Exponential stability of non-linear stochastic evolution equations

The aim of this paper is to investigate exponential stability of paths for a class of Hilbert space-valued non-linear stochastic evolution equations. The analyses consist in using exponential martingale formula, Lyapunov functional and some special inequalities derived for our stability purposes. Various sufficient conditions are obtained to ensure the stability of the strong solutions. Several applications to stochastic partial differential equations are studied to illustrate our theory. In particular, by means of our results we loosen the conditions of certain stochastic evolution systems from Haussmann (1978) or Ichikawa (1982).