Boundary value problems on a domain of the Wiener space

Ichiro Shigekawa (Kyoto University)

1. Hodge-Kodaira Laplacian on a domain of the Wiener space

We consider the Hodge-Kodaira Laplacian on a domain of an abstract Wiener space (B, H, μ). Suppose we are given a smooth non-degenerate Wiener functional F in the sense of Malliavin calculus. The domain is given as

M = {x ∈ B; F (x) ≤ 0}.  (1.1)

M is noting but a subset of B but we regards it as a smooth manifold with boundary. The boundary ∂M of M is naturally defined by

∂M = {x ∈ B; F (x) = 0}.  (1.2)

The inner normal vector field N on the boundary is given by

N = −\frac{DF}{|DF |}  (1.3)

and the surface measure σ on ∂M is given by

σ = |DF |δ_{0}(F)  (1.4)

where δ_{0}(F) is a composite of the Dirac measure and F in the sense of Watanabe. The Gauss’ divergence formula is formulated as follows:

Proposition 1.1. For smooth 1-form θ, it holds that

∫_{M} D^{∗}θdμ = ∫_{∂M}⟨θ, N⟩dσ.  (1.5)

Using this identity, we can have the integration by parts formula for p-forms as follows.

Proposition 1.2. Let θ be a p-form and η be a p + 1-form. Then it holds that

∫_{M}(dθ, η)dμ = ∫_{M}(θ, d^{∗}η)dμ − ∫_{∂M}⟨θ, i(N)η⟩dσ.  (1.6)

Next we introduce the Hodge-Kodaira operator. To do this, we use the bilinear form. We consider the following bilinear form:

ℰ_{a}^{(p)}(θ, η) = ∫_{M}(dθ, dη)dμ + ∫_{M}(d^{∗}θ, d^{∗}η)dμ  (1.7)

with the domain

Dom(ℰ_{a}^{(p)}) = {θ ∈ W^{∞,∞−}|i(N)θ = 0 on ∂M}.  (1.8)

The associated self-adjoint operator is called the Hodge-Kodaira Laplacian with the absolute boundary condition, denoted by □_{a}^{(p)}. We can introduce different kind of boundary condition (relative boundary condition) but we concentrate on the absolute boundary condition because it corresponds to the Neumann boundary condition.

Theorem 1.3. Take any θ, η ∈ Dom(ℰ_{a}^{(p)}) and suppose that i(N)dθ = 0 and i(N)d^{∗}θ = 0 on ∂M. Then it holds that

ℰ_{a}^{(p)}(θ, η) = ∫_{M}((dd^{∗} + d^{∗}d)θ, η)dμ.  (1.9)

This theorem show that ℰ_{a}^{(p)} is closable and the associated generator is dd^{∗} + d^{∗}d with the boundary condition i(N)θ = 0, i(N)dθ = 0 and i(N)d^{∗}θ = 0 on ∂M.

We can introduce a different kind of boundary condition that is called the relative boundary condition. The bilinear form ℰ_{r}^{(p)} is defined by

ℰ_{r}^{(p)}(θ, η) = ∫_{M}(dθ, dη)dμ + ∫_{M}(d^{∗}θ, d^{∗}η)dμ

with the domain

Dom(ℰ_{r}^{(p)}) = {θ ∈ W^{∞,∞−}|N ∧ θ = 0 on ∂M}.

The bilinear form itself is the same as that of absolute boundary condition but the domain is different.

We will give a different expression of ℰ_{a}^{(p)} and ℰ_{r}^{(p)} in terms of covariant differentiation. To do this, we need the second fundamental form α on ∂M. It is defined as follows. Let X, Y be vector fields tangential to ∂M. The following bilinear form α is called the second fundamental form:

α(X, Y) = (∇_{X}Y, N).  (1.10)

The second fundamental form is given as follows:

α(X, Y) = \frac{D^{2}F (X, Y)}{|DF |}.

Note that α is symmetric.

From now on we assume p = 1. We have the following identity.

ℰ_{a}^{(1)}(θ, η) = ∫_{M}(∇θ, ∇η)dμ + ∫_{M}(θ, η)dμ + ∫_{∂M} α(θ, η)dσ  (1.11)

and

ℰ_{r}^{(1)}(θ, η) = ∫_{M}(∇θ, ∇η)dμ + ∫_{M}(θ, η)dμ + ∫_{∂M} d^{∗}N(θ, N)(η, N)dσ.  (1.12)

Thus, if α is non-negative definite in the absolute boundary condition case, or d^{∗}N is non-negative in the relative boundary condition case, we can have that there is no harmonic 1-form. Since α = D^{2}F/|DF |, the positivity of α corresponds to the positivity of D^{2}F, i.e., the convexity of F. Hence, if {F ≤ 0} is convex, then it follows that α is non-negative. So the vanishing theorem holds in this case for the absolute boundary condition.