Forced harmonic oscillators, waves on a forced string and changes of measure

• Published in: Statistics & probability letters Vol. 179; p. 109232
• Main Author: Gzyl, Henryk
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.12.2021
• Subjects: Cameron–Martin-Girsanov–Maruyama change of measure; Gauss–Markov process; Harmonic oscillator; Wave equation; Wave equation; Gauss–Markov process; Cameron–Martin-Girsanov–Maruyama change of measure; Harmonic oscillator
• ISSN: 0167-7152; 1879-2103
• DOI: 10.1016/j.spl.2021.109232

There is a connection between deterministic Gaussian processes, that is, flows that leave a Gaussian measure invariant and waves on a string. The connection appears when the solution to the wave equation is expanded in the eigenvalues of the Laplacian. The Fourier components behave as one dimensional harmonic oscillators. It so happens that if the initial conditions of each deterministic oscillator have an appropriate Gaussian distribution, then the motion of the oscillator in its phase space leaves it invariant. When we consider oscillations in a string that is subject to an external force, another interesting connection appears. Fourier components can be thought of as forced harmonic oscillators. The motion of each oscillator does not generate a time homogeneous flow in its phase space, but if the initial Gaussian measure is changed in time by means of an analogue of the Cameron–Martin-Girsanov–Maruyama transformation, the time flow of the forced oscillator preserves the initial Gaussian measure. It is the aim of this work to make this construction explicit.