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Space-time continuous solutions to SPDE's driven by a homogeneous Wiener process

Zdzisław BrzeźniakSzymon Peszat — 1999

Studia Mathematica

Stochastic partial differential equations on d are considered. The noise is supposed to be a spatially homogeneous Wiener process. Using the theory of stochastic integration in Banach spaces we show the existence of a Markovian solution in a certain weighted L q -space. Then we obtain the existence of a space continuous solution by means of the Da Prato, Kwapień and Zabczyk factorization identity for stochastic convolutions.

A note on γ-radonifying and summing operators

Zdzisław BrzeźniakHongwei Long — 2015

Banach Center Publications

In this note, we discuss certain generalizations of γ-radonifying operators and their applications to the regularity for linear stochastic evolution equations on some special Banach spaces. Furthermore, we also consider a more general class of operators, namely the so-called summing operators and discuss the application to the compactness of the heat semi-group between weighted L p -spaces.

A quantitative asymptotic theorem for contraction semigroups with countable unitary spectrum

Charles BattyZdzisław BrzeźniakDavid Greenfield — 1996

Studia Mathematica

Let T be a semigroup of linear contractions on a Banach space X, and let X s ( T ) = x X : l i m s T ( s ) x = 0 . Then X s ( T ) is the annihilator of the bounded trajectories of T*. If the unitary spectrum of T is countable, then X s ( T ) is the annihilator of the unitary eigenvectors of T*, and l i m s T ( s ) x = i n f x - y : y X s ( T ) for each x in X.

Stochastic convolution in separable Banach spaces and the stochastic linear Cauchy problem

Zdzisław BrzeźniakJan van Neerven — 2000

Studia Mathematica

Let H be a separable real Hilbert space and let E be a separable real Banach space. We develop a general theory of stochastic convolution of ℒ(H,E)-valued functions with respect to a cylindrical Wiener process W t H t [ 0 , T ] with Cameron-Martin space H. This theory is applied to obtain necessary and sufficient conditions for the existence of a weak solution of the stochastic abstract Cauchy problem (ACP) d X t = A X t d t + B d W t H (t∈ [0,T]), X 0 = 0 almost surely, where A is the generator of a C 0 -semigroup S ( t ) t 0 of bounded linear operators on...

Continuity of stochastic convolutions

Zdzisław BrzeźniakSzymon PeszatJerzy Zabczyk — 2001

Czechoslovak Mathematical Journal

Let B be a Brownian motion, and let 𝒞 p be the space of all continuous periodic functions f with period 1. It is shown that the set of all f 𝒞 p such that the stochastic convolution X f , B ( t ) = 0 t f ( t - s ) d B ( s ) , t [ 0 , 1 ] does not have a modification with bounded trajectories, and consequently does not have a continuous modification, is of the second Baire category.

Ergodicity for a stochastic geodesic equation in the tangent bundle of the 2D sphere

Ľubomír BaňasZdzisław BrzeźniakMikhail NeklyudovMartin OndrejátAndreas Prohl — 2015

Czechoslovak Mathematical Journal

We study ergodic properties of stochastic geometric wave equations on a particular model with the target being the 2D sphere while considering only solutions which are independent of the space variable. This simplification leads to a degenerate stochastic equation in the tangent bundle of the 2D sphere. Studying this equation, we prove existence and non-uniqueness of invariant probability measures for the original problem and obtain also results on attractivity towards an invariant measure. We also...

Stochastic evolution equations driven by Liouville fractional Brownian motion

Zdzisław BrzeźniakJan van NeervenDonna Salopek — 2012

Czechoslovak Mathematical Journal

Let H be a Hilbert space and E a Banach space. We set up a theory of stochastic integration of ( H , E ) -valued functions with respect to H -cylindrical Liouville fractional Brownian motion with arbitrary Hurst parameter 0 < β < 1 . For 0 < β < 1 2 we show that a function Φ : ( 0 , T ) ( H , E ) is stochastically integrable with respect to an H -cylindrical Liouville fractional Brownian motion if and only if it is stochastically integrable with respect to an H -cylindrical fractional Brownian motion. We apply our results to stochastic evolution equations...

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