Persistence of a continuous stochastic process with discrete-time sampling

Abstract

We introduce the concept of 'discrete-time persistence', which deals with zero-crossings of a continuous stochastic process, X(T ), measured at discrete times, T = n∆T . For a Gaussian Markov process with relaxation rate µ, we show that the persistence (no crossing) probability decays as [ρ(a)] n for large n, where a = exp(−µ∆T ), and we compute ρ(a) to high precision. We also define the concept of 'alternating persistence', which corresponds to a < 0. For a > 1, corresponding to motion in an unstable potential (µ < 0), there is a nonzero probability of having no zero-crossings in infinite time, and we show how to calculate it.PACS numbers: 05.70. Ln, 05.40.+j, 81.10.Aj Persistence of a continuous stochastic process has generated much recent interest in a wide variety of nonequilibrium systems including various models of phase ordering kinetics, diffusion, fluctuating interfaces and reactiondiffusion processes [1]. Persistence has also been recently used in fields as diverse as ecology [2] and seismology [3]. Persistence is simply the probability P (t) that a stochastic process x(t) does not change sign up to time t. In most of the systems mentioned above, P (t) ∼ t −θ for large t, where the persistence exponent θ is nontrivial. Apart from various analytical and numerical results, this exponent has also been measured experimentally in systems such as breath figures [4], liquid crystals [5], soap bubbles [6], and more recently in laser-polarized Xe gas using NMR techniques [7].Persistence has also remained a popular subject among applied mathematicians for many decades [8]. They are most interested in the probability of 'no zero crossing' of a Gaussian stationary process (GSP) between times T 1 and T 2 [9]. It is well known that this probability usually decays as ∼ exp(−θT ) for large T = |T 2 − T 1 | where θ is nontrivial [9,8]. The persistence of some of the nonstationary processes mentioned in the previous paragraph such as the diffusion processes, can be mapped to that of a corresponding GSP [10]. This makes the two sets of problems related to each other and the power law exponent in the former problem becomes the inverse decay rate in the latter. Even though θ is, in general, hard to compute analytically, it is very easy to evaluate numerically in most cases. Given this fact, and the combined interest of both statistical physicists and applied mathematicians, much recent effort has been devoted to computing θ numerically to extremely high precision.This raises a natural question: How accurately can one measure θ? Is there a natural limitation and if so, can it be overcome? This issue arises from the following simple observation. All the stochastic processes mentioned above occur in continuous time. However, when one performs numerical simulations or experiments on persistence, one has to discretize time in some way and sample the data only at these discrete time points to check if the process has retained its sign. Due to this discretization, some information is lost. For example, the process may have cro...

Persistence exponents and the statistics of crossings and occupation times for Gaussian stationary processes

“…For finite n, one can write E n (x) = nλ + u n (x). An explicit expression for u n (x) can be obtained from that of E n (x) in equation (23). Using the explicit value of λ and after a few steps of algebra we get,…”

“…This is the Ornstein-Uhlenbeck process whose persistence exponent for discrete sampling was calculated in [23]. Integrating equation (6), we get…”

“…The largest eigenvalue ρ p (a) and the corresponding eigenfunction can then be determined either by the matrix method or by the variational method as in our previous paper [23]. Using the matrix method, we get,…”

“…1. Plot of u(x) = En(x) − nλ for a = 1/2 and n = 1000, calculated using equation (23). u(x) is symmetric about x = 0 .…”

“…This is, of course, completely equivalent to studying a discrete process with the same correlators C(j∆T ). In [23,24] we have studied the persistence of a discretely sampled random walk and of a randomly accelerated particle, both of which can be mapped to a GSP by a change of variables. In [25] we calculated, using a series expansion in the correlator C(j∆T ), the persistence probability for an arbitrary discretely-sampled GSP.…”