Computing Lyapunov constants for random recurrences with smooth coefficients

摘要

In recent years, there has been much interest in the growth and decay rates (Lyapunov constants) of solutions to random recurrences such as the random Fibonacci sequence x_(n+1) = ±x_n±x_(n-1). Many of these problems involve nonsmooth dynamics (nondifferentiable invariant measures), making computations hard. Here, however, we consider recurrences with smooth random coefficients and smooth invariant measures. By computing discretised invariant measures and applying Richardson extrapolation, we can compute Lyapunov constants to 10 digits of accuracy. In particular, solutions to the recurrence x_(n+1) = x_n + c_(n+1)x_(n-1), where the {c_n} are independent standard normal variables, increase exponentially (almost surely) at the asymptotic rate (1.0574735537…)~n. Solutions to the related recurrences x_(n+1) = c_(n+1)x_n + x_(n-1) and x_(n+1) = c_(n+1)x_n + d_(n+1)x_(n-1) (where the {d_n} are also independent standard normal variables) increase (decrease) at the rates (1.1149200917…)~n and (0.9949018837…)~n, respectively.