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 amp
litude and orientation. Such models are particular instances of piecewise d
eterministic Markov processes and they arise naturally in the study of a nu
mber 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 forc
ed problem is proved. Further structure is then added to the deterministic
semigroup, which enables the proof of ergodic theorems. Characteristic func
tions 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 st
ructure. (C) 1999 Academic Press.