*(English)*Zbl 1031.60061

Let $K$ and $H$ be real separable Hilbert spaces. Assume that $W$ is a $K$-valued Wiener process with covariance operator $Q$ and ${x}_{0}$ is an $H$-valued random variable which is independent of $W$. Consider the initial value problem of semilinear functional integro-differential stochastic evolution equations

with values in $H$, where $A:H\to H$ represents a linear operator, $G:C([0,T],H)\to C([0,T],{L}^{2}({\Omega},BL(K,H)))$, $F:C([0,T],H)\to {L}^{p}([0,T],{L}^{2}({\Omega},H))$ with $1\le p<+\infty $ and $h:C([0,T],H)\to {L}_{0}^{2}({\Omega},H)$. The authors discuss global existence results concerning mild and periodic solutions under several growth and compactness conditions. Weak convergence of induced probability measures belonging to the family of finite-dimensional distributions of certain sequences of such stochastic equations is treated too. Basic proof-tools include Schaefer’s fixed point theorem, techniques of linear semigroups and probability measures as well as results from infinite-dimensional SDEs. Conceivable applications to electromagnetic theory, population dynamics and heat conduction in materials with memory underline the importance of their work. An example of a nonlocal integro-partial SDE illustrates some thoughts of related abstract theory. Some necessary preliminaries compiled from probability theory and functional analysis ease the process of understanding by lesser experienced readership.

##### MSC:

60H25 | Random operators and equations |

34F05 | ODE with randomness |

37H10 | Generation, random and stochastic difference and differential equations |

37L55 | Infinite-dimensional random dynamical systems; stochastic equations |

60B05 | Probability measures on topological spaces |

60H15 | Stochastic partial differential equations |

60H20 | Stochastic integral equations |

60H30 | Applications of stochastic analysis |

60H10 | Stochastic ordinary differential equations |

34K30 | Functional-differential equations in abstract spaces |