Divergence Theorems in Path Space. Denis Bell University of North Florida
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1 Divergence Theorems in Path Space Denis Bell University of North Florida
2 Motivation Divergence theorem in Riemannian geometry Theorem. Let M be a closed d-dimensional Riemannian manifold. Then for any smooth function and C 1 vector field Z on M, we have Z ZM Z( )dx = DivZdx DivZ is given in local co-ordinates by M DivZ = 1 p det g d X q (a i det i where g is the metric tensor and P d i=1 a local representation of Z. is
3 The Laplace-Beltrami operator is defined Divr This L-B operator and its extension to di erential 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.
4 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 measure d defined on X. Look for vector fields Z on X having an integration by parts formula with respect to : ZX Z( )d = Z X DivZd Say such Z is admissible
5 The (n-dimensional) Wiener space Let X be the space of continuous paths n w : [0, T ] 7! R n / w(0) = 0 o. and the Wiener measure on X. Let H denote the Cameron-Martin space, i.e. the subspace consisting of paths in X with finte energy h 2 X. Z T 0 h0 t 2 dt < 1. Theorem. Let h : X 7! H be a bounded random adapted path. Then h is admissible and Z T Divh = 0 h0 dw where the integral is the Itô integral. Follows from the Girsanov theorem.
6 There exists another class of admissible vector fields. Theorem. Let a be a continuous adapted process taking values in so(n) (the set of n n skew-symmetric matrices). Define Z = adw. 0 Then Z is admissible and DivZ = 0. Follows from the infinitessimal rotation invariance 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).
7 Combining the two previous results yields: Theorem. Processes of the following form are admissible Z = 0 adw + 0 bds where a is a continuous adapted so(n)-valued process and b is a continuous adapted R n - valued process. Furthermore DivZ = Z T 0 b dw We will refer to the space of such processes as the Cameron-Martin-Driver space. It constitutes the tangent bundle T X.
8 . Measures induced by stochastic di erential equations Let M denote a closed d-dimensional manifold and A 1,..., A n smooth vector fields on M and o a point in M. Consider the (Stratonovich) SDE dx t = nx i=1 x 0 = o. A i (x t ) dw i, t 2 [0, T ] where (w 1,..., w n ) is a Wiener process in R n. x t o
9 Let X be the space of continuous paths from [0, T ] to M with initial point o, and define the measure on X to be the law of the process x. T x X {V : [0, T ] 7! T M/ V 0 = 0, V t 2 T xt M, 8t 2 [0, T ]}. The objective is to construct a class of admissible vector fields on (X, ). There are two approaches to this type of problem. They both rely upon lifting the problem to the flat Wiener space, then using the divergence theorems in the previous section.
10 1. The Malliavin approach (1976) Recall the SDE dx t = nx i=1 A i (x t ) dw i, t 2 [0, T ]. Malliavin studied the regularity of the law T of x T. (endpoint problem). He established results of the form Z M Z( )d T = Z M DivZd T for smooth vector fields Z on M. The basic idea is to lift the problem to the Wiener space by the map w 7! x T. This works under very weak nondegeneracy conditions on x (Hörmander condition on A 1,..., A n, and weaker).
11 2. The Driver approach (1991) 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 di usion process x: the vector fields A 1,..., A n span T M 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.
12 Outline of the method dx t = nx i=1 A i (x t )dw i, t 2 [0, T ]. Let g denote the Itô map g : C 0 (R n ) 7! C o (M) w 7! x. The idea is to construct a vector field Z on X that lifts via g to an admissible vector field r on the Wiener space C 0 (R n ). Lifting means that the following diagram commutes dg T C 0 (R n )! T C o (M) r " " Z C 0 (R n )! C o (M) g
13 Let be a test function in C 0 (M). Then E[(Z( )(x)] = E h r( g)(w)] E[ g(w)divr] = E h (x)e[divr/x] i where Div denotes the divergence operator in the classical Wiener space. Thus Z is admissible with divergence given by DivZ = E[Divr/x] where apple Z T = E 0 bdw. x. r = 0 adw + 0 bds
14 Digression: The endpoint problem Let g T (w) = x T and suppose Z is a C 1 vector field on M. Then r is lift of Z if dg T (w)r = Z. Now it can be shown that if A 1,..., A n satisfy Hörmander s condition, then dg T (w) : H 7! T xt M is a.s. surjective (i.e. g T is a submersion). We choose r = dg T (dg T dg T ) 1 Z. The operator dg T dgt covariance matrix. is known as the Malliavin This construction does not work on the path space level.
15 We consider first the elliptic case. In this case the vector fields A i induce a Riemannian metric on M, defined as follows: Let a r be a local representation of A i, 1 apple i apple n. (Note that here and in all subsequent formulas, we use the summation convention.) The metric tensor [g jk ], is defined by g jk = a ij a ik, 1 apple j, k apple d. Let r denote the Levi-Civita covariant derivative corresponding to the metric g.
16 Theorem (Lifting Theorem) Let h and r be adapted processes in R n. Then r a lift of the vector field Z t h i (t)a i (x t ) (1) if and only if r and h are related by the SDE h k = r k + 0 < [A j, A i ], A k > (x t )h j dw i. (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, define h as the solution to (2) and Z by (1). However, in this case Z will depend explicitly on w and, since w is generically not a function of x, the process h will not be well-defined as a function of x. (w 1, w 2 ) 7! x h = h(w) 6= h(x) The answer is construct (r, Z) as a pair.
17 Observe that the is problem is that the di usion coe cient in the SDE h k = r k + 0 < [A j, A i ], A k > (x t )h j dw i is non-tensorial in A i. We address this by decomposing the di usion coe cient into a term that is a tensor in A i and a term that is skewsymmetric in the i and k indices, then absorbing the skew-symmetric part into the lift. Write < [A j, A i ], A k >=< r Aj A i, A k > < r Ai A j, A k > =< r Aj A i, A k > < r Aj A k, A i > + < r Aj A k, A i > < r Ai A j, A k >.
18 Introduce the notation G ik j (t) = < r Aj A i, A k > < r Aj A k, A i > (x t ) and T jk =< r Aj A k, > < r A j, A k >
19 Let r be a path in H (or CMD). Write (2) as h k = r k + h k = r k + 0 < [A j, A i ], A k > (x t )h j dw i 0 Gik j h j dw i + 0 T jk (A i )h j dw i = r k + 0 Gik j h j dw i + T jk ( dx)h j (3) 0 Let h = h(x) denote the solution to h k = r k + T jk ( dx)h j (4) 0 and define a process by k = r k 0 Gik j h j dw i. Substituting for r k in (4) we have h k = k + 0 Gik j h j dw i + T jk ( dx)h j 0 and (3) holds with r replaced by. Thus Z h i A i (x)is a vector field on C o (M), is a lift of Z to C 0 (R n ) and is admissible.
20 Degenerate di usions Again consider the di usion process dx = nx i=1 A i (x t ) dw i. Define E x span {A 1 (x),..., A n (x)}. We allow the possibility that E x T x M but assume the spaces E x have constant dimension. Define E [ x2m E x. Then A 1,..., A n induce a metric <.,. > on E, as before. Define A(x) : (h 1,..., h n ) 2 R n 7! h i A i (x) 2 E x
21 There is a metric connection r on E introduced by Elwothy-Le Jan-Li (Le Jan-Watanable connection) defined as follows r V W A(x)d V (A W ), W 2 (E), V 2 T x M where d is the standard derivative applied to the function x 2 M 7! A(x) W (x) 2 R n. Lemma. have For all V 2 T x M and W 2 E x, we nx j=1 < r V A j, W > A j = 0. We suppose that M is a Riemannian manifold and let r the Levi-Civita covariant derivative on M. Define T (U, V ) r V U r V U, U 2 E x, V 2 T x M. Note that T is a tensor in both arguments.
22 Let r : [0, T ] 7! R n be an Itô semimartingale. Di erentiating the original SDE in the direction r gives the following covariant equation for the path dg(w)r D t = r t A i dw i + A i dr i = r t A j dw j + T (A i, t ) dw i + A i dr i (5) In view of the fact that r t A i 2 E, we may write (5) as < r t A j, A i > A i dw j + T (A i, t ) dw i + A i dr i Define G ji V < r V A j, A i > < r V A i, A j >. Using the Lemma, we may then write the previous equation in the form D t = T ( dx, t ) + A i (dr i + G ji t dw j ) (6) This leads to the following result:
23 Theorem. Let r be a path in H (n-dim C-M space) and define by the SDE D t = T ( dx, t ) + A i ṙ i dt. Then is an admissible vector field on C o (M). Proof. Eq. (6) implies that the path r i = r i is an admissible lift of. 0 Gji dw j (7) In order to compute Div( ), it is necessary to convert the Stratonovich integral in (7) into Itô form.
24 The gradient system case Suppose M is isometrically embedded in a Euclidean space R n (This is possible by the Nash embedding theorem.) Define A i to be the orthogonal projection of the i-th standard orthonormal basis vector in R n onto T x M. Then the process dx t = nx i=1 A i (x t ) dw i defines a Brownian motion on M (i.e. generator 1 2 ). x has We verify that in this case, our method yields Driver s divergence theorem for the Wiener measure on a Riemannian manifold (Driver, 1991), which involves Ricci curvature.
25 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 < r V A i, W >=< r W A i, V >, i = 1,..., n. This is a consequence of the fact that the corresponding 1-forms A i are closed. Using these facts, we obtain the following result. Theorem. Let h be a Cameron-Martin path in T o M and define t = u t h t, where u t denotes stochastic parallel translation along x. Then is admissible and the divergence of is given by Z T 0 < Aj, D dt > 1 2 < r A i [A j, A i ], t > dw j
26 Lemma 3.9 (B. Driver) r Ai [A i, A j ] = Ric(A j ) A(LQ)Qe j where L nx A 2 i i=1 and Q is the orthogonal projection from R n onto the normal bundle of M. We note that the path r given by r j = 0 < P (LQ)Q(x s)e j, s > ds lies in ker dg(w). So with a little reverse engineering, we obtain
27 The process r defined by r j = 0 D D dt Ric( t), A j (x t ) E dt 0 Gij dw i is a lift of the vector field t u t h t to C 0 (R n ). This yields Driver s formula Div( ) = Z T 0 D D dt Ric( t), A j (x t ) E dw j
28 References Paul Malliavin, Stochastic calculus of variations and hypoelliptic operators. Proceedings of the International Conference on Stochastic Di erential Equations, Kyoto, Wiley, Bruce Driver, A Cameron-Martin type quasiinvariance theorem for Brownian motion on a compact manifold. J. Funct. Anal. 109 (1992), D.B., Divergence theorems in path space. J. Funct. Anal. 218, (2005) , Divergence thorems in path space II: degenerate di usions. C. R. Acad. Sci. Paris, Ser. I 342 (2006), , Divergence theorems in path space III: hypoelliptic di usions and beyond. J. Funct. Anal. 251 (2007),
29 An alternative form of the lifting theorem Let Y t : T o M 7! T xt M denote the derivative of the flow of the SDE (4), i.e. Y t = dg t (o) where g t : x 0 7! x t and define Z t Y 1 t. Let r : [0, T ] 7! R n be an Itô semimartingale. Then the process dg(w)r is given by t = Y t Z t 0 Z sa i (x s ) dr i.
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