Stability of stochastic neutral partial functional differential equations under Hölder type conditions. (Q637001)

A stochastic neutral partial functional differential equation in a Hilbert space \[ d[x(t)+f(t,\pi _{t}x)]=[Ax(t)+a(t,\pi _{t}x)]\,dt+b(t,\pi _{t}x)dw(t),\quad t>0, \] \[ x(t)=\varphi (t),\quad t\in [-r,0], \] driven by a Hilbert space valued Wiener process is considered. The linear operator \(A\) is assumed to generate an analytic exponentially stable semigroup, \(\pi _{t}x(s)=x(t-r+s)\), \(f\) is Lipschitz, \(a\) and \(b\) may be little less than Lipschitz continuous and \(a\), \(b\) and \(f\) grow at most linearly (in appropriate norms). Sufficient conditions for existence, uniqueness, mean square and almost sure exponential stability of global mild solutions are addressed by means of the method of successive approximations.