Wyniki wyszukiwania

1

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

100%

Brzeźniak Z., Peszat S.

Studia Mathematica

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1999

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tom 137

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nr 3

261-299

EN

Stochastic partial differential equations on $ℝ^d$ 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.

2

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

100%

Brzeźniak Z., van Neerven J.

Studia Mathematica

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2000

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tom 143

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nr 1

43-74

EN

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) $dX_t = AX_tdt + BdW_t^H$ (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 E and B ∈ ℒ(H,E) is a bounded linear operator. We further show that whenever a weak solution exists, it is unique, and given by a stochastic convolution $X_t = ∫^{t}_{0} S(t-s)BdW_{s}^{H}$.

3

A note on γ-radonifying and summing operators

100%

Brzeźniak Z., Long H.

Banach Center Publications

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2015

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tom 105

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nr 1

43-57

EN

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.

4

A quantitative asymptotic theorem for contraction semigroups with countable unitary spectrum

81%

J. K. Batty C., Brzeźniak Z., Greenfield D. A.

Studia Mathematica

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1996

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tom 121

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nr 2

167-183

EN

Let T be a semigroup of linear contractions on a Banach space X, and let $X_{s}(T) = {x ∈ X : lim_{s→∞} ∥T(s)x∥ = 0}$. 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 $lim_{s} ∥T(s)x∥ = inf{∥x-y∥ : y ∈ X_{s}(T)}$ for each x in X.

Ograniczanie wyników

3 Studia Mathematica

1 Banach Center Publications

4 Brzeźniak Z.

1 Greenfield D. A.

1 J. K. Batty C.

1 Long H.

1 Peszat S.

1 van Neerven J.

1 2015

1 2000

1 1999

1 1996

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

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

A note on γ-radonifying and summing operators

A quantitative asymptotic theorem for contraction semigroups with countable unitary spectrum