Persistence of Gaussian processes: non-summable correlations

Abstract

Abstract. Suppose the auto-correlations of real-valued, centered Gaussian process Z(·) are nonnegative and decay as ρ(|s − t|) for some ρ(·) regularly varying at infinity of order −α ∈ [−1, 0). With Iρ(t) =´t 0 ρ(s)ds its primitive, we show that the persistence probabilities decay rate of − log P(sup t∈[0,T ] {Z(t)} < 0) is precisely of order (T /Iρ(T )) log Iρ(T ), thereby closing the gap between the lower and upper bounds of [NR], which stood as such for over fifty years. We demonstrate its usefulness by sharpening recent results of [Sak] about the dependence on d of such persistence decay for the Langevin dynamics of certain ∇φ-interface models on Z d .

Persistence Probabilities of Two-Sided (Integrated) Sums of Correlated Stationary Gaussian Sequences

“…There are sufficient conditions in the literature for truly exponential decay (cf. [11,16,18,19]), but in our understanding none of them are both necessary and sufficient.…”

Persistence and Ball Exponents for Gaussian Stationary Processes

“…In the late 2010's, Dembo and the third author [18,19], seeking to study both the solutions of the heat equation initiated by white noise and the probability that a random polynomial has no roots, revisited Slepian's method. They observed that in the restricted case of GSPs with nonnegative covariance, it is possible to use probabilistic arguments and tools from linear algebra to extend the method and obtain the exact rate of the decay of the persistence up to sub-exponential factors.…”

“…Here, we go beyond establishing exponential-type behavior of persistence: we show the existence of a persistence exponent and provide several new continuity results. We do so by combining and expanding the spectral method of [27] and the covariance method of [19].…”

“…The validity of this conjecture has often been taken for granted in the physics literature [21,23]. Prior to this work, the only cases in which an exponent was shown to exist were non-negative correlated processes [19] (using a subadditivity argument), Markov processes (using Perron-Frobenius), and m-dependent processes (using independence properties) [6].…”

Persistence probabilities of mixed FBM and other mixed processes

“…Thus, our result contributes to the amount of rather rare persistence results for stochastic processes violating both the properties of self-similarity and stationary increments. Self-similarity is a valuable property in the context of persistence as in this case, one is able to apply the so-called Lamperti transformation to get a stationary process and concerning persistence, many powerful tools are available for the class of stationary centred Gaussian processes, see [14], [18], [15], [10], [20], [8] and [19]. In particular, combined with non-negative (and non-degenerate) covariances, self-similarity always guarantees the existence of the persistence exponent.…”