Integration theory on infinite-dimensional manifolds

Abstract:

The purpose of this paper is to develop a natural integration theory over a suitable kind of infinite-dimensional manifold. The type of manifold we study is a curved analogue of an abstract Wiener space. Let $H$ be a real separable Hilbert space, $B$ the completion of $H$ with respect to a measurable norm and $i$ the inclusion map from $H$ into $B$. The triple $(i,H,B)$ is an abstract Wiener space. $B$ carries a family of Wiener measures. We will define a Riemann-Wiener manifold to be a triple $(\mathcal {W},\tau ,g)$ satisfying specific conditions, $\mathcal {W}$ is a ${C^j}$-differentiable manifold $(j \geqq 3)$ modelled on $B$ and, for each $x$ in $\mathcal {W},\tau (x)$ is a norm on the tangent space ${T_x}(\mathcal {W})$ of $\mathcal {W}$ at $x$ while $g(x)$ is a densely defined inner product on ${T_x}(\mathcal {W})$. We show that each tangent space is an abstract Wiener space and there exists a spray on $\mathcal {W}$ associated with $g$. For each point $x$ in $\mathcal {W}$ the exponential map, defined by this spray, is a ${C^{j - 2}}$-homeomorphism from a $\tau (x)$-neighborhood of the origin in ${T_x}(\mathcal {W})$ onto a neighborhood of $x$ in $\mathcal {W}$. We thereby induce from Wiener measures of ${T_x}(\mathcal {W})$ a family of Borel measures ${q_t}(x, \cdot ),t > 0$, in a neighborhood of $x$. We prove that ${q_t}(x, \cdot )$ and ${q_s}(y, \cdot )$, as measures in their common domain, are equivalent if and only if $t = s$ and ${d_g}(x,y)$ is finite. Otherwise they are mutually singular. Here ${d_g}$ is the almost-metric (in the sense that two points may have infinite distance) on $\mathcal {W}$ determined by $g$. In order to do this we first prove an infinite-dimensional analogue of the Jacobi theorem on transformation of Wiener integrals.