Survival probability of stochastic processes beyond persistence exponents

Abstract

For many stochastic processes, the probability of not-having reached a target in unbounded space up to time follows a slow algebraic decay at long times, . This is typically the case of symmetric compact (i.e. recurrent) random walks. While the persistence exponent has been studied at length, the prefactor , which is quantitatively essential, remains poorly characterized, especially for non-Markovian processes. Here we derive explicit expressions for for a compact random walk in unbounded space by establishing an analytic relation with the mean first-passage time of the same random walk in a large confining volume. Our analytical results for are in good agreement with numerical simulations, even for strongly correlated processes such as Fractional Brownian Motion, and thus provide a refined understanding of the statistics of longest first-passage events in unbounded space.

Computation of time probability distributions for the occurrence of uncertain future events

“…As the system condition randomly evolves in time (i.e., the condition indicator is a stochastic process), a probability distribution for the threshold first hitting time is therefore induced: the First-Passage Time (FPT) [1][2][3][4] or First-Hitting Time (FHT) [5,6] probability distribution. On the one hand, this concept is equivalent to duration models [7,8] and, although understood in a different context, it is also analogous to survival probability in statistics [9][10][11]. On the other hand, in the engineering discipline of Prognostics and Health Management (PHM) these concepts are related to the Remaining Useful Life (RUL), End-of-Life (EoL), Time-of-Failure (ToF) and Time-to-Failure (TtF) probability distributions [12][13][14].…”

Anomalous persistence exponents for normal yet aging diffusion

“…The survival probability S(t) of a 1-dimensional unbounded stochastic process x(t) is defined as the probability that x(t) has not reached a threshold value up to time t. This observable has proved to be very useful to quantify the dynamics of a broad range of complex systems in contexts as varied as diffusion controlled reactions, finance, search processes, or biophysics [1][2][3][4][5]. In many examples of symmetric stochastic processes, the large time behaviour of the survival probability is characterised by a power law decay S(t) ∝ t −θ that defines the persistence exponent θ.…”

“…In fact, all available examples of processes that are diffusive (H = 1/2), even if only asymptotically for t → ∞, display the universal exponent θ = 1/2. This is illustrated by the example where x(t) is the position of a given monomer of a finite 1-dimensional ideal Rouse chain of N monomers ; this non Markovian process is diffusive at times larger than the Rouse time (that is the slowest relaxation time of the internal degrees of freedom of the chain) and is shown to be characterised by θ = 1/2 [5]; note that in this example the increments x(t + T ) − x(T ) are stationary at long times, i.e. have statistics independent of the observation time T , for T larger than the Rouse time.…”

“…Based on this example, we now aim at determining a general criterion that allows to identify the asymptotically diffusive processes that lead to anomalous persistence exponents. It has been proposed that for continuous scale invariant processes with stationary increments the universal relation θ = 1 − H should hold [4,13]; this relation can in fact be extended to processes whose time dependent increments [x(t+T )−x(T )] 2 are only asymptotically stationary after a finite characteristic timescale [5]. We review in [35] examples of such processes, which all lead to θ = 1/2.…”

Everlasting impact of initial perturbations on first-passage times of non-Markovian random walks

“…Our theory covers the case of spatial dimension higher than one, opening the way to characterize non-trivial reaction kinetics for complex systems with non-equilibrium initial conditions.The persistence S(t) is the probability that a random process x(t) has not reached a threshold up to time t 1,2 . This quantity is a natural tool in non equilibrium statistical physics to probe the history of various systems undergoing phase ordering 10,11,19 or reaction diffusion dynamics 2 , or to quantify the efficiency of target search problems [20][21][22][23][24][25][26][27][28][29] .It has been recognized that the long time decay of persistence is often algebraic, S(t) ∼ t −θ , where the persistence exponent θ is non trivial as soon as the process is non-Markovian (i.e. displays memory effects).As a matter of fact, even for seemingly simple Gaussian dynamics where all correlation functions are known, θ is generally non-trivial and not known in closed form.…”