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Coupling constructions for hypoelliptic diffusions: Two examples. (English) Zbl 0823.60068

Cranston, Michael C. (ed.) et al., Stochastic analysis. Proceedings of the Summer Research Institute on stochastic analysis, held at Cornell University, Ithaca, NY, USA, July 11-30, 1993. Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 57, 193-212 (1995).
Summary: Coupling constructions are given for two examples of hypoelliptic diffusions with smooth coefficients: the two-dimensional diffusion formed by scalar Brownian motion and its time integral; and the three- dimensional diffusion formed by planar Brownian motion and its stochastic area. In both cases it is shown that one can construct co-adapted copies of Brownian motion such that the corresponding copies of the diffusion will couple in finite time. The first case uses stopping time arguments; the second is computationally more involved and the solution was found using a computer algebra implementation of stochastic calculus (though the final proof has been checked completely by hand). Questions are formulated about coupling problems for more general hypoelliptic diffusions with smooth coefficients.
For the entire collection see [Zbl 0814.00017].

MSC:

60J65 Brownian motion
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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