On stochastic delay evolution equations with non-Lipschitz nonlinearities in Hilbert spaces. (Q636978)

A stochastic delay equation in a Hilbert space \[ dx(t)=[Ax(t)+f(t,x(\rho (t)))]\,dt+g(t,x(\rho (t)))\,dw(t),\quad x(t)=\varphi (t),\quad t\in [-r,0], \] driven by a Hilbert space valued Wiener process is considered. The linear operator \(A\) is assumed to generate a \(C_0\)-semigroup, \(\rho (t)\leq t\) is a deterministic delay and \(f\) and \(g\) may grow little more than linearly and be little less than Lipschitz continuous. Sufficient conditions for existence, uniqueness, mean square and almost sure exponential stability of global mild solutions are addressed in case \(A\) generates a \(C_0\)-semigroup, and analogous results are proved in fractional spaces, provided \(A\) generates an analytic semigroup and the delay is constant. The method of successive approximations and a comparison principle are employed.