The ergodic behaviour of homogeneous strong Feller irreducible Markov processes in Banach spaces is studied; in particular, existence and uniqueness of finite and $\sigma $-finite invariant measures are considered. The results obtained are applied to solutions of stochastic parabolic equations.

Existence, uniqueness and regularity of mild solutions to semilinear nonautonomous stochastic parabolic equations with locally lipschitzian nonlinear terms is investigated. The adopted approach is based on the factorization method due to Da Prato, Kwapień and Zabczyk.

For a wide class of Markov processes on a Hilbert space H, defined by semilinear stochastic partial differential equations, we show that their transition semigroups map bounded Borel functions to functions weakly continuous on bounded sets, provided they map bounded Borel functions into functions continuous in the norm topology. In particular, an Ornstein-Uhlenbeck process in H is strong Feller in the norm topology if and only if it is strong Feller in the bounded weak topology. As a consequence,...

A rather general class of stochastic evolution equations in Hilbert spaces whose transition semigroups are Feller with respect to the weak topology is found, and consequences for existence of invariant measures are discussed.

The paper presents a review of some recent results on uniqueness of invariant measures for stochastic differential equations in infinite-dimensional state spaces, with particular attention paid to stochastic partial differential equations. Related results on asymptotic behaviour of solutions like ergodic theorems and convergence of probability laws of solutions in strong and weak topologies are also reviewed.

Using unitary dilations we give a very simple proof of the maximal inequality for a stochastic convolution $${\int}_{0}^{t}S(t-s)\psi \left(s\right)\mathrm{d}W\left(s\right)$$
driven by a Wiener process $W$ in a Hilbert space in the case when the semigroup $S\left(t\right)$ is of contraction type.

We revisit the proof of existence of weak solutions of stochastic differential equations with continuous coeficients. In standard proofs, the coefficients are approximated by more regular ones and it is necessary to prove that: i) the laws of solutions of approximating equations form a tight set of measures on the paths space, ii) its cluster points are laws of solutions of the limit equation. We aim at showing that both steps may be done in a particularly simple and elementary manner.

Space-time regularity of stochastic convolution integrals
J = 0 S(-r)Z(r)W(r)
driven by a cylindrical Wiener process $W$ in an ${L}^{2}$-space on a bounded domain is investigated. The semigroup $S$ is supposed to be given by the Green function of a $2m$-th order parabolic boundary value problem, and $Z$ is a multiplication operator. Under fairly general assumptions, $J$ is proved to be Holder continuous in time and space. The method yields maximal inequalities for stochastic convolutions in the space of continuous...

Zbyněk Šidák, the chief editor of the Applications of Mathematics, an outstanding Czech statistician and probabilist, died on November 12, 1999, aged 66 years. This article is devoted to memory of him and outlines his life and scientific work.

In the present paper integral continuity theorems for solutions of stochastic evolution equations of parabolic type on unbounded time intervals are established. For this purpose, the asymptotic stability of stochastic partial differential equations is investigated, the results obtained being of independent interest. Stochastic evolution equations are treated as equations in Hilbert spaces within the framework of the semigroup approach.

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