On local stability for a nonlinear difference equation with a non-hyperbolic equilibrium and fading stochastic perturbations

Abstract

We consider the nonlinear stochastic difference equation Here, (ξ n ) n ∈ ℕ is a sequence of independent random variables with zero mean and unit variance and with distribution functions F n . The function f : ℝ → ℝ is continuous, f(0) = 0, xf(x)>0 for x ≠ 0. We establish a condition on the noise intensity σ and the rate of decay of the tails of the distribution functions F n , under which the convergence of solutions to zero occurs with probability zero. If this condition does not hold, and f is bounded by a linear function with slope 2 − γ, for γ ∈ (0, 2), all solutions tend to zero a.s. On the other hand, if f grows more quickly than linear function with slope 2 + γ, for γ>0, the solutions tend to infinity in modulus with arbitrarily high probability, once the initial condition is chosen sufficiently large. Such equations can still demonstrate local stability; for a wide class of highly nonlinear f, it is shown that solutions tend to zero with arbitrarily high probability, once the initial condition is chosen appropriately. Results which elucidate the relationship between the rate of decay of the noise intensities and the rate of decay of the tails, and the necessary condition for stability, are presented. The connection with the asymptotic dynamical consistency of the system, when viewed as a discretisation of an Itô stochastic differential equation, is also explored.