Integration by parts formulae for Wiener measures restricted to subsets in R d

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1 Journal of Functional Analysis ) Integration by parts formulae for Wiener measures restricted to subsets in R d Yuu Hariya Graate School of Mathematics, Kyushu University, Fukuoka , Japan Received 11 November 25; accepted 21 June 26 Available online 8 August 26 Communicated by Paul Malliavin Abstract We study an integration by parts formula for a pinned Wiener measure restricted to a space of paths staying within a subset in R d. The result presented here generalizes the formula in [L. Zambotti, Integration by parts formulae on convex sets of paths and applications to SPDEs with reflection, Probab. Theory Relat. Fields ) 579 6] for the case of a half-line in R. 26 Elsevier Inc. All rights reserved. Keywords: Integration by parts; Wiener measure; Hitting distribution 1. Introction and the main result Let W = C[, 1]; R d ), d N, and H W the Cameron Martin subspace. Let R d be an open region. We denote by W the set of paths staying within : W = { w W ; ws), s 1 }. In this paper, we study an infinite-dimensional) integration by parts formula IbP formula) for a pinned Wiener measure restricted to W, which is formulated in the following form. For a smooth functional F on W and h H, W a,b h Fw)dW[,1] w) = W a,b Fw)[h]w) dw[,1] w) + BC) IbP) address: hariya@math.kyushu-u.ac.jp /$ see front matter 26 Elsevier Inc. All rights reserved. doi:1.116/j.jfa

2 Y. Hariya / Journal of Functional Analysis ) with a,b. Here h F denotes the Gâteaux derivative of F, W a,b [,1] is the law on W of ddimensional) pinned Brownian motion with boundary conditions w) = a,w1) = b, [h]w) denotes the Wiener integral: [h]w) = 1 h s) dws), and BC) represents the boundary contribution, which is an analogue to that in Gauss divergence formula of finite dimension. Zambotti [12] firstly explored this problem for the case =, ) R, in connection with stochastic partial differential equations with reflection. In this paper, we present an explicit form of BC) for more general s in R d, in terms of hitting times of Brownian motion; in particular, we specify the infinite-dimensional boundary measure appearing in BC). To state the result, we prepare several notation: For x = x k ) 1 k d,y = y k ) 1 k d R d, we denote by x y the inner proct in R d : x y = d k=1 x k y k, and by x the Euclidean norm: x =x x) 1/2. We denote by the closure of, and by the boundary of : = \. Throughout this paper, is assumed to be a bounded region for which Gauss divergence theorem holds. For a,b,letb, B be independent d-dimensional Brownian motions with B = a, B = b. Let τ B) and τ B) be the first exit times from of B and of B, respectively. Conditionally on τ B) + τ B) = 1, B τ B) = x, and B τ B) = x, define the process X ={X t, t 1} by { Bt, t τ B), X t = B τ B)+τ B) t, τ B) t τ B) + τ B). We denote by P a,x,b [,1] the law on C[, 1]; ) of X. For each element w suppp a,x,b [,1], we denote by S x w), 1) the time at which ws x w)) = x. Let be the Dirichlet Laplacian for. We assume: A) there exists a fundamental solution p t; x,y) to the Cauchy equation / t + 1/2) =. Note that p t; x,y), if it exists, is unique by the maximum principle; moreover, it is nonnegative. We also note that it admits the following probabilistic representation: p t; x,y) = pt; x,y)w x,y [,t] ws), s t ), 1.1) where pt; x,y) denotes the Gaussian kernel pt; x,y) = { } 1 exp x y 2, 2πt) d 2t and W x,y [,t] is the law of pinned Brownian motion over the interval [,t] starting at x and ending at y. We assume A) so that, for every f C), fy)p t; x,y)dy = 1 t 2 fy)p t; x,y)dy, and that the following commutations are also valid: for 1 i, j d,

3 596 Y. Hariya / Journal of Functional Analysis ) x i fy)p t; x,y)dy = fy) x i p t; x,y)dy, 1.2) 2 2 fy)p t; x,y)dy = fy) p t; x,y)dy. 1.3) x i x j x i x j Following [1, Theorem A.3.2], we also assume: A1) for each fixed t> and y, p t;,y)is C 1 up to the boundary; A2) the restrictions to of functions that are harmonic on, and C 1 up to boundary, are dense in C ). Under these conditions, they proved that the joint distribution of τ B) and B τ B) is given by, for every a, P a τ B) dt,b τ B) dx ) = 1 p t; a,x)σdx)dt, 1.4) 2 where σ is the surface measure on, n x is the inward normal vector and / denotes the normal derivative at x. This formula will often be referred to in this paper. We endow W with the sup-norm: w W := sup t 1 wt), w W.LetBW ) be the corresponding Borel σ -field. Let W be the topological al of W and, the natural coupling between W and W.Forl W, we denote by l W its operator norm. Let FCb 1 be the set of the functionals F of the form Fw)= f l 1,w,..., l N,w ), w W, for N N, l i W 1 i N) and f C 1 b RN ).HereC 1 b RN ) denotes the set of the bounded, continuously differentiable functions on R N with bounded derivatives. Finally, as was mentioned above, we denote by H the Cameron Martin subspace; that is, H consists of the elements h = h k ) 1 k d W that are absolutely continuous and satisfy 1 h) = h1) = and h 2 H := h s) 2 ds <. Here h s) = h k s)) 1 k d. We recall that H is continuously embedded in W ; indeed, h W h H for all h H. We are now prepared to state the main result of this paper. Theorem 1.1. Assume A) A2). Then, for every F FCb 1 and h H, IbP) holds with BC) given by 1 BC) = 2p1; a,b) σdx)e a,x,b [{ [,1] nx h S x w) )} Fw) ]

4 Y. Hariya / Journal of Functional Analysis ) p u; a,x) p 1 u; b,x). 1.5) Remark 1.1. i) The assumptions A) and A1) are fulfiled if is of class C 3 ; we refer to [6,7], where the validity of the commutative relations 1.2) and 1.3) is also shown. ii) For the assumption A2), the following fact is known. If is of class C and f C 2,α ) for some α, 1), then the Dirichlet problem u = in and u = f on possesses a unique solution u of class C 2,α ).HereC 2,α ) denotes the set of functions in C 2 ) whose second derivatives are Hölder continuous of order α. This is a particular case of Kellogg s theorem; see, e.g., [9, Theorem 1.3.1]. Remark 1.2. The expression 1.5) of BC) would also be true even if has a finite number of corners, at least if it satisfies a certain cone condition; for instance, we may easily see that 1.5) holds when is a box in R d. The existence of the fundamental solution p t; x,y) for such s is discussed in, e.g., [8, Chapter 3, Section 16]. We may compare 1.5) with the formula in [12] for the case =, ) R; we refer to it with a slight generalization of boundary conditions: when =, ), BC) is expressed as, for a,b >, 1 2abe a b) 2 /2 { hu) πu 3 1 u) exp a2 3 2u b 2 } E a,,+ 21 u) [,u] E[,1 u][ b,,+ Fw1 w 2 ) ], 1.6) where E x,y,+ [,t] denotes the expectation with respect to P x,y,+ [,t], the law of pinned 3-dimensional Bessel process over [,t] starting at x and ending at y, and for w 1 C[,u]; R), w 2 C[, 1 u]; R) with w 1 u) = w 2 1 u), the notation w 1 w 2 stands for the path given in 3.2) below. See [12, formula 3)]. Note that in this case, ={}. We recall the fact that, for a>, P a τ B) ) = ) a exp a2, 2πu 3 2u and that, conditionally on τ B) = u, {B t, t τ B)} has the same law as P a,,+ [,u].from these, we see that 1.6) is rewritten as, in our notation, 2a + b)e 2ab E a,,b [ [,1] h S w) ) Fw) ]. 1.7) We may find that the formula 1.5) gives a generalization of 1.7). In [12], the formula 1.6) was proven in the case a = b; the proof given there relied upon an explicit formula relating a pinned 3-dimensional Bessel process to a pinned Brownian motion, known as Biane s theorem. Recently in [4], a generalization of Zambotti s result has been

5 598 Y. Hariya / Journal of Functional Analysis ) obtained in the case where 1-dimensional) Wiener measures are restricted to a space of paths staying between two curves. Their method is based on polygonal approximations of Brownian motions. Our result gives examples of infinite-dimensional Caccioppoli sets for which boundary measures are explicitly written. In the setting of an abstract Wiener space, several description of Caccioppoli sets are given in detail in [2,3]. IbP formulae in fact, divergence formulae) in infinite dimensions have also been studied by Goodman [5], Shigekawa [11] and other researchers in various contexts. The organization of this paper is as follows. In the next section, we prove Theorem 1.1 by using Proposition 2.1. We prove Proposition 2.1 in Section 2.2, by introcing Propositions , whose proofs are given in Sections 3 and 4. In Section 5, we give concluding remarks. Throughout this paper, we write = / x k ) 1 k d and =, the Laplacian. To indicate a variable, we sometimes write x and x with x = x k ) 1 k d. 2. Proof of Theorem 1.1 In this section, we prove Theorem Proof of Theorem 1.1 We consider the family of functionals F of the form: n 1 Fw)= f i wti ) ) 2.1) i=1 for n N, f i C 1 b Rd ), and a partition { = t <t 1 < <t n 1 <t n = 1} of the interval [, 1]. Proposition 2.1. Assume A) A2). Then, for every F of the form 2.1), and for every h C 2, 1)), IbP) holds with BC) given by 1.5). We give a proof of this proposition in the next subsection. Once this proposition is shown, then Theorem 1.1 is proven by approximations. Proof of Theorem 1.1. We prove Theorem 1.1 in two steps. Step 1 Approximation of elements in FCb 1 ). In this step, by using Proposition 2.1, we prove the assertion of Theorem 1.1 for F FCb 1 and h C2, 1)). Note that, by the Riesz Markov theorem, the mapping l : W R belongs to W if and only if there exists λ = λ k ) 1 k d with λ k : [, 1] R of finite variation, such that lw) = 1 ws) dλs). Therefore, by approximating lw) l,w by Riemann Stieltjes sums, it suffices to prove the assertion for F given by Fw)= f wt 1 ),..., wt n 1 ) ), f C 1 b R d ) n 1).

6 Y. Hariya / Journal of Functional Analysis ) Moreover, for any f = fx 1,...,x n 1 ) Cb 1Rd ) n 1 ), we may find a family f i,j Cb 1Rd ), 1 i n 1, j N, such that lim M M j=1 n 1 i=1 f i,j x i )) = f uniformly on ) n 1. Since IbP) is linear in F, it thus suffices to prove the assertion for F of the form 2.1), and for h C 2, 1)), which is nothing but the assertion of Proposition 2.1. Step 2 Approximation of elements in H ). In this step, using the conclusion obtained in the previous step, we complete the proof of Theorem 1.1. We pick h H and fix it. Note that there exists a sequence {h m } m N C 2, 1)) such that h m h H,m. Since H is continuously embedded in W, h m h W, from which we see that, as m, BC) for h m given by 1.5) converges to that for h. Therefore, in order to complete the proof, it suffices to show: for every F FCb 1, lim lim m W m W a,b hm Fw)dW[,1] w) = W a,b Fw)[h m ]w) dw[,1] w) = W a,b h Fw)dW[,1] w), 2.2) Fw)[h a,b ]w) dw[,1] w). 2.3) Let F FC 1 b be expressed as Fw)= f l 1,w,..., l N,w ). Then we have h Fw) d dε Fw+ εh) ε= = N i=1 f x i l1,w,..., l N,w ) l i,h, hence hm Fw) h Fw) h m h W max 1 i N sup x R N f x) x i max l i W, 1 i N which implies 2.2). Since h m converges to h in H, [h m ]w) converges to [h ]w) in L 2 W ; W a,b [,1]), from which 2.3) follows. So the proof is complete Proof of Proposition 2.1 In this subsection, we prove Proposition 2.1 in a sequence of propositions whose proofs are given in a separate section. For notational simplicity, we denote by IbP) 1 and IbP) 2, the left-hand side of IbP) and the first term on the right-hand side of IbP), respectively. Then IbP) is restated as IbP) 1 = IbP) 2 + BC). IbP ) Let F be of the form 2.1). For x = x,x 1,...,x n 1,x n ) R d ) n+1,weset

7 6 Y. Hariya / Journal of Functional Analysis ) Fx) = n f i x i ) f = f n 1), i= ν a,b dx) = δ a dx ) dx 1... dx n 1 δ b dx n ), a,b, where δ x denotes the Dirac measure at x. Proposition 2.2. For F given by 2.1), and for h C 2, 1)), we have where 1 IbP) k = ν a,b dx) Fx)I k x), k = 1, 2, p1; a,b) n+1 n 1 I 1 x) = ht i ) x i i=1 { n 1 p tj+1 t j ; x j,x j+1)}, j= n 1 { I 2 x) = p tj+1 t j ; x j,x j+1)} i= t i+1 t i j i dx x h u) ) p u ti ; x i,x ) p ti+1 u; x i+1,x ). For BC), we have the following expression. Proposition 2.3. For F given by 2.1), we have where 1 BC) = ν a,b dx) Fx)I 3 x), 2.4) p1; a,b) n+1 I 3 x) = 1 n 1 { p tj+1 t j ; x j,x j+1)} 2 i= t i+1 t i j i σdx) n x hu) ) p u ti ; x i,x ) p ti+1 u; x i+1,x ). By these two propositions, one may see that Proposition 2.1 follows once the following assertion is shown.

8 Y. Hariya / Journal of Functional Analysis ) Proposition 2.4. It holds that, for all x R d ) n+1, Now we are ready to prove Proposition 2.1. I 1 x) = I 2 x) + I 3 x). Proof of Proposition 2.1. Combining Propositions leads to IbP ), which shows the proposition. For the rest of the paper, we prove the above three propositions: Propositions 2.2 and 2.3 are proven in the next section; Proposition 2.4 is proven in Section Proof of Propositions 2.2 and Proof of Proposition 2.2 In this subsection, we prove Proposition 2.2. Proof. Note that, for F of the form 2.1), n 1 h Fw)= hti ) f i wti ) )) f j wtj ) ). i=1 Recall that the joint distribution W a,b [,1] wt 1) dx 1,...,wt n 1 ) dx n 1 ) is given by { n 1 p t i t i 1 ; x i 1,x i)} dx 1... dx n 1, x = a, x n = b. 3.1) p1; a,b) i=1 The expression for IbP) 1 follows from these, the relation 1.1) and the divergence theorem. For h C 2, 1)), note that the Wiener integral [h]w) can be defined pathwisely via: [h]w) = 1 h s) ws)ds. The expression for IbP) 2 follows from this, and again from 3.1) and the relation 1.1) Proof of Proposition 2.3 In this subsection, we prove Proposition 2.3. For a Brownian motion B starting from a, we denote by P a,x [,u] {B s, s τ B)} given τ B) = u and B τ B) = x, and by E a,x [,u] to P a,x [,u]. Lemma 3.1. i) It holds that, for A BW ), j i P a,x,b [,1] A S x = u) = P a,x [,u] Pb,x [,1 u] w 1 w 2 A), the conditional law of the expectation with respect

9 62 Y. Hariya / Journal of Functional Analysis ) where, for w 1 C[,u]; R d ) and w 2 C[, 1 u]; R d ) with w 1 u) = w 2 1 u), w 1 w 2 denotes the path defined by { w1 s), s u, w 1 w 2 )s) = w 2 1 s), u s ) ii) The distribution of S x under P a,x,b [,1] is given by P a,x,b [,1] S x ) = Z x ) 1 p u; a,x) p 1 u; b,x), <u<1, with normalizing constant 1 Z x = p u; a,x) p 1 u; b,x). Proof. The assertion i) is obvious from the definition of P a,x,b [,1]. The assertion ii) follows from 1.4). By Lemma 3.1, the integrand in 1.5) relative to σdx) may be written as 1 n x hu) ) E a,x [,u] [ Eb,x [,1 u] Fw1 w 2 ) ] p u; a,x) p 1 u; b,x). 3.3) To rewrite further 3.3) for F of the form 2.1), we also prepare the following. Lemma 3.2. For <r<u,letgw) = Gws), s r) be a bounded, measurable functional on C[,u]; R d ). Then it holds that E a,x [ ] [,u] Gw) p u; a,x) = dy pr; a,y)w a,y [ ] [,r] Gw)1{ws), s r} p u r; y,x). Here W a,y [,r][ ] denotes the expectation relative to Wa,y [,r], and the notation 1 A stands for the indicator function of an event A. An intuition regarding Lemma 3.2 is that we may identify E a,x [,u][gw)] with lim z x z W a,z [,u] [Gw)1 {ws), s u}] W a,z [,u] = lim ws), s u) z x z pu; a,z)w a,z [,u] [Gw)1 {ws), s u}], p u; a,z)

10 Y. Hariya / Journal of Functional Analysis ) where the equality follows from the relation 1.1). The assertion of Lemma 3.2 may be deced from this, conditioning on wr), and L Hospital s rule. Proof of Lemma 3.2. For every non-negative, measurable function f on, and for every t>r, we have, by 1.4) and the definition of P a,x [,u], [ E a GBs,s r)1 {τ B)>t}f )] B τ B) = 1 2 t σdx)fx)e a,x [ ] [,u] Gws), s r) p u; a,x). 3.4) Since {τ B) > t}={b s, s t}, the left-hand side of 3.4) may also be written as, by the Markov property of Brownian motion, E a [ GBs,s r)1 {Bs, s r}vt r, B r ) ]. 3.5) Here, for s> and x, Vs,x):= E x [fb τ B))1 {τ B)>s}]. Note that, by 1.4), Vt r, B r ) = 1 2 t σdx)fx) p u r; B r,x). Plugging this into 3.5) and using the law of B at time r, we see that 3.5) can be rewritten as P a B r dy)w a,y [ ) ] [,r] G ws),s r 1{ws), s r} = t t σdx)fx) p u r; y,x) σdx)fx) dy pr; a,y) W a,y [ ) ] [,r] G ws),s r 1{ws), s r} p u r; y,x). Here we used Fubini s theorem for the equality. Comparing this with the right-hand side of 3.4), we obtain the lemma. We are now in a position to prove Proposition 2.3: Proof of Proposition 2.3. We decompose 3.3) into the sum of J i, i n 1, defined by t i+1 J i = t i nx hu) ) E a,x [,u] Eb,x [,1 u] [ Fw1 w 2 ) ] p u; a,x) p 1 u; b,x).

11 64 Y. Hariya / Journal of Functional Analysis ) So we rewrite 1.5) as 1 BC) = 2p1; a,b) n 1 σdx) J i. 3.6) i= For F given by 2.1), we develop each J i as where K i,1 = E a,x [,u] t i+1 J i = t i [ i j= [ n K i,2 = E b,x [,1 u] By Lemma 3.2, K i,1 is rewritten as dx i f i x i ) p t i ; a,x i) W a,xi [,t i ] [ i 1 nx hu) ) K i,1 K i,2, f j wtj ) )] p u; a,x), j=i+1 f j w1 tj ) )] p 1 u; b,x). f j wtj ) ) ] 1 {ws), s ti } p u ti ; x i,x ). 3.7) j= Note that, by using the joint distribution of wt ),...,wt i 1 )) under W a,xi [,t i ], and the relation 1.1), p t i ; a,x i) W a,xi [,t i ] = i δ a [ i 1 Plugging this into 3.7), we have Similarly we have K i,1 = f j wtj ) ) ] 1 {ws), s ti } j= dx ) { i 1 dx 1... dx i 1 f j x j ) p tj+1 t j ; x j,x j+1)}. i+1 δ a { i 1 j= dx ) { i dx 1... dx i f j x j )} j= j= p tj+1 t j ; x j,x j+1)} p u ti ; x i,x ).

12 K i,2 = n i Y. Hariya / Journal of Functional Analysis ) dx i+1... dx n 1 δ b dx n ){ n { n 1 j=i+1 Combining these, we obtain j=i+1 f j x j )} p tj+1 t j ; x j,x j+1)} p ti+1 u; x i+1,x ). t i+1 J i = t i nx hu) ) { ν a,b dx) Fx) p tj+1 t j ; x j,x j+1)} n+1 j i p u ti ; x i,x ) p ti+1 u; x i+1,x ). Summing up these over i n 1 and noting 3.6), we arrive at 2.4), and the proposition is proven. 4. Proof of Proposition 2.4 In this section we prove Proposition 2.4. For this purpose, we prepare the following proposition. Proposition 4.1. For κ = κ k ) 1 k d,κ k C 2 b, )) C1 [, )), it holds that, for all t> and y,z, t dx x κ u) ) p u; y,x)p t u; z, x) t σdx) n x κu) ) p u; y,x) p t u; z, x) = z κ t) y κ ) ) p t; y,z) + κ) y p t; y,z) + κt) z p t; y,z). 4.1) Once this proposition is shown, then Proposition 2.4 follows readily. Proof of Proposition 2.4. By Proposition 4.1, I 2 x) + I 3 x) = { n 1 p ti+1 t i ; x i,x i+1)} n 1 x i+1 h t i+1 ) x i h t i ) ) i= i=

13 66 Y. Hariya / Journal of Functional Analysis ) n 1 { + p tj+1 t j ; x j,x j+1)} i= j i { ht i ) x i p ti+1 t i ; x i,x i+1) + ht i+1 ) x i+1p ti+1 t i ; x i,x i+1)}. Since h ) = h 1) = by assumption, the first term on the right-hand side vanishes. Rearranging the second term, we see that this is equal to I 1 x). The rest of this section is devoted to proving Proposition 4.1. For f C ), we write U f t, x) = dy f y)p t; x,y), t >, x. By definition, t U f t, x) = 1 2 U f t, x) 4.2) and lim t U f t, x) = fx)for every x. Let f,g C ) be arbitrarily fixed. We multiply the first term on the left-hand side of 4.1) by fy)gz), and integrate it over with respect to dy dz. Then it is written as, by using Fubini s theorem and the notation introced above, t dx x κ s) ) U f t, x)u g t u, x). 4.3) Now we take the Laplace transform of 4.3) in t with parameter γ>, which may be written as dsdt e γs e γt dx x κ s) ) U f s, x)u g t, x). 4.4) Lemma ) is equal to I 1 + I 2.Here I 1 = I 2 = dt e γt dt e γt dx { x κ t) ) gx)u f t, x) x κ ) ) fx)u g t, x) }, dx { κ) U g t, x) ) fx)+ κt) U f t, x) ) gx) } dsdt e γs e γt dx { κs) U g t, x) ) U f s, x) + κs) U f s, x) ) U g t, x) }.

14 Y. Hariya / Journal of Functional Analysis ) Proof. Using integration by parts in the variable s, we see that 4.4) is rewritten as dt e γt dx x κ ) ) fx)u g t, x) + γ dsdt e γs e γt dx x κ s) ) U f s, x)u g t, x) dsdt e γs e γt dx x κ s) ) U g t, x) s U f s, x). 4.5) We use the resolvent equation γ dt e γt U g t, x) = gx) + 1 dt e γt U g t, x) 4.6) 2 for the second term, and the relation 4.2) for the third term, to see that 4.5) is rewritten as I 1 +II with II = 1 2 dsdt e γs e γt dx x κ s) ){ U f s, x) U g t, x) U g t, x) U f s, x) }. So, in order to prove the lemma, it suffices to show II = I ) Using the divergence theorem in the definition of II and the fact that U f s, x) = U g t, x) = for x, we see that II = dsdt e γs e γt dx κ s) U g t, x) ) U f s, x). By integration by parts in the variable s, this is rewritten further as II = dt e γt dx κ) U g t, x) ) fx) γ dsdt e γs e γt dx κs) U g t, x) ) U f s, x)

15 68 Y. Hariya / Journal of Functional Analysis ) dsdt e γs e γt dx κs) U g t, x) ) s U f s, x). 4.8) By the divergence theorem, the second term is equal to γ = dsdt e γs e γt dse γs dx κs) U f s, x) ) U g t, x) dx κs) U f s, x) ) gx) dsdt e γs e γt dx κs) U f s, x) ) U g t, x), where we used 4.6) for the equality. Replacing by this the second term on the right-hand side of 4.8), and using the relation 4.2) for U f s, x) in the third term, we get 4.7). This ends the proof. Lemma 4.3. Let f,g be functions of class C 2 ) which satisfy f = g = and are C 1 up to the boundary. Then we have, for all v R d, dxv f) g+ Proof. Noting the identity dxv g) f + σdx)v n x ) f g =. v f) g+ v g) f = div { v f) g + v g) f f g)v }, we see that the assertion follows immediately from the divergence theorem, and from the fact that f = f/ )n x and g = g/ )n x at x, by assumption. Combining Lemmas 4.2 and 4.3 leads to Proposition 4.1. Proof of Proposition 4.1. By Lemma 4.2, and by taking the inverse Laplace transform, we have, for all t>, t dx x κ u) ) U f u, x)u g t u, x) = dx { x κ t) ) gx)u f t, x) x κ ) ) fx)u g t, x) }

16 + Y. Hariya / Journal of Functional Analysis ) dx { κ) U g t, x) ) fx)+ κt) U f t, x) ) gx) } t dx { κu) U g t u, x) ) U f u, x) + κu) U f u, x) ) U g t u, x) }. Since f,g C ) are arbitrary, we obtain, for a.e. y,z, t dx x κ u) ) p u; y,x)p t u; z, x) = z κ t) y κ ) ) p t; y,z) + κ) y p t; y,z) + κt) z p t; y,z) t dx { κu) x p t u; z, x) ) x p u,y,x) + κu) x p u; y,x) ) x p t u; z, x) }. Here we used the commutative relations 1.2) and 1.3). By continuity, this relation holds for all y,z. Note that, by Lemma 4.3, the second term on the right-hand side is equal to 1 2 t σdx) n x κu) ) p u; y,x) p t u; z, x). This shows the proposition. 5. Concluding remarks i) The formula 1.5) asserts that BC) is given by the integral of E a,x,b [ [,1] Fw)h Sx w) )] 5.1) with respect to the vector-valued measure mdx) on given by { 1 1 mdx) = 2p1; a,b) } p u; a,x) p 1 u; b,x) n x σdx). As can be seen, the normal derivative of p in fact, p itself) is not involved in 5.1). So, if we obtain another expression of the measure mdx) in which, at least the normal derivative of p is not involved, then we may extend our result to more general s such as, say, Caccioppoli sets. ii) In [12], integration by parts formulae were applied to the study of stochastic partial differential equations with reflection introced by Nualart and Pardoux [1]; in particular, he gave

17 61 Y. Hariya / Journal of Functional Analysis ) a characterization of local times in equations in terms of Revuz measures, to which boundary measures in BC) were associated. Our result might have a similar application. References [1] M. Aizenman, B. Simon, Brownian motion and Harnack inequality for Schrödinger operators, Comm. Pure Appl. Math ) [2] M. Fukushima, BV functions and distorted Ornstein Uhlenbeck processes over the abstract Wiener space, J. Funct. Anal ) [3] M. Fukushima, M. Hino, On the space of BV functions and a related stochastic calculus in infinite dimensions, J. Funct. Anal ) [4] T. Funaki, K. Ishitani, Integration by parts formulae for Wiener measures on a path space between two curves, Probab. Theory Related Fields, in press. [5] V. Goodman, A divergence theorem for Hilbert space, Trans. Amer. Math. Soc ) [6] S. Itô, The fundamental solution of the parabolic equation in a differentiable manifold, II, Osaka Math. J ) [7] S. Itô, A boundary value problem of partial differential equations of parabolic type, Duke Math. J ) [8] S. Itô, Diffusion Equations, Kinokuniya, Tokyo, 1979 in Japanese). [9] J. Jost, Partial Differential Equations, Springer, New York, 22. [1] D. Nualart, E. Pardoux, White noise driven quasilinear SPDEs with reflection, Probab. Theory Related Fields ) [11] I. Shigekawa, Vanishing theorem of the Hodge Kodaira operator for differential forms on a convex domain of the Wiener space, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6 23) [12] L. Zambotti, Integration by parts formulae on convex sets of paths and applications to SPDEs with reflection, Probab. Theory Related Fields )