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We obtain lower bounds for ℙ(ξ ≥ 0) and ℙ(ξ > 0) under assumptions on the moments of a centered random variable ξ. The estimates obtained are shown to be optimal and improve results from the literature. They are then applied to obtain probability lower bounds for second order Rademacher chaos.

In this paper we prove that a Gaussian white noise on the d-dimensional torus has paths in the Besov spaces $B_{p,\infty }^{-d/2}{(}^d)$ with p ∈ [1,∞). This result is shown to be optimal in several ways. We also show that Gaussian white noise on the d-dimensional torus has paths in the Fourier-Besov space $b{̂}_{p,\infty }^{-d/p}{(}^d)$. This is shown to be optimal as well.

We study the non-autonomous stochastic Cauchy problem on a real Banach space E, $dU\left(t\right)=A\left(t\right)U\left(t\right)dt+B\left(t\right)d{W}_H\left(t\right)$, t ∈ [0,T], U(0) = u₀. Here, $W_H$ is a cylindrical Brownian motion on a real separable Hilbert space H, ${\left(B\left(t\right)\right)}_{t\in [0,T]}$ are closed and densely defined operators from a constant domain (B) ⊂ H into E, ${\left(A\left(t\right)\right)}_{t\in [0,T]}$ denotes the generator of an evolution family on E, and u₀ ∈ E. In the first part, we study existence of weak and mild solutions by methods of van Neerven and Weis. Then we use a well-known factorisation method in the setting of evolution...

We consider generalized square function norms of holomorphic functions with values in a Banach space. One of the main results is a characterization of embeddings of the form $L^p\left(X\right)\subseteq \gamma \left(X\right)\subseteq L^q\left(X\right)$, in terms of the type p and cotype q of the Banach space X. As an application we prove $L^p$-estimates for vector-valued Littlewood-Paley-Stein g-functions and derive an embedding result for real and complex interpolation spaces under type and cotype conditions.

We consider function spaces of Besov, Triebel-Lizorkin, Bessel-potential and Sobolev type on ${ℝ}^d$, equipped with power weights $w\left(x\right)={\left|x\right|}^{\gamma }$, γ > -d. We prove two-weight Sobolev embeddings for these spaces. Moreover, we precisely characterize for which parameters the embeddings hold. The proofs are presented in such a way that they also hold for vector-valued functions.

In stochastic partial differential equations it is important to have pathwise regularity properties of stochastic convolutions. In this note we present a new sufficient condition for the pathwise continuity of stochastic convolutions in Banach spaces.