Quasi-invariant Measures on Path Space. Denis Bell University of North Florida

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1 Quasi-invariant Measures on Path Space Denis Bell University of North Florida

2 Transformation of measure under the flow of a vector field Let E be a vector space (or a manifold), equipped with a finite Borel measure and let be a vector field on E. Consider the corresponding flow on E, i.e. the maps s : x 7! x s where dx s ds = (xs ) x 0 = x We want to study the transformation of under s and establish conditions under which the measures s s 1 are mutually absolutely continuous (say is quasi-invariant under the flow of ).

3 Example (Cameron-Martin Theorem) Let E be the Wiener space (space of continuous paths starting at 0, equipped with the Wiener measure ). Let be a deterministic path of finite energy: 1 0 (Cameron-Martin path) Ż 2 s ds < 1. Since h is constant we have s (w) = w + s. The measures s and are equivalent and d s 1 d (w) = exp s Ż t dw s Żt 2 dt.

4 Definition. We say that is admissible if there exists an L 1 function Div such that E D d = E for all test functions on E. Divd The two properties - quasi-invariance and admissibilty - are obviously related.

5 Supppose quasi-invariance holds and denote Then we have d s d = X s. E s d = E d s = E X sd Di erentiating wrt s and setting s = 0 gives E D d = E dx s d ds s=0 i.e. is admissible and Div = dx s. ds s=0 Does the converse hold?

6 No. Let be a measure on R n with a compactly supported C 1 density F and let be a (non-zero) constant vector. a b Then is admissible. R n D d = R n D F dx = D R n R n F dx D F F d Quasi-invariance under the flow (x 7! x + s) would imply that and (.+s) have the same null-sets for all s. This is obviously not the case.

7 D F Note that Div = F blows up at the boundary of the support of F.

8 a b

9 Under what conditions does admissibility imply quasi-invariance? In order to address this question we need to develop the relationship between the two properties further.

10 Suppose is admissible and also the family of measures s s are alsolutely continuous wrt. Denote as before d s d = X s. For test functions E s d = E d s = E X sd (1) Replacing by 1 s E d = E we have 1 s X s d. Thus the RHS is independent of s. Di erentiating wrt s gives E D ( s 1 (x)) d ds s 1 (x)x s + 1 s X 0 s d = 0.

11 This relation can be rewritten in the form. E D( 1 s )(x)(x s ) + 1 s X 0 s d = 0. Using the defining property of divergence we have E s 1 Xs 0 Div(X s ) d = 0. (2) Since (2) holds for all test functions, we conclude that X s is the (unique) solution to the IVP Xs 0 = Div(X s ) (3) We can show that satisfies (3). X 0 = 1. X s (x) = exp s 0 Div(x u )du Working the previous argument backwards (under suitable conditions), we arrive at (1) and prove the following result.

12 Theorem 1. Suppose is admissible and there exists a set B E with s (B) = 1 for all s such that Div is defined and di erentiable along on B. Suppose further that the function u 7! Div(x u ) is continuous for x 2 B. Then is quasi-invariant under and d s d = exp s 0 (Div)(x u )du. Note: The continuity hypothesis precludes the singular behavior seen in the previous example.

13 Example. Let (E, ) be the Wiener space and be a (deterministic) Cameron-Martin path. Then w s = w + s. It can be shown that Div = 1 0 Żdw. Applying Thm 1 we see that under and is quasi-invariant d s d = exp = exp = exp (C-M Theorem) s s (Div)(w u )du 0 Żd(w 1 s Żdw s2 0 2 u) du 1 0 Ż 2 dt.

14 Application to measures induced by stochastic di erential equations Let M denote a closed compact manifold and X 0,..., X n smooth vector fields on M. Let o be a fixed point in M. Consider the Stratonovich SDE dx t = nx i=1 with initial point x 0 = o. X i (x t ) dw i, t 2 [0, T ] x t o

15 We consider two measures associated with this equation (1) The law T of x T, a measure on M (the endpoint problem). (2) The law of x, a measure on C o (M), the space of paths { : [0, T ] 7! M/ (0) = o} (the pathspace problem). The objective is to construct admissible vector fields and to establish quasi-invariance under the corresponding flows.

16 The endpoint problem Let T denote the law of x T where x is the solution of the SDE dx t = nx i=1 X i (x t ) dw i, t 2 [0, T ]. Deote by µ denote the Wiener measure on C 0 (R n ). Note that T is the image of µ under the map g T : (w 1,..., w n ) 7! x T Suppose the vector fields X i satisfy the Hormander s condition (HC): X 1,..., X n together with the collection of iterated Lie brackets [X i, X j ], [[X i, X j ], X k ],... span T M at every point. Then we can show that every C 1 vector field on M is admissible wrt the measure T :

17 Let be a test function on M. Consider the integral M D d T = C 0 (R n ) D (g T (w))dµ (4) We now perform integration by parts on the Wiener space in the style of the Malliavin calculus. This involves constructing a lift of to Wiener space via the map g T, i.e. a function satisfying Dg T (w) = (possible under HC). THe RHS of (4) may now be expressed in the form ( C 0 (R n ) g T )(w))dµ = C 0 (R n ) (x T )Div( )dµ where Div denotes the divergence operator in Wiener space: Div( ) = (, w) T race H D (w).

18 Writing this as an integral over the original measure space (M, T ), we have M D d T = M (x)e[div( )/x T = x]d T We conclude that is admissible with divergence Div(x) = E[Div( )/x T = x]. This implies that has a density (in the Euclidean case), a result that is related to the hypoellipticity of the generator of x. However, it is seems very di cult to establish the regularity of Div and hence to verify the continuity condition in Theorem 1. In fact, quasi-invariance in the Euclidean case implies everywhere positivity of the density of T and this is known not to hold in certain cases.

19 Example in R dx = x t dw x 0 = 1 Then x T = e w T, so x t > 0. In this case, we have X Div(1)(x) = 1 ln x + T, x > 0 xt which blows up as x! 0 +. We can compute the density F of x T from X via F (x) = C exp Xdx = 1 p 2 xt exp (ln x)2 2T

20 The pathspace problem The space E under study here is the set of paths { : [0, T ] 7! M with (0) = o}. The measure on E is the law of x. The tangent space T x {V : [0, T ] 7! T M/ V (0) = 0, V t 2 T xt M}. Assume the vector fields X 1,..., X n span T M at each point of M. We construct admissible vector fields on the pathspace in the form where the h i processes. t = nx i=1 X i (x t )h i (t) are suitably chosen real-valued

21 We introduce a Riemannian structure [g jk ] on M, by defining g jk = a ij a ik X i = a r is a local representation of X i. Note that here, and from this point on, we are using the summation convention.

22 A class of admissible vector fields on C 0 (M) Let r denote the Levi-Civita covariant derivative. Define a set of 1-forms on M! jk =< r Xj X k, > < r X j, X k > and functions B jk = 1 2 < Lji X i, X k > < L ij X k, X i > < r Xj X k, r Xi X i > + < r Xl X i, X k >< r Xj X l, X i > where L ij is the (Hessian) operator on vector fields H(X i, X j ) = r Xi r Xj r rxi X j. In the case of a gradient system terms defining B jk other than the first two vanish.

23 Theorem 2. Let r be an adapted C-M path in R n and define h by the following linear system of SDE s dh i t =! ji ( dx t )h j t + B ji (x t )h j t + ṙi t dt h i 0 = 0. Define t X i (x t )h i t. Then there is the integrationby-parts formula E[D (x)] = E[ (x)r] = E[ (x)e[r/x]] where R = T 0 ṙ i t < Ric( t), X i (x t ) > dw i By choosing r appropriately, we can ensure that R is x-measurable. In this case we have Div = R.

24 For example, let V be a vector field on M and is a real-valued continuous x-measurable process and define Then R = = T 0 T 0 and converting ṙ i t =< X i (x t ), V (x t ) > (t) ṙ i t < Ric( t), X i (x t ) > dw i V (x t ) t Ric( t), X i (x t )dw i dx t = X i (x t ) dw to Ito form, we have X i (x t )dw i = dx t 1 2 r X i X i (x t )dt.

25 Theorem 3. There exists a solution in C o (M) to the flow equation dx s ds = (xs ). x 0 = x. The processes x s are semi-martingales of the form dx s t = X i (s, t) dw i (t) + X 0 (s, t)dt where X 0 (s, ),..., X n (s, ) are adapted process in T M such that s 7! X j (s, ) are continuous into the space L 2 [0, T ].

26 In particular Div(x u ) = T 0 V (x u t ) t u Ric((x u ) t ), X i ( u, t)dw i (t). We can verify continuity in u. Hence we obtain Theorem 4. The measure under the flow of and is quasi-invariant d s d = exp s 0 (Div)(x u )du.