The Annals of Probability

The Limits of Stochastic Integrals of Differential Forms

Terry Lyons and Lucretiu Stoica

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This paper is concerned with the integration (of 1-forms) against the Markov stochastic process associated with a second-order elliptic differential operator in divergence form. It focuses on the limiting behavior of the integral as the process leaves a fixed point or goes to infinity. This extends previous work in the area where advantage was usually taken of the fact that the operator was self adjoint and started with the associated invariant measure. Applications are given. For example, it is a trivial consequence that the diffusion associated to a uniformly elliptic operator on a negatively curved Cartan–Hadamard manifold has an asymptotic direction (recovering and strengthening the previous arguments of Pratt, Sullivan and others). The approach can also be used to construct a Lévy area for such processes, to study the thinness of sets for the elliptic operator, and probably has wider applications.

Article information

Ann. Probab., Volume 27, Number 1 (1999), 1-49.

First available in Project Euclid: 29 May 2002

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Zentralblatt MATH identifier

Primary: 60J60: Diffusion processes [See also 58J65] 60H05: Stochastic integrals 31C25: Dirichlet spaces

Stochastic integral Dirichlet process path integral singular integral martingale decomposition Lyons-Zheng decomposition


Lyons, Terry; Stoica, Lucretiu. The Limits of Stochastic Integrals of Differential Forms. Ann. Probab. 27 (1999), no. 1, 1--49. doi:10.1214/aop/1022677253.

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