SURVIVAL PROBABILITY OF STOCHASTIC PROCESSES BEYOND PERSISTENCE EXPONENTS

Maxim Dolgushev (Personal webpage)

Laboratoire de Physique Théorique de la Matière Condensée, CNRS/Sorbonne Université, Paris, France

How long does it take a random walker to find a "target"? This time, called the first-passage time (FPT), appears in various domains: time taken by a predator to find its prey or by a transcription factor to find a specific sequence on the DNA, time taken by a virus to infect a cell or by a financial asset to exceed a certain threshold, time taken for the cyclization of a polymeric chain, etc.

From a theoretical point of view, a crucial parameter to evaluate FPTs is the possible presence of a geometrical confinement. For a symmetric random walk in a confined domain, the mean FPT \(\langle T\rangle\) is in general finite. The opposite case of unconfined random walks is radically different. In this case, either the walker has a finite probability of never finding the target (transient random walks), or he reaches it with probability one (recurrent random walks) and the probability of survival of the target decreases algebraically with time, \(S(t)\sim S_0/t^\theta\), where \(\theta\) is the persistence exponent which does not depend on the initial distance to the target. While the persistence exponent \(\theta\) has been intensively studied, the analysis of the prefactor \(S_0\), which is essential for assessing the time to observe a first-passage event with a given likelihood, or for determining the dependence of the survival probability on the initial distance to the target, has essentially been neglected in the literature.

Our main result is a general exact relation for a process with stationary increments (more generally, for a process whose increments become stationary at long time only), Markovian or not, between this prefactor \(S_0\) (defined in the absence of confinement) and the mean FPT \(\langle T\rangle\) for the same process in a large confinement volume. This relation provides in particular an exact expression of the prefactor \(S_0\) for processes with finite memory, and in very good agreement with numerical simulations for processes with infinite memory like the famous fractional Brownian motion.