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**Functional integro-differential stochastic evolution equations in Hilbert space.**
*(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
\[
x^\prime (t) = A x(t) + F(x)(t) + \int^t_0 G(x)(s) dW(s),\;0 \leq t \leq T, \quad x(0) = h(x) + x_0
\]
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 \(1\leq p < + \infty\) and \(h: C([0,T],H) \to L^2_0(\Omega,H)\). 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.

Reviewer: Henri Schurz (Carbondale)

### MSC:

60H25 | Random operators and equations (aspects of stochastic analysis) |

34F05 | Ordinary differential equations and systems 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 (aspects of stochastic analysis) |

60H20 | Stochastic integral equations |

60H30 | Applications of stochastic analysis (to PDEs, etc.) |

60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |

34K30 | Functional-differential equations in abstract spaces |