Displaying similar documents to “Continuity of stochastic convolutions”

Typical multifractal box dimensions of measures

L. Olsen (2011)

Fundamenta Mathematicae

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We study the typical behaviour (in the sense of Baire’s category) of the multifractal box dimensions of measures on ℝ d . We prove that in many cases a typical measure μ is as irregular as possible, i.e. the lower multifractal box dimensions of μ attain the smallest possible value and the upper multifractal box dimensions of μ attain the largest possible value.

On the k -Baire property

Alessandro Fedeli (1993)

Commentationes Mathematicae Universitatis Carolinae

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In this note we show the following theorem: “Let X be an almost k -discrete space, where k is a regular cardinal. Then X is k + -Baire iff it is a k -Baire space and every point- k open cover 𝒰 of X such that card ( 𝒰 ) ≤ k is locally- k at a dense set of points.” For k = ℵ 0 we obtain a well-known characterization of Baire spaces. The case k = ℵ 1 is also discussed.

Continuity versus nonexistence for a class of linear stochastic Cauchy problems driven by a Brownian motion

Johanna Dettweiler, J.M.A.M. van Neerven (2006)

Czechoslovak Mathematical Journal

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Let A = d / d θ denote the generator of the rotation group in the space C ( Γ ) , where Γ denotes the unit circle. We show that the stochastic Cauchy problem d U ( t ) = A U ( t ) + f d b t , U ( 0 ) = 0 , ( 1 ) where b is a standard Brownian motion and f ∈ C ( Γ ) is fixed, has a weak solution if and only if the stochastic convolution process t ↦ ( f * b ) t has a continuous modification, and that in this situation the weak solution has a continuous modification. In combination with a recent result of Brzeźniak, Peszat and Zabczyk it follows that (1) fails to have a weak solution...

Extension of functions with small oscillation

Denny H. Leung, Wee-Kee Tang (2006)

Fundamenta Mathematicae

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A classical theorem of Kuratowski says that every Baire one function on a G δ subspace of a Polish (= separable completely metrizable) space X can be extended to a Baire one function on X. Kechris and Louveau introduced a finer gradation of Baire one functions into small Baire classes. A Baire one function f is assigned into a class in this hierarchy depending on its oscillation index β(f). We prove a refinement of Kuratowski’s theorem: if Y is a subspace of a metric space X and f is a...

On the closure of Baire classes under transfinite convergences

Tamás Mátrai (2004)

Fundamenta Mathematicae

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Let X be a Polish space and Y be a separable metric space. For a fixed ξ < ω₁, consider a family f α : X → Y ( α < ω ₁ ) of Baire-ξ functions. Answering a question of Tomasz Natkaniec, we show that if for a function f: X → Y, the set α < ω ₁ : f α ( x ) ≠ f ( x ) is finite for every x ∈ X, then f itself is necessarily Baire-ξ. The proof is based on a characterization of Σ ⁰ η sets which can be interesting in its own right.

Functions of Baire class one

Denny H. Leung, Wee-Kee Tang (2003)

Fundamenta Mathematicae

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Let K be a compact metric space. A real-valued function on K is said to be of Baire class one (Baire-1) if it is the pointwise limit of a sequence of continuous functions. We study two well known ordinal indices of Baire-1 functions, the oscillation index β and the convergence index γ. It is shown that these two indices are fully compatible in the following sense: a Baire-1 function f satisfies β ( f ) ≤ ω ξ ₁ · ω ξ ₂ for some countable ordinals ξ₁ and ξ₂ if and only if there exists a sequence (fₙ) of Baire-1...