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January 1999 The Limits of Stochastic Integrals of Differential Forms
Terry Lyons, Lucretiu Stoica
Ann. Probab. 27(1): 1-49 (January 1999). DOI: 10.1214/aop/1022677253


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.


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Terry Lyons. Lucretiu Stoica. "The Limits of Stochastic Integrals of Differential Forms." Ann. Probab. 27 (1) 1 - 49, January 1999.


Published: January 1999
First available in Project Euclid: 29 May 2002

zbMATH: 0969.60078
MathSciNet: MR1681146
Digital Object Identifier: 10.1214/aop/1022677253

Primary: 31C25 , 60H05 , 60J60

Keywords: Dirichlet process , Lyons-Zheng decomposition , martingale decomposition , path integral , singular integral , stochastic integral

Rights: Copyright © 1999 Institute of Mathematical Statistics

Vol.27 • No. 1 • January 1999
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