Linear Control Systems on Unbounded Time Intervals and Invariant Measures of Ornstein--Uhlenbeck Processes in Hilbert Spaces

We consider linear control systems in a Hilbert space over an unbounded time interval of the form $$ y_\alpha'(t)=(A-\alpha I)y_\alpha(t)+Bu(t), \qquad t\in (-\infty, T], $$ with bounded control operator B, under appropriate stability assumptions on the operator A. We study how the space of states reachable at time T depends on the parameter $\alpha\geq 0$. We apply the results to study the dependence on $\alpha$ of the Cameron--Martin spaces of the invariant measures of the Ornstein--Uhlenbeck processes $X_\alpha$ defined by the equation driven by the Wiener process W: $$ dX_\alpha(t) = (A-\alpha I) X_\alpha(t)\; dt + B\; dW(t),\qquad t\geq 0.