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

Zabczyk, J.

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

Keywords:

structural and asymptotic behaviour; Markov processes on a Hilbert space; stochastic evolution equation; stationary measures

Citations:

Zbl 0568.00025