Approximation of the Ornstein-Uhlenbeck local time by harmonic oscillators
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 \textit{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.
scientific article; zbMATH DE number 1455542