## Structural properties and limit behaviour of linear stochastic systems in Hilbert spaces.(English)Zbl 0573.93076

Mathematical control theory, Banach Cent. Publ. 14, 591-609 (1985).
[For the entire collection see Zbl 0568.00025.]
The structural and asymptotic behaviour of a class of Markov processes on a Hilbert space H are studied. The class is the solution x(t) of the stochastic evolution equation $$dx=Axdt+Bdw$$; $$x(0)=x_ 0\in H$$, where A is the infinitesimal generator of a $$C_ 0$$-semigroup on H, w is a Wiener process with values in a Hilbert space U and $$B\in L(U,H)$$. Properties concerning degeneracy smoothness of the transition probabilities and stationary measures are investigated.
Reviewer: R.Curtain

### MSC:

 93E15 Stochastic stability in control theory 60J25 Continuous-time Markov processes on general state spaces 93C05 Linear systems in control theory 46C99 Inner product spaces and their generalizations, Hilbert spaces 60H25 Random operators and equations (aspects of stochastic analysis) 93C25 Control/observation systems in abstract spaces 47D03 Groups and semigroups of linear operators 93B05 Controllability 93E20 Optimal stochastic control

Zbl 0568.00025