[For the entire collection see Zbl 0563.00010.]
The article describes asymptotic stability results for solutions of stochastic functional (hereditary) equations of the form \(dx(t)=A(t,x^ t)dt+B(t,x^ t)dW(t)\), \(x=\phi \in X\). Here \(X=R\times L^ P_{\rho}\) is a ”fading memory space” and \(\rho\) is an influence function with relaxation property as introduced in connection with the study of hereditary phenomena in continuum mechanics by B. D. Coleman and the first author [Arch. Ration. Mech. Anal. 23, 87-123 (1966; Zbl 0146.461)], W is a Wiener process, A and B are functionals on \(R\times X\) and \(x^ t\in X\) is the history process \(x^ t(s)=(x(t),x(t-s)).\)
The first part of the article reviews this framework. Concentrating on an example, it describes the deterministic hereditary equations of the form \(x(t)=A(x^ t)\), states and discusses some asymptotic stability results, obtained by using free energy type Lyapunov functionals. The second part describes the stochastic equations which are related to deterministic hereditary equations in the same way as (Ito) stochastic differential equations are related to ordinary differential equations. Using this analogy the corresponding stability properties are introduced and an asymptotic stability result is given for the stochastic perturbation of the example treated in the first part. No proofs are given. Details appeared in the authors’ paper [J. Integral Equations 7, 1-72 (1984; Zbl 0539.60052)].