Transience of stochastically perturbed classical hamiltonian systems and random wave operators

• Published in: Stochastics and stochastics reports Vol. 60; no. 1-2; pp. 41 - 55
• Main Authors: Albeverio, S., Hilbert, A., Kolokoltsov, V.
• Format: Journal Article
• Language: English
• Published: Abingdon Gordon and Breach Science Publishers, Inc 01.02.1997; Taylor & Francis
• Subjects: AMS 1991 Subject Classifications: Primary: 60H10; Secondary: 60H30, 34C35, 70H05, 60J60, 70L05; Distribution theory; Exact sciences and technology; Girsanov density estimates; Hamiltonian systems; Mathematical analysis; Mathematics; Operator theory; Potential theory; Probability and statistics; Probability theory and stochastic processes; random wave operators; recurrence and transience; Sciences and techniques of general use; Stochastic perturbations; Stochastic processes; Transient process; Hamiltonian system; Random wave operator; Girsanov density; Potential; Lyapunov function
• ISSN: 1045-1129
• DOI: 10.1080/17442509708834098

A classical Hamiltonian system perturbed by a white noise force is considered. Under suitable smallness assumptions on the deterministic force transience of the solution process is proven in all dimensions. The existence of random wave operators (almost surely with respect to Wiener measure) is proven for potentials which decrease at infinity. A remark on recurrence in one dimension when the force grows stronger than linearly at infinity is also given.