Persistence Probability for a Class of Gaussian Processes Related to Random Interface Models

Abstract

We consider a class of Gaussian processes which are obtained as height processes of some (d + 1)-dimensional dynamic random interface model on Z d . We give an estimate of persistence probability, namely, large T asymptotics of the probability that the process does not exceed a fixed level up to time T . The interaction of the model affects the persistence probability and its asymptotics changes depending on the dimension d.where η(x, t) is a space-time white noise. This describes the time evolution of a (1 + 1)dimensional random interface and h(x, t) represents the height at position x ∈ R and time t ≥ 0.

Persistence and Ball Exponents for Gaussian Stationary Processes

“…Persistence is sometimes regarded more generally, as the event that a given stochastic process takes values in a specific set over a long time interval. Such events have received significant attention for a large class of examples, including random walks [11,28] (see references there-in), Lévy processes [10,22], Markov processes [16,17], random polynomials [20,40] and (spatial) processes driven by either stochastic differential equations, or by partial differential equations with random initial configuration [50,51].…”

“…This has been studied in particular for the critical case ℓ = 0 (c.f. [26,27,33,50]), with applications to statistical mechanics (c.f. [24,41,47,61]).…”