Global existence and asymptotic stability of mild solutions for stochastic evolution equations with nonlocal initial conditions (Q2011925)

The authors prove theorems on the global existence, uniqueness and asymptotic stability of mild solutions for a class of semilinear stochastic evolution equations with nonlocal initial conditions on the infinite interval \[ du(t) + Au(t) dt = f(t, u(t)) dW(t), \quad t\in J, \quad u(0) = \sum_{k=1}^{\infty}c_{k}u(t_{k}), \] where the state \(u(\cdot)\) takes values in the real separable Hilbert space \(H\), \(A:\;D(A)\subset H \to H\) is a positive definite self-adjoint operator; \(K\) is another separable Hilbert space and \(\{W(t):\;t\geq 0\}\) is a cylindrical \(K\)-valued Wiener process with a finite trace nuclear covariance operator \(Q\geq0\) defined on a filtered complete probability space. An illustrative example for the main results is presented.