Existence and stability of stationary solutions of nonlinear difference equations under random perturbations

Existence and stability of stationary solutions of nonlinear random difference equations are studied in this note. Firstly, we give the weak conditions that guarantee the continuity of Lypanunov exponents under small random perturbations. Secondly, we find out the conditions under which the ratio of the random norm and the standard Euclidean norm has deterministic bounds. Based on these new results, we provide easyto- use conditions that guarantee the existence and almost sure stability of a stationary solution. In addition, we also prove that the stationary solution converges with probability one to the fixed point of the corresponding deterministic system as the noise intensity tends to zero.