[PDF] - Divergence Theorems in Path Space Denis Bell University of North PDF Document

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Divergence Theorems in Path Space

Denis Bell University of North Florida

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Motivation Divergence theorem in Riemannian geometry Theorem. Let M be a closed d-dimensional Riemannian manifold. Then for any smooth function Φ and C1 vector field Z on M, we have

DivZ is given in local co-ordinates by DivZ = 1 pdet g

∂ ∂xi (ai

det g) where g is the metric tensor and Pd

a local representation of Z.

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The Laplace-Beltrami operator ∆ is defined ∆ ⌘ Divr This L-B operator and its extension to differ- ential forms - the Hodge de-Rham operator, have given rise to a vast body of theory that includes harmonic functions, spectral theory, harmonic forms, Hodge theory, and the heat kernel approach to index theory.

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One would like to generalize the divergence theorem (and hopfully the associated theory) to an infinite-dimensional setting. There is no analogue of a volume form dx for an infinite-dimensional manifold X. A natural approach is to replace dx by a mea- sure dγ defined on X. Look for vector fields Z on X having an integration by parts formula with respect to γ:

Say such Z is admissible (wrt γ).

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The (n-dimensional) Wiener space Let X be the space of continuous paths

w : [0, T] 7! Rn/ w(0) = 0

and γ the Wiener measure on X. Let H denote the Cameron-Martin space, i.e. the subspace consisting of paths in X with fi- nite energy

h 2 X

dom adapted path. Then h is admissible and Divh =

where the integral is the Itˆ

Follows from the Girsanov theorem.

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There exists another class of admissible vector fields.

cess taking values in so(n) (the set of n ⇥ n skew-symmetric matrices). Define Z =

Then Z is admissible and DivZ = 0. Follows from the infinitessimal rotation invari- ance of the Wiener measure. The use of the result in the present context is a fundamental insight due to B. Driver. Note the previous two theorems do not require smoothness of Z (in w).

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Combining the two previous results yields:

admissible Z =

where a is a continuous adapted so(n)-valued process and b is a continuous adapted Rn- valued process. Furthermore DivZ =

We will refer to the space of such processes as the Cameron-Martin-Driver space. It constitutes the tangent bundle TX.

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. Measures induced by stochastic differential equations Let M denote a closed d-dimensional manifold and A1, . . . , An smooth vector fields on M and

Consider the (Stratonovich) SDE dxt =

Ai(xt) dwi, t 2 [0, T] x0 = o. where (w1, . . . , wn) is a Wiener process in Rn.

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Let X be the space of continuous paths from [0, T] to M with initial point o, and define the measure γ on X to be the law of the process x. TxX ⌘ {V : [0, T] 7! TM/ V0 = 0, Vt 2 TxtM, 8t 2 [0, T]}. The objective is to construct a class of admis- sible vector fields on (X, γ). There are two approaches to this type of prob-

to the flat Wiener space, then using the diver- gence theorems in the previous section.

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Recall the SDE dxt =

Ai(xt) dwi, t 2 [0, T]. Malliavin studied the regularity of the law γT of

for smooth vector fields Z on M. The basic idea is to lift the problem to the Wiener space by the map w 7! xT. This works under very weak nondegeneracy con- ditions on x (H¨

and weaker).

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This method involves lifting the problem via the stochastic development (rolling) map. Produces admissible vector fields on the full path space X but requires ellipticity of the diffusion process x: the vector fields A1, . . . , An span TM at every point of M. The goal of this work is to obtain divergence theorems on the path space by the Malliavin type lifting, without the ellipticity assumption.

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Outline of the method dxt =

Ai(xt)dwi, t 2 [0, T]. Let g denote the Itˆ

w 7! x. The idea is to construct a vector field Z on X that lifts via g to an admissible vector field r

Lifting means that the following diagram com- mutes dg TC0(Rn) ! TCo(M) r " " Z C0(Rn) ! Co(M) g

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Let Φ be a test function in C0(M). Then E[(Z(Φ)(x)] = E

r(Φ g)(w)] E[Φ g(w)Divr] = E

Φ(x)E[Divr/x]

where Div denotes the divergence operator in the classical Wiener space. Thus Z is admissible with divergence given by DivZ = E[Divr/x] = E

x

where r =

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Digression: The endpoint problem Let gT(w) = xT and suppose Z is a C1 vector field on M. Then r is lift of Z if dgT(w)r = Z. Now it can be shown that if A1, . . . , An satisfy H¨

dgT(w) : H 7! TxT M is a.s. surjective (i.e. gT is a submersion). We choose r = dg⇤

The operator dgTdg⇤

covariance matrix. This construction does not work on the path space level.

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We consider first the elliptic case. In this case the vector fields Ai induce a Rie- mannian metric on M, defined as follows: Let air∂/∂xr be a local representation of Ai, 1  i  n. (Note that here and in all subsequent formulas, we use the summation convention.) The metric tensor [gjk], is defined by gjk = aijaik, 1  j, k  d. Let r denote the Levi-Civita covariant deriva- tive corresponding to the metric g.

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Theorem (Lifting Theorem) Let h and r be adapted processes in Rn. Then r a lift of the vector field Zt ⌘ hi(t)Ai(xt) (1) if and only if r and h are related by the SDE hk = rk +

(2) If we choose a path h and define r by (2), then r will not generally lie in the CMD space. Alternatively, we could choose r in CMD, de- fine h as the solution to (2) and Z by (1). However, in this case Z will depend explicitly

a function of x. (w1, w2) 7! x h = h(w) 6= h(x) The answer is construct (r, Z) as a pair.

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Observe that the is problem is that the diffu- sion coefficient in the SDE hk = rk +

is non-tensorial in Ai. We address this by de- composing the diffusion coefficient into a term that is a tensor in Ai and a term that is skew- symmetric in the i and k indices, then absorb- ing the skew-symmetric part into the lift. Write < [Aj, Ai], Ak >=< rAjAi, Ak > < rAiAj, Ak > =< rAjAi, Ak > < rAjAk, Ai > + < rAjAk, Ai > < rAiAj, Ak > .

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Introduce the notation Gik

< rAjAi, Ak > < rAjAk, Ai >

(xt) and T jk =< rAjAk, · > < r·Aj, Ak >

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Let r be a path in H (or CMD). Write (2) hk = rk +

as hk = rk +

= rk +

(3) Let h = h(x) denote the solution to hk = rk +

(4) and define a process ρ by ρk = rk

Substituting for rk in (4) we have hk = ρk +

and (3) holds with r replaced by ρ. Thus Z ⌘ hiAi(x)is a vector field on Co(M), ρ is a lift of Z to C0(Rn) and ρ is admissible.

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Degenerate diffusions Again consider the diffusion process dx =

Ai(xt) dwi. Define Ex ⌘ span {A1(x), . . . , An(x)}. We allow the possibility that Ex ⇢ TxM but assume the spaces Ex have constant dimension. Define E ⌘

Ex. Then A1, . . . , An induce a metric < . , . > on E, as before. Define A(x) : (h1, . . . , hn) 2 Rn 7! hiAi(x) 2 Ex

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There is a metric connection r on E intro- duced by Elwothy-Le Jan-Li (Le Jan-Watanable connection) defined as follows rV W ⌘ A(x)dV (A⇤W), W 2 Γ(E), V 2 TxM where d is the standard derivative applied to the function x 2 M 7! A(x)⇤W(x) 2 Rn. Lemma. For all V 2 TxM and W 2 Ex, we have

< rV Aj, W > Aj = 0. We suppose that M is a Riemannian manifold and let ˜ r the Levi-Civita covariant derivative

Define T(U, V ) ⌘ ˜ rV U rV U, U 2 Ex, V 2 TxM. Note that T is a tensor in both arguments.

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Let r : Ω ⇥ [0, T] 7! Rn be an Itˆ

gale. Differentiating the original SDE in the direction r gives the following covariant equa- tion for the path η ⌘ dg(w)r ˜ Dtη = ˜ rηtAi dwi + Ai dri = rηtAj dwj + T(Ai, ηt) dwi + Ai dri(5) In view of the fact that rηtAi 2 E, we may write (5) as < rηtAj, Ai > Ai dwj + T(Ai, ηt) dwi + Ai dri and we have ˜ Dtη = T(Ai, ηt)dwi+Ai

dri+ < rηtAj, Ai > dwj

Gji

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Using the Lemma, we may then write the pre- vious equation in the form ˜ Dtη = T(dx, ηt) + Ai (dri + Gji

(6) This leads to the following result:

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space) and define η by the SDE ˜ Dtη = T(dx, ηt) + Ai ˙ ridt. Then η is an admissible vector field on Co(M).

˜ ri = ri

(7) is an admissible lift of η. In order to compute Div(η), it is necessary to convert the Stratonovich integral in (7) into Itˆ

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The gradient system case Suppose M is isometrically embedded in a Eu- clidean space Rn (This is possible by the Nash embedding theorem.) Define Ai to be the orthogonal projection of the i-th standard orthonormal basis vector in Rn onto TxM. Then the process dxt =

Ai(xt) dwi defines a Brownian motion on M (i.e. x has generator 1

We verify that in this case, our method yields Driver’s divergence theorem for the Wiener mea- sure on a Riemannian manifold (Driver, 1991), which involves Ricci curvature.

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It turns out that, in this case the L-W and the L-C coincide, so the tensor T defined earlier vanishes. There is also the symmetry relation < rV Ai, W >=< rWAi, V >, i = 1, . . . , n. This is a consequence of the fact that the cor- responding 1-forms A⇤

Using these facts, we obtain the following re- sult. Theorem. Let h be a Cameron-Martin path in ToM and define ηt = utht, where ut denotes stochastic parallel translation along x. Then η is admissible and the divergence of η is given by

< Aj, Dη dt > 1 2 < rAi[Aj, Ai], ηt >

dwj

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Lemma 3.9 (B. Driver) rAi[Ai, Aj] = Ric(Aj) A(LQ)Qej where L ⌘

A2

and Q is the orthogonal projection from Rn

Hence Div(η) =

< Aj, Dη dt > 1 2

Ric(Aj) A(LQ)Qej

We note that the path r given by rj =

lies in ker dg(w). So with a little reverse engineering, we obtain

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The process ˜ r defined by ˜ rj =

dt + 1 2Ric(ηt), Aj(xt)

dt

is a lift of the vector field ηt ⌘ utht to C0(Rn). This yields Driver’s formula Div(η) =

dt + 1 2Ric(ηt), Aj(xt)

dwj

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References Paul Malliavin, Stochastic calculus of varia- tions and hypoelliptic operators. Proceedings

Differential Equations, Kyoto, 195-263. Wiley, 1976. Bruce Driver, A Cameron-Martin type quasi- invariance theorem for Brownian motion on a compact manifold.

Anal. 109 (1992), 276-376. D.B., Divergence theorems in path space. J.

erate diffusions. C. R. Acad. Sci. Paris, Ser. I 342 (2006), 869-872.

poelliptic diffusions and beyond. J. Funct. Anal. 251 (2007), 232-253.

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An alternative form of the lifting theorem Let Yt : ToM 7! TxtM denote the derivative of the flow of the SDE (4), i.e. Yt = dgt(o) where gt : x0 7! xt and define Zt ⌘ Y 1