Quadratic Lyapunov's functionals and the second moments for stochastic systems with an after-effect

We introduce two positive semigroups of class $C_0$ in a locally convex space of self-adjoint bounded operators defined on a Hilbert space. This semigroups are connected with the quadratic functionals and second moments for stochastic systems with an after-effect. The formulas for the infinitesimal operators of this semigroups are obtained. Our results permit us to formulate the criterions for the mean square stability within the framework of the second Lyapunov's method.