BV FUNCTIONS, CACCIOPPOLI SETS AND DIVERGENCE THEOREM OVER WIENER SPACES*
BENIAMIN GOLDYS1, XICHENG ZHANG1,2
1School of Mathematics and Statistics The University of Sydney , Sydney, 2006, Australia 2Department of Mathematics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, P.R.China Emails: B. Goldys: [email protected], X. Zhang: [email protected]
Abstract. Using finite dimensional approximation, we give a version of the definition of BV functions on abstract Wiener space introduced in Fukushima-Hino [17]. Then, we study Caccioppoli sets in the classical Wiener space and pinned Wiener space, and provide concrete examples of Caccioppoli sets, such as the balls and the level sets of solutions to SDEs. Moreover, without assuming the ray Hamza conditions in [16], we prove the infinite dimensional divergence theorem in any Caccioppoli set for any bounded continuous and H-Lipschitz continuous vector field in the classical Wiener space. In particular, the isoperimetric inequality holds true for Caccioppoli sets.
1. Introduction Let U be an open and bounded subset of the Euclidean space Rd, with C1-boundary. 1 d d The classical divergence theorem states that for any ϕ ∈ Cc (R ; R ) Z Z d−1 divϕ(x)dx = ϕ(x) · nU (x)H (dx), (1) U ∂U d−1 where nU (x) is the unit exterior normal to U at x, and H is the Hausdorff measure on ∂U. This fundamental formula holds also for any Borel set U with finite perimeter; in this case ∂U is replaced by the measure theoretic boundary(cf. [13, p.209, Theorem 1] and [14]). In [2], Airault-Malliavin established an infinite-dimensional version of the formula (1) in the classical Wiener space. Therein, they used the finite dimensional approximation, and considered the level sets of functionals that are non-degenerate and smooth in the sense of Malliavin calculus. We remark that infinite dimensional versions of the divergence theorem (1) have been also investigated in other contexts, see for example [26, 18, 15]. Fukushima [16] and Fukushima-Hino [17] studied BV functions over the abstract Wiener space (X, H, µ). Let L(log L)1/2 be the Orlicz space over X. According to [17], a function f ∈ L(log L)1/2 is of bounded variation if Z V (f) := sup divf(x) · f(x)µ(dx) < +∞, (2) X where the supremum is taken over all X∗-valued smooth cylindrical vector fields f with ∗ kf(x)kH 6 1 for each x ∈ X. Here, X is the dual space of X. The set of all functions
Key words and phrases. BV function; Caccioppoli set; Divergence theorem; Pinned Wiener space; Isoperimetric inequality. *This work is supported by the Australian Research Council Discovery Project DP120101886. 1 f : X → R of bounded variation is denoted by BV (X). In particular, they also showed that 1,1 D BV (X), where D1,1 is the first order Mallavin-Sobolev space on X. Basing on the theory of their established BV functions, Fukushima and Hino extended the infinite dimensional like-formula (1) to the Caccioppoli set Γ (i.e. 1Γ is a BV function). That is, for a Caccioppoli set Γ ⊂ X, if 1Γ satisfies the ray Hamza condition, Fukushima [16, Theorem 4.2] proved that for any smooth cylindrical vector field f Z Z divf(x)µ(dx) = hf(x), n (x)i k∂Γk(dx), (3) Γ H Γ ∂Γ where k∂Γk denotes the corresponding surface measure of bounded variation function 1Γ. The proof of this formula in [16, 17] depends on the theory of Dirichlet form. We may put forward the following three questions that are of some importance for the analysis of stochastic partial differential equations and the related deterministic Dirichlet considered in a bounded set of a Banach space, see for example [9]. (I) In the definition of BV (X), can one start from a broader space rather than the Orlicz space L(log L)1/2? (II) Can we give some concrete Caccioppoli sets such as the balls in X? (III) Is the ray Hamza condition satisfied by 1Γ necessary for (3), and can the range of vector fields be enlarged to be such that the formula (3) holds? For the first question, we note recent work [3], where there no additional integrability condition (except that of L1) is imposed but the space W 1,1 is defined a somewhat non- 1 standard way. In this paper we shall show that it is enough to require f ∈ Lw(X, µ) 1 in the definition of (2), where Lw(X, µ) is the integrable functions space with weight 1/2 1 kxkX + 1. By Fernique theorem, it is clear that L(log L) ⊂ Lw(X, µ). Using finite 1 dimensional approximation, we finally prove that for f ∈ Lw(X, µ), if V (f) is finite, then f ∈ L(log L)1/2. This will be given in Section 2. Thus, in a priori position, we can work in a bigger space. The second question was recently an object of intense study, see for example [4, 6, 10, 25]. We shall give an affirmative answer in the case of classical Wiener space. It should be noted that in the finite dimensional case, the indicator function of a Lipschitz domain is a BV function(cf. [13]). Naturally, we would ask if the indicator functions of a “smooth domain” in Wiener space such as the ball, is also a BV function so that the infinite dimensional divergence theorem holds. In [16] and [17], the authors used the coarea formula to assert the existence of Caccioppoli sets in X. Moreover, in the pinned Wiener space, Zambotti [27] and Hariya [19] gave some concrete Caccioppoli sets. In their papers, the surface measures are explicitly given by some analytic method. In Section 3, we shall first give a criterion for Caccioppoli sets. Then, we prove that the balls in the classical Wiener space and the level sets of solutions to SDEs with constant diffusion coefficients are Caccioppoli sets. It is worthy to say that in [1], the authors proved that the indicator function of the ball in the classical Wiener space (resp. does not) belongs to Dp,α provided pα < 1 (resp. pα > 1). However, it is not known in the critical case of pα = 1. Little is known about the third question, except some special cases, see for example [24, 27, 19]. In Section 4 we will approximate a general vector field by smooth cylindrical vector fields and will prove that without the ray Hamza condition assumption on 1Γ, the formula (3) still holds true for any bounded continuous and H-Lipschitz continuous 2 vector fields on the classical Wiener space. The proof depends on the structure of the classical Wiener space. It is not known whether this is also true for an abstract Wiener spaces. As a simple conclusion, using the result of Bobkov [8](see also [5]), we obtain the isoperimetric inequality for Caccioppoli sets. In Section 5, we also study the integration by parts formula in the pinned Wiener space, and give a more general integration by parts formula than Hariya [19]. The price to pay is that we can not explicitly give the surface measure.
2. BV Functions
Let (X, H, µ) be an abstract Wiener space. Namely, (X, k · kX) is a separable Banach space, (H, k · kH) is a separable Hilbert space densely and continuously embedded in X, and µ is the Gaussian measure over X. If we identify the dual space H∗ with itself, then ∗ may be viewed as a dense linear subspace of so that `(x) = h`, xi whenever ` ∈ ∗ X H H E and x ∈ , where h·, ·i denotes the inner product in . H H H p p For p > 1, let L (X, µ) be the usual L -space over (X, µ), the norm is denoted by k · kp. 1 1 Let Lw(X, µ) be the following weighted L -space: 1 1 Lw(X, µ) := f ∈ L (X, µ): kfk1,w < +∞ , where Z kfk1,w := |f(x)| · (kxkX + 1)µ(dx). X Define Z s p Φ1(s) := log(1 + r)dr, s > 0, 0 and Z s r2 Φ2(s) := (e − 1)dr, s > 0. 0 Then Φ1 and Φ2 are a pair of complementary Young functions. The corresponding Orlicz spaces LΦi (X, µ), i = 1, 2 are defined by the following norms Z |f(x)| kfkΦi := inf α > 0 : Φi µ(dx) 6 1 , i = 1, 2. X α We also write LΦ1 (X, µ) as L(log L)1/2. ∗ In what follows, we shall fix an orthogonal basis E := {`k, k ∈ N} ⊂ X of H. Let ∞ F Cb be the set of all smooth cylindrical functions with the following form: ∞ m f(x) := F (`i1 (x), ··· , `im (x)), `ij ∈ E ,F ∈ Cb (R ), ∞ ∗ ∗ and F Cb (X ) be the set of all cylindrical X -valued vector fields with the following form: l X ∞ f(x) = fj(x)`lj , `lj ∈ E , fj ∈ F Cb . j=1 ∞ p ∞ ∗ It is well known that for any p > 1, F Cb is dense in L (X, µ) and F Cb (X ) is dense in Lp(X, µ; H). ∞ For f ∈ F Cb , the Malliavin derivative of f is defined by(cf. [22]) m X Df(x) := (∂jF )(`i1 (x), ··· , `im (x))`ij , (4) j=1 3 ∞ ∗ and for f ∈ F Cb (X ), the divergence of f is defined by l X h i divf(x) := f (x) · ` (x) − D f (x) . (5) j lj `lj j j=1 where D` fj(x) = h`l , Dfj(x)i . It is well known that lj j H Z Z hDf(x), f(x)i µ(dx) = f(x) · divf(x)µ(dx), H X X which means that div is the dual operator of D. The following lemma is direct from the definitions.
∞ ∗ ∞ Lemma 2.1. For any f ∈ F Cb (X ), we have divf ∈ F Cb and for some C > 0
|divf(x)| 6 C(kxkX + 1), ∀x ∈ X. (6) p,1 ∞ For p > 1, the Malliavin Sobolev space D is defined as the completion of F Cb with respect to the norm:
kfkp,1 := kfkp + kDfkp. Then
1,1 1/2 1 Proposition 2.2. D ⊂ L(log L) ⊂ Lw(X, µ), i.e., for some C1,C2 > 0
kfk1,w 6 C1kfkΦ1 6 C2kfk1,1.
Proof. The second inequality was proved in Proposition 3.2 [17]. Put g(x) := kxkX + 1. Then, by Fernique’s theorem(cf. [22]), one has
kgkΦ2 < +∞. Thus, by the generalized H¨olderinequality Z kfk1,w = |f(x)| · (kxkX + 1)µ(dx) 6 kfkΦ1 · kgkΦ2 6 CkfkΦ1 . X The result follows.
1 For f ∈ Lw(X, µ), define Z V (f) := sup divf(x) · f(x)µ(dx). (7) ∞ ∗ f∈F Cb (X );kf(x)kH61 X By (6), one knows that the above integral is well defined. Thus, we can introduce the following space of bounded variation functions.
1 Definition 2.3. A function f ∈ Lw(X, µ) is called bounded variation if V (f) < +∞. All such functions are denoted by BV (X). We have
1 Theorem 2.4. The map f 7→ V (f) is lower semi-continuous in Lw(X, µ), and BV (X) is a Banach space under the norm:
kfkBV := kfk1,w + V (f). 4 1 ∞ ∗ Proof. Let fn → f in Lw(X, µ). For any f ∈ F Cb (X ) with kf(x)kH 6 1, and all x ∈ X, we have Z Z divf(x) · f(x)µ(dx) = lim divf(x) · fn(x)µ(dx) lim V (fn). n→∞ 6 X X n→∞ So,
V (f) 6 lim V (fn). (8) n→∞
It is clear that BV (X) is a linear normed space under k · kBV . We now prove the completeness of BV (X) with respect to k · kBV . Let fn be a Cauchy sequence under 1 1 k · kBV . Since Lw(X, µ) is complete, there is an f ∈ Lw(X, µ) such that
lim kfn − fk1,w = 0. n→∞ By (8), we know f ∈ BV (X) and
lim V (f − fn) 6 lim lim V (fn − fk) = 0. n→∞ n→∞ k→∞ The proof is complete.
Let {Tt}t>0 be the OU semigroup defined by the Mehler formula Z √ −t −2t Ttf(x) := f(e x + 1 − e y)µ(dy). X Then
1 1 Lemma 2.5. Tt is a bounded operator from Lw(X, µ) to Lw(X, µ). More precisely, for 1 some universal constant C > 0 and any t > 0, f ∈ Lw(X, µ) √ −t −2t kTtfk1,w 6 (e + C 1 − e ) · kfk1,w. (9) Moreover,
lim kTtf − fk1,w = 0. (10) t↓0 Proof. Notice that µ ⊗ µ is invariant under the rotation: √ e−t, 1 − e−2t √ . − 1 − e−2t, e−t
1 Thus, we have for f ∈ Lw(X, µ) Z kTtfk1,w = |Ttf(x)| · (kxkX + 1)µ(dx) 6 X Z Z √ −t −2t 6 |f(e x + 1 − e y)| · (kxkX + 1)µ(dy)µ(dx) X X Z Z √ −t −2t = |f(x)| · (ke x − 1 − e ykX + 1)µ(dy)µ(dx) X X √ Z −t −2t 6 e kfk1,w + 1 − e kykXµ(dy) · kfk1, X which gives (9). ∞ 1 The limit (10) is clearly true for f ∈ F Cb . For general f ∈ Lw(X, µ), it follows by using (9) and a simple approximation. 5 1/2 1,1 It was proved in [17, Proposition 3.5 ] that Tt maps L(log L) into D . It is not known 1 1,1 whether Tt maps Lw(X, µ) into D . Thus, we can not simply use Ttf as a mollifier to approximate the function f ∈ BV (X). So, we need to introduce a finite dimensional approximation operator(cf. [22]). For n ∈ N, let Hn := span{`1, `2, ··· , `n}, and Πn the orthogonal projection from H to ⊥ Hn, Hn the orthogonal complement of Hn in H. It is clear that Πn can be extended to X by defining Πn(x) := `1(x) · `1 + ··· + `n(x) · `n. −1 Then µn := µ ◦ Πn is the finite dimensional Gaussian measure on (Hn, B(Hn)). We have the following decomposition ⊥ ⊥ (X, H, µ) := (Hn, Hn, µn) ⊕ (Yn, Hn , µn ), ⊥ ⊥ where (Hn, Hn, µn) is the finite dimensional Gaussian space, (Yn, Hn , µn ) is still an ab- stract Wiener space and X = Hn ⊕ Yn. In particular, we have the following disintegrated formula(cf. [21, 22]): Z Z Z ⊥ f(x)µ(dx) = f(` + y)µn (dy)µn(d`), X Hn Yn and we shall write Z ⊥ Hn 3 ` 7→ f(` + y)µn (dy) = Pnf(`). Yn −1 We remark that (Pnf) ◦ Πn actually equals to E(f|Bn(X)), where Bn(X) := σ{Πn (B): B ∈ B(Hn)}. In the following, if there are no dangers of confusions, we shall not distin- guish Pnf with (Pnf) ◦ Πn. We have Lemma 2.6. For any f ∈ L1(X, µ), it holds that kPnfk1 6 kfk1 (11) and
lim kPnf − fk1 = 0. (12) n→∞
Proof. Since Bn(X) ↑ B(X), {E(f|Bn(X)), n ∈ N} is a martingale. The result follows from the martingale convergence theorem. Define the following approximation operator
Anf := PnT1/nf. The following lemma is easy to verify. ∞ ∞ ∗ Lemma 2.7. For any f ∈ F Cb and f ∈ F Cb (X ), we have ∞ ∞ ∗ Pnf, T1/nf ∈ F Cb ,PnΠnf = ΠnPnf ∈ F Cb (X ), and PnΠn(Df) = D(Pnf),Pn(divf) = div(PnΠnf).
We now prove the following mollifying property of An. ∞ 1 Proposition 2.8. The operator DAn : F Cb → L (X, µ; H) can be extended to a bounded 1 1 linear operator from Lw(X, µ) to L (X, µ; H). In particular, An is a bounded linear oper- 1 1,1 ator from Lw(X, µ) to D . 6 ∞ Proof. Let f ∈ F Cb . Noting that Anf is in fact a function on Hn, we have by Lemma 2.7 n !1/2 n X 2 X kDAnf(x)kH = |D`i Anf(x)| 6 |PnD`i T1/nf(x)|. i=1 i=1 On the other hand, a direct calculation in finite dimensional Gaussian space leads to(cf. [22, 23]) −1/n Z e −1/n p −2/n D` T1/nf(x) = √ f(e x + 1 − e y)`i(y)µ(dy). i −2/n 1 − e X Hence, Z Z Z −1/n p −2/n kDAnf(x)kHµ(dx) 6 |f(e x + 1 − e y)| · kykXµ(dy)µ(dx) X X X n e−1/n X ×√ k`ik ∗ , −2/n X 1 − e i=1 and by (9) we get Z kDAnf(x)kHµ(dx) 6 Cnkfk1,w. X The proof is complete. Basing on this proposition, we can prove the following characterization for BV (X). 1/2 Theorem 2.9. If f ∈ BV (X), then f ∈ L(log L) and there exists a sequence {fn}n∈N ⊂ 1,1 1 D such that fn → f in L (X, µ) and
lim kDfnk1 = V (f). (13) n→∞ 1 1,1 Conversely, for f ∈ L (X, µ), if there exists a sequence {fn}n∈N ⊂ D such that fn → f 1 in L (X, µ) and supn∈N kDfnk1 < +∞. Then f ∈ BV (X) and V (f) 6 lim kDfnk1. (14) n→∞ In particular, 1/2 f ∈ BV (X) if and only if f ∈ L(log L) and there exists a sequence 1,1 1 {fn}n∈N ⊂ D such that fn → f in L (X, µ) and supn∈N kDfnk1 < +∞. Proof. Put fn := Anf. It is clear by (10) (12) and Proposition 2.8 that 1,1 lim kfn − fk1 = 0 and fn ∈ . (15) n→∞ D
Noticing that Anf is indeed a function defined on the finite dimensional Gaussian prob- ability space (Hn, µn), we know
V (fn) = kDfnk1. (16) ∞ ∗ For any f ∈ F Cb (X ), we have by Lemma 2.7 Z Z divf(x) · fn(x)µ(dx) = Pndivf(x) · T1/nf(x)µ(dx) X X Z = divPnΠnf(x) · T1/nf(x)µ(dx) X 7 Z = T1/ndivPnΠnf(x) · f(x)µ(dx) X Z −1/n = e divT1/nPnΠnf(x) · f(x)µ(dx), X which implies that −1/n V (fn) 6 e V (f). (17) So,
sup kDfnk1 = sup V (fn) 6 V (f). (18) n∈N n∈N In particular, this also produces that f ∈ L(log L)1/2 by Proposition 2.2 and (15). On the other hand, we have for n sufficiently large Z Z divf(x) · fn(x)µ(dx) = Pndivf(x) · T1/nf(x)µ(dx) X X Z = divf(x) · T1/nf(x)µ(dx). X So, by (10) we have Z Z lim divf(x) · fn(x)µ(dx) = divf(x) · f(x)µ(dx), n→∞ X X which then gives
V (f) 6 lim V (fn). (19) n→∞ The limit (13) thus follows by combining (16) (17) and (19). We now look at the converse part. For m, n ∈ N, set m fn (x) := (−m) ∨ (fn(x) ∧ m), f m(x) := (−m) ∨ (f(x) ∧ m). m 1,1 m Then fn ∈ D , and kDfn k1 6 kDfnk1, m m m lim kfn − f k1 = 0, lim kf − fk1 = 0. n→∞ m→∞ ∞ ∗ Thus, for any f ∈ F Cb (X ) with kf(x)kH 6 1 for all x ∈ X, we have by the dominated convergence theorem Z Z m m divf(x) · f (x)µ(dx) = lim divf(x) · fn (x)µ(dx) n→∞ X X Z m = lim hf(x), Dfn (x)i µ(dx) n→∞ H X Z m 6 lim kDfn (x)kHµ(dx) n→∞ X 6 lim kDfnk1. n→∞ This implies by Theorem 2.4 that m V (f) 6 lim V (f ) 6 lim kDfnk1 < +∞. m→∞ n→∞ The proof is complete. 8 Remark 2.10. From this theorem, one sees that our definition is equivalent to the one in [17]. In particular, for f ∈ BV (X) we have Z V (f) = sup divf(x) · f(x)µ(dx), (20) X
where the supremum runs over all the following vector fields f with kf(x)kH 6 1: n X f(x) = F (h (x), ··· , h (x))` , h , ` ∈ ∗,F ∈ C∞( mj ). (21) j n1 nmj j nj j X j b R j=1 ˜ ∞ ∗ All such vector fields will be denoted by F Cb (X ). Indeed, (20) follows from the above characterization. Using this characterization, we have the following useful proposition. Proposition 2.11. Let f ∈ BV (X) ∩ L∞(X, µ) and g ∈ D1,1 ∩ L∞(X, µ). Then fg ∈ BV (X) ∩ L∞(X, µ), and V (fg) 6 V (f) · kgk∞ + kfk∞ · kDgk1.
Proof. Let fn := Anf. Then
kD(fng)k1 6 kDfnk1 · kgk∞ + kfnk∞ · kDgk1 6 V (f) · kgk∞ + kfk∞ · kDgk1. and lim kfng − fgk1 kgk∞ lim kfn − fk1 = 0. n→∞ 6 n→∞ The result follows by Theorem 2.9. We also have the following isoperimetric inequality for BV functions(cf. [5, 8]). Theorem 2.12. Let f ∈ BV (X) with 0 6 f 6 1. Then Z Z U fdµ − U (f)dµ 6 V (f), X X s 2 where (s) := Φ0 ◦ Φ−1(s) and Φ(s) := √1 R e−r /2ds. U 2π −∞ 1,1 Proof. Set fn := An1Γ. Then fn ∈ D and 0 6 fn 6 1. Noting that fn is in fact a function defined on Hn, we thus have by using [8, Corollary 2] and mollifying technique Z Z Z U fndµ − U (fn)dµ 6 kDfn(x)kHµ(dx). X X X Taking the limits n → ∞ yields the desired inequality by Theorem 2.9. The following result was proved in [17, Theorem 3.9]. Theorem 2.13. For each f ∈ BV (X), there exists a positive finite measure ν(also written as kDfk) on (X, B(X)) and an H-valued Borel function nf with knf kH = 1 ν– a.e. such ˜ ∞ ∗ that for all f ∈ F Cb (X )(see Remark 2.10) Z Z divf(x) · f(x)µ(dx) = hf(x), n (x)i ν(dx), (22) f H X X and V (f) = kDfk(X). 0 0 Moreover, ν and nf are uniquely determined; namely, if ν and nf are another pair of 0 0 satisfying (22), then ν = ν and nf = nf ν– a.e.. 9 The following co-area formula can be proved by using the same method as in [13, p.185 Theorem 1](see [16, 17]). Theorem 2.14. Let f ∈ BV (X). Then Z ∞ V (f) = V (1{f>t})dt. −∞
In particular, for a.e. t, 1{f>t} ∈ BV (X). 3. Caccioppoli Sets
Definition 3.1. A Borel set Γ ⊂ X is called a Caccioppoli set if 1Γ ∈ BV (X). The total of all the Caccioppoli sets is denoted by C (X). The following proposition is obvious.
c Proposition 3.2. Let Γ1, Γ2 ∈ C (X) with Γ1 ∩ Γ2 = ∅. Then, Γ1, Γ1 ∪ Γ2 ∈ C (X). For a Borel set Γ ⊂ X, the lower Minkowski content of the boundary of Γ is defined by
µ(Γ) − µ(Γ) µs(∂Γ) := lim , (23) →0
where Γ := {x ∈ X : dis(x, Γ) < }. We have the following simple proposition for a Borel set to be a Caccioppoli set.
Proposition 3.3. Let Γ ∈ B(X). If µs(∂Γ) < +∞, then Γ ∈ C (X). Proof. Noting that x 7→ dis(x, Γ) is a Lipschitz continuous function on X with Lipschitz constant 1, we have by [12]
ess sup kD[dis(·, Γ)](x)kH 6 cH,→X, x∈X
where cH,→X is the embedding constant of H ,→ X. Let χ(s) be defined by 1, s ∈ [0, /2], χ (s) := 1 − [(2s − ) ∧ ]/, s ∈ [/2, ∞). Then 2c 2c kDχ (dis(·, Γ))k H,→X [µ(Γ ) − µ(Γ )] H,→X [µ(Γ ) − µ(Γ)], 1 6 /2 6 and kχ(dis(·, Γ)) − 1Γk1 6 µ(Γ \ Γ) = µ(Γ) − µ(Γ). The result now follows by (14) (23) and µs(∂Γ) < +∞. From this proposition, it is immediate that the balls in the classical Wiener space are Caccioppoli sets. However, it seems difficult to verify that µs(∂Γ) < +∞ for a general set Γ. We now give a more useful criterion for the level set of a functional to be a Caccioppoli set.
2,1 d Theorem 3.4. Let f := (f1, ··· , fd) ∈ D (R ) be a d-dimensional real valued random variable on X. Let U be a Borel subset of Rd with compact and Lipschitz boundary ∂U. Assume that the law of f has a bounded density ρ on some neighbourhood U of ∂U U0 0 with respect to the Lebesgue measure, and
Cf := ess sup kDf(x)kH < +∞, (24) −1 x∈f (U0 ) 10 where for > 0 d U := {z ∈ R : dis(z, ∂U) < }. Then, the set Γ := {x : f(x) ∈ U} belongs to C (X). Proof. First of all, since ∂U is Lipschitz, noticing the following Minkowski content for- mula(cf. [14, Theorem 3.2.39]) Vol(U ) lim = Hd−1(∂U), ↓0
we have for < 0
Vol(U) 6 C∂U · , (25)
where the constant C∂U is independent of . Define 1, dis(z, U c) ∈ [, ∞), χ (z) := dis(z, U c)/, dis(z, U c) ∈ [0, ].
1 Then z 7→ χ(z) is a Lipschitz function with Lipschitz constant , and by (25) Z
lim kχ(f) − 1Γk1 lim k1U (f)k1 = lim ρU (z)dz ↓0 6 ↓0 ↓0 0 U
sup ρU (z) · lim Vol(U) = 0. 6 0 ↓0 z∈U0 On the other hand, by the local property of the operator D(cf. [23, Proposition 1.3.7]), we know d X j D(χ(f))(x) = 1U (f(x)) · (∂jχ)(f(x)) · Df (x). j=1
Hence, by (24) and (25) we have for < 0 d Z X j kD(χ(f))k1 6 1U (f(x)) · |(∂jχ)(f(x))| · kDf (x)kHµ(dx) j=1 X 1 n o ess sup(1 (f(x)) · kDf(x)k ) · µ x : f(x) ∈ U 6 U0 H x∈X 1 Z = C · ρ (z)dz f U0 U C · sup ρ (z) · C , 6 f U0 ∂U z∈U0 and by (14) V (1Γ) 6 lim kD(χ(f))k1 < +∞. ↓0 The proof is thus finished. Remark 3.5. It was proved in [12] that (24) is equivalent to the local µ-a.e. H-Lipschitz continuity. Moreover, in place of the assumption that the law density of f is uniformly bounded on a neighbourhood of ∂U, we can only require the following estimation: n o µ x : f(x) ∈ U 6 C. 11 In what follows, we shall study the sets in the classical Wiener space (W, H, µ). Namely, W is the space of all Rd-valued continuous functions on [0, 1] starting from 0 at 0 with the norm d !1/2 X 2 kwkW := kwik∞ , kwik∞ := sup |wi(s)|, i=1 s∈[0,1] where w = (w1, ··· , wd) denotes a generic element in W; H ⊂ X is the Cameron-Martin space, in which the elements have absolutely continuous and square integrable derivatives, and endowed with the norm, Z 1 1/2 ˙ 2 k`kH := |`(s)| ds ; 0 and µ is the Wiener measure. 1 1 1 Let 0 < r < 2 , q > 2r ∨ 1−2r . We also consider the subspace Wq,r of W with the following Sobolev pseudo-norms(cf.[2]):
d !1/2 X 2 kwkq,r := kwikq,r < ∞, i=1 where 1 Z 1 Z 1 |w(t) − w(s)|2q 2q kwikq,r := 1+2qr dtds . 0 0 |t − s| Using Theorem 3.4, we have
Theorem 3.6. Let U ∈ B(Rd) have compact and Lipschitz boundary. Then we have
{w ∈ W :(kw1k∞, ··· , kwdk∞) ∈ U} ∈ C (W), 1 1 1 and for 0 < r < 2 , q > 2r ∨ 1−2r ,
{w ∈ Wq,r :(kw1kq,r, ··· , kwdkq,r) ∈ U} ∈ C (Wq,r).
In particular, the balls in W and Wq,r are Caccioppoli sets. Proof. Without loss of generality, we assume d = 1. It is well known that for any a > 0(cf. [20, p. 30]) +∞ Z (2n+1)a n o 1 X n −r2/2 µ w : sup |w(s)| 6 a = √ (−1) e dr. s∈[0,1] 2π n=−∞ (2n−1)a
That is, the random variable k · k∞ has a smooth density. 2q ∞ p,α Moreover, Airault and Malliavin [2] proved that k · kq,r ∈ D = ∩p,αD , and Fang [11] 2q proved that k · kq,r is non-degenerate in the Malliavin calculus sense. Thus, k · kq,r has an infinitely differentiable density. On the other hand, k · k∞ and k · kq,r are Lipschitz continuous, the condition (24) holds by [12]. The result thus follows by Theorem 3.4. We now consider the following SDE: d dXt = b(Xt)dt + dw(t),X0 = x ∈ R . Then
Theorem 3.7. Let b : Rd 7→ Rd be a Lipschitz continuous function, U a Borel subset of Rd with compact and Lipschitz boundary. Then for any t ∈ [0, 1], {w : Xt(w) ∈ U} ∈ C (W). 12 Proof. First of all, Xt has a continuous law density with respect to the Lebesgue measure i because the generator L := ∆ + b (x)∂i is strictly elliptic. So, we only need to check (24). By Theorem 2.2.1 in [23], we know Z t k DrXt = I + Bk(s) · DrXs ds, r
where Bk(s) is a uniformly bounded and adapted d-dimensional process. Thus Z 1 1/2 Z t 2 2 kDXtkH = |DrXt| dr 6 1 + C kDXskHds. 0 0 By the Gronwall inequality, we get
ess sup kDXt(w)kH 6 C, ∀t ∈ [0, 1]. w∈W By Theorem 3.4, we complete the proof. 4. Divergence Theorem In this section, we still work in the classical Wiener space. For any n ∈ N, define a family of functions in W∗ by n/2 n h˙ (s) := 2 1 n n (s), k = 0, 1, ··· , 2 − 1, nk (tk ,tk+1] n −n where tk = k2 and the dot denotes the derivative with respect to s. It is easy to see that hh , h i = δ , k, m = 0, 1, ··· , 2n − 1, (26) nk nm H km where δkm is the Kronecker symbol. For n ∈ N and w ∈ W, define 2n−1 X h n n n n i π w(t) := w(t ) + w(t ) − w(t )(2 t − k) · 1 n n (t). n k k+1 k [tk ,tk+1] k=0 P2n−1 Then πnw = k=0 hnk(w) · hnk and
lim kπnw − wk 2 lim sup |w(t) − w(s)| = 0. (27) n→∞ W 6 n→∞ |t−s|62−n We now prove the following generalized Green formula:
Theorem 4.1. Let Γ ∈ C (W). Then the support of k∂Γk := kD1Γk is contained in the boundary ∂Γ := Γ¯ − Γ◦, where Γ¯(resp. Γ◦) denotes the closure(resp. interior) of Γ in W. Moreover, for any bounded continuous and H-Lipschitz continuous vector field f on W, Z Z divf(w) µ(dw) = hf(w), n (w)i k∂Γk(dw), (28) Γ H Γ ∂Γ
where nΓ := n1Γ and k∂Γk := kD1Γk are from Theorem 2.13. Proof. In the first three steps, we shall prove Z Z divf(w) µ(dw) = hf(w), n (w)i k∂Γk(dw). (29) Γ H Γ W In the last step, we prove (28). (Step 1): In this step, we assume that f has the form:
f(w) := F (hn0(w), ··· , hn(2n−1)(w))`, 13 n where ` ∈ W∗ and F : R2 → R is a bounded and Lipschitz continuous function. ∞ 2n Let ϕ ∈ Cc (R ) be a positive smooth function with compact support and satisfy R n ϕ(x)dx = 1. Define for > 0 R2 Z 2n −1 F(x) := F (y)ϕ( (x − y))dy, n R2 and
f(w) := F(hn0(w), ··· , hn(2n−1)(w))`, (30) ∞ ∗ Then f ∈ F Cb (W ) and by (22) Z Z divf (w)µ(dw) = hf (w), n (w)i k∂Γk(dw). (31) Γ H Γ W Since for each w ∈ W
lim kf(w) − f(w)k = 0, (32) ↓0 H the right hand side of (31) converges to the right hand side of (29) by the dominated convergence theorem. Note that
divf(w) = F(hn0(w), ··· , hn(2n−1)(w)) · `(w)
−D`F(hn0(w), ··· , hn(2n−1)(w)). Since F is Lipschitz continuous, we know
kDF(hn0(w), ··· , hn(2n−1)(w))kH 6 Lip(F) 6 Lip(F ). So sup kdivfk2 < +∞. ∈(0,1] Thus, thanks to (32) 2 divf → divf weakly in L (W, µ) possible a subsequence, and hence, the left hand side of (31) converges to the left hand side of (29). (Step 2): In this step, we assume that f has the following form: ∗ f(w) = f(w)`, ` ∈ W , where f is a bounded continuous and and H-Lipschitz continuous function on W, i.e.
|f(w + h) − f(w)| 6 LipH(f) · khkH, ∀w ∈ W, h ∈ H. (33) Define 2n−1 X Fn(x0, ··· , x2n−1) := f xkhnk , xk ∈ R, k=0 and the approximation of f by
fn(w) := Fn(hn0(w), ··· , hn(2n−1)(w))`. It is clear that by (26) and (33)
2n−1 !1/2 0 0 X 0 2 |Fn(x0, ··· , x2n−1) − Fn(x0, ··· , x2n−1)| 6 LipH(f) |xk − xk| (34) k=0 and Fn(hn0(w), ··· , hn(2n−1)(w)) = f(πn(w)), fn(w) = f(πn(w)). 14 Thus, by Step 1 we have Z Z divf (w) µ(dw) = hf (w), n (w)i k∂Γk(dw). (35) n n Γ H Γ W Moreover, it is easy to see that
sup kfn(w)kH 6 sup kf(w)kH, (36) w∈W w∈W and by (27), for each w ∈ W
lim kfn(w) − f(w)k = 0, (37) n→∞ H Therefore, the right hand side of (35) converges to the right hand side of (29) as n → ∞. On the other hand, noting that 2n−1 X divf (w) = (∂ F )(h (w), ··· , h n (w))h`, h i + f(π (w)) · `(w), n k n n0 n(2 −1) nk H n k=0 we have by (34) and the Rademacher theorem(cf. [13])
|divfn(w)| 6 LipH(f) · k`kH + |`(w)| · sup |f(w)|, w∈W Therefore, it is the same reason as in Step 1 that the left hand side of (35) converges to the left hand side of (29). (Step 3:) We now assume that f is a bounded continuous and H-Lipschitz continuous vector field f on W. For n ∈ N, define n X f (w) = hf(w), ` i ` . n k H k k=1 Then clearly, (36) and (37) hold. Moreover, by Kr´ee-Meyer inequality(cf. [22]), we have by [12]
kdivfnk2 6 C(kDfnk2 + kfnk2) 6 C · LipH(f) + Ckfk2, where the constant C is independent of n. Thus, as above, using the result obtained in Step 2 we prove (29). + (Step 4:) For any > 0, let ϕ > 0 be a smooth function on R satisfying
ϕ(s) = 0, s ∈ [0, /2], ϕ(s) = 1, s ∈ [, ∞). Set ¯ χ(w) := ϕ(dis(w, Γ)) and ¯ Γ := {w ∈ W : dis(w, Γ) < }. ∞ ∗ Then for any f ∈ F Cb (W ), f · χ is a bounded and W-Lipschitz continuous vector field on W, and div(f · χ)(w) = 0, µ-a.e. on Γ/2. Hence, by (29) Z hf · χ , n i k∂Γk(dw) = 0. (38) Γ H W Set for j = 1, 2, ··· nj (w) := hn (w), ` i . Γ Γ j H 15 ∞ As in Fukushima [16, p.240], for each n we can find {vj,m, j = 1, ··· , n, m ∈ N} ⊂ F Cb such that j lim vj,m(w) = n (w), k∂Γk − a.e., j = 1, ··· , n. m→∞ Γ Define vj,m(w) ∞ gj,m(w) := ∈ F Cb , qPn 2 k=1 vj,m(w) + 1/m and n X ∞ ∗ fn,m(w) := gj,m(w)`j ∈ F Cb (X ). j=1 Then
kfn,m(w)kH 6 1, ∀w ∈ W, and by (38) Z hf · χ , n i k∂Γk(dw) = 0. n,m Γ H W Taking the limits m → ∞ and n → ∞ yields by the dominated convergence theorem that Z χ(w)k∂Γk(dw) = 0. W By the arbitrariness of > 0, we obtain
k∂Γk(W − Γ)¯ = 0. (39) In view of Z divf µ(dw) = 0, W we have Z Z hf, n i k∂Γk(dw) = − divf µ(dw). Γ H W Γc Using the same argument as above, we also have
c k∂Γk(W − Γ¯ ) = 0. which together with (39) gives
k∂Γk(W − ∂Γ) = 0. The proof is complete. As a corollary of Theorems 4.1 and 3.6, we have
Corollary 4.2. Let a > 0 and Ba := {w ∈ W : kwkW < a}. Then for any > 0 and any bounded continuous and H-Lipschitz continuous vector field f on W Z Z divf(w)µ(dw) = hf(w), n (w)i k∂B k(dw). Γ H a Ba ∂Ba As a corollary of Theorems 2.12 and 4.1, we have the following isoperimetric inequality.
Corollary 4.3. For any Γ ∈ C (W), U (µ(Γ)) 6 k∂Γk(∂Γ). 16 5. Application to pinned Wiener space Fix an a ∈ Rd, the pinned Wiener space is a closed linear subspace of W in which each path ends a at time 1, Wa := {w ∈ W : w(1) = a}, and the pinned Wiener measure is defined by the regular conditional probability of µ with respect to w(1):
µa(·) := µ(· | w(1) = a). More precisely, for any bounded measurable function F on Rdn Z p1(0, a) · F (w(t1), ··· , w(tn))µa(w) Wa Z n = F (x1, ··· , xn)Πi=0pti+1−ti (xi+1, xi)dx1 ··· dxn, (40) Rdn 2 where x = 0, x = a, t = 1 and p (x, y) = √1 e−|x−y| /(2t). The corresponding 0 n+1 n+1 t 2πt Cameron-Martin space is given by H0 := H ∩ W0. Notice that the pinned Wiener measure µa can also be regarded as the law of pinned Brownian motion w(t)−t(w(1)−a) under µ. It is clear that (W0, H0, µ0) forms an abstract Wiener space, and for a 6= 0, the translation w 7→ w − ·a establishes an isomorphism between (Wa, µa) and (W0, µ0). Therefore, it suffices to consider the pinned Wiener space n (W0, H0, µ0). In this case, for any n ∈ N, let {`nk, k = 1, ··· , 2 − 1} be a linearly ∗ independent subset of W0 given by n `˙ (s) := 1 n n (s) − 1 n n (s), k = 1, ··· , 2 − 1, nk (tk−1,tk ] (tk ,tk+1] n −n n where tk = k2 . Let {hnk, k = 1, ··· , 2 − 1} be the Gram-Schmidt orthogonalization of n {`nk, k = 1, ··· , 2 − 1}. Then, for any w ∈ W0 2n−1 2n−1 X X h n n n i h (w) · h (t) = w(t ) + ∆w(t )(2 t − k) · 1 n n (t) =: π w(t), nk nk k k [tk ,tk+1] n k=1 k=0 n n n where ∆w(tk ) := w(tk+1) − w(tk ). Indeed, it follows from hnk(w) = hnk(πnw) and πnw ∈ ˜ n Hn := span{hnk, k = 1, ··· , 2 − 1}. Thus, using the same method as in proving Theorem 4.1, we find that the conclusions of Theorem 4.1 still holds for the pinned Wiener measure. On the other hand, for d = 1, it is well known that for any r > a(cf. [7, (4.12)]) ∞ X n µa(w ∈ Wa : kwk∞ < r) = (−1) exp{−2nr(nr − a)}. n=−∞ Thus, combining Theorems 3.4 and 4.1 yields that
Theorem 5.1. Let U ∈ B(Rd) have compact and Lipschitz boundary. Define
Γ := {w ∈ Wa :(kw1k∞, ··· , kwdk∞) ∈ U}.
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