Jerzy Zabczyk

Ergodicity for Infinite Dimensional Systems

Part I. Markovian Dynamical Systems: 1. General dynamical systems 2. Canonical Markovian systems 3. Ergodic and mixing measures 4. Regular Markovian systems Part II. Invariant Measures For Stochastics For Evolution Equations: 5. Stochastic differential equations 6. Existence of invariant measures 7. Uniqueness of invariant measures 8. Densities of invariant measures Part III. Invariant Measures For Specific Models: 9. Ornstein-Uhlenbeck processes 10. Stochastic delay systems 11. Reaction-diffusion equations 12. Spin systems 13. Systems perturbed through the boundary 14. Burgers equation 15. Navier-Stokes equations Appendices.

Second order partial differential equations in Hilbert spaces

Part I. Theory in the Space of Continuous Functions: 1. Gaussian measures 2. Spaces of continuous functions 3. Heat equation 4. Poissons equation 5. Elliptic equations with variable coefficients 6. Ornstein-Uhlenbeck equations 7. General parabolic equations 8. Parabolic equations in open sets Part II. Theory in Sobolev Spaces with a Gaussian Measure: 9. L2 and Sobolev spaces 10. Ornstein-Uhlenbeck semigroups on Lp(H, mu) 11. Perturbations of Ornstein-Uhlenbeck semigroups 12. Gradient systems Part II. Applications to Control Theory: 13. Second order Hamilton-Jacobi equations 14. Hamilton-Jacobi inclusions.

Stochastic evolution equations with a spatially homogeneous Wiener process

A semilinear parabolic equation on d with a non-additive random perturbation is studied. The noise is supposed to be a spatially homogeneous Wiener process. Conditions for the existence and uniqueness of the solution in terms of the spectral measure of the noise are given. Applications to population and geophysical models are indicated. The Freidlin-Wentzell large deviation estimates are obtained as well.

Controllability of stochastic linear systems

In the paper necessary and sufficient conditions for various types of stochastic controllability of the linear stochastic system of the form d x = Ax d t + Bu d t + C d w r , x(0) = x 0 are given. Some possible extensions are indicated as well.

Evolution equations with white-noise boundary conditions

The paper is devoted to nonlinear evolution equations with nonhomogenous boundary conditions of white noise type. Necessary and sufficient conditions for the existence of solutions in the linear case are given. It is also shown that if the nonlinearity satisfies appropriate dissipativity conditions the nonlinear equation has a solution as well. The results are applied to equations with polynomial nonlinearities

Densities for Ornstein–Uhlenbeck processes with jumps

We consider an Ornstein-Uhlenbeck process with values in ℝ n driven by a Levy process (Z t ) taking values in ℝ d with d possibly smaller than n. The Levy noise can have a degenerate or even vanishing Gaussian component. Under a controllability rank condition and a mild assumption on the Levy measure of (Z t ), we prove that the law of the Ornstein-Uhlenbeck process at any time t > 0 has a density on ℝ n . Moreover, when the Levy process is of α-stable type, α ∈ (0, 2), we show that such density is a C ∞ -function.

Null controllability with vanishing energy

Linear, null controllable systems, for which an arbitrary initial state can be transferred to the origin with arbitrarily small energy, are characterized. Theorems are stated in terms of an associated algebraic Riccati equation and in terms of the spectrum of the linear part of the system. The results so obtained allow us to determine Ornstein--Uhlenbeck operators for which the Liouville theorem about bounded harmonic functions holds.

A note on stochastic convolution

We give a new result on stochastic convolution and an application to stocastic differential equations in Hilbert spaces

Regularity of Ornstein–Uhlenbeck Processes Driven by a Lévy White Noise

The paper is concerned with spatial and time regularity of solutions to linear stochastic evolution equation perturbed by Lévy white noise “obtained by subordination of a Gaussian white noise”. Sufficient conditions for spatial continuity are derived. It is also shown that solutions do not have in general cádlág modifications. General results are applied to equations with fractional Laplacian. Applications to Burgers stochastic equations are considered as well.

On Decomposition of Generators

Let