Stability-like properties of stochastic differential equations (Q1780206)

Generalizing some results of \textit{R. F. Curtain} [J. Math. Anal. Appl. 79, 352--369 (1981; Zbl 0452.60072)], \textit{P.-L. Chow} [J. Math. Anal. Appl. 89, 400--419 (1982; Zbl 0496.60059)] and \textit{H. J. Kushner} [J. Differ. Equations 4, 424--443 (1968; Zbl 0169.11601)], the author obtains sufficient conditions of Lyapunov functional type for asymptotical stability in probability and asymptotical stability almost surely of the zero solution of the equations in Banach space \( {dx(t) \over dt }= F(t,x, \zeta(t)) \) under the noise \( \zeta\) with differentiable realizations, of the nonlinear stochastic equations \(dx = A(x) dt + B(x) dw_t \) and of stochastic functional-differential equations \( x(t) = x(0) + \int_0^t f(x_s) ds + \int_0^t g(x_s) dW(s), x_t( \theta) := x(t+ \theta) \) for all \( \theta \in [ - \rho,0]\).