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Strong Feller solutions to SPDE's are strong Feller in the weak topology

Bohdan MaslowskiJan Seidler — 2001

Studia Mathematica

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,...

On sequentially weakly Feller solutions to SPDE’s

Bohdan MaslowskiJan Seidler — 1999

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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.

Stochastic affine evolution equations with multiplicative fractional noise

Bohdan MaslowskiJ. Šnupárková — 2018

Applications of Mathematics

A stochastic affine evolution equation with bilinear noise term is studied, where the driving process is a real-valued fractional Brownian motion with Hurst parameter greater than 1 / 2 . Stochastic integration is understood in the Skorokhod sense. The existence and uniqueness of weak solution is proved and some results on the large time dynamics are obtained.

Invariant measures for nonlinear SPDE's: uniqueness and stability

Bohdan MaslowskiJan Seidler — 1998

Archivum Mathematicum

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.

An averaging principle for stochastic evolution equations. II.

Bohdan MaslowskiJan SeidlerIvo Vrkoč — 1991

Mathematica Bohemica

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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