Bouncary 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 abstrct 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 norman 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 will give a different expression of ℰ_{a}^{(p)} in terms of covariant differentiation. To do this, we need the second fundamantal 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, T) = \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)

If the third term ∫_{∂M} α(θ, η)dσ is relatively small, we may have a chance to have the spectral gap for □_{a}^{(1)}. We can construct such an example.

2. Path group

Let G be a Lie group with the Lie algebra 𝔤. We suppose that Ad(G)-invariant inner product is given on 𝔤. Let (B_{t}) be a Brownian motion on 𝔤 and we consider the following SDE on G:

dγ_{t} = γ_{t} ∘ dB_{t}.

The solution defines a measure on the path space

P G := {γ: [0, T] → G, γ is continuous and γ(0) = e}.

We denote the measure by ν. We identify (P G, ν) with the Wiener space. Let us define a domain in (P G, ν). Let π: P G ↦ G by π(γ) := γ(T). Let log: G → 𝔤 be the inverse function of exp : 𝔤 → G. log is well-defined only on a neighborhood of e. We define F: P G ↦ ℝ by F = | log ∘π|_{𝔤}. We define a domain by M = {F ≤ δ} for small δ.

Now we can consider the Hodge-Kodaira operator on M with the asolute boundary condition. We are interested in the spectral gap of □_{a}^{(1)} and can obtain the following theorem.

Theorem 2.1. If T is small, then ℰ_{a}^{(1)} has the spectral gap.