[Paper Review] Special Itô maps and an $L^2$ Hodge theory for one forms on path spaces

K. D. Elworthy, Xue-Mei Li|arXiv (Cornell University)|Nov 21, 2019

Algebraic Geometry and Number Theory34 references4 citations

TL;DR

This paper establishes an $L^2$ Hodge theory for one-forms on path spaces $C_{x_0}M$ by introducing special Itô maps and defining a closed exterior derivative $\bar{d}$ on $L^2$-sections of the dual Cameron-Martin bundle. The key contribution is a Hodge decomposition theorem showing that every $L^2$ cohomology class in $H^1(C_{x_0}M)$ has a unique harmonic representative in the kernel of the Laplacian $\Delta^1 = \bar{d}^*\bar{d} + \bar{d}\bar{d}^*$, with $L^2\Gamma(\mathcal{H}^1_\cdot^*) = \ker \Delta^1 \oplus \overline{\text{Image}\, \bar{d}} \oplus \overline{\text{Image}\, d^{1*}}$. This extends finite-dimensional Hodge theory to infinite-dimensional path spaces using Malliavin calculus and stochastic analysis.

ABSTRACT

We prove a Kodaira-Hodge decomposition on differential 1-forms on the space of non-smooth paths over a Riemannian manifold, allowing us to define the corresponding first cohomology group. This uses the Itô map of a Brownian system and damped stochastic parallel translation.

Motivation & Objective

• To develop an $L^2$ Hodge theory for one-forms on the path space $C_{x_0}M$ of a compact Riemannian manifold $M$.
• To define a closed exterior derivative $\bar{d}$ on $L^2$-sections of the dual Cameron-Martin bundle $\mathcal{H}^1_\cdot^*$.
• To establish a Hodge decomposition for $L^2$ one-forms using the Laplacian $\Delta^1 = \bar{d}^*\bar{d} + \bar{d}\bar{d}^*$.
• To show that every $L^2$ cohomology class in $H^1(C_{x_0}M)$ has a unique representative in $\ker \Delta^1$, extending finite-dimensional Hodge theory to infinite dimensions.
• To use special Itô maps and stochastic analysis to construct the $L^2$-theory despite the lack of smooth partitions of unity on path spaces.

Proposed method

• Define the path space $C_{x_0}M$ as the space of continuous paths $\sigma: [0,T] \to M$ with $\sigma(0) = x_0$, equipped with a $C^\infty$ Banach manifold structure.
• Equip tangent spaces $T_\sigma C_{x_0}M$ with the sup-norm $\|v\|_\sigma^\infty = \sup_t |v(t)|_{\sigma(t)}$, inducing a Finsler structure.
• Define the Cameron-Martin space $H^1_\sigma$ as the space of vector fields $v$ along $\sigma$ such that $v_t = //_t(\sigma) h_t$ for $h_t \in L^2$-H"older paths with $h_0 = 0$, forming the 'admissible directions' for $H$-differentiability.
• Construct the dual bundle $\mathcal{H}^1_\cdot^*$ of the Cameron-Martin bundle and consider $L^2$-sections $L^2\Gamma(\mathcal{H}^1_\cdot^*)$ as the space of $L^2$ one-forms.
• Define the closure $\bar{d}$ of the exterior derivative $d$ on smooth cylindrical one-forms, using the $L^2$-closure of the standard exterior derivative on $C^1$ vector fields.
• Define the adjoint $d^*$ and the Laplacian $\Delta^1 = \bar{d}^*\bar{d} + \bar{d}\bar{d}^*$ on $L^2\Gamma(\mathcal{H}^1_\cdot^*)$, and prove it is self-adjoint with dense domain.
• Use Driver’s integration by parts formula and the closed range property of $\bar{d}$ to establish the Hodge decomposition $L^2\Gamma(\mathcal{H}^1_\cdot^*) = \ker \Delta^1 \oplus \overline{\text{Image}\, \bar{d}} \oplus \overline{\text{Image}\, d^{1*}}$.

Experimental results

Research questions

• RQ1 Can an $L^2$ Hodge theory be developed for one-forms on the infinite-dimensional path space $C_{x_0}M$ despite the absence of smooth partitions of unity?
• RQ2 What is the correct definition of the exterior derivative $\bar{d}$ on $L^2$-sections of the dual Cameron-Martin bundle $\mathcal{H}^1_\cdot^*$, and is it closed?
• RQ3 Does the Laplacian $\Delta^1 = \bar{d}^*\bar{d} + \bar{d}\bar{d}^*$ on $L^2\Gamma(\mathcal{H}^1_\cdot^*)$ admit a Hodge decomposition into harmonic, exact, and coexact components?
• RQ4 Is every $L^2$ cohomology class in $H^1(C_{x_0}M)$ represented by a unique harmonic one-form in $\ker \Delta^1$?
• RQ5 How do special Itô maps relate to the $L^2$-theory of differential forms on path spaces?

Key findings

• The exterior derivative $\bar{d}$ on $L^2$-sections of $\mathcal{H}^1_\cdot^*$ is a closed operator, obtained as the $L^2$-closure of the standard exterior derivative on smooth cylindrical one-forms.
• The adjoint $d^*$ of $\bar{d}$ is well-defined on a dense domain containing smooth cylindrical one-forms, and $\bar{d}^*\bar{d} + \bar{d}\bar{d}^*$ defines a nonnegative self-adjoint Laplacian $\Delta^1$ on $L^2\Gamma(\mathcal{H}^1_\cdot^*)$.
• The space $L^2\Gamma(\mathcal{H}^1_\cdot^*)$ admits a Hodge decomposition: $L^2\Gamma(\mathcal{H}^1_\cdot^*) = \ker \Delta^1 \oplus \overline{\text{Image}\, \bar{d}} \oplus \overline{\text{Image}\, d^{1*}}$, with orthogonal summands.
• Every $L^2$ cohomology class in $H^1(C_{x_0}M)$ has a unique representative in $\ker \Delta^1$, the space of harmonic one-forms.
• The operator $\tilde{d} + \tilde{d}^*$ on the complex $L^2(C_{x_0}M; \mathbb{R}) \oplus L^2\Gamma(\mathcal{H}^1_\cdot^*) \oplus L^2\Gamma(\mathcal{H}^2_\cdot^*)$ is self-adjoint, and its square is also self-adjoint, confirming the consistency of the Hodge theory.
• The proof relies on the closed range of $\bar{d}$ and the orthogonality of $\overline{\text{Image}\, \bar{d}}$ and $\overline{\text{Image}\, d^{1*}}$, established via integration by parts and the vanishing of $\psi$ on the image of $\wedge^2 T\mathcal{I}$.