THE ABSTRACT RIEMANNIAN PATH SPACE

On the Wiener space Ω, we introduce an abstract Ricci process A t and a pseudo-gradient F ! F ] which are compatible through an integration by parts formula. They give rise to a ] -Sobolev space on Ω, logarithmic Sobolev inequalities, and capacities, which are tight on H¨older compact sets of Ω. These are then applied to the path space over a Riemannian manifold.


I. Introduction
Since the construction of the Riemannian Brownian motion by Itô and the introduction of the Malliavin calculus in the Wiener space Ω, a great amount of recent work has been devoted to the extension of stochastic calculus to the path space Σ of a Riemannian manifold [1,2,3,4,7,8,9,10,11,12,13,14,15,25,27,30]. The stochastic development map I introduced by Itô takes the Wiener measure µ onto an isomorphic measure ν = I(µ) on Σ.
One is guided by the flat case of the Wiener space of R m where the Girsanov integration by parts formula plays a dominant part. In the case of the path space of a Riemannian manifold, this formula is replaced by the Bismut-Driver formula which introduces the Ricci curvature of M .
These considerations encounter two kinds of difficulties, the first one due to the use of stochastic calculus, the second due to differential geometry.
Our first goal to show that these two problems can be dealt with separately. We begin by describing a stochastic framework without involving differential geometry.
On the Wiener space Ω we introduce a pseudo-differential operator F → F depending on a so-called "Ricci process" A which is set a priori. This pseudo-differential operator F (ω, ) on Ω × Ω is linear in the second variable, as F (ω, ) which has been introduced for the classical flat case [18,19]. It should be noticed that in the flat case we have A = 0 and F = F .
With the help of a damped derivation, that is a modified derivation F → F , we easily obtain a pseudo-Clark formula (which is equivalent to an integration by parts formula), a closed Dirichlet form, a spectral gap inequality and even a logarithmic Sobolev inequality (by using the Maurey-Ledoux method of the classical Gaussian case).
The closable Dirichlet form gives rise to -Sobolev spaces W 1,2, and, with natural supplementary hypotheses, to -Sobolev spaces W 1,p, .
In fact, it turns out that the i.b.p. (integration by parts formula) for the damped derivation and the pseudo-Clark formula do not depend on the Ricci process A t . This last process is only involved for the link between and -derivations. Therefore, in Section IV, we deal with the problem of constructing some "concrete" -derivation. For this purpose , we generalize an idea of [16], by using some kind of rotation of µ ⊗ µ instead of translation as we do usually for the Girsanov formula. It is to be noticed that the problem to generalize the Cameron-Martin space is avoided by this construction.
In Section V, we apply these results to the path space Σ of a compact Riemannian manifold M (endowed with a Driver connection): first we choose a -derivation in the sense of Section IV, and after that, we choose the convenient A t to obtain the true i.b.p. Bismut-Driver formula.
It should be noticed that for simplicity, M is embedded in a finite dimensional space E, but the Riemannian structure of M is not necessarily induced by an euclidean structure on E. The link between M and E is explicited by a Weingarten type tensor field on M .
In Section VI, we return to the general situation (without Riemannian manifold), and define the natural capacity C 1,p, associated on the Wiener space Ω, to the -Sobolev spaces. We show that under a natural hypothesis, these capacities are tight on Hölder compact sets of Ω.
In the case of a Riemannian manifold, the Itô map transforms the -Sobolev space W 1,p, (Ω, µ) onto a Sobolev space W 1,p, (Σ, ν) for ν = I(µ). The associated capacity is tight on the Hölder compact sets of Σ (and this improves the results of [10]). We can then specify the Itô map as a quasi-isomorphism from Ω onto Σ.
In conclusion, we can say that the abstract setting presented here must be considered as an attempt to simplify rather than to generalize the stochastic Riemannian path theory. Nevertheless, this setting allows us to see that there are many different ways to do a reasonable differential calculus on the Wiener space, or on the Riemannian path space.

II. Preliminaries and notations
Let µ be the Wiener measure on Ω = C 0 ([0, 1], R m ) with its natural filtration F t . We denote by W t (ω) the canonical Brownian motion. If F is an elementary functional (cylindrical functional) F = f (W t 1 , . . . , W tn ), the differential F (ω, ) is defined on Ω × Ω by the formula The norm of the Gaussian Sobolev space W 1,2 is defined by where the second expectation is taken with respect to µ ⊗ µ.
For F ∈ W 1,2 one has where D t F is the square integrable rough Borel process with values in R m which is worth where E stands for the partial expectation w.r. to .
Note that we have two independent Brownian motions W t (ω) and W t ( ), in short we shall denote them respectively W t and W t . We can easily get the Clark-Ocone-Haussmann formula where F t is the conditional expectation operator w.r. to F t . It is enough to check the formula for F (ω) = exp f (ω) where f is a continuous linear functional on Ω (cf. [20]). Let a zero mean value random variable, where g t is a predictable process. Define and the Girsanov derivative of F in the direction of G From the Clark formula we can deduce the Cameron-Martin-Girsanov integration by parts formula

III. Generalizations
Let us now be given an arbitrary predictable process A t with values in the linear operators of R m . For some reasons which will be clarified later on (Section V), it will be called the Ricci process.
With every g ∈ L p (µ ⊗ dt, R m ) is associated g which is the solution of the ordinary differential equation One has where C t is the resolvant, that is the solution of where K is the uniform norm of the process A t . Then g → g is an automorphism of L p (µ ⊗ dt, R m ), which preserves the predictable subspace.
Proof: It is easy to get On the other hand and next by Gronwall lemma The right hand-side is a convolution, so that Now, let L 2 0 be the zero mean value subspace of L 2 , and let G ∈ L 2 which consists in all the Wiener functionals on Ω × Ω which are linear in the second variable.
3 Definition: The pseudo-gradient or pseudo-differential is any operator F → F (ω, ) ∈ L 2 (µ ⊗ µ) which satisfies the following conditions: As above, we define the rough process D t F ∈ R m such that and the Girsanov pseudo-gradient of F in the direction of G by the formula 4 Definition: Let F be in the domain of the pseudo-gradient, we define the damped pseudogradient F by the formulae It turns out that it is the solution of the ODE where A * s is the adjoint operator of A s . Indeed, consider the adjoint J * of the operator J of L 2 (µ ⊗ dt) defined by J(g) = g. It is easily seen that D · F = J * (D · F ). Moreover we get the estimates

Now the integration by parts formula writes
where D G F is the damped pseudo-gradient of F in the direction of G. Finally, F and F have equivalent norms and are defined on the same domain.
Note that the damped pseudo-gradient F is easily seen to be also a derivation.
Proof: It is straightforward, thanks to the integration by parts formula for F .
7 Theorem: F and F are closable in L 2 .
Proof: Assume that F n and F n respectively converge to 0 and H. One gets for every G ∈ L 2 0 . Let γ be a C 1 bounded Lipschitz function vanishing at 0 and such that γ (0) = 1. Replace F n with Φγ(F n ) where Φ is a bounded element of the domain. The damped pseudo-gradient F n = Φ γ(F n ) + Φγ (F n )F n converges to ΦH in L 2 . So we get E (HΦG) = 0, for every G. Take G as the Wiener integral As Φ ⊗ g runs through a total set in L 2 (µ ⊗ dt), we get H = 0.
are local. More precisely, we have The proof is the same as the one in the classical case ( [5]).

Theorem: (Logarithmic Sobolev inequality) We have
Proof: We follow the idea of [3]. There is a simplification thanks to the use of the damped pseudo-gradient F . It is sufficient to consider the case Applying the Cauchy-Schwarz inequality to the conditional expectation, we get Hence, Extension to L p 10 Proposition: Assume that W t belongs to L p for every t ∈ [0, 1]. If F is an elementary So the pseudo-gradient extends to L p by this formula. The same property holds for the damped pseudo-gradient F .
Proof: The L p -domain contains elementary functions, so it is dense in L p . Closability is obtained, via Burkholder's inequality, in the same way as in the L 2 case. The L p norm equivalence for the two pseudo-gradients comes from the fact that J is also an automorphism of L p (µ⊗ dt).
Notice that W 1,p, is a subspace of L p for the norm is closable. The damped norm F 1,p, which is equivalent to the previous one is defined in the same way, so the -Sobolev space is the the same as the one.
12 Remarks: 1) It follows that it may happen many different Ricci processes generating the same -Sobolev space, since the -Sobolev space only depends on the -derivation.
2) It follows from a result of [8] an example of a concrete Ricci process (defined by a Riemannian manifold,cf. Section V) generating a -Sobolev space different from the Gaussian Sobolev space. Nevertheless the difference D t F −D t F is singular to the predictable σ-algebra when F belongs to the two domains.
3) The choice of the J operator may seem to be quite arbitrary (consider for example the same formula but with the Ricci process under the integration sign). The same properties as above would hold. In fact the formula that we have taken was motivated by the example of the path space of a Riemannian manifold.

IV. Some concrete derivations
Choosing F Let β t (ω, ) and γ t (ω, ) two predictable processes which are square integrable on Ω×Ω×[0, 1], with values in skew-symmetric operators of R m . We assume that β is linear in the second variable (first Wiener chaos in the second variable). Generalizing an idea of [16], we put for ε ∈ R, It is easily seen that the couple (W ε t , W t ε ) is an R m × R m -Brownian motion under µ ⊗ µ, so that its distribution does not depend on ε. For a regular F (ω), definė Note that properties a), b), d) of definition 2 are satisfied. It remains to prove an integration by parts formula to see that in fact F →Ḟ is a damped pseudo-gradient, that is We have, and for every ε so that for a bounded regular g The last two terms have zero expectation since they are ends of martingales w.r. to . Hence, by density of regular g, we are done.

Remarks:
1) In fact,Ḟ depends on β but not on γ as it is seen below, so that we can take γ = 0.
2) In the case β = 0, one hasḞ = F , so that we exactly get the Cameron-Martin-Girsanov integration by parts formula.
3) If F = f (W t 1 , . . . , W tn ) is an elementary function, we havė Hence, for every G, where β G is defined by Choosing F Now, take an arbitrary Ricci process A t in order to define F in such a way that F =Ḟ . So, putḞ It is easy to verify that F is a pseudo-gradient in the sense of Definition 2, and that F =Ḟ .

V. The Riemannian manifold paths
Let M be an m-dimensional compact submanifold of a finite dimensional vector space E. First we assume that M is endowed with a Riemannian structure G. Second we assume that we are given a Driver connection ∇, that is [7] a) ∇ is G-compatible, i.e. ∇G = 0 b) For every tangent vectors ξ and η we have where T is the torsion tensor of ∇. It is known [7,9] that this implies ∇ = ∇ + 1 2 T where ∇ is the Levi-Civita connection.
Consider the natural connection D on the vector space E. We define a Weingarten type tensor by writing This is an E-valued tensor field which is not symmetric as we have V (ξ, η) − V (η, ξ) = −T (ξ, η). Let Σ be the space of continuous paths starting at a point o ∈ M , and let W t be the canonical Brownian motion of R m . According to Itô and Driver, we get an M -Brownian motion X t starting at o by solving the Itô-Stratonovich SDE where X 0 = o, and H 0 a fixed isometry of R m onto T o (M ). These initial conditions are in force all over the section. It is known that X t is M -valued and is an M -Brownian motion. Moreover H t is an isometry of R m onto T Xt (M ). The second equation means that H t is a stochastic parallel field over X t .
This system has a unique solution (X t , H t ) for t ∈ [0, 1], and X t is an M -Brownian motion.

The Bismut-Driver formula
14 Theorem: There exists a process β in the sense of Section IV such that for the Ricci process where Ric is the Ricci tensor field of ∇, and Θ = Trace(∇T ) = i ∇ i T (e i , · ).
Proof: First we search for the damped pseudo-gradient (i.e. F ) in the form of Section IV. So, take β t and γ t as in Section IV. We have X ε t (ω, ) = X t (ω ε ), and we get the new Itô-Stratonovich system with the same initial conditions as I(0). Taking the derivative with respect to ε at 0, we geṫ where V is a suitable tensor field. Obviously the vectorẊ t belongs to T Xt (M ), so that it writeṡ Hence, In the same way, we havė where α t is a skew-symmetric operator of R m since H t is an isometry. By identification with the first line ofİ(0), we get where τ t = H −1 t T H t is the stochastic parallel transport of T , and where W t stands for W t ( ). On the other hand, H t α t is the stochastic covariant derivative of H t w.r. to ε, so that we have where Riem is the curvature tensor of the connection ∇ at X t . By comparison with the second line of I(0) we get where r t = H −1 t Riem H t is the stochastic parallel transport of Riem. Then we obtain the Itô-Stratonovich system By stochastic contraction, we get t Ric H t is the stochastic parallel transport of the Ricci tensor at X t . In the same way we get So we obtain the new Itô-Stratonovich system   where θ t = dτ t (dW t , ·) = H −1 t ΘH t is the stochastic parallel transport of Θ = Trace(∇T ). Observe that the coefficient τ t (·, ξ t ) of dW t is skew-symmetric valued, thanks to the Driver condition; and that ξ(ω, ) is linear in .
At this time, we already obtained a -derivation and even a family of -derivation (one for each process β), with the good i.b.p. formula. In addition, if we take an arbitrary bounded Ricci process, we get also a -derivation with a good i.b.p.
Nevertheless, we want to have X t = H t W t . In order to obtain such a -derivation, put A t = ric t + θ t , we get where α t = α t + τ t (·, ξ t ) and where A t = ric t + θ t .
Introduce a priori the solution η t of Observe that η t is linear in , and now choose the particular β process by putting which is skew-symmetric valued (∇ is a Driver connection), and which is also linear in . It turns out that the couple (ξ t , α t ) = (η t , t 0 r s (•dW s , η s )) is the solution of the last system. The proof of Theorem 14 is completed.
Proof: We haveḞ , as it can be easily seen from the obvious relation

Remarks:
a) As we have X t = H t W t and X t = H t η t , we can verify that with the notations of definition 4, we have for τ ∈ [0, 1] two lines of vectors or in terms of columns of covectors ∈ (R m ) * , which is better b) Notice that the solution ξ t of the above system is an affine function of β. For the choice of Bismut we get the i.b.p. of Bismut-Driver (modulo the good Ricci process), for β = 0 we get F = F that is the ordinary flat derivation on the Wiener space. For an arbitrary β, we get with a suitable skew-symmetric valued predictable process β depending on G. Hence we have again the i.b.p. and c) More generally, if ζ satisfies the preceding SDE with an arbitrary β, one can prove in the same way that this last expectation vanishes.

VI. Capacities
In the first part of this section we return to the general case (without manifold). We assume that every W t belongs to the space W 1,p, = W 1,p, . It is equivalent to say that every W t or W t belongs to L p (µ ⊗ µ).
A functional capacity is defined on the Wiener space. Put for every l.s.c. function g ≥ 0 on Ω; and put for every numerical function h, In the same way we define the functional capacity C 1,p . Clearly these two capacities are equivalent.
17 Theorem: Suppose that the process W t satisfies the inequality with p > 2, 1/p < α < 1/2 for a given constant k. Then for 0 < γ < α − 1/p, the capacities C 1,p and C 1,p are tight on γ-Hölder compact sets of Ω.
Proof: The hypothesis means that t → W t is a W 1,p, valued α-Hölder function. Let β such that γ < β < α − 1/p, consider the Hölder norm Denote H α the space of α-Hölder continuous functions with its natural norm. We have the inclusions H α (L p ) ⊂ L p (H β ) ( [22], proof of Theorem 5), hence the function q belongs to L p .

Application of capacities to the Riemannian case
First observe that X and H are solutions of the system I(0), so by [21] they belong to H α (L p ) for 1/2 > α > 1/p. Now we have and

So for various constants
19 Corollary: C 1,p, and C 1,p, are tight on Hölder compact sets of Ω.
The Itô map ω → σ defined by σ(t) = X t (ω) exchanges the measurable function classes on Ω (resp. Σ). It also exchanges the -Sobolev spaces W 1,p, (Ω, µ) constructed on Ω with the -Sobolev spaces W 1,p, (Σ, ν) constructed on Σ. More precisely, we have 20 Theorem: Let p > 1. We can refine the Itô map into a C 1,p -quasi-continuous map with values in a separable subset of Σ ∩ H α for 1/2 > α > 0. The image capacity Γ 1,p is associated with W 1,p, (Σ, ν) and is tight on Hölder compact sets of Σ and then the Itô map is a quasiisomorphism.
Proof: As both capacities C 1,p and Γ 1,p are increasing with p, we can suppose p as great as we want, and take α > 1/p. The Itô map I is an isomorphism of the Wiener measure µ onto its image ν which is carried by Σ, so that f → f • I is an isomorphism of L p (ν) on L p (µ), and also of W 1,p, (Σ, ν) onto W 1,p, (Ω, µ). Let us show first that Γ 1,p is tight on compact sets of Σ. Consider, as above, for α − 1/p > β, Then,