Mean first-passage time for systems driven by pre-Gaussian noise: Natural boundary conditions

Abstract

In order to determine the mean first-passage time for systems driven by a superposition of suitably scaled independent dichotomous Markovian processes (pre-Gaussian noise), the well-known absorbing boundary conditions must be complemented in the generic case by a novel type of natural boundary conditions. We treat explicitly a linear stochastic Aow for the superposition of up to five dichotomous processes and compare the analytic results with a digital simulation for these processes and an Ornstein-Uhlenbeck process.

Drift by dichotomous Markov noise

“…As was pointed out in detail in [19], the situation is entirely different, both physically and mathematically, when one or both of the "±" dynamics have unstable fixed points, depending on whether the system can cross (or not) these unstable fixed points in the long time limit. In fact, while this issue was also mentioned in [15] and [16], a full and detailed discussion was first given for a specific example with multiplicative dichotomous noise in [19]. In the following we focus on the presentation and discussion of the final results for the asymptotic probability density and drift velocity, relegating the technical details to the Appendix.…”

“…However, contrary to the claims made in some of these papers, the problem of the first passage time moments, and the related issue of finding the asymptotic drift velocity [14], was not solved in the most general case. Indeed, with a few exceptions [15,16,17,18], all the results that have been obtained exclude the cases when one or both of the "±" dynamics have unstable fixed points, and thus do not consider the possibility of crossing unstable fixed points in the long time dynamics. The technical subtleties were first highlighted and discussed in detail in a recent paper [19].…”

Dichotomous Markov Noise: Exact Results for Out-of-Equilibrium Systems

“…This is the main reason why, with a few exceptions, see Refs [17,61,62]. (and also Refs [63,64]. for the related problem of the mean-first passage time), the problem of dichotomous flows with unstable critical points was generally not approached.…”

Mean First-Passage Time in a Bistable Kinetic Model with Stochastic Potentials

“…In recent years several papers have dealt with the derivation of exact expressions for the MFPT of the bistable system (Behn et al, 1993;Jia and Li, 1996;Kus and Wodkiewicz, 1993;Laio et al, 2001;Mei et al, 1999;Porra and Lindenberg, 1995;Venkatesh and Patnaik, 1993). Behn and his co-workers (Behn et al, 1993)studied the MFPT of the system driven by a superposition of suitably scaled independent dichotomous Markovian processes with natural boundary condition. Jia and his co-workers (Jia and Li, 1996) investigated the properties of bistable system with correlations between additive and multiplicative noise terms.…”