Continuity of stochastic convolutions

Zdzisław Brzeźniak; Szymon Peszat; Jerzy Zabczyk

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 $card (𝒰) \leq k$ 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 $f \in C (Γ)$ is fixed, has a weak solution if and only if the stochastic convolution process $t \mapsto {(f * b)}_{t}$ 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 $f_{α} : X \to Y (α < ω ₁)$ of Baire-ξ functions. Answering a question of Tomasz Natkaniec, we show that if for a function f: X → Y, the set $α < ω ₁ : f_{α} (x) \neq f (x)$ 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 $β (f) \leq ω^{ξ ₁} \cdot ω^{ξ ₂}$ for some countable ordinals ξ₁ and ξ₂ if and only if there exists a sequence (fₙ) of Baire-1...

Extending $n$ times differentiable functions of several variables

Hajrudin Fejzić, Dan Rinne, Clifford E. Weil (1999)

Czechoslovak Mathematical Journal

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It is shown that $n$ times Peano differentiable functions defined on a closed subset of $ℝ^{m}$ and satisfying a certain condition on that set can be extended to $n$ times Peano differentiable functions defined on $ℝ^{m}$ if and only if the $n$ th order Peano derivatives are Baire class one functions.

Descriptive properties of elements of biduals of Banach spaces

Pavel Ludvík, Jiří Spurný (2012)

Studia Mathematica

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If E is a Banach space, any element x** in its bidual E** is an affine function on the dual unit ball $B_{E *}$ that might possess a variety of descriptive properties with respect to the weak* topology. We prove several results showing that descriptive properties of x** are quite often determined by the behaviour of x** on the set of extreme points of $B_{E *}$ , generalizing thus results of J. Saint Raymond and F. Jellett. We also prove a result on the relation between Baire classes and intrinsic Baire...

A model-theoretic Baire category theorem for simple theories and its applications

Ziv Shami (2013)

Fundamenta Mathematicae

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We prove a model-theoretic Baire category theorem for $τ ̃_{l o w}^{f}$ -sets in a countable simple theory in which the extension property is first-order and show some of its applications. We also prove a trichotomy for minimal types in countable nfcp theories: either every type that is internal in a minimal type is essentially 1-based by means of the forking topologies, or T interprets an infinite definable 1-based group of finite D-rank or T interprets a strongly minimal formula.

Rudin-like sets and hereditary families of compact sets

Étienne Matheron, Miroslav Zelený (2005)

Fundamenta Mathematicae

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We show that a comeager Π₁¹ hereditary family of compact sets must have a dense $G_{δ}$ subfamily which is also hereditary. Using this, we prove an “abstract” result which implies the existence of independent ℳ ₀-sets, the meagerness of ₀-sets with the property of Baire, and generalizations of some classical results of Mycielski. Finally, we also give some natural examples of true $F_{σ δ}$ sets.

On Borel reducibility in generalized Baire space

Sy-David Friedman, Tapani Hyttinen, Vadim Kulikov (2015)

Fundamenta Mathematicae

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We study the Borel reducibility of Borel equivalence relations on the generalized Baire space $κ^{κ}$ for an uncountable κ with $κ^{< κ} = κ$ . The theory looks quite different from its classical counterpart where κ = ω, although some basic theorems do generalize.

Non-separable Banach spaces with non-meager Hamel basis

Taras Banakh, Mirna Džamonja, Lorenz Halbeisen (2008)

Studia Mathematica

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We show that an infinite-dimensional complete linear space X has: ∙ a dense hereditarily Baire Hamel basis if |X| ≤ ⁺; ∙ a dense non-meager Hamel basis if $| X | = κ^{ω} = 2^{κ}$ for some cardinal κ.

The Covering Principle for Darboux Baire 1 functions

Piotr Szuca (2007)

Fundamenta Mathematicae

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We show that the Covering Principle known for continuous maps of the real line also holds for functions whose graph is a connected $G_{δ}$ subset of the plane. As an application we find an example of an approximately continuous (hence Darboux Baire 1) function f: [0,1] → [0,1] such that any closed subset of [0,1] can be translated so as to become an ω-limit set of f. This solves a problem posed by Bruckner, Ceder and Pearson [Real Anal. Exchange 15 (1989/90)].