Series Expansion Calculation of Persistence Exponents

Abstract

We consider an arbitrary Gaussian Stationary Process X(T) with known correlator C(T), sampled at discrete times T_n = n ΔT. The probability that (n+1) consecutive values of X have the same sign decays as P_n \sim \exp(-θ_D T_n). We calculate the discrete persistence exponent θ_D as a series expansion in the correlator C(ΔT) up to 14th order, and extrapolate to ΔT = 0 using constrained Padé approximants to obtain the continuum persistence exponent θ. For the diffusion equation our results are in exceptionally good agreement with recent numerical estimates.