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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

${x}^{\text{'}}\left(t\right)=Ax\left(t\right)+F\left(x\right)\left(t\right)+{\int }_{0}^{t}G\left(x\right)\left(s\right)dW\left(s\right),\phantom{\rule{4pt}{0ex}}0\le t\le T,\phantom{\rule{1.em}{0ex}}x\left(0\right)=h\left(x\right)+{x}_{0}$

with values in $H$, where $A:H\to H$ represents a linear operator, $G:C\left(\left[0,T\right],H\right)\to C\left(\left[0,T\right],{L}^{2}\left({\Omega },BL\left(K,H\right)\right)\right)$, $F:C\left(\left[0,T\right],H\right)\to {L}^{p}\left(\left[0,T\right],{L}^{2}\left({\Omega },H\right)\right)$ with $1\le p<+\infty$ and $h:C\left(\left[0,T\right],H\right)\to {L}_{0}^{2}\left({\Omega },H\right)$. 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