Transience of stochastically perturbed classical hamiltonian systems and random wave operators

Abstract

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.

Keywords:

• Stochastic perturbations
• Hamiltonian systems
• random wave operators
• recurrence and transience
• Girsanov density estimates
• AMS 1991 Subject Classifications: Primary: 60H10; Secondary: 60H30, 34C35, 70H05, 60J60, 70L05

^*BiBoS, SFB 237, CERFIM (Locarno)

^†BiBoS

^‡On leave from Moscow Institute of Electronics and Mathematics (MIEM); supported by A.v. Humboldt Stiftung

^*BiBoS, SFB 237, CERFIM (Locarno)

^†BiBoS

^‡On leave from Moscow Institute of Electronics and Mathematics (MIEM); supported by A.v. Humboldt Stiftung

Notes

^*BiBoS, SFB 237, CERFIM (Locarno)

^†BiBoS

^‡On leave from Moscow Institute of Electronics and Mathematics (MIEM); supported by A.v. Humboldt Stiftung