[Paper Review] A class of integration by parts formulae in stochastic analysis I

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

Spectral Theory in Mathematical Physics15 references21 citations

TL;DR

This paper establishes a class of integration by parts formulae on path spaces over compact Riemannian manifolds using stochastic flows and derivative processes. By extending the basic formula of Elworthy and Li (1994), it derives a general integration by parts identity involving the derivative of cylindrical functionals along adapted processes in the Cameron-Martin space, leading to intrinsic formulae via Ricci curvature and connection geometry, with applications to free path spaces and quasi-invariance properties.

ABSTRACT

An integration by parts formula is the foundation for stochastic analysis on path spaces over a (finite dimensional) Riemannian manifold or over $R^n$, from which we may deduce the operator $d$ is closable and define the Laplacian operator on path spaces. A useful formula on the Riemannian manifold is $$dP_tf(v)=(1/t)E f(x_t) \int_0^t \langle d\{x_s\}, v_s angle ,$$ for $P_t$ the heat semi-group, $x_t$ the BM, $v_t$ the derivative flow or its conditional expectation (which is a damped parallel translation), $d\{x_s\}$ is the martingale part of $x_t$. As a meta theorem, this leads to the Clark-Ocone formula (martingale representation theorem with specific integrand) and Logrithmic Sobolev inequalities. Interpreted appropriately, the latter formula is obviously a special case of the integration by parts formula. Here we show by the Markov property and by induction that the latter formula implies the integration by parts formula. WE also use Bismut's original approach to prove an integration by parts formula, using a connection with torsion and one on the free path space.

Motivation & Objective

• To generalize the integration by parts formula of Elworthy and Li (1994) to a broader class of stochastic processes on compact Riemannian manifolds.
• To establish a connection between integration by parts on path spaces and the geometry of the underlying manifold via the Ricci curvature of the LeJan-Watanabe connection.
• To provide a geometric interpretation of the integration by parts formula using parallel translation and stochastic flows.
• To extend the results to free path spaces and derive intrinsic formulae without requiring torsion skew-symmetry of the connection.
• To demonstrate that the derived formulae are consistent with known results such as Driver’s formula, while offering a new, geometrically motivated derivation.

Proposed method

• Derives a general integration by parts formula via Itô’s formula and martingale calculus, applying to cylindrical functionals on path space.
• Uses the derivative of the stochastic flow $ T\xi_t $ to define vector fields on path space, leading to the vector field $ \bar{V}^h $ and its divergence $ \delta \bar{V}^h $.
• Applies the Girsanov-Maruyama theorem to relate measures induced by different initial flows, enabling change-of-measure arguments.
• Introduces a time-continuous family of flows $ H_t^\tau $ on the manifold to transport initial points, allowing differentiation in the flow parameter.
• Applies the classical Stokes theorem on the manifold $ M $ to relate the derivative of the functional to the divergence of the vector field $ h_0 $.
• Uses regularization via stopping times $ \tau_R $ and $ L^{1+\epsilon} $ integrability to extend results from bounded to general adapted processes.

Experimental results

Research questions

• RQ1 How can the integration by parts formula on path space be derived from finite-dimensional stochastic calculus on a compact Riemannian manifold?
• RQ2 What is the geometric role of the Ricci curvature of the LeJan-Watanabe connection in determining the divergence of vector fields on path space?
• RQ3 Can integration by parts formulae be extended to free path spaces using stochastic flows and geometric flows on the base manifold?
• RQ4 How does the choice of connection (metric or not) affect the form of the integration by parts formulae?
• RQ5 What is the relationship between the derived formulae and known results such as Driver’s integration by parts formula?

Key findings

• The paper derives a general integration by parts formula on path space: $ \mathbb{E} dF(T\xi_\cdot(h_\cdot)) = \mathbb{E} F(\xi_\cdot(x)) \int_0^T \langle T\xi_s(\dot{h}_s), X(\xi_s(x)) dB_s \rangle $, valid for cylindrical functionals.
• The formula is shown to reduce to Driver’s integration by parts formula under adaptedness and metric connection assumptions, without requiring torsion skew-symmetry.
• The divergence term $ \delta \bar{V}^h $ is explicitly expressed as $ \mathbb{E} \left[ \int_0^T \langle T\xi_s(\dot{h}_s), X(\xi_s(x)) dB_s \rangle \middle| \xi_\cdot(x) = \gamma \right] $, providing a probabilistic interpretation.
• For the free path space, the formula extends to $ \mathbb{E} \int_M dF(T_x\xi_\cdot(h_\cdot(x))) dx = \mathbb{E} \int_M F(\xi_\cdot(x)) \left( -\text{div}\, h_0(x) + \int_0^T \langle T_x\xi_s(\dot{h}_s(x)), X(\xi_s(x)) dB_s \rangle \right) dx $.
• The results are robust under regularization via stopping times, ensuring convergence for general $ L^{1+\epsilon} $-integrable processes.
• The intrinsic formulae are consistent with known geometric structures, such as Ricci flow and Dohrn-Guerra parallel translation, highlighting the role of curvature in stochastic analysis.