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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.

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.

Let T be a semigroup of linear contractions on a Banach space X, and let $X_s\left(T\right)=x\in X:li{m}_{s\to \infty }\parallel T\left(s\right)x\parallel =0$. Then $X_s\left(T\right)$ is the annihilator of the bounded trajectories of T*. If the unitary spectrum of T is countable, then $X_s\left(T\right)$ is the annihilator of the unitary eigenvectors of T*, and $li{m}_s\parallel T\left(s\right)x\parallel =inf\parallel x-y\parallel :y\in X_s\left(T\right)$ for each x in X.

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 ${W_t^H}_{t\in [0,T]}$ 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) $d{X}_t=A{X}_{t}dt+Bd{W}_t^H$ (t∈ [0,T]), $X_0=0$ almost surely, where A is the generator of a $C_0$-semigroup ${S\left(t\right)}_{t\ge 0}$ of bounded linear operators on...

Let $B$ be a Brownian motion, and let ${𝒞}_{\mathrm{p}}$ be the space of all continuous periodic functions $fℝ\to ℝ$ with period 1. It is shown that the set of all $f\in {𝒞}_{\mathrm{p}}$ such that the stochastic convolution $X_{f,B}\left(t\right)={\int }_0^{t}f(t-s)\mathrm{d}B\left(s\right)$, $t\in [0,1]$ does not have a modification with bounded trajectories, and consequently does not have a continuous modification, is of the second Baire category.

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...

Let $H$ be a Hilbert space and $E$ a Banach space. We set up a theory of stochastic integration of $ℒ(H,E)$-valued functions with respect to $H$-cylindrical Liouville fractional Brownian motion with arbitrary Hurst parameter $0<\beta <1$. For $0<\beta <\frac{1}{2}$ we show that a function $\Phi :(0,T)\to ℒ(H,E)$ 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...