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Approximation of the Ornstein-Uhlenbeck local time by harmonic oscillators. (English) Zbl 0955.60074
The trajectories of a particle of mass $$1/\beta$$ submitted to the action of a harmonic oscillator with an external white noise force converge, as $$\beta\to\infty$$, to an Ornstein-Uhlenbeck process [see E. Nelson, “Dynamical theories of Brownian motion” (1967; Zbl 0165.58502)]. The authors of the present paper show that the number of crossings of the particle with a fixed level, suitably normalized, converge in $$L^2$$ to the local time of the Ornstein-Uhlenbeck process. They also obtain a weak limit theorem indicating the speed of the convergence. A key to these results is the interpretation of the approximating process as a regularization of the Ornstein-Uhlenbeck process by convolution with a non-compactly supported kernel.

##### MSC:
 60J60 Diffusion processes 60J55 Local time and additive functionals
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