The author obtains sufficient and necessary conditions for the existence of a \(\tau\)-periodic distribution of the linear equation \[ dx_ t=A(t)x_ t dt+dz_ t,\quad t\geq 0,\tag{1} \] where \(A(\cdot)\) is an \({\mathcal L}(R^ d)\)-valued, \(\tau\)-periodic function, integrable on \([0,\tau]\) and \((z_ t)_{t\in\mathbb{R}}\) is a process with \(\tau\)-periodic independent-increments. These conditions are simply expressed in a form by using Lévy characteristics (or some equivalent conditions) and the well-known characteristics of \(U(t,s)\), \(U(\tau,0)\), where \(U(t,s)\) is the Cauchy operator of the deterministic differential systems (of periodic type) \(\dot x(t)=A(t)x(t)\). The result of the author is a natural analogue of the result of J. B. Graveraux (1982) on the existence of a stationary distribution of solutions of (1) in the case \(A(t)\equiv A\), but with different proof (which are of interest for further generalizations to more general equations such as nonlinear, as the reviewer thinks).
Reviewer’s notes: This article is related to the recent work of E. Zehnder (1990/91) on periodic stochastic Hamiltonian systems, and results of the school of L. Arnold (Bremen) on stochastic ODE.