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

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′ α(t) = (A− αI)yα(t) +Bu(t), t ∈ (−∞, 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 α ≥ 0. We apply the results to study the dependence on α of the Cameron–Martin space...

Invariant Measures for Passive Tracer Dynamics in Ornstein-uhlenbeck Flows

The structure of entrance laws for time-inhomogeneous Ornstein-Uhlenbeck Processes with Lévy Noise in Hilbert spaces

This paper is about the structure of all entrance laws (in the sense of Dynkin) for time-inhomogeneous Ornstein-Uhlenbeck processes with Lévy noise in Hilbert state spaces. We identify the extremal entrance laws with finite weak first moments through an explicit formula for their Fourier transforms, generalising corresponding results by Dynkin for Wiener noise and nuclear state spaces. We then ...

The Stationary Distributions of Doubly Skew Ornstein-Uhlenbeck Processes and Markov-modulated Skew Ornstein-Uhlenbeck Processes

In this paper, we consider the stationary density function of the doubly skew Ornstein-Uhlenbeck process. We present the explicit formula for the stationary density function and show that this process is positive Harris recurrent and geometrically ergodic. We expand our method to the more general cases in which the multiple parameters are present and we try to consider the stability of the skew...

Multivariate Generalized Ornstein-Uhlenbeck Processes

De Haan and Karandikar [12] introduced generalized Ornstein–Uhlenbeck processes as one-dimensional processes (Vt)t≥0 which are basically characterized by the fact that for each h > 0 the equidistantly sampled process (Vnh)n∈N0 satisfies the random recurrence equation Vnh = A(n−1)h,nhV(n−1)h + B(n−1)h,nh, n ∈ N, where (A(n−1)h,nh, B(n−1)h,nh)n∈N is an i.i.d. sequence with positive A0,h for each ...