A TYPE OF GAUSS' DIVERGENCE FORMULA ON WIENER SPACES

We will formulate a type of Gauss' divergence formula on sets of functions which are greater than a specific value of which boundaries are not regular. Such formula was first established by L. Zambotti in 2002 with a profound study of stochastic processes. In this paper we will give a much shorter and simpler proof for his formula in a framework of the Malliavin calculus and give alternate expressions. Our approach also enables to show that such formulae hold in other Gaussian spaces.


Introduction
Gauss' divergence formula plays a fundamental role in a wide range of fields in mathematics and other sciences. The formula is simply stated in the case of Euclidian space and Lebesgue measure; where D is a smooth domain, div f (x) is a divergence of the vector field f (x), n(x) denotes the exterior normal vector at x ∈ ∂ D, and σ D (d x) denotes the surface measure on ∂ D. In the case that f is a scalar field and the vector field is given by, using a vector h, f (x)h, and if the measure has a Gaussian density g(x) = 1 divergence formula still holds in the infinite dimensional spaces. To our knowledge, such formulae were first studied by Goodman [6], and, among others, recently by Shigekawa [15] to establish a Hodge-Kodaira vanishing theorem in infinite dimensional spaces. They, however, assume that the domains on which divergence formulae are stated have a regular boundary in the sense of Malliavin calculus. In this paper we are concerned with a divergence formula for a subset W D of a Banach space The (topological) boundary ∂ W D is clearly given by the set of functions whose minima are −a. We will show that, though w ∈ W D may hit −a many times, the metrical boundary consists of the set that w ∈ W D hits −a exactly once. We also note that the minimum m(w) of each continuous function is not H-C 1 but H-Lipschitz (see [1,15] for these notions), and is degenerate in terms of Malliavin calculus. The divergence formula for W D was first established by Zambotti [18] and the proof based on profound insights of a theory of stochastic processes. He then applied his formula to analyze solutions to stochastic partial differential equations with reflection of Nualart-Pardoux type [13] through a theory of Dirichlet forms. His formula, which has its own interest, has been extended by Funaki-Ishitani [5] for domains { f ∈ W ; h 1 < f < h 2 } with some continuous functions h 1 and h 2 using a random walk approximation for Brownian paths, and by Hariya [7] for multidimensional cases applying a concrete representation of the heat kernel. Such approaches rely on a Markovian property of the Brownian motions. Closely related studies to such divergence formulae have been treated by Fukushima, Hino, and Uchida [3,4,8,9]. The aim of the present paper is to give a simple and shorter proof, and an alternate representation for Zambotti's divergence formula in a frame work of Malliavin calculus. After that we will recover his original formula applying simple and well-known probabilistic formulae. Our approach here relies on the Gaussian property of Brownian motions and enables to extend his result to other Gaussian spaces. We are grateful to the referees for several suggestions which improve the paper.

Framework and main result
Our main subjects in the present paper are divergence formulae for two types of Wiener spaces consisting of a Brownian motion starting from zero and a pinned Brownian motion starting from and ending at zero on time interval [0, 1]. We will, however, begin with a slight general setting and introduce a Hilbert space E = L 2 (0, 1). Let Q be an operator E → E given by a symmetric, nonnegative, and factorizable integral kernel ρ(s, t), namely, Q f (t) : It is well-known that there exists a Gaussian measure on E with covariance operator Q if and only if Q is symmetric nonnegative nuclear operator. With above settings, let us introduce a mean zero Gaussian measure µ on E with covariance operator Q. Then H := Q 1/2 E is a reproducing kernel Hilbert space with an inner product f , g H := Q −1/2 f , Q −1/2 g E . We will always identify H and its dual H * hereafter. We shall denote by W ⊂ C([0, 1], ) a Banach space which actually supports µ. We note that the above setting also includes the canonical probability space induced by the fractional Brownian motion [10]. We introduce here several notions which are used later before proceeding. The space s p is the completion of the set of stochastic polynomials with respect to the norm F s p := (I − L) s/2 F p , where L is the Ornstein-Uhlenbeck operator and · p denotes the usual L p -norm on W with respect to µ for p > 1 and s ∈ . And the space ∞ ∞− is defined by p>1,s∈ s p . For every F ∈ 1 p we can define the Gross-Shigekawa derivative D : 1 p → L p (W, H) which is the set of H-valued L p -functions on W as usual, see, e.g., [12,17] for detailed discussions and analysis on such spaces. Before stating our result, we will introduce a concept of locally nondegenerate Wiener functional following Florit-Nualart [2].
It is known that the minimum m(w) is locally nondegenerate on (−∞, −α) for every α > 0 for the Brownian motion and the fractional Brownian motion [10] starting from zero.
We note that the fact that w(t) attains its minimum at a unique t ∈ [0, 1] can be also proved in the framework of Malliavin calculus (see [1]) and 〈Dm(w), h〉 H = h(τ m (w)).
Since δ −a (m(w)) is a positive generalized Wiener functional, it determines a positive measure on W by Sugita's theorem [16]. Moreover, by virtue of Theorem 6.1 in [16], we obtain the following integral representation.

Corollary 2.2. With the settings of the above theorem, we have
whereF is (p, 1)-quasi-continuous version of F and σ −a (d w) := δ −a (m(w))µ(d w) in the sense of Sugita.

Watanabe compositions for locally nondegenerate functionals
Florit and Nualart [2] showed that the maximum of Brownian sheet has a C ∞ density by introducing a concept of locally nondegenerate functionals. To show the result, they used a standard method in the Malliavin calculus, namely, they showed Malliavin's integration by parts formula still holds for locally nondegenerate functionals [12, Proposition 2.1.4]. Another well-known approach to show the smoothness of the law of nondegenerate functionals is the Watanabe composition method [17], which is also based on Malliavin's integration by parts. We will show that Watanabe composition still holds for locally nondegenerate functionals with a little restricted situations that are not essential for our purpose. Before stating the result, let us recall that −2 p is the completion of the set of stochastic polynomials with respect to F −2 p := (I − L) −1 F p . It is also known that −2 p is the dual space of 2 q with 1/p + 1/q = 1 under identification of L 2 (E, µ) with its dual.

Corollary 3.2.
For each locally nondegenerate functional F ∈ L ∞− (W ) on A and a Schwartz distribution T ∈ −2 with supp T ⊂ A, we can uniquely define a composition T • F ∈ −2 p for every 1 < p < ∞.
Since δ x ∈ −2 if n = 1, we have Corollary 3.3. Let m(w) be the minimum of w. If m(w) is locally nondegenerate, then δ x (m(w)) has a rigorous meaning for every x < 0 as an element of −2 p for every 1 < p < ∞. Hence we can derive the regularity of the density for the maximum/minimum of some Gaussian processes in this framework (cf. [17]). We do not describe them since they are entirely standard.

Proof of the theorem and other representations
Once the Watanabe composition (Corollary 3.3) is established, the proof goes easily. We first note Note that, for every ϕ ∈ C 1 b ( ), G ∈ 1 q and F ∈ 1 p with 1/p + 1/q = 1, we have the integration by parts formula: and f , g H is defined by Q −1/2 f , Q −1/2 g E . Therefore we can approximate 1 (−a,∞) by smooth functions and obtain where 〈·, ·〉 of the last term is a natural coupling between 2 p and −2 q with 1/p+1/q = 1. However it is known that 〈Dm(w), h〉 H = h(τ m (w)), we have the conclusion. We have now proved the theorem which was stated in a framework of infinite dimensional analysis. We here reformulate the formula using some concrete expressions for several random variables related to Brownian motions and pinned Brownian motions. The following representation corresponds to those obtained by Hariya [7]. 1)). Then we have: (A) The case of Brownian motion starting from zero: (B) The case of pinned Brownian motion starting from and ending at zero: Proof. Since Watanabe composition is a dual operator of the conditional expectation, we have Then it is well known that p m (−a) = 2 π e −a 2 /2 for Brownian motion and p m (−a) = 4ae −2a 2 for pinned Brownian motion.
We can derive one more expression which corresponds to Zambotti's representation. Proof. The proof can be done similarly to the above proposition. Therefore it is sufficient to recall that, for a Brownian motion starting from a > 0, where H(t) := inf{s < t; W (s) = m(t)}. Hence we have P a m(w) = 0, τ m (w) = u du = a du.
Moreover it is also known that P a H(t) ∈ d v, inf 0≤s≤t W (s) ∈ d y, W (t) ∈ dz = (a − y)(z − y) so that the pinned case is also derived.