[Paper Review] Intertwining and the Markov uniqueness problem on path spaces

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

Geometric Analysis and Curvature Flows10 references4 citations

TL;DR

This paper investigates the Markov uniqueness problem for Malliavin calculus on path spaces using Itô maps to establish intertwining relations between Wiener and Brownian motion measures. It proves that weak differentiability and Markov uniqueness are preserved under these maps, resolving gaps in prior work and showing that divergence operators and Sobolev spaces on path spaces are consistently defined via stochastic integration and conditional expectations.

ABSTRACT

There are two open problem on the analysis of continuous paths on a Riemannian manifold, the Markov uniqueness and the independence of the closure of the differential operator $d$ on its initial domain. The operator $d$ acts naturally on $BC^1$ functions, one is concerned with its extensions to the $L^2$ spaces. With a suitable choice of an initial domain we denote by $D^{2,1}$ its closure under the graph norm. For the Wiener space, the domain of $d$ can be classified, as a consequence its extension is unique whether the initial domain is smooth cylindrical or in $BC^1$ etc. This has not shown to be the same when the measure is the probability distribution of any smooth elliptic diffusion. In an earlier paper, we have shown that the closure of $BC^\infty$ functions agree with that of smooth cylindrical functions, leaving an undesirable gap. The Markov uniqueness is essentially concerned with the problem whether there exists a unique Markov process on the path space whose Markov generator agrees with the infinite-dimensional Laplacian on $C^\infty$ cylindrical functions. Here we reduce Markov uniqueness to whether the pull back of $D^{2,1}$ by the ito map is $ D^{2,1}$ (i.e. a surjection). We also propose a possible approach for tackle this problem.

Motivation & Objective

• To resolve the Markov uniqueness problem for the Laplacian on path spaces $C_{x_0}M$ by analyzing the interplay between stochastic processes and Malliavin calculus.
• To correct and extend earlier claims in [Elworthy-Li3] regarding the equivalence of different Sobolev space constructions on path spaces.
• To establish that the divergence operator on $C_{x_0}M$ is well-defined and continuous via the pushforward of vector fields under Itô maps.
• To demonstrate that weak differentiability on $C_{x_0}M$ arises naturally from conditional expectations of derivatives on $C_0bR^m$.
• To show that the Itô map serves as a substitute for local charts in path space analysis, preserving key differentiability structures.

Proposed method

• Uses Itô maps $\mathcal{I}_t$ to pull back vector fields and derivatives from $C_0\bbR^m$ to $C_{x_0}M$, enabling transfer of differentiability properties.
• Defines the Bismut tangent space $\mathcal{H}_\sigma$ as the image of the Cameron-Martin space under parallel transport, with a canonical isometry to its dual.
• Introduces the operator $\overline{T\mathcal{I}}_\sigma$, the conditional expectation of the derivative of the Itô map, which acts as a projection onto $\mathcal{H}_\sigma$.
• Constructs the right inverse $\boldsymbol{Y}_\sigma$ of $\overline{T\mathcal{I}}_\sigma$ using the adjoint of the SDE's diffusion coefficient $X(x)$.
• Applies the Cameron-Martin integration by parts formula to extend the Malliavin derivative $d^H$ to a closed operator $d$ on $L^2(C_{x_0}M; \bbR)$, defining $\mathbb{D}^{2,1}$.
• Uses the identity $\operatorname{div} V = \mathbb{E}[\operatorname{div}(\mathcal{I}^*({\bf Y}_-V(-))) \mid \mathfrak{F}^{x_0}]$ to transfer the divergence operator from $C_0\bbR^m$ to $C_{x_0}M$.

Experimental results

Research questions

• RQ1 Does the choice of initial domain for the Malliavin derivative affect the resulting $\mathbb{D}^{2,1}$ space on $C_{x_0}M$?
• RQ2 Is the Laplacian on $C_{x_0}M$ essentially self-adjoint, or does it possess Markov uniqueness?
• RQ3 Can the Itô map serve as a substitute for local charts in path space Malliavin calculus, preserving weak differentiability?
• RQ4 How is the divergence operator on $C_{x_0}M$ related to the divergence on $C_0\bbR^m$ via the Itô map?
• RQ5 What is the precise relationship between weak derivatives on $C_{x_0}M$ and conditional expectations of derivatives on $C_0\bbR^m$?

Key findings

• The space $\mathbb{D}^{2,1}(C_{x_0}M; \bbR)$ is independent of the choice of initial domain $\operatorname{Dom}(d^H)$ when using $C^\infty$ cylindrical functions or $BC^\infty$ functions.
• The Itô map $\mathcal{I}$ induces a continuous linear map $\overline{T\mathcal{I}}_\sigma: H \to \mathcal{H}_\sigma$, which is an orthogonal projection onto the Bismut tangent space.
• For any $V \in \mathbb{D}^{2,1}{\mathcal{H}}$, the pushforward $\overline{T\mathcal{I}(V)}$ lies in $\operatorname{Dom}(\operatorname{div})$ on $C_{x_0}M$, and $\operatorname{div} V = \mathbb{E}[\operatorname{div}(\mathcal{I}^*({\bf Y}_-V(-))) \mid \mathfrak{F}^{x_0}]$ almost surely.
• Weak differentiability on $C_{x_0}M$ is characterized by $ (df)_\sigma = \mathbb{E}[d(\mathcal{I}^*f)_\omega \mid x_\cdot(\omega) = \sigma] \cdot \boldsymbol{Y}_\sigma $, linking it to the conditional derivative on $C_0\bbR^m$.
• The set of vector fields $V$ on $C_0\bbR^m$ such that $\overline{T\mathcal{I}(V)} \in \mathbb{D}^{2,1}{\mathcal{H}}$ is dense in $\mathbb{D}^{2,1}$, ensuring the closure of the divergence operator is well-behaved.
• The construction confirms that the Markov uniqueness of the Laplacian on $C_{x_0}M$ holds if the generator agrees with $\Delta$ on smooth cylindrical functions, as per [Eberle].