Functional integro-differential stochastic evolution equations in Hilbert space
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.
semilinear stochastic evolution equations
weak convergence of probability measures
scientific article; zbMATH DE number 2002245