Ergodicity of Dissipative Differential Equations Subject to Random Impulses

Abstract

Differential equations subject to random impulses are studied. Randomness is introduced both through the time between impulses, which is distributed exponentially, and through the sign of the impulses, which are fixed in amplitude and orientation. Such models are particular instances of piecewise deterministic Markov processes and they arise naturally in the study of a number of physical phenomena, particularly impacting systems. The underlying deterministic semigroup is assumed to be dissipative and a general theorem which establishes the existence of invariant measures for the randomly forced problem is proved. Further structure is then added to the deterministic semigroup, which enables the proof of ergodic theorems. Characteristic functions are used for the case when the deterministic component forms a damped linear problem and irreducibility measures are employed for the study of a randomly forced damped double-well nonlinear oscillator with a gradient structure.