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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.

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
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