs)dBs〉∫ T

0

〈vs, X(xs)dBs〉

= E∫ T

0

〈∇PT−sf (xs), vs〉ds.

A choice vt [Li92], as inspired by Elliott-Kohlman, can be taken to be Tξs(v). Theright hand side equals dpT f (v) by the Markov property and so,

dPT f (v) =1T

Ef (xT )∫ T

0

〈vs, X(xs)dBs〉.

Conditioning on xt giving an intrinsic formula using the damped stochastic paralleltranslation Wt, defined by (1.6),

dPT f (v) =1T

Ef (xT )∫ T

0

〈Ws(v), X(xs)dBs〉.

This technique has many applications in stochastic partial differential equations.

Example 2.4 Consider the statement “If a process σt is a Brownian motion then f (σt)is a martingale for all harmonic function f .” Take the Brownian motion measure on thepath space and let σt be the canonical process. The Clark-Ocone integral representationTheorem says that:

f (σt) = Ef (σt) +∫ 1

0

〈E˙︷ ︸︸ ︷

min(r, t)//−1t ∇f (σt)|Fr, dbr〉

= Ef (σt) +∫ t

0

〈E//−1t ∇f (σt)|Fr, dbr〉,

where b· is the stochastic anti-development map of σ·. Form this identity we see thatf (σt) is a martingale if and only if Ef (σt) = Ef (σs) and E//−1

t ∇f (σt)|Fr =E//−1

s ∇f (σs)|Fr.The first identity folows from ∆f = 0 and the second identity that //−1

t ∇f (σt) isa martingale comes for free. The latter is interesting of its own right, c.f. [34, 47].

Example 2.5 Bounds on the variance of a function in the domain of d can be deducedfrom the Clark-Ocone formula,

f (σt) = Ef (σt) +∫ t

0

〈E d

dr//−1

t ∇f (σt)|Fr, dbr〉

where b· is the stochastic development map. we see that

Var(f (σt)) ≤ E|∇f |2(σt) = Pt(|∇f |2).

If Pt(|∇f |2) ≤ e−Ct|∇f |2L2, then Var(f (σt)) ≤ e−Ct|∇f |2L2 . The variance estimate

is a Poincare inequality.

EXAMPES 7

Example 2.6 Intertwining and Gradient Estimates. Under suitable conditions, for f ∈BC1(M ),

dPtf = Edf (Tξt(v)) = Edf (Wt(v)).

This is one of the crucial point for the simple proof of Bismut type formula. The firstidentity require the so-called strong 1-completeness condition, a weaker assumptionthan the existence of smooth stochastic flow and some integrability condition on thederivative flow [37, 38]. In particular the first identity holds for stochastic differen-tial equations, which are not necessarily elliptic, with coefficients reasonably smooth,bounds on the first two derivatives of the diffusion and first derivative of the drift term.The second identity follows from taking conditional expectation with respect to x· andit is valid if the connection used to define Wt is the same as that induced by the SDE[16].

Example 2.7 We discuss the relation between Bismut type formula, intertwinning re-lation and integration by parts and Differentiation formula. Let h ∈ Hσ and in par-ticular h0 = 0. Driver’s integration by parts formula says that if ∇ is torsion skewsymmetric and F in the closure of d,∫Cx0M

dF (//·(σ)h·) dµx0 (σ) =∫Cx0M

F (σ)∫ T

0

〈hs+12//−1s Ric#

σs (//shs), dbs〉 dµx0 (σ)

where bt is the stocchastic anti-development map of the Brownian motion.

hs +12//−1s Ric#

σs (//shs) = //−1s

IDds

(//shs),

Let h be a function in L2,1, we write:

//shs = //s(hs − h0) + (//sh0 −Wsh0) +Wsh0.

To the first two parts we apply the integration by part formula above, note that

IDds

(//sh0 −Wsh0) =D

ds(//sh0 −Wsh0) +

12

Ric#(//sh0 −Wsh0) =12

Ric#(//sh0).

It follows that

Edf (//s(hs − h0) + (//sh0 −Wsh0)) = E∫ T

0

〈hs +12//−1s Ric#

σs (//shs), dbs〉.

To understand what is Edf (W·h0), recall the intertwinning relation in example 2.6:Edf (Wt(v)) = dPtf (v) where f is a suitably smooth function from M → R. LetF (σ) = g(σt1 , . . . , σtk ) be a smooth cylindrical function where g : M × . . .×M → Ris smooth with compact support. Induction on k we see that for any v ∈ Tx0M ,∫

dF (W·v)dµx0 (σ) = d

(∫F (σ)dµx0

)(v),

see [32]. Adding the two identities together taking v = h0 leads to the differentiationformula below, see Theorem 7.36 in [10] ,

dEx0F (v) = EdF (W·h0) + EF∫ T

0

〈//−1s

IDds

(//shs), dbs〉.

BACKGROUND 8

3 Background

If M is a general metrizable space and d a complete metric which is compatible withthe topology of M and T > 0 a real number. The set C([0, T ];M ) of all continuousfunctions from [0, T ] to M is given the compact open topology. Open sets are of theform σ : σ(K) ⊂ U where K is a compact subset of [0, T ] and U an open subset ofM . The metric

ρ(σ1, σt) = suptd(σ1(t), σ2(t)),

induces the compact open topology and is hence independent of the choice of the dis-tance function d on the base space. IfM is separable then C([0, T ],M ) is separable (Ifσn is a sequence of curves in the path space limn→∞ σn = σ means limn→∞ σn(t) =σ(t) for every t).

Let M be a smooth Riemannian manifold endowed with metric connection ∇which are not necessarily torsion free. The following manifold structure on Cx0Mwas considered by J. Eells (for the Levi-Civita connection). If σ is a C1 smooth curvein Cx0M , consider the pull back tangent bundle σ∗(TM ).

Γσ∗(TM ) := v : [0, T ]→ TM continuous | v(0) = 0, v(t) ∈ TσtM, .

It can be identified through parallel translation, with

u | u(0) = 0, u(t) ∈ Tx0M,u : [0, T ]→ Tx0M is continuous .

The tangent space is isomorphic to the Wiener space over Rn by a chosen linear framer : Rn → Tx0M and we have a Finsler manifold, with ‖v‖ = supt |//−1

t vt| =supt |vt|. Let σ be a C1 curve in Cx0M . Let 2a > 0 be the injectivity radius ofthe image of σ. Consider

Sa := N ∈ Γσ∗(TM ) : supt|v|t ≤ a.

Define φσ : Sa → Uσ ⊂ Cx0M by φσ(v)(t) = expσt (vt). Here Uσ = Image(φσ). Thecollection Uσ, φσ) : σ is C1 is an atlas for Cx0M . As every continuous curve on afinite time interval can be approximated in the uniform norm by a smooth curve, theopen sets Uσ is a cover of the path space. The map φσβ ≡ φ−1

σ φβ is smooth as a mapon Γσ∗(TM ): for each u there is a linear map Tu : Γσ∗(TM )→ Γσ∗(TM ) such that

‖φσβ(u+ εh)− φσβ(u)− εTu(h)‖ε‖h‖

→ 0.

For each t, let φσβ(u)(t) = exp−1σ(t)(expβt (ut)) and note that

φσβ(u+ h)(t) = φσβ(u)(t) + d(φσβ)(u)(t)(εht) +Rt(ε),

where limt→0Rt(ε)ε|h| = 0. Now

supt|d(φσβ)(u)(t)(ht)| ≤ sup

t|d(φσβ)(ut)||ht| ≤ sup

t|d(φσβ)(ut)|‖h‖.

As the curves α and β has compact range, d(φσβ)(ut)(ε(ht)) is continuous in t. Itfollows that Rt(ε) is continuous in t and hence converges uniformly. If Tu(h)(t) =d(φσβ)(u)(t)(ht), Tu is a bounded linear map.

BACKGROUND 9

3.1 The geometry of stochastic differential equationsOne one hand the manifold structure on the path space may depend on the Riemannianmetric and the linear connection through which the exponential map is defined. On theother hand any metric connection onM can be defined through a stochastic differentialequation as explained below.The following is taken from [15, 16].

Let M be a smooth manifold. A linear connection∇ is adapted or metric if for anyv ∈ TxM and U,W ∈ ΓM ,

∇v〈U,W 〉 = 〈∇vU,W 〉+ 〈U,∇vW 〉.

Consider the stochastic differential equation (1.4) on M . An SDE is cohesive if thelinear maps X(x), x ∈ M have constant rank and that X0(x) ∈ Ex ≡ Im[X(x)].For simplicity we assume that for each x ∈ M , X(x) : Rm → TxM is elliptic, i.e.X(x) is a surjection. The map X induces a Riemannian structure on M in the ellipticcase and a sub-Riemannian structure on E in the case of cohesiveness.

At each point x ∈M the linear map X(x) : Rm = kerX(x)⊕ [kerX(x)]⊥ → Exsplits Rm and defines a right inverse map:

Y (x) : Ex → [kerX(x)]⊥.

which induces a metric on Ex and Y (x) is the adjoint of X(x): 〈X(x)(e), u〉Ex =〈e, Y (x)v〉Rm .

In the case of M = Rn and the SDE is elliptic, each X(x) : Rm → Rn is an n×mmatrices and Y (x) is the m×n matrix determined by Y (x) = XT (x)[X(x)X(x)T ]−1.The Riemannian metric is: 〈u, v〉x = 〈Y (x)(u), Y (x)(v)〉 for u, v ∈ Ex.

If v ∈ Tx0M is a tangent vector and U ∈ Γ t Ex an E valued vector field define

(∇vU )(x0) = X(x0)D(Y (x)U (x))(v). (3.1)

Then ∇ defines a metric semi-connection (connection in the elliptic case) characterisedby the property:

∇vX(e) ≡ 0, ∀e ∈ [kerX(x0)]⊥, v ∈ Tx0M. (3.2)

In the case of X is induced from an isometric embedding this connection is the Levi-Civita connection. In general the torsion does not vanish. All metric adapted linearconnection can be obtained in this way: a map X exists to give rise to the given con-nection. This last facts uses a theorem of Narasimhan and Ramanan on universal con-nections [41].

If ∇ is the connection determined by (1.4) its adjoint connection can be viewed as

3.2 Torsion skew symmetric connectionsLet M be a manifold with a Riemannian metric (gij) and ∇ a compatible connection.The corresponding torsion tensor field T : TM×TM → TM is defined by T (U, V ) =∇V U−∇UV −[U, V ] where U and V are two vector fields. In terms of the Christoffel

BACKGROUND 10

symbols, T kij = (Γkij − Γkji). It is anti-symmetric in i, j and if it is furthermore anti-symmetric in all three indices

〈T (u, v), w〉 = −〈T (u,w), v〉

then 〈Tx(u, v), w〉 defines a differential 3-form and we say that T is (totally) skew-symmetric. Skew symmetric connections are interesting mathematical physics objects.

A metric connection ∇ has skew symmetric torsion if and only if its adjoint con-nection ∇ := ∇ − T is a metric connection with respect to this same Riemannianmetric. The stochastic parallel translation defined by ∇ is an isometry, while we havelittle control over the norm of the parallel transport defined by the adjoint connectionwithout further assumptions. If the adjoint connection is adapted to some Riemannianmetric, which is not necessarily the original metric then the stochastic parallel trans-port //· along the paths of the stochastic prcoesses xT,y0t , obtained by conditioning xtto be at y0 at time T , is a bounded random variable with valued in L(Tx0 , Ty0M ). Theconverse is also true. See Teorem 1.3.8 in [16] for detail.

Any connection∇ differs from the Levi-Civita connection∇LC by a tensor whichsplits into a symmetric tensor S and an antisymmetric tensor determined by the torsion:

∇V U = ∇LCV U + S(V,U ) +12T (V,U )

and for vector fields V and U . The connection ∇ is a metric connection if and only if

〈S(w, u), v〉+ 〈S(w, v), u〉+12

(〈T (w, u), v〉+ 〈T (w, v), u〉) = 0,

In this case〈S(w, u), v〉+ 〈S(w, v), u〉 = −〈S(u, v), w〉

and S is determined by T . So ∇ is t.s.s. if and only if S = 0. On the other handsince the geodesic equation is symmetric in the two lower indices the connection withChristoffel symbol Γkij and Γkij = Γkji determine the same set of geodesics. They aresaid to be adjoint to each other. That they determine the same family of geodesicsmeans that the manifold structure on the space of continuous paths on M determinedby the exponential maps of these connections are the same. Torsion skew symmetricconnections were introduced into the context of path space analysis by B. Driver [10].

3.3 Relating two norms on Bismut tangent spacesWe compare the two norms on Bismut tangent spaces. Assume that Z = 0 for sim-plicity so the measure µx0 on the path space is the Brownian motion measure. Let∇1 : L2(Cx0M ; R)→ ΓH be the gradient operator using the norm

〈U, V 〉Hσ =∫ T

0

〈 dds//−1s (σ)Us,

d

ds//−1s (σ)Vs〉 =

∫ T

0

〈DdsUs,

D

dsVs〉 ds

and ∇2 : L2(Cx0M ; R)→ γH that by

〈U, V 〉Hσ =∫ T

0

〈Wsd

dsW−1s Us,Ws

d

dsW−1s Vs〉 =

∫ T

0

〈 IDdsUs,

IDdsVs〉 ds.

BACKGROUND 11

Let F be a function in the domain of the definition of the operator d, see §4.2. Forexample take F to be a BC1 function or a cylindrical functions. For h ∈ L1,2

0 .∫〈Dds∇1F,

D

dsh〉 =

∫〈 IDds∇2F,

IDdsh〉.

Since∫〈 IDds∇2F,

IDdsh〉 =

∫ T

0

〈//−1s

IDds∇2F, hs〉ds+

12

∫ T

0

〈Ric#(IDds∇2F ), hs〉ds.

D

ds∇1F (s) =

IDds∇2F +

12

∫ T

s

//−1r Ric#(

IDdr∇2F )dr.

The two norms are related as following:

〈U, V 〉Hσ =∫ T

0

〈(Dds

+12

#

Ric)Us, (D

ds+

12

#

Ric)Vs〉

= 〈U, V 〉Hσ +∫ T

0

Ric

(14Us, Ric

#Vs

)+∫ T

0

12Ric(Us,

D

dsVs) +

∫ T

0

12Ric(

D

dsUs, Vs)

It follows that

|U |2Hσ ≤ 3|U |2Hσ +∫ T

0

|Ric#|2σs |Us|2ds

|U |2Hσ ≤ 3|U |2Hσ +∫ T

0

|Ric#|2σs |Us|2ds

If the Ricci curvature is bounded the two norms are equivalent.

|U |2Hσ ≤ 3|U |2Hσ +∫ T

0

|Ric#|2 |//s∫ s

0

(d

dr//−1r Ur)dr|2σsds ≤ (3 +

∫ T

0

|Ric#|2σs ds)|U |2Hσ .

For the other way around, we consider the norms as a function of T ,

|U |2Hσ (T ) ≤ 3|U |2Hσ (T ) +∫ T

0

|Ric#|2 |//s∫ s

0

(d

dr//−1r Ur)dr|2σsds

≤ 3|U |2Hσ (T ) +∫ T

0

|Ric#|2σs |U |2Hσ (s)ds.

For the equivalence of the induced norms on vector fields, E|U |2H and E|U |2H wesuppose that

∫ ∫ T0|Ric#|2σsdsdµx0 (σ) < ∞. Then the two norms are equivalent on

vector fields which are uniformly bounded or we may assume that the Ricci curvatureis bounded.

THE ITO MAP AND BASIC ASSUMPTIONS 12

4 The Ito map and basic assumptions

Let ∇ and ∇ ≡ ∇ be a pair of linear connections adjoint to each other. Take v ∈σ∗(TM ) define covariant differentiation along a C1 curve σ: D

dtvt = //tddt (//t)

−1Vt.In local coordinates (Dvt)k = d

dtvkt + Γkji(σt)v

jt dσit for Γki,j the Christoffel symbols

For v ∈ TxM let Txξt be the space derivative of ξt(·, ω) in probability. The derivativeflow vt = Tξt(v0) satisfies the SDE:

Dvt = ∇Xj(vt) dBjt + ∇X0(vt)dt.

Take ∇ to be that defined in (3.1), Let vt = ETξt(v)| xs : 0 ≤ s ≤ T. Then vtsatisfies

Dvt = −12

(Ric)#(vt)dt+ ∇Z(vt)dt (4.1)

The Ito map I induced by the SDE (1.4), I(ω)(t) = xt(ω) is a measurable map fromthe path space Ω over Rn to the path space Cx0M over M . Let Vt = TIt(h) be itsMalliavin dertivative for h a Cameron Martin vector then Vt satisfies

DVt = ∇Xεj (Vt) dBjt + ∇X0(Vt)dt+X(xt)(ht)dt.

In terms of the derivative flow, TωI(h)(t) = Tξt∫ t

0Tξ−1

s (X(hs))ds.The corresponding conditional expectation of the vector field Vt satisfies

Vt = −12

(Ric)#(Vt)dt+ ∇Z(Vt)dt+X(xt)(ht)dt, (4.2)

which means that //t−1Vt is of bounded variation and a vector in theHσ .

The following are fundamental observations that lead us to the choice of torsionskew symmetric connections and the basic assumptions.

Proposition 4.1 [16] For compact M ,

sup0≤s≤T

|Ws|L(Tx0M ;TxsM ) and sup0≤s≤T

∣∣W−1s

∣∣L(TxsM ;Tx0M )

lie in Lp for all 1 ≤ p < ∞. If also ∇ is metric, then both are in L∞ with respect toany Riemannian metric on M .

Let ∇ be a torsion skew symmetric connection on a complete manifold M . IfV ∈ LpΓH, the space of Lp H-vector fields on Cx0M . Then, for any inner product onTx0M and almost all σ ∈ Cx0M

supt|W−1

t Vt(σ)| = supt

∣∣∣∣∫ t

0

W−1s

IDdsVs(σ)ds

∣∣∣∣≤ T

12 sup

t|W−1

t |L(Tσ(t)M ;Tx0M )‖V ‖Hσ

By Proposition 4.1, forM compact supt |W−1t Vt(σ)| is inLp. The above consideration

leads to the following standard assumptions.

THE ITO MAP AND BASIC ASSUMPTIONS 13

• Condition (M0) : The damped stochastic parallel transport map Wt satisfies:

(i) sup0≤t≤T∣∣(W−1

t

∣∣L(Txt ;Tx0M ) ∈ L

∞ and

(ii) sup0≤t≤T |∇Wt(−)X|L(Tx0M ;L(Rm;Txt )) ∈ L∞

• Condition (M ) : The adjoint connection ∇′ is metric for some Riemannianmetric on TM , (which we will denote by 〈·, ·〉′).

Condition (M) holds if∇ is torsion skew symmetric. For compact manifolds conditionM implies condition M0. The following proposition was given in [23].

Proposition 4.2 Assume condition (M0) holds. Then for all 1 ≤ p < ∞ there is aconstant αp with

E

sup0≤s≤T

∣∣W−1s TIs(h)

∣∣pTx0M

∣∣∣ Fx0

≤ αp ‖h‖pH , all h ∈ H a.s.

4.1 Conditioning vector fieldsFor suitable vector field h : C0Rm → H define a measurable vector field TI(h) onCx0M by conditioning the derivative of the Ito map with respect to the measure µx0

TI(h)(σ) = E TI(h(·)) |I(·) = σ .

If h ia L2 vector field the induced vector field on the path space is L2, given conditionM0. Furthermore all L2 sections ofH are pushed froward vector fields.

Proposition 4.3 [Corollary 3.7 [23]] Assume that Condition (M0) holds. For 1 < q <∞ the map h 7→ TI(h) gives a continuous linear map

TI(−):Lq(C0Rm;H)→ LqΓH.

Furthermore the map is surjective.

Furthermore if h ∈ C0Rm → H is adapted to the filtration of I, there is explicitformulae for the pushed forward map and its right inverse.

Proposition 4.4 [19] The map TI : L2 (C0(Rm),Fx0 ,P|Fx0 ;H)→ L2ΓH· given byTI(h·)(σ) = TI(h·(σ))(σ) is

h 7→W·

∫ ·0

W−1s X(σ(s))h(σ)sds (4.3)

and is a Hilbert submersion with inverse and adjoint given by

v 7→∫ ·

0

Y (xs(·))D∂svsds (4.4)

THE ITO MAP AND BASIC ASSUMPTIONS 14

for Y (x) : TxM → Rm the adjoint of X(x), x ∈ M . The co-joint of TI(−) definedby, ∫

C0RmI∗(φ)(h)dP =

∫C0Rm

φ(TI(h)

)dP (4.5)

can be written I∗(−) : L2Γ(H·∗)→ L2 (C0(Rm),Fx0 ;H∗) in the sense that it agreeswith φ 7→ EI∗(φ) |Fx0 for φ a 1-form on Cx0M .

4.2 Pull back by Ito mapLet Ω ⊂ Rn be a bounded domain. A locally integrable function v is called a weakderivative of u if it satisfies:∫

Ω

φvdx = (−1)|α|∫

Ω

uDαφdx,∀φ ∈ C |α|0 (Ω).

Write v = Dαu. WriteW k the linear space of k-times weakly differentiable functions.Let u and v be locally integrable in Ω then Dαuh(x) = (Dαu)h(x). Then v = Dαuif and only if there exists a sequence of C∞(Ω) functions um converging to u inL1

loc(Ω) whose derivatives Dαum converges to v in L1loc(Ω). This last property is one

of the basic property we seek to prove on the path space.Denote by Cyl the space of smooth cylindrical functions on Cx0M . Let Dom(d) be

a linear subspace of L∞(Cx0M ; R) with Cyl ⊂ Dom(d) ⊂ BC2. The linear operatord : Dom(d) → ∩1≤p<∞L

pΓH∗, where the d is the usual differentia when restrictedto differentiable functions, is closable and its closure more precisely the domain of theclosure.

By definition if f is a measurable L2 function on Cx0M then I∗f is an L2 functionon C0Rm and there is a correspondence between L2 functions on C0Rm which aremeasurable with respect to I and measurable functions on Cx0M . The closure of d, onfunctions on C0Rm, can be classified by a theorem of Sugita [46]. We are interested inthe basics questions such as: is d closable? does it have closed range? How to classifythe domain of d? Is the closure of d independent of the choice of the initial domain?Does it hold dI∗ = I∗d?

If f is a smooth with bounded derivatives then d(f I) makes sense and dI∗ = I∗dassuming condition under which the conditional expectation of TI behaves well. If wecan find an SDE which induces the correct linear connection and such that there is noredundant noise then dI∗ = I∗d. For general manifolds this is an unknown question:the closure of d is independent of the domain when the domain is such that Cyl ⊂Dom(d) ⊂ BC2. The stronger version that uniqueness holds within the class of initialdomains Cyl ⊂ Dom(d) ⊂ BC1 remain unknown. The problem of independenceof domains is equivalent to the Markov uniqueness problem and the problem whetherdI∗ = I∗d, We refer to [21, 22] for detail. The following theorems from[23], Theorem3.4, summarizes this in a nutshell.

Theorem 4.5 Suppose condition (M0) hold. The operators

I∗d : Dom(d) ⊂ L2(Cx0M ; R) −→ L2(C0Rm;H)dI∗ :

f ∈ L2

∣∣ I∗f ∈ Dom(d2)⊂ L2(Cx0M ; R) −→ L2(C0Rm;H)

A HODGE DECOMPOSITION THEOREM 15

are densely defined closed operators with I∗d ⊂ dI∗. The map φ 7→ I∗φ := φ TI defined on measurable geometric forms on Cx0M extends to a continuous linearinjective map

I∗ : L0ΓH∗ −→ L0 (C0Rm;H∗)

from measurableH-one forms on Cx0M to measurable H-one -forms on C0Rm, usingthe topology of convergence in probability. Furthermore

I∗(φ)(h) =∫ T

0

⟨IDφ#

s

ds,∇TIs(h)X(//sdβs) +X(xs)(hs) ds

⟩xs

, h ∈ H (4.6)

This is an Ito integral using the filtration Gt := Fβt ∨ Fx0 , 0≤ t ≤ T . The map I∗restricts to a continuous linear map

I∗ : L2ΓH∗ −→ L2 (C0Rm;H∗) (4.7)

Furthermore E I∗(φ(−)) |Fx0 = φ(TIx·−) and ‖φ‖L2 ≤ ‖I∗(φ)‖L2 .

5 A Hodge decomposition theorem

Given v1, . . . , vq in V we use the convention:

v1 ∧ . . . ∧ vq =1q!

∑π

(−1)πvπ(1) ⊗ . . .⊗ vπ(q) (5.1)

where the summation is over all permutations π of 1, 2 . . . , q and (−1)π is the signof the permutation.

For the purpose of establishing a Hodge decomposition theorem we define admissi-ble tangent q-vectors by the tensor product of the derivative of the Ito map. For generalBanach spaces Ei, if v is in the algebraic tensor product E1 ⊗0 ... ⊗0 Eq , define thecross norm

‖v‖π = inf n∑i=1

Πqk=1‖a

ki ‖, where v =

n∑i=1

⊗aki , aki ∈ Ek, n <∞.

Denote by ⊗q0TσCx0M and ∧q0TσCx0M the q-th algebraic tensor products of the tan-gent space at σ of the path space. Their completions using the largest cross norm,i.e. the projective tensor products ‖ − ‖π are denoted by ⊗qTσCx0M and ∧qTσCx0Mrespectively. For our Hilbert tangent spaces, ⊗qH and ∧qH will however denote thestandard Hilbert space completions.

Given a linear operator S of Banach spaces E and F there is the functorial con-struction of a linear map on the tensor products: (d⊗q) (S)⊗q0E → ⊗

q0F so on primitve

vectors

((d⊗q) (S)) (e1 ⊗ . . .⊗ eq)= S(e1)⊗ e2 ⊗ . . .⊗ eq + e1 ⊗ S(e2)⊗ . . .⊗ eq + . . .+ e1 ⊗ e2 ⊗ . . .⊗ S(eq)

and is extended by linearity. These maps also extend over the relevant completions[24].

A HODGE DECOMPOSITION THEOREM 16

The restriction to ∧q0E is denoted by (dΛq(S)). For q = 2

(dΛ2(S)

)(v1 ∧ v2) =

12Sv1 ⊗ v2 + v1 ⊗ Sv2 − Sv2 ⊗ v1 − v2 ⊗ Sv1

.

5.1 q-vector fieldsWe are now in a position to define the space of admissible q-vectors:

H(q)σ := Image(∧q

(TI)σ

) ⊂ ∧qTσCx0 . (5.2)

together with the inner product induced by the linear bijection:

∧qTIσ|[ker∧qTIσ ]⊥: [ker∧q (TI)σ]⊥ → Hqσ.

ThusHqσ is a Hilbert space with natural continuous linear inclusions into ∧qTσCx0 .

5.2 Characterisation ofH2

LetR : ∧2TM → ∧2TM be the curvature operator andRq ∈ Hom(∧qTM ;∧qTM )the Weitzenbock curvature term defined byR = ∆−trace∇2. For a q-vector v ∈ ∧qTx0M ,define W (q)

t (V ) ∈ ∧qTxtM to be the random q-vector satisfying

D

dtW (q)t (V ) = −1

2RqW (q)

t (V ), 0 ≤ t ≤ T (5.3)

Note thatR1 = Ric# and the second Weitzenbock curvatureR2 is

R2 = d ∧2(

Ric#)− 2R.

By conditioning the relevant stochastic differential equation we may characterizethe H(2) space. A vector u of ∧2TσCx0M is in ∧2Hσ if and only if there exists k ∈∧2L2TσCx0M so that

us,t =(Wt

∫ t

0

(Wr1 )−1(−)dr1 ∧Wt

∫ t

0

(Wr2 )−1(−)dr2

)kr1,r2 . (5.4)

where ks,t = ID∂s ⊗

ID∂tu

We define a linear map Qσ on ∧2εTσCx0 given by

Qσ(G)s,t = (1⊗W st )W (2)

s

∫ s

0

(W (2)r

)−1(Rσr (Gr,r)) dr, s ≤ t. (5.5)

The map Q relates to another map which we now define. Let ⊗qεE and ∧qεE refer tothe completions of the algebraic tensor products of Banach spaces E with itself withrespect to the smallest reasonable cross norm, i.e. the inductive cross norm,

‖w‖ε = sup‖u∗k‖E∗≤1,u∗k∈E∗

|(u∗1 ⊗ . . .⊗ u∗q

)(w)|.

A HODGE DECOMPOSITION THEOREM 17

Define a linear map IR on ∧2εTσCx0M

IR(Z)s,t = (Ws ⊗Wt)∫ s

0

(∧2Wr

−1)

(Rσr (Zr,r)) dr, s ≤ t, (5.6)

It satisfies (1⊗ ID

dt )IR(Z)s,t = 0, s < t,ID(2)

ds IR(Z)s,s = Rσs (Zs,s − IR(Z)s,s) .(5.7)

In fact 1 +Q and 1− IR are inverse of each other and we have the following character-isation theorem.

Theorem 5.1 For any h1, h2 ∈ L2,10 Rm, set h = h1 ∧ h2. Then

∧2TI(h) = (1 +Q) ∧2 TI(h). (5.8)

In particular the space H2σ = ∧2TIσ(h), h ∈ ∧2H can be characterised by either

of the following:

(i)

H2σ =

u ∈ D(∧2TσCx0 ), such that there exists G ∈ H1

σ ∧H1σ,with((

1⊗ IDd·)u)s,t

=((

1⊗ IDd·)G)s,t, s < t,

and ID(2)

ds us,s =(((

d∧2) IDd·)G)s,s

0 6 s 6 T

.

(ii)

H2σ =

u ∈ ∧2

εTσCx0 , such that u = v +Qσ(v), some v ∈ H1σ ∧H1

σ

,

and for v1, v2 ∈ ∧2H1σ , by definition,

〈v1 +Qσ(v1), v2 +Qσ(v2)〉H2σ

= 〈v1, v2〉∧2H1σ. (5.9)

(iii) u ∈ H2 if and only if u− IR(u) ∈ ∧2H1. If so

‖u‖H2 = ‖u− IR(u)‖∧2H1 .

5.3 Integration by parts on differential formsWe state a simple integration by parts formula for q-forms [20]. Let ψ be a smoothcylindrical q-form on Cx0M . Then for h ∈ ID2,1(∧q+1H) of the form h = h1 ∧. . . ∧ hq+1 with each hi adapted, the q+1-vector field ∧q+1TI(h) on Cx0M has adivergence:∫

Cx0M

dψ(∧q+1TI(h)

)dµx0 = −

∫Cx0M

ψ(div ∧q+1TI(h)

)dµx0 (5.10)

anddiv(∧q+1TI(h)

)= ∧qTI(div h). (5.11)

A HODGE DECOMPOSITION THEOREM 18

This follows from Shigekawa [44] integration by parts formula and that the pull backform I∗(ψ) is a ID1,2 ((∧qH)∗) form on Ω :∫

Cx0M

dψ(∧q+1TI(h)

)dµx0 =

∫C0Rm

d(I∗ψ)(h) dP

= −∫C0Rm

I∗ψ (divh) dP = −∫Cx0M

ψ(∧qTI(divh)

)dµx0 .

5.4 Hodge Decomposition TheoremFor a C1 smooth one form φ on Cx0M let dφ be the exterior differential defined byPalais’s formula, which says that, for V j , j = 1 to q + 1 C1 vector fields, [V i, V j] theLie bracket and φ ∈ C1Ωq

dφ(V 1 ∧ . . . ∧ V q+1

)=∑q+1i=1 (−1)i+1LV i

[φ(V 1 ∧ . . . ∧ V i ∧ . . . ∧ V q+1

)]+

∑1≤i<j≤q+1

(−1)i+jφ(

[V i, V j] ∧ V 1 ∧ . . . V i ∧ . . . V j . . . ∧ V q+1) (5.12)

where V j means omission of the vector field V j . This can be restricted to give anH-2-form which we denote by d1

Hφ. Thus d1Hφσ is the composition of dφσ with the

continuous inclusion ofH2σ in∧2TσCx0M . We use smooth cylindrical functions as ini-

tial domain of d0 = d and cylindrical forms as initial domains for d1. As for functionswe consider the operator:

d1H : Dom(d1

H) ⊂ L2Γ(H1)∗ → L2Γ(H2)∗.

Theorem 5.2 [19] The exterior derivative considered as an operator

d1H : Dom(d1

H) ⊂ L2Γ(H1)∗ → L2Γ(H2)∗

is closable.

This can be proved by first obtaining a simple integration by parts formula for cylin-drical forms by considering their pull backs, and that of their exterior derivatives toWiener space by the Ito map. The pull back operation commutes with exterior differ-entiation, and a simple integration by parts formula for Wiener space can be appliedto give the standard closability argument when combined with Proposition 4.3. Let d1

denote the closure of d1H.

Theorem 5.3 [19]. The derivative d0f of any function f ∈ ID2,1 lies in the domain ofd1 and

d1d0f = 0.

Furthermore there is the derivation property d1(fφ) = fd1φ+ d0f ∧ φ.

Define ∆1 by ∆1 = d1∗d1 + d0d0∗. Taking into consideration that d0 has closedrange, following from the spectral gap theorem of Fang [?],

A HODGE DECOMPOSITION THEOREM 19

Theorem 5.4 [19]. There is the orthogonal decomposition

L2ΓH = Image(d0) + Image(d1∗) + ker41

where Image(d1∗) denotes the closure of the image of the adjoint of d1.

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