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Central limit theorems and random currents. (English) Zbl 0589.60066
Stochastic differential systems, Proc. 3rd Bad Honnef Conf. 1985, Lect. Notes Control Inf. Sci. 78, 195-205 (1986).
[For the entire collection see Zbl 0579.00014.]
Under several mild conditions, we can regard a trajectory of a continuous semi-martingale as a random current, by using the duality between domains of integration and the integrands. Hence we can reformulate various limit theorems for continuous stochastic processes in terms of random currents.
In this articles, we show that in case of reflecting diffusions on a compact manifold with boundary, various central limit theorems can be discussed in a unified way by using the framework mentioned above. This is closely related to homogenization problems for periodic diffusions and central limit theorems for the homological position of diffusions on a compact surface of constant negative curvature.

MSC:
60J60 Diffusion processes
60F05 Central limit and other weak theorems
58J65 Diffusion processes and stochastic analysis on manifolds